Pentane, Hexane, and Heptane - in a Supersonic Nozzle

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Investigating the Phase Transitions of lower n-alkanes – pentane, hexane, and heptane - in a supersonic nozzle

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Kehinde Emeka Ogunronbi, M.S.

Graduate Program in Chemical Engineering

The Ohio State University 2019

Dissertation Committee:

Barbara E. Wyslouzil, Advisor

Isamu Kusaka

Nicholas Brunelli

Abstract n-Alkanes play important roles in our everyday lives, and they are basic constituents of many complex lipid molecules. A knowledge of the phase behavior of these aliphatic hydrocarbons provides insight into the behavior of any given lipid class, as there are interesting similarities between chain packing in normal alkanes and other aliphatic lipids. The importance of saturated aliphatic hydrocarbons is not limited to terrestrial applications, as interplanetary studies have shown that they are both major and minor components of the giant planet atmospheres and Saturn’s moon -Titan. Thus, in these situations, short chain alkanes can play a role similar to that of water on earth.

Nucleation is a phenomenon that initiates many phase transitions, and supersonic nozzles are characterized by a large temperature gradient that results in high supersaturations and nucleation rates. Therefore, the goal of this work is to study the phase transitions of lower n-alkanes in a supersonic nozzle. First, we investigate and expand the vapor to liquid nucleation and condensation database for lower n-alkanes – pentane, hexane, and heptane - over a broad range of temperatures and partial pressures. Secondly, we apply the first and second nucleation theorems to determine the properties of the critical clusters of these chain molecules and advance our understanding of the nucleation physics.

Thirdly, we study the freezing behavior of n-pentane, n-hexane, and n-heptane droplets and advance our understanding of the surface-templating effect in these short chain n-alkanes.

The experimental vapor to liquid nucleation rates, at temperatures between 109 K and

168 K, for n-pentane, n-hexane and n-heptane were obtained by combining data from pressure trace measurements and small angle x-ray scattering experiments. For all n-

ii alkanes, the nucleation rates increase with increase in supersaturation and decrease in temperature. Using two nozzles, the critical cluster sizes of n-heptane ranged from ~ 8 to

~12 and increased with temperature. Overall, the molecular contents of the critical clusters determined from experiments are higher than predictions from classical nucleation theory.

Motivated by the success of the first nucleation rates experiments, we determined the properties of the critical clusters of short chain n-alkanes. Additional nucleation measurements (pressure trace measurements and small angle x-ray scattering) were made in another nozzle with higher expansion rates. Again, for all n-alkanes, the molecular content of the critical clusters was found to be from ~ 2 to ~ 9. Remarkable consistency was found between the critical cluster sizes determined using two and three supersonic nozzles.

Furthermore, we delved into answering the question of when the surface-templating effect in the crystallization of a homologous series of n-alkanes slows down or vanishes.

The freezing behavior of n-hexane and n-heptane are consistent with those of n-octane through n-decane. However, complementary results from pressure trace measurements, small angle x-ray scattering experiments, and Fourier transform infrared ray experiments suggested that freezing did not occur in n-pentane nanodroplets despite being supercooled by ~ 60 K. We resorted to molecular dynamics to rationalize these results. Thus, molecular dynamics simulations, using a slab geometry, supported our hypothesis that if the rearrangement of molecules at the surface becomes less pronounced as chain length decreases then freezing of the whole droplet may not occur on the timescale of our experiment.

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To my father, late mother, hungry kids on the planet, homeless humans, and the universe.

Acknowledgments

My first utmost sincere and unparalleled appreciation goes to my academic advisor,

Dr. Barbara Ellen Wyslouzil, who has been very supportive and kind. Working with her for the past five years has shaped my approach to scientific inquiries and honed my skills in the field of chemical engineering. I have been imparted with the virtue of doggedness via working with her as she always offered help in making sure I achieved my goals.

I would also like to thank my committee members - Dr. Isamu Kusaka and Dr.

Nicholas Brunelli - for their constructive criticism of my research. Words cannot express how much I am happy to have them on my committee. I am also forever grateful for the inestimable assistance I received on molecular dynamics simulations from Dr. Sherwin

Singer. He was always patient to explain to me the nitty-gritty of molecular dynamics simulations.

The completeness of my work has been possible with the help of Dr. Soenke Seifert and Randall Winans at the Argonne National Laboratory in Illinois. Contributions of Dr.

Judith Wolk in running small angle x-ray scattering (SAXS) experiments are also appreciated.

My gratitude also goes to my former colleagues, Dr. Viraj Modak and Dr. Andrew

Amaya, for getting me started with pressure trace measurements and Fourier transform infrared ray experiments in the aerosols research laboratory. They were always there to answer all my questions, both the smart and dumb ones. I would also like to appreciate my

v former colleague (Dr. Yensil Park) and current colleagues – Kayane Dingilian, Sun Tong, and Jiaqi Luo – for making the office environment a non-toxic and conducive one for me.

My appreciation also goes to my Nigerian friends at the Ohio state university – Dr.

Olumuyiwa Adesoye, Ayo Olugbuyiro, Kayode Odumboni, and Sylvester Odonnell. It was fun spending some Christmas and new year holidays together with them.

Finally, I want to deeply appreciate my father, my late mother of blessed memory, my siblings, my aunt, uncle, and first cousins (The Oyebodes) for their support throughout my academic journey in three continents on this planet. Knowing that I have family members who would have sleepless nights over me makes me think life might have a meaning and the universe might have a purpose.

Vita

May 2003 - Nov 2008 ………………… Bachelor of Chemical Engineering, Obafemi Awolowo University, Ile-ife, Nigeria.

September 2011 – June 2013 …………… Master of Science in Chemical Engineering King Fahd University of Petroleum and Minerals, Dhahran, Kingdom of Saudi Arabia.

August 2013 – June 2014 ………………. Graduate Research Associate University of South Carolina

August 2014 – May 2017 ………………… Master of Science in Chemical Engineering The Ohio State University

August 2014 – present ……………………. Graduate Research Associate The Ohio State University

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Publications

K.E. Ogunronbi, A. Sepehri, B. Chen, and B.E. Wyslouzil “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, Volume 148, pp. 144312 (2018).

Fields of Study

Major Field: Chemical Engineering

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Table of Contents

Abstract ...... ii

Dedication ...... iv

Acknowledgement ...... v

Vita ...... vii

List of Tables ...... xiii

List of Figures ...... xvi

Chapter 1: Introduction ...... 1

1.1 Overview and Motivation ...... 1

1.2 Research Objectives ...... 7

1.3 Thesis Outline ...... 8

Chapter 2: Nucleation Theory ...... 14

2.1 Vapor-Liquid Nucleation Theory ...... 14

2.2 Liquid-Solid Nucleation Theory ...... 20

Chapter 3: Nucleation of highly Supersaturated Vapors of lower n-alkanes ...... 25

3.1 Introduction ...... 26

3.2 Experiments and Data Analysis ...... 29

3.2.1 Chemicals and Physical Properties ...... 29 ix

3.2.2 Flow Apparatus and Supersonic nozzle ...... 30

3.2.3 Pressure Trace Measurements (PTM) ...... 31

3.2.4 Aerosol Size Distributions via SAXS ...... 31

3.2.5 Peak Nucleation rate and Characteristic time ...... 32

3.2.6 Simulation Methods ...... 33

3.3 Results and Discussion ...... 34

3.3.1 Pressure Trace Experiments ...... 34

3.3.2 Aerosol SAXS Experiments ...... 39

3.3.3 Experimental nucleation rates ...... 41

3.3.4 Determining the critical cluster size of n-heptane ...... 44

3.4 Comparisons with theoretical predictions and simulations ...... 46

3.4.1 Comparisons with Classical Nucleation Theories ...... 46

3.4.2 Insights from molecular simulations ...... 48

3.4.3 Comparisons with nucleation rates from simulations ...... 52

3.5 Summary and Conclusions ...... 56

Chapter 4: Properties of the Critical Clusters of the lower n-alkanes ...... 77

4.1 Introduction ...... 78

4.2 Experiments and Data Analysis ...... 80 x

4.2.1 Chemicals ...... 80

4.2.2 Supersonic Nozzles, PTM, and SAXS ...... 80

4.3 Results and Discussion ...... 82

4.3.1 PTM and Characterizing the onset of condensation ...... 82

4.3.2 Particle Sizes, Number densities, and Nucleation rates ...... 85

4.3.3 Size and Excess Internal Energy of critical lower n-alkane clusters ...... 89

4.4 Conclusions ...... 96

Chapter 5: Freezing of Short Chain n-alkanes: Experiments and Molecular Dynamics

Simulation ...... 113

5.1 Introduction ...... 114

5.2 Experiments and Data Analysis ...... 118

5.2.1 Chemicals and Physical Properties ...... 118

5.2.2 Particle Production and Characterization Methods ...... 118

5.2.3 Integrated Data Analysis ...... 120

5.3 Molecular Simulations ...... 121

5.4 Results and Discussion ...... 123

5.4.1 The case for Surface Induced Freezing in n-hexane and n-heptane ...... 124

5.4.2 Limit of n-pentane crystallization ...... 134

5.4.3 MD Simulation...... 140

5.4.3.1 Bulk Melt Temperature ...... 141

5.4.3.2 Surface Freezing Temperature ...... 144

5.5 Conclusions ...... 149

Chapter 6: Conclusions and Future Work ...... 156

Bibliography ...... 162

Appendix A: Fortran Inversion Code to determine flow properties using p and g as input

...... 188

Appendix B: Fortran Code for determining characteristic time ...... 233

Appendix C: Mathematica code to produce n-alkane crystal ...... 242

Appendix D: Fortran code to divide top and bottom frozen layers into halves ...... 244

Appendix E: Fortran code that divides the slab into halves in the y-direction ...... 248

Appendix F: GROMACS topology file for n-pentane ...... 252

Appendix G: GROMACS topology file for n-hexane ...... 255

Appendix H: GROMACS topology file for n-heptane ...... 258

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List of Tables

SJmax, and ∆푡퐽푚푎푥, denote the pressure, temperature, supersaturation, and characteristic time corresponding to the maximum nucleation rate. NZ/VV is the ratio between the density at the nucleation zone and the point of observation, N is the number density, and J is the maximum nucleation rate. All experiments used argon as the carrier gas and the stagnation pressure was p0= 30.1kPa...... 36

Table 3.2: Thermophysical properties of n-pentane, n-hexane, n-heptane, n-octane, n- nonane, and argon. The properties are 휇, the molecular weight; Cpl, heat capacity of the liquid; Tc, critical temperature; Ttriple point, the triple point; Cp (T), molar heat capacity of the vapor and carrier gas; Pe (T), equilibrium vapor pressure; ∆Hvap, heat of vaporization; ρl, bulk liquid density; ζ, surface tension...... 59

Table 3.3: Aerosol size distribution parameters 7 cm downstream of the physical throat for the n-alkanes: 〈푟〉 is the average radius of the droplets,  is the spread in radius, N is the number density of the droplets, P is the polydispersity of the droplets, and ∅SAXS is the volume fraction of the droplets. T0 is the stagnation temperature and pv0 is the partial pressure of the condensable at the inlet to the supersonic nozzle...... 62

Table 3.4: Comparison between the experimental critical cluster sizes of n-heptane and critical cluster sizes determined using the Gibbs-Thomson equation. Cluster sizes were determined at temperatures of 150K, 155K, 160K, and 165K...... 63

Table 4.1: Experimental values of the critical clusters of n-pentane through n-heptane at temperatures ranging from 110 K through 155 K. At each temperature, supersaturations,

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SJmax, corresponding to the peak nucleation rates, Jmax, in nozzles C, T1, and T3, critical cluster sizes from experiments and capillarity approximation are shown...... 91

Table 4.2: Summary of the stagnation conditions, pressure pJmax, temperature TJmax, and distance downstream of the throat xJmax, corresponding to peak nucleation rates in nozzles T1 and T3...... 98

∅SAXS is the volume fraction of the droplets. T0 is the stagnation temperature and pv0 is the partial pressure of the condensable at the inlet to the supersonic nozzle...... 100

TJmax, pJmax, SJmax, and ∆푡퐽푚푎푥, denote the pressure, temperature, supersaturation, and characteristic time corresponding to the maximum nucleation rate. NZ/VV is the ratio between the density at the nucleation zone and the point of observation, N is the number density, and J is the maximum nucleation rate. All experiments used argon as the carrier gas and the stagnation pressure was p0 = 60.5kPa...... 102

푝퐽푚푎푥 and 푇퐽푚푎푥 characterize the vapor-liquid phase transition. 푇푚 is the equilibrium melting point of the n-alkanes. In all cases the flow temperature is below the melt temperature when condensation starts, although the droplet temperature can be significantly higher...... 123

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Table 5.2: Values of the fit parameters determined from fitting Eq.1 (dashed lines) to the data in Fig. 3. The first step of freezing is termed surface freezing, while the second step is termed bulk freezing...... 129

Table 5.3: Summary of the degree of supercooling, ∆T, required for n-alkane nanodroplets from n-hexane through n-decane to initiate “bulk” freezing. Data from n-C8, n-C9, and n-

C10 are from Modak et al. pv0 is the stagnation partial pressure of the n-alkane, Tdrop and

Tflow are the average temperatures of the droplets and the gas mixture, respectively, at the corresponding t0V. Tm is the equilibrium melt temperature...... 131

Table 5.4: Details of simulation for determining the melt temperature for each n-alkane.141

Table 5.5: Simulation details for calculating the surface freezing temperature for each n- alkane and the values of the surface freezing temperatures...... 144

List of Figures

Figure 1.1: Droplet profiles at various stages in a 130 ns run. The surface is completely frozen within 4 ns. This state persists till 20 ns. This is followed by layer-by-layer freezing. (Source: culled from Modak) ...... 7

Figure 2.1: Classical nucleation theory-based free energy of cluster formation of n-pentane molecules at temperature, T = 129 K and supersaturation S = 7.5 × 105...... 18

Figure 3.1: (a) Experimental set-up for the pressure trace measurements used to measure the pressure profile along the axis of the nozzle (Adapted from Tanimura et al.). The Laval nozzle used in the experiments is machined from aluminum. (b) Unassembled nozzle blocks and (c) a side walls incorporating a CaF2 window...... 29

Figure 3.2: The legend is the same for both figures. (a). Pressure and temperature profiles for condensing flows of n-pentane (n-C5), n-hexane (n-C6), and n-heptane (n-C7) at comparable inlet partial pressures of 429 Pa, 426 Pa, and 440 Pa, respectively. All expansions start at a total stagnation pressure, p0 = 30.1 kPa and the noted values of T0. For clarity, only the isentropic pressure ratio corresponding to the n-C5 mixture is shown. For n-heptane, the strong increase in temperature near x= 5 cm is the signature of freezing. (b) The normalized nucleation rates, corresponding to the data in (a), were calculated using Classical Nucleation Theory. Time t = 0 corresponds to the nozzle throat...... 34

Figure 3.3: Change in pJmax with 1/TJmax for n-decane (C10) through n-pentane (C5). Grey and open symbols are data from Ghosh et al. for the noted carrier gas, N2 or Ar, while black symbols are data from current work. The n-heptane data from Ghosh et al. (grey triangles) lie at higher temperature/lower pressure than the current n-heptane data consistent with the higher expansion rates used here. All data sets lie along reasonably parallel straight lines.37

Figure 3.4: One-dimensional SAXS spectra for n-pentane measured as a function of pv0 at constant stagnation conditions p0 = 30.1kPa and T0 = 293.15 K respectively. The spectrum xvi at pv0=880.4Pa is on the absolute intensity scale, while spectra at lower pv0 were scaled down by decreasing factors of 10. The size distribution parameters in the top right corner correspond to the best fit parameters for the pv0=880.4Pa data assuming scattering is from a polydisperse collection of spheres that follow a Schultz distribution...... 39

Figure 3.5: The variation in particle mean radius (〈푟〉), size distribution spread (σ), and number density (N) with the partial pressure of condensable at the inlet to the nozzle. The dashed lines passing through each data are to guide the reader’s eye. Data in Nozzle C are those of Ghosh et al...... 40

Figure 3.6: Peak nucleation rates, Jmax, versus the supersaturation corresponding to the peak nucleation rates, SJmax for n-pentane, n-hexane, and n-heptane from the current work. The heptane data of Ghosh et al. for measured in nozzle C are also included. The uncertainty in Jmax is represented by a vertical bar with a factor of 2, while the uncertainty in SJmax, primarily from temperature measurements, is within ±10%...... 43

Figure 3.7: (a) Isothermal peak nucleation rates, J, as a function of supersaturation, S. The lines through each isothermal data are power-law fits through which we determine the molecular content of the critical clusters. The values of n*exp and n*GT noted for each temperature. (b) The values of n*exp and n*sim are compared to those of n*GT. For our data, the vertical error bars correspond to the standard error of the slopes, and the horizontal error bars reflect the uncertainty in the supersaturation. The solid line corresponds to perfect agreement and the dashed lines correspond to limits...... 46

Figure 3.8: Ratio of the measured and predicted nucleation rates versus reduced inverse temperature, TJmax. The C5, C6, and C7 data are from the current work while the C8 and C9 data are from Ghosh et al. The straight lines through each set of data are only meant to guide the reader’s eyes...... 48

16 Figure 3.9: (a) The 훿∆퐺(푛) values for a n-heptane simulation at 휌v1 = 2.62 × 10 molecules/cm3 and T = 159K.The dashed red line is the fit to the simulation data for clusters containing more than 14 molecules. (b) The ratio of nucleation rates from simulations (Jsim) and the predicted rates from classical nucleation theory (JCNT). The lines through each data xvii set is meant to guide the reader’s eye. The simulation data exhibit almost universal behavior when plotted as a function of scaled temperature...... 51

Figure 3.10: Snapshots of clusters of n-pentane through n-nonane at approximately constant Tc/TJmax ~ 3.6. As the chain length increases, the clusters exhibit more order at the same scaled temperature...... 52

Figure 3.11: Rates from simulation agree quite well with the data if the former (solid lines) are reduced by factor of 1000 (dashed lines). (a) For the scaling exponent w = 1.5, the simulation rates do not scale perfectly at low values of of lnS/(Tc/T-1)1.5. (b) The scaling exponent of w = 1.53 does a better job of scaling the simulations but broadens the spread in the TDCC experimental data...... 54

Figure 3.12: (a) In a Hale plot, simulation data follow the trend of the experimental data when Ω = Ωexp. (b) Using Ωsim to scale Jsim yields a slope that is more consistent with scaling but introduces a substantial offset. The variable C0 is related to Ω via 퐶0 = (16휋⁄3 ln 10)Ω3...... 56

Figure 3.13: (a). Side view of one of the shaped aluminum blocks, showing the dimensions in inches. (b). A view of the whole nozzle assembly from the flow entry point...... 64

Figure 3.14: Rates from simulation and experiments as a function of scaled supersaturation over a wide range of temperature. For both exponents 1.5 and 1.53, the simulated rates at 159K and 169K scale perfectly over a wide range of supersaturation. The dashed line corresponds to the simulated rate at 159 K divided by 1000...... 65

Figure 3.15: Snapshots of n-heptane clusters show that as the temperature decreases, the structure of clusters becomes more ordered. For T = 142 K, the cluster appears quite ordered and it was not possible to generate a smooth G(n) curve...... 65

Figure 3.16: Solid lines correspond to Jsim, dashed lines to Jsim/1000. The MC simulation results agree quantitatively with the experimental data when they are reduced by a factor

xviii of 1000. (a) n-pentane (b) n-hexane. For these alkanes the scaling parameter w = 1.5 works well over the limited (20 K) temperature range of the simulations...... 66

Figure 4.1: Effective flow area ratios in nozzles C, T1, and T3 as a function of distance downstream of the throat. In all nozzles, Ar is the carrier gas. The stagnation conditions are given in the main text. In general, changes in stagnation temperatures (T0) do not affect d(A/A*)/dz. Although d(A/A*)/dz can change much with changes in stagnation pressure

(푝0), for a given nozzle p0 was held constant...... 81

Figure 4.2: Pressures and inverse temperatures corresponding to point of maximum nucleation in three nozzles characterized by different expansion rates. Grey closed symbols are data measured in nozzle T3, dark closed symbols are measured in nozzle T1, and open symbols are data measured in nozzle C. Data from nozzles T1 and C are from Ogunronbi et al. and Ghosh et al., respectively. The lines on each n-alkane data are meant to guide the reader’s eyes...... 84

Figure 4.3: Filled square and diamond symbols correspond to temperatures at condensable pressure of 40 Pa in nozzles C and T3, respectively. Systematic deviation, with respect to carbon number, of the acentric factor and thermodynamic properties reflects that these data are extremely consistent...... 85

Figure 4.4: Number density, average size of droplets, and width of size distribution as a function of condensable partial pressure. Experiments in nozzles T1 and T3 started at p0 = 30.2kPa and 60.5kPa, respectively. For clarity, we omit data for n-heptane in nozzle C. The crossed square symbols are data with high uncertainty, where the spread of the average droplet size is unacceptably large...... 87

Figure 4.5: Maximum nucleation rate, J, versus supersaturation at maximum nucleation, S, for n-pentane, n-hexane, and n-heptane in three supersonic nozzles (nozzles T3, T1, and C) characterized by different expansion rates. The vertical error bars correspond to a factor of 2 while the horizontal error bars represent a ±10% change in supersaturation. The open symbols for n-heptane are a result of the high unacceptable uncertainty in the number density...... 88 xix

Figure 4.6: (a). Fits to nucleation rate versus supersaturation used to derive n* for n-pentane at temperatures (from right to left) 110 K, 113 K, 125K, and 129 K. (b). Fits used to derive n* for n-hexane at temperatures (from right to left) 128.8 K, 136.5 K, and 143.5 K. (c). Fits for n-heptane at temperatures (from right to left) 155 K and 150 K. The vertical error bars correspond to a factor of 2, and horizontal error bars reflect the uncertainty in the supersaturation. The critical cluster sizes at different temperatures are shown in boxes...... 91

Figure 4.7: Except for sizes less than four, the critical cluster sizes from experiment are underpredicted by the capillarity approximation. The vertical error bars are from the standard error of the J-S slopes; the horizontal error bars, although less visible, reflect the uncertainty in supersaturation. The solid line corresponds to perfect agreement between experiment and theory...... 92

Figure 4.8: Nucleation rate as a function of temperature at constant supersaturation for (a) n-pentane, (b) n-hexane, and (c) n-heptane. The error bars on J correspond to a factor of 2. The solid lines through each constant-supersaturation data set are power-law fits through which we determine the slope used in evaluating Ex,cl. The dashed lines are from correlations between Jmax and TJmax in nozzles of different expansion rates discussed in the supporting information...... 93

Figure 4.9: Normalized excess internal energy as a function of the molecular content of the critical clusters of (a) n-pentane (b) n-hexane and (c) n-heptane. The solid lines are the predictions from the theory of capillarity approximation at T = 110 K, 143.5 K, and 165 K for pentane, hexane, and heptane, respectively. The dashed line is the prediction for heptane at T = 275 K. For n-heptane, current values are compared to derived values from the data of Rudek et al.7 For all n-alkanes in this work, the vertical error bars correspond to the standard error of the slopes of Fig.5 and the error bars on the number of molecules are from the standard error of the slope from the first nucleation theorem. Open symbols correspond to data from the correlations described in the text. The value of T0 is 273.15 K...... 96

Figure 4.10: Correlations of temperature and supersaturation with mass flow rate for n- pentane, n-hexane, and n-heptane. These correlations are used to determine the adjusted

TJmax corresponding to the averaged SJmax in each nozzle...... 104

Figure 4.11: Correlations of nucleation rates vs temperature and supersaturation; and supersaturation vs temperature for n-pentane. This is an alternative procedure we use to analyze our pentane data, from which we evaluate the first and second nucleation theorems...... 105

Figure 4.12: Correlations of nucleation rates vs temperature and supersaturation; and supersaturation vs temperature for n-hexane. This is an alternative procedure we use to analyze our n-hexane data, from which we evaluate the first and second nucleation theorems...... 106

Figure 4.13: Parameters, in particular the width of the average size distribution σ, from the fits to the scattering data for inlet conditions 98.1 푃푎 ≤ 푝푣0 ≤ 266.3 푃푎 are unacceptably high...... 108

Figure 5.1: Temperature profiles of the expanding flows of a mixture of carrier gas- condensable (n-hexane, and n-heptane) and the corresponding mean droplet size. The open circles represent the temperature of the droplets and the labels indicate the initial partial pressure of the condensable. In (a) there is a second bump in the n-hexane flow at ~ 6.5cm (noticeable for the high flow rate), a signature of the release of latent heat of fusion. Near this position in (b) there is a gradual decrease in the size of the droplets, signifying an increase in density and a decrease in volume of the droplets as they start to freeze. The arrows indicate where freezing starts. The same behavior is seen (c), (d) in the n-heptane flow at inlet partial pressure pv0 = 323 Pa...... 125

Figure 5.2: Mass fractions of the condensable (vapor) and condensate (liquid and solid) as a function of position in the nozzle for (a) n-hexane at inlet partial pressure of 279 Pa (b) n-hexane at inlet partial pressure of 138 Pa (c) n-heptane at inlet partial pressure of 323 Pa and (d) n-heptane at inlet partial pressure of 171.6 Pa. In each graph, M.B is the material balance i.e. the summation of the mass fractions of the three phases at positions xxi downstream of the nozzle throat. (The lines through the points are to guide the reader’s eye)...... 127

Figure 5.3: Freezing kinetics of n-hexane at pv0 = 138 Pa and 279 Pa and n-heptane at pv0 = 172 Pa and 323 Pa. As the nanodroplets cool down after growth is complete, crystallization occurs via a two-step process. The black dashed lines are the fits to the fraction of solid in the aerosol. The green dashed line is the hypothetical fraction of solid if the surface of every droplet has a fully developed solid monolayer. Plots for (a), (b), (c), and (d) have been shifted by 0.096 ms, 0.114ms, 0.070ms, and 0.067ms, respectively, to present the analysis from all experiments on the same figure. The parameters for the black dashed lines are listed in Table 2...... 130

Figure 5.4: Odd-even variation of the difference between the equilibrium melting temperature and bulk freezing temperature of short-chain n-alkanes nanodroplets...... 132

Figure 5.5: A summary of (a) surface, (b) volume-based, and (c) heterogeneous nucleation rates of solid in supercooled n-alkane droplets as a function of chain length. Current data for n-hexane and n-heptane are plotted with data for n-octane, n-nonane, and n-decane from Modak et al...... 133

Figure 5.6: Temperature profiles, (a), and size distributions, (b), as a function of position for n-pentane at inlet partial pressures of 326 Pa and 688 Pa. As expected, the onset of condensation is earlier in the higher inlet partial pressure experiment. The size distributions from both experiments suggest that growth is complete, and droplets are fully developed by the exit of the nozzle ...... 135

Figure 5.7: (a). The normalized absorptivities for n-pentane droplets at T = 117 K, and 137

K for expansions starting at pv0 = 326 Pa and 688 Pa, respectively. (b). Peak locations of normalized absorptivity of n-heptane from current experiment are consistent with those of n-nonane in Modak et al.20 Ideally, a spectrum from frozen n-pentane droplets should exhibit the same features as the spectra in (b) that were measured near the exit of the nozzle.

For the spectra in (b), the shape and intensity of the CH3 peaks are quite close, while the intensity of the CH2 peak clearly decreases with chain length...... 136 xxii

Figure 5.8: Mass fraction of vapor and liquid n-pentane, as a function of position downstream of the throat, calculated based on PTM and FTIR. Here, M.B. = (gv + gl)/ginf...... 137

Figure 5.9: (a). Pressure and temperature profiles for condensing flow of n-pentane. The inlet conditions are noted in the legend. The effect of boundary layer has not been accounted for. However, accounting for this will only change the exit temperature by ~ 6K.

The dashed grey line is the equilibrium melting point for n-pentane, and the arrows guide the eyes to read the right axis. (b). Absorbance spectra of n-pentane droplets characterized by different temperatures. Despite a further supercooling of ~ 26 K, the spectra do not differ from each other. The warmer spectrum was scaled by a factor of 1.8 ...... 139

Figure 5.10: Images of n-heptane slabs at different configurations. (a). Starting configuration of a slab (b). final configuration of a half-melted and half-frozen slab. (c). T = 188 K, where growth of the solid phase is evident. (d). T = 189 K. (e). T = 190 K, where the whole slab is completely melted...... 143

Figure 5.11: (a). Potential energy of n-heptane systems with half-melted and half-frozen partitions as a function of time at different temperatures. The potential energy at 189 K remains approximately stationary for 10 ns. Therefore, the melt temperature for n-heptane is estimated as 189 ± 1 K. (b). Melt temperatures of the real n-alkanes from experiments,

Tm, and united atom model, Tm, UA...... 143

Figure 5.12: Images of n-heptane slabs under different conditions (a). Initial configuration of a slab with half-melted and half-frozen surfaces. Configurations after 5 ns for (b). T =

184 K where growth of the frozen portion is evident at the top and bottom (c). T = 185 K.

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(d). T = 186 K, and (e). T = 187 K, the whole slab is completely melted even though the slab is supercooled...... 145

Figure 5.13: This is as discussed in relation to Fig. 5.12, except that the configurations of (a) through (e) are at 3 ns. The interface stays roughly the same at a temperature of 185 K as depicted in (c)...... 146

Figure 5.14: Potential energy of n-heptane systems with half-melted and half-frozen surfaces as a function of time at different temperatures. The potential energy at 185 K stays approximately stationary for at least 3 ns. The same applies to 184 K. Therefore, taking this uncertainty into consideration, the onset of surface freezing temperature is approximately 185 ± 1 K...... 147

Figure 5.15: Degree of supercooling as a function of n-alkane chain length...... 148

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CHAPTER 1

INTRODUCTION

1.1. Overview and Motivation

Saturated aliphatic hydrocarbons, also known as normal alkanes, are basic constituents of many complex lipid molecules. A knowledge of the phase behavior of these aliphatic hydrocarbons gives insight into the behavior of any given lipid class, as there are interesting similarities between chain packing in normal alkanes and chain packing in aliphatic lipids1. Models of n-alkane molecules have been used in attacking the difficulties inherent in the evaluation of experimental data on the structure of polymeric glasses and melts2. This is based on the assumption that the nature of the ordering in n-alkane systems will reflect the typical features of organization of polymer molecules.3

The importance of saturated aliphatic hydrocarbons is not limited to terrestrial applications, as interplanetary studies have shown that these molecules are both major and minor components of the giant planet atmospheres and Saturn’s moon -Titan4. Thus, in these situations, short chain alkanes can play a role like that of water on earth.

Aliphatic hydrocarbons are also important constituents of petroleum deposits and raw natural gas. Consumption of natural gas as a source of energy has been on the increase and is forecast to keep increasing5. Its use, above coal, in new power plants stems from its clean burning nature. However, raw natural gas is not completely clean when produced, and

requires some purification before being transported. Conventional separation methods involve large plants comprising of different sections where chemicals are used to extract unwanted components. These chemicals are often disposed or accidentally spilled thereby creating pollution in the immediate environment5.

Recently, supersonic gas separators have been developed that use the high velocity coupled with a controlled drop in pressure and temperature to separate water and higher boiling point components from the main stream of natural gas6, 7. After condensing, the heavier hydrocarbons can also freeze as the temperature continues to drop. The freezing points of n-pentane, n-hexane, and n-heptane, -129.8oC, -95.3oC, and -90.5oC respectively

8, are higher than that of liquefied natural gas (LNG). If not removed, they would freeze during the liquefaction of natural gas and could damage the process equipment. The smaller sizes of these separators and their minimal waste production have proven to be an advantage over conventional equipment.

Alkanes are technologically important materials, and vapor-phase nucleation plays a critical role in many alkane separation processes. Phase transitions that are commonly observed in both industrial and natural processes include crystallization, evaporation and condensation 9. Scientific studies of the nucleation phenomena that initiate these phase transitions are necessary to further our understanding of the overall processes because our ability to predict nucleation rates over a wide range of temperatures with any degree of confidence is poor and hampers development of the separation process models.

An attempt to predict the nucleation rates and understand the thermodynamics and kinetics of the nucleation of supersaturated vapor was made by Volmer and Weber 10, and improved by Becker and D표̈ring11. Developing the Classical Nucleation Theory (CNT),

Volmer and Weber described the onset of vapor to liquid nucleation as the formation of liquid-like aggregates from vapor molecules. These aggregates, consisting of small number of molecules, are unstable and stochastically increase in size due to the free energy barrier.

However, high supersaturations in the vapor phase lower the free energy barrier thereby making growth to supercritical aggregate or ‘droplet’ size faster and easier.

The inconsistency of the Becker-D표̈ring theory (CNT) to quantitatively predict the nucleation rates of different substances has led to several modifications of the theory.

However, none of the modifications 12,13,14 provides accurate and consistent predictions of the experimental nucleation rates for different substances. Despite this drawback, experiments on vapor-to-liquid nucleation have continued to advance our understanding of this phenomenon.

Due to their technological importance, researchers have investigated the vapor- liquid nucleation of n-alkanes, in particular n-nonane and its mixtures, using a variety of experimental devices with characteristic nucleation rates ranging from 102 to 1017 cm-3s-1

15, 16, 17, 18, 19. Overall, they found that experimental rates were higher than those predicted by Classical Nucleation Theory (CNT) over the measurement ranges. In an earlier work,

Ghosh et al. 20 investigated the vapor-phase homogeneous nucleation of n-heptane, n- octane, n-nonane and n-decane using argon and nitrogen as carrier gases in a supersonic

nozzle. Condensation is noticed by an increase in the pressure and temperature of the flow.

They found maximum nucleation rates ranging from 4 × 1015 to 2 × 1018 푐푚−3푠−1 at temperatures between 150K and 200K. Also, maximum supersaturations on the order of

105 were found at the lowest temperatures. Comparing their nucleation rates with predictions of classical nucleation theory they found their results were 4.5-8 orders of magnitude higher than the predictions. This discrepancy increased with increasing chain length and decreasing temperature. Since the capillarity approximation – properties of the small clusters are the same as the bulk properties of the new phase – inherent in the CNT is thought to mainly contribute to the discrepancy between experiment and theory, using simulations to test the validity of this aspect of the theory as a function of chain length could lead to useful insight. This observation has partially motivated the first portion of the current studies.

Despite the large amount of research done on understanding the vapor-phase behavior of n-alkanes in general, there is much less information available on the homogeneous nucleation of lower n-alkanes (퐶푛퐻2푛+2, 푛 ≤ 6) relative to the higher

21 alkanes (퐶푛퐻2푛+2, 7 ≤ 푛 ≤ 10). However, very recently, Signorell and co-workers investigated the nucleation and growth of n-propane and n-propane/n-ethane mixtures in the postnozzle region of a supersonic Laval nozzle where there is uniform flow due to the pressure in the postnozzle region being matched with the chamber pressure. The molecular content of critical clusters was determined by resolving the masses of molecular aggregates via single photon ionization and mass spectrometry in a postnozzle flow 22. Using three supersonic nozzles characterized by different expansion rates they found the average 4

critical cluster sizes to be between 8 and 14. However, experimentally measured nucleation rates were not determined. Thus, there is a strong need to bridge the gap by carrying out experiments for determining the vapor-phase nucleation rates of lower n-alkanes

(퐶푛퐻2푛+2, 4 ≤ 푛 ≤ 6).

Not only have studies on the vapor to liquid phase transitions of n-alkanes been done, a few researchers have also studied the crystallization of supercooled n-alkane liquid droplets. Bartell and co-workers 23,24,25 investigated the freezing of liquid nanodroplets in supersonic nozzles for several organic and inorganic molecules. Interestingly, for all substances investigated, including an isomer of heptane, they assumed that nucleation starts randomly within the supercooled droplets. However, they posited that for linear chain molecules nucleation might have been initiated by segments of the molecular chains.

Clearly, there has been an overwhelmingly large amount of literature on freezing of simple molecules. However, there is a dearth of experimental contribution to the understanding of the phase transition from liquid to crystal of chain molecules under extreme conditions of supercooling.

More recently, Modak et al. 26, 27 showed that supersonic Laval nozzles could also be used to study the liquid to solid phase transitions in supercooled n-alkane (n-octane, n- nonane, and n-decane) nanodroplets. They characterized the freezing of these nanodroplets by using pressure trace measurements (PTM), small angle x-ray scattering measurements

(SAXS) and Fourier transform infra-red (FTIR) spectroscopy measurements and observed that the decrease in size of the nanodroplets, from 9 nm to ~7.8 nm for n-octane and from

8.4 nm to ~8nm for n-nonane, coincided with a second heat addition due to freezing.

Similar decrease was found upon the crystallization of n-decane nanodroplets for three of the four experimental conditions. The distribution, as a function of position, of the condensable between the three phases was determined by using FTIR. The increasing fraction of solid in the aerosol was determined as a function of time and then fitted to a functional form to determine the surface-based and volume-based nucleation rates. Surface and bulk nucleation rates for both n-octane and n-nonane were found to be on the order of

1015 cm-2 s-1 and 1022 cm-3s-1 respectively. Their work 26, 27 advanced our understanding of surface freezing in short-chain alkanes despite prior experimental reports 28, 29, 30 that had concluded that the surface freezing phenomenon was restricted to longer chain-like molecules.

Given the apparent drawback of experiments in directly resolving the molecular interactions involved in the crystallization of chain molecules, powerful tools, e.g. MD simulation, have been used to an extent in overcoming this challenge. Molecular dynamics simulation captures the atomistic details of the crystal nucleation events in n-alkane systems. Yi and Rutledge31 used MD simulation to capture the critical nucleus in a crystallizing n-octane system. They determined the dependence of free energy on nucleus size using the CNT but with cylindrical molecules. Similarly, in trying to understand the two-step behavior of the freezing of supercooled n-octane nanodroplet, Modak et al.26 used a united-atom force field model in a MD simulation to support their claim of surface freezing. As shown in Fig. 1.1, freezing of the n-octane droplet proceeded in a layer-by-

layer manner, with the surface fully frozen by 4 ns. The fully-frozen layer serves as a template for the subsequent layer. This would continue till the whole droplet freezes.

Therefore, this interesting phenomenon of surface freezing has raised a question on the effect of chain length on the alignment of the molecules at the surface of n-alkane droplets.

Fig. 1.1: Droplet profiles at various stages in a 130 ns run. The surface is completely frozen within 4 ns. This state persists till 20 ns. This is followed by layer-by-layer freezing. (Source: culled from Modak32)

1.2. Research Objectives

The first goal of this scientific research is to build on earlier efforts of Ghosh and co- workers20 by extending the investigation of vapor-phase nucleation to relatively lower n- alkanes, n-pentane and n-hexane, and provide additional nucleation data for n-heptane. The additional data for n-heptane in a different nozzle characterized by a higher expansion rate will be combined with earlier data and used to determine the number of molecules in the critical clusters of n-heptane.

The second goal of this work is to conduct an extensive investigation on the crystallization of relatively shorter n-alkanes - pentane, hexane, and heptane. In addition, we would like to provide insight from both experiments and MD simulation into when surface-templated freezing stops or at least slows down on the timescale of our experiment.

The final goal is to determine the sizes of the critical clusters of n-pentane and n- hexane, as studies of critical clusters of lower n-alkane molecules generated in a supersonic nozzle could lead to a more complete characterization of vapor-liquid phase transition in chain molecules. This investigation will also provide insight into the systematic effect of temperature on the molecular content of critical clusters in a homologous series of n- alkanes.

1.3. Thesis Outline

The outline of this thesis is as follows: Chapter 2 explains in detail the classical nucleation theory (CNT), showing the derivation of the Becker-Doring expression used in

predicting the nucleation rate of the liquid critical clusters from the metastable vapor phase.

In addition, modifications to improve CNT will be discussed.

Chapter 3 discusses the study of the vapor-phase nucleation of n-pentane, n-hexane, and n-heptane via experiments complemented by insights from Monte Carlo simulations.

Chapter 4 presents an overview of the sizes and excess internal energy of the critical clusters of lower n-alkanes – n-pentane, n-hexane, and n-heptane.

In chapter 5 we present quantitative and qualitative investigation of the crystallization of lower n-alkanes – n-pentane, n-hexane, n-heptane via experiment and complementary molecular dynamics simulations. Here, we provide evidence for the crystallization of n- hexane and n-heptane nanodroplets and discuss the inability of n-pentane nanodroplets to freeze under the timescale of our experiment.

Finally, chapter 6 summarizes conclusions and possible future work on short chain n- alkanes in the homologous series.

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CHAPTER 2

NUCLEATION THEORY

2.1. Vapor-Liquid nucleation theory

In the absence of impurities, the vapor molecules of a substance being cooled become supersaturated before nucleation occurs. Vapor to liquid nucleation involves the formation of liquid-like aggregates from vapor molecules. These aggregates either continue to grow or disintegrate depending on their ability to overcome the free energy barrier to nucleation. The free energy barrier reduces as supersaturation increases. That is, nucleation occurs easily in more supersaturated vapor systems. Difficulty in observing subcritical, critical, and supercritical clusters involved in vapor-liquid nucleation has led to the development of the classical nucleation theory (CNT). The CNT, originated from the fundamental work of J.W. Gibbs 1 in 1878, later developed by Volmer and Weber 2, and refined by Becker and Döring 3, is based on the assumption that the free energy of forming a cluster of n molecules is a sum of the energy required to move the n molecules from the supersaturated vapor to a cluster and the energy cost in creating the surface of the cluster.

That is,

∆퐺 = 푛∆휇 + 휑 (2.1) where ∆휇 is the change in chemical potential between the new liquid phase and the saturated vapor phase (mother phase).

휑 is the energy cost in creating the spherical surface of the new phase.

In other words, the free energy of cluster formation is the summation of volume and surface terms. In a supersaturated vapor, the first term in Equation 2.1 has a negative value because

휇푙𝑖푞 < 휇푠푎푡푣푎푝 , that is the thermodynamic potential of the liquid phase is less than that of the metastable vapor phases. This is the driving force pushing the system towards a spontaneous phase transition.

Assuming the gas mixture is ideal, and ignoring a small pressure-volume term, ∆휇 is related to the equilibrium vapor pressure by

푝 ∆휇 = −푘푇푙푛( ⁄ ) (2.2) 푝∞

where 푝 is the pressure of the supersaturated vapor, 푝∞ is the equilibrium vapor pressure on an infinitely large condensed phase 4, and 푘 is Boltzmann constant.

Eq. 2.2 is generally written as

∆휇 = −푘푇푙푛(푆) (2.3) where S is the supersaturation of the metastable vapor.

Since one of the basic assumptions of the theory is that clusters are spherical in shape, we can express the energy cost in creating the surface of a cluster, 휑, as

휑 = 4휋푟2훾 (2.4) where, 훾 is the vapor-liquid interfacial free energy or surface tension.

Combining equations 2.1, 2.3, and 2.4, (2.1) now becomes

∆퐺(푛) = −푛푘푇푙푛(푆) + 4휋푟2훾 (2.5)

Since the cluster size n is related to the radius r by

4 휋푟3 i.e. 푛 = 3 (2.6) 푉푚

where 푉푚 is the molecular volume , then the surface term can also be written as

2 2 1/3 2/3 4휋푟 훾 = (36휋푉푚) 푛 훾 (2.7)

Thus,

2 1/3 2/3 ∆퐺(푛) = −푛푘푇푙푛(푆) + (36휋푉푚) 푛 훾 (2.8)

Alternatively, putting (2.6) in (2.5) to make the change free energy a function of the cluster radius,

4휋푟3푘푇푙푛푆 ∆퐺(푟) = − + 4휋푟2훾 (2.9) 3푉푚

We can plot this function for a n-alkane at conditions relevant to expansions in supersonic nozzles. Figure 2.1 shows the variation of free energy with radius of cluster. The gradual increase and subsequent decrease in free energy point to an interesting fact that at small radius, the free energy barrier to nucleation is governed by the surface term, and at relatively large radius, the volume term takes over. The competition between these two terms reaches its maximum when the cluster reaches a particular size - the critical cluster size denoted by the dashed vertical line.

At the critical cluster size, ∆퐺 is at a maximum point. Mathematically, this is represented as

푑∆퐺 푑∆퐺 ( ) = 0 or ( ) = 0 (2.10) 푑푛 푇,푆 푑푟 푇,푆

And at the critical size,

32휋푉2 훾3 2푉 훾 푛∗ = 푚 or 푟∗ = 푚 (2.11) 3(푘푇푙푛푆)3 푘푇푙푛푆

Putting equation (2.11) in (2.5), we have the expression for the free energy change of the critical cluster,

16휋푉2 훾3 ∆퐺(푛∗) = ∆퐺(푟∗) = 푚 (2.12) 3(푘푇푙푛푆)2

Figure 2.1: Classical nucleation theory-based free energy of cluster formation of n-pentane molecules at temperature, T = 129 K and supersaturation S = 7.5 × 105.

For a supersaturated vapor in equilibrium, the concentration of a population of clusters obeys Boltzmann distribution function 5, 6

푒 ∆퐺(푛)⁄ 푁푛 = 푁1exp (− 푘푇) (2.13)

푒 Here, 푁푛 is the number, per unit volume, of clusters of size n. 푁1 is the total number of molecules in the vapor.

From (2.12), at maximum ∆퐺,

16휋푉2 훾3 푁푒∗ = 푁 exp (− 푚 ) (2.14) 푛 1 3(푘푇푙푛푆)2

This expression is approximate for the critical region because it fails to vanish when 푁1 =

Now, according to Becker and Döring3, the net rate of formation of clusters containing n molecules from clusters containing n-1 molecules is

퐽푛 = 푣푎푛−1푁푛−1 − 푤푎푛푁푛 (2.15)

Where 푣 is the number of molecules striking the surface per unit time and 푎푛−1 is the area of a cluster of size n-1. 푤 is the number of molecules leaving the surface per unit time and

푎푛 is the area of a cluster of size n. 푁푛 and 푁푛−1 are the non-equilibrium concentrations of clusters of sizes n and n-1, respectively.

From the kinetic theory of gas, we can estimate 푣 as7

푝1 푣 = 1/2 (2.16) (2휋휇1푘푇)

Where 푝1 is the pressure of the supersaturated vapor and 휇1 is the mass of a molecule.

To estimate 푤, a constrained equilibrium condition is placed on (2.15), i.e 퐽푛 = 0.

Obtaining 푤 and substituting back in (2.15), we obtain,

푒 푁푛−1 푁푛 퐽푛 = 푣푎푛−1푁푛−1 [ 푒 − 푒] (2.17) 푁푛−1 푁푛

Applying the pseudo-steady state approximation (퐽푛 = 퐽푛+1 = ⋯ = 퐽) will make J a constant independent of n. Thus,

푒 푁푛−1 푁푛 퐽 = 푣푎푛−1푁푛−1 [ 푒 − 푒] (2.18) 푁푛−1 푁푛

Equation 2.17 can be summed from 푛 = 2 to a large value of 푛. This cancels successive terms and leads to:

1 퐽 = 푛−1 1 (2.19) ∑푛=1 푒 휐푎푛푁푛

Equation 2.19 can be simplified by approximating the summation by an integral and assuming that contribution only comes from the region around 푛∗.

The first simplification yields:

푛−1 푑푛 −1 퐽 = [∫ 푒] (2.20) 1 휐푎푛푁푛

푒 For the second approximation, we expand 푁푛 from Equation 2.13 using Taylor series around 푛∗,

1/3 Δ퐺∗ 1 2(36휋푉2 ) 훾 푁푒 = 푁 푒푥푝 [− + ( 푚 푛∗−4/3(푛 − 푛∗)2)] (2.21) 푛 1 푘푇 2 9푘푇

Substituting 2.21 into 2.20 and changing the limits to 0 and ∞ reduces the expression to an error function7

2 1/3 1 ∞ 1 2(36휋푉푚) 훾 ∗−4/3 ∗ 2 퐽 = 푒 ∫ 푒푥푝 [− ( 푛 (푛 − 푛 ) )] 푑푛 (2.22) 휐푎푛∗푁푛∗ 0 2 9푘푇

∗ Solving this and using (2.14), (2.16), and 푎푛∗ = 4휋푟 , we obtain:

2 2 3 2훾 푝1 −16휋푉푚훾 퐽 = √ 푉푚 ( ) exp [ 3 2] (2.23) 휋휇1 푘푇 3(푘푇) (푙푛푆)

2.2. Liquid-Solid nucleation theory

The experimental and theoretical analyses of crystal nucleation and growth kinetics of supercooled liquids have been treated using various techniques and models. Typically, models use viscosity to describe the diffusion of molecules at the interface of the supercooled liquid and crystal.

The usefulness of the classical nucleation theory depends on whether the spontaneous appearance of crystal nuclei occurs anywhere within a supercooled liquid (homogeneous nucleation) or the formation of the critical crystal nucleus occurs at the surface of the supercooled liquid (heterogeneous nucleation). According to the classical nucleation theory, the rate of crystal nucleation from the liquid phase, 퐽푙푠, is expressed by

∗ ∆퐺푙푠 퐽푙푠 = 퐾푙푠 푒푥푝 (− ) (2.24) 푘퐵푇

Where 퐾푙푠 is the prefactor usually based on a viscous model; the free energy of forming a

∗ spherical critical crystal nucleus, ∆퐺푙푠, is given as

3 ∗ 16휋휎푙푠 ∆퐺푙푠 = 2 (2.25) 3(∆퐺푣+Ψ)

Where,

∆퐺푣 is the free energy of freezing of the liquid per unit volume, assuming a spherical liquid droplet, Ψ is the work done that accompanies the change in the surface area of the droplet as the volume changes due to the formation of the solid nucleus.휎푙푠 is the free energy per unit area of the liquid-solid nucleus interface. Ψ is a function of the densities of the phases 21

and at highly supercooled conditions, this variable is typically negligible because the densities of the mother and new phases are assumed to be close 8.

For a small degree of supercooling or for moderate departures from equilibrium, ∆퐺푣 may be expressed as 9, 10

∆퐺푣 ≈ ∆퐻푣(∆푇⁄푇푚) (2.26)

Where ∆퐻푣 is the heat of fusion per unit volume at the equilibrium melting point 푇푚, ∆푇 is the supercooling which is equal to 푇푚 − 푇.

The viscous flow model, a model which assumes that the energy of activation for diffusion of a molecule from the liquid to the crystal nucleus is proportional to the energy of activation in Eyring’s coefficient of viscosity11, 12 is commonly used in approximating the prefactor. The coefficient of viscosity, 휂, is given by

휂 = (ℎ⁄푣푚) 푒푥푝 (퐸⁄푘퐵푇) (2.27)

Where E is the activation energy for the movement of molecules from the liquid to the growing crystal nucleus, 푣푚 is the molecular volume of the liquid, and h is the Planck constant.

On the basis of the viscous flow model, the pre-exponential factor, 퐾푙푠, is expressed by

1/2 푒푥푝(−퐸⁄푘퐵푇) 퐾푙푠 = 2(휎푙푠푘퐵푇) . 2/3 (2.28) ℎ푣푚

Combining equations 2.22 and 2.23, equation 2.23 transforms into

1/2 2(휎푙푠푘퐵푇) 퐾푙푠 = 5/3 (2.29) 푣푚 휂

Overall, the main advantage or attractiveness of the classical nucleation theory (CNT) that describes the vapor-liquid transition is the user-friendly nucleation rate expression that contains terms relating to easily accessible bulk properties. This will be applied in Chapter

3. In contrast, it is challenging to implement the liquid-solid CNT because quantifying liquid-solid interfacial tension under highly supercooled conditions is impractical.

Moreover, applying the liquid-solid CNT to the crystallization of n-alkanes is not reasonable because of the underlying assumption of random appearance of crystal nuclei in the bulk of a crystallizing system.

References

1. Gibbs, J. W., On the equilibrium of heterogeneous substances. American Journal of Science 1878, (96), 441-458.

2. Volmer, M. Kinetics of Phase Formation (Kinetik der Phasenbildung); Kinetik der

Phasenbildung ( Theodor Steinkopff Verlag, Dresdenund Leipzig): 1939.

3. Becker, R.; Döring, W., Kinetische behandlung der keimbildung in übersättigten dämpfen. Annalen der Physik 1935, 416 (8), 719-752.

4. Nanev, C. N., Theory of nucleation. In Handbook of Crystal Growth, Elsevier:

2015; pp 315-358.

5. Kashchiev, D., Nucleation: basic theory with applications. 2000. Butterworth

Heinemann: Boston.

6. McDonald, J. E., Homogeneous nucleation of vapor condensation. I.

Thermodynamic aspects. American Journal of Physics 1962, 30 (12), 870-877.

7. Sienfeld, J. H., Atmospheric chemistry and physics of air pollution. Willey

Interscience, New York 1986, 738.

8. Huang, J.; Lu, W.; Bartell, L. S., Isomeric differences in the nucleation of crystalline hydrocarbons from their melts. The Journal of Physical Chemistry 1996, 100

(34), 14276-14280.

9. Uhlmann, D.; Kritchevsky, G.; Straff, R.; Scherer, G., Crystal nucleation in normal alkane liquids. The Journal of Chemical Physics 1975, 62 (12), 4896-4903.

10. Yi, P.; Rutledge, G. C., Molecular simulation of crystal nucleation in n-octane melts. The Journal of chemical physics 2009, 131 (13), 134902.

11. Bartell, L. S.; Dibble, T. S., Electron diffraction studies of the kinetics of phase changes in molecular clusters: freezing of carbon tetrachloride in supersonic flow. The

Journal of Physical Chemistry 1991, 95 (3), 1159-1167.

12. Eyring, H., Viscosity, plasticity, and diffusion as examples of absolute reaction rates. The Journal of chemical physics 1936, 4 (4), 283-291.

CHAPTER 3

Nucleation of highly supersaturated vapors of lower n-alkanes

This chapter is part of the manuscript titled “Vapor phase nucleation of the short-chain n- alkanes (n-pentane, n-hexane and n-heptane): Experiments and Monte Carlo simulations” authored by Kehinde Ogunronbi, Aliasghar Sepehri, Bin Chen, and Barbara Wyslouzil which was published in The Journal of Chemical Physics 148, 144312 (2018). The experiments in this chapter were performed by Kehinde Ogunronbi and Barbara Wyslouzil.

Analysis of experimental data was done by Kehinde Ogunronbi under the guidance of

Barbara Wyslouzil. Monte Carlo simulations were performed by Aliasghar Sepehri and

Bin Chen. All figures in this chapter and the publication were plotted by Kehinde

Ogunronbi.

3.1 Introduction

n-Alkanes are interesting molecules, both because of their chain-like shapes and because the interactions between molecules are governed purely by dispersion forces.

These compounds are also important players in our everyday lives: C1-C4 are used in heating, cooking, and electric production; C5-C8 are good solvents for nonpolar substances;

C9-C16 are important diesel and aviation fuels; higher alkanes are usually cracked to produce lower alkanes. Apart from crude oil, a primary source of the shorter alkanes is raw natural gas. Separation processes that recover natural gas liquid (NGL) from raw natural gas streams often involve expanding and cooling the natural gas stream across

Joule-Thompson throttling valves, turbo-expanders, or supersonic nozzles,1 to condense the higher alkane. Thus, understanding vapor-liquid transitions of n-alkanes is technologically important.

The simplicity of the interactions between n-alkane molecules also makes them interesting candidates for vapor-liquid nucleation studies from a fundamental experimental and theoretical point of view. In particular, n-alkanes do not exhibit the strong vapor phase association that complicates the study of n-alcohols, especially methanol, 2 3 4 5 6 nor does dissociation or reaction play a role during nucleation as it can in acid-water-base systems

7 8 9 10 11 12 13. Investigating the vapor phase nucleation for a homologous series of molecules like the n-alkanes, that interact in the same way and differ only by an increasing number of -CH2 groups, remains an attractive way to test our understanding of this important physical phenomenon. A survey of the extensive experimental studies that have measured

critical supersaturations or nucleation rates for alkanes was presented in Ghosh et al.14 and will not be repeated here. In the broadest terms, experimental rates were generally higher, or critical supersaturations were lower, than those predicted by Classical Nucleation

Theory (CNT) over the measurement ranges, and disagreement between experiment and theory increased with chain length. Furthermore, the temperature dependence of CNT was significantly stronger than that of the experiments, suggesting that there are experimental conditions where CNT can predict experimental results. On a more positive note, both

Rusyniak15 and El-Shall and Ghosh et al.14 showed that Hale’s scaled nucleation theory16

17 could predict the observed nucleation rates to within ±2 orders of magnitude over a wide range of temperatures and supersaturations using only a single adjustable parameter. Since the disparity between experiment and theory is thought to primarily stem from CNT’s assumption that the physical properties of the small clusters are the same as the bulk properties of the new phase18 19 20 21, using simulations to test the validity of this aspect of the theory as a function of chain length and critical cluster size could lead to useful insight.

Molecular simulations of vapor-liquid nucleation have moved beyond relatively small atoms and molecules22 23 24, and, in some cases, molecular dynamics simulations can achieve system sizes and timescales that overlap with results from supersonic nozzle (SSN) experiments25 26. To date, n-alkane nucleation simulation studies include the MC work of

Chen and coworkers27, as well as the MD simulations of Horsch et al.28 Although the n- alkanes considered here are relatively large molecules, they are a good class of compounds to study in silico because their interactions can be described reasonably well using relatively simple united atom potentials. In particular, n-alkane phase behavior is 27

reasonably well described by the transferable potentials for phase equilibria-united atom

(TraPPE-UA) force field.29 30 31. One useful feature of this model is that the parameters are fit to ensure that the critical temperature, Tc, of the model is close to that of the real molecule. Given the importance of scaling both temperature and supersaturation when comparing experiments to simulation results, or simulation results determined using different intermolecular models32, using the TraPPE-UA potential ensures the absolute and scaled temperatures are equivalent in the simulations and the experiments.

In this paper we report homogeneous nucleation rates measured for the lower n-alkanes

– pentane, hexane and heptane – in a supersonic nozzle. We determine the critical cluster sizes of n-heptane by using isothermal nucleation rate data from nozzles characterized by different expansion rates. We compare our results with the predictions of different variants of Classical Nucleation Theory as well as Monte Carlo (MC) simulations using the aggregation-volume-biased method coupled with umbrella sampling and histogram reweighting.

The outline of this chapter is as follows: Section 3.2 presents an overview of the experimental set-up and data analysis, as well as the simulation approach and Section 3.3 summarizes the key experimental results. Section 3.4 compares experimental results with theoretical predictions and simulations. A summary of the work and our conclusions are found in Section 3.5.

3.2 Experiments and Data Analysis

3.2.1 Chemicals and Physical Properties

n-Pentane and n-heptane, each with purity of 99%, were purchased from Sigma-Aldrich whereas n-hexane with purity of 99% was purchased from ChemSampCo. All chemicals were used as received. Liquid argon (Ar) with purity of 99.998% was purchased from

Praxair. The thermophysical properties of these chemicals are listed in Table S3.1 of the supporting information.

Mass flow controller Heater Water bath 121mm 31.5mm 32mm Nozzle Mesh Regulator Hole (b) Liquid Vapor Ar 1 Gas generator P Pressure probe 2 2 : Heating tape A* = 59mm Ae = 224mm Water Plenum Flow controller T Heater P Mass flow Regulator . . Nozzle Liquid controller . . .. Ar 2 (c) Peristatic Balance pump (a) pumps

CaF2window

FIGURE 3.1(a) Experimental set-up for the pressure trace measurements used to measure the pressure profile along the axis of the nozzle (Adapted from Tanimura et al.33 [ J. Chem. Phys. 127, 034305 (2007)], with the permission of AIP Publishing). The Laval nozzle used in the experiments is machined from aluminum. (b) Unassembled nozzle blocks and (c) a side walls incorporating a CaF2 window.

3.2.2 Flow Apparatus and Supersonic nozzle

The experimental set-up used in the position-resolved pressure measurements is shown in Fig. 3.1. Briefly, argon gas is drawn from two Dewars and flows through heaters where it is warmed to room temperature. About 80% of the argon is drawn from tank 1, while the remaining is drawn from tank 2. Liquid n-alkane is pumped into the vapor generator where it is atomized and the droplets evaporate as they mix with the heated argon. After combining the condensable-rich stream with the main flow, the mixture (carrier gas + condensable vapor) is brought to the desired stagnation temperature in the plenum via a heat exchanger with water circulating from the water bath. As the mixture flows through the nozzle it expands, cools, and particles form and grow. The pressure drop required to maintain steady state flow is provided by two rotary vane vacuum pumps.

The temperature required to condense an n-alkane decreases with chain length. Thus, the current work requires a nozzle with a higher expansion rate than the nozzle used by Ghosh et al.14 Moreover, since an additional goal of our research is to observe freezing beyond the point of condensation, that is the subject of a separate paper, a longer nozzle was necessary. Hence, we designed a conventional Laval nozzle with a nominal opening half angle of 3.19o and a 12 cm long supersonic region. A schematic diagram of the nozzle is given in Fig. 14 of Tanimura et al.34 and photographs are shown in Fig.3.1(b) and 3.1(c).

A more detailed drawing of the nozzle block contour and a view of the nozzle assembly from the flow entry point are shown in Fig. S1 of the supplementary information.

All experiments start at a constant stagnation pressure, 푝0 = 30.1 kPa, that is maintained by adjusting the flow of argon from tank 1. The stagnation temperature 푇0, was 303.15 K for experiments with n-hexane and n-heptane, and 293.15 K for experiments with n- pentane.

3.2.3 Pressure trace measurements (PTM)

To determine the pressure variation of the gas mixture in the nozzle, a 76 cm long and

0.92 mm OD static pressure probe with holes ~ 31 cm from the tip, is moved along the nozzle centerline of the nozzle. In a typical dry or wet trace measurement, the pressure is measured from ~ 1.20 cm upstream to ~ 10.9 cm downstream of the effective throat. Values of p0, T0, p, and the mass flow rates of the carrier gas and condensable are recorded in a data acquisition system. The dry pressure trace yields the effective area ratio of the expansion (A/A*)dry. The wet trace is analyzed by integrating the equations that describe the continuity, momentum, and energy of the flow together with an equation of state. Input data are (A/A*)dry, the measured static pressure profile, and stagnation conditions as input.

This yields the remaining position resolved flow parameters, i.e. temperature T, density ρ, velocity u, and mass fraction of condensate g. The Fortran code used is in Appendix A.

3.2.4 Aerosol size distributions via SAXS

To characterize the sizes and number densities of the n-alkane droplets we conducted

SAXS experiments at the Advanced Photon Source at Argonne National Laboratory using the 12-ID_C beam line.35 We made measurements 7 cm downstream of the physical throat where, for these alkanes and the conditions examined, particle formation and growth are 31

complete. The nozzle used in SAXS experiments has the same expansion rate as the nozzle used for PTMs but each sidewall has a 12 cm x 1 mm slit covered by 25 µm thick mica windows. The 2-D scattering patterns were averaged and converted to 1-D spectra using the APS data inversion program. Data were placed on an absolute scale by following the calibration procedure outlined in Manka et al.36 We then fit the spectra assuming the droplets follow a Schultz polydisperse distribution of spheres.

3.2.5 Peak Nucleation rate and Characteristic time

We determine the maximum nucleation rates of n-pentane, n-hexane, and n-heptane by evaluating37 38:

푁 휌NZ (3.1) 퐽max = , ∆푡Jmax 휌vv where the droplet number density, N, is determined from SAXS data analysis and the density ratio 휌NZ⁄휌VV corrects for the expansion of the gas mixture density between the nucleation zone (NZ) and the viewing volume (vv). The characteristic time ∆푡Jmax, corresponds to the time required to produce the droplets in the nucleation zone. Since the ratio

퐽theory(푡) (3.2) ∫ 푑푡 퐽max,theory is relatively insensitive to the nucleation theory used, we assume that

퐽exp(푡) 퐽theory(푡) (3.3) ∫ 푑푡 = ∫ 푑푡 = ∆푡Jmax . 퐽max,exp 퐽max,theory

The flow time t is given by d푡 = d푥/푢, where 푥 is the distance downstream of the actual throat, and 푢 is the local velocity of flow at x, and Jtheory(t) is evaluated using Classical

Nucleation Theory. In particular, we use the time-resolved temperature and supersaturation profiles determined from the PTM to evaluate the Becker Döring nucleation rate18

2 2 3 2휎푣푙 푝푣 −16휋푣푚휎푣푙 퐽퐵퐷 = √ 푣푚 ( ) exp [ 3 2] (3.4) 휋휇푣 푘퐵푇 3(푘퐵푇) (푙푛푆)

as a function of time. In Eq. (3.4) 휎푣푙 is the vapor-liquid surface tension, 휇푣 is the mass of a condensable molecule, 푣푚 is the molecular volume of the liquid condensate, 푘퐵 is the

Boltzmann constant, and the supersaturation 푆 = 푝푣/푝푒푞(푇). Here 푝푣 is the pressure of the condensable vapor and 푝푒푞(푇) is the equilibrium vapor pressure of the condensable at temperature T. Although aerosol number densities are high enough that some coagulation is observed, this effect changes the measured nucleation rates by less than ~ 15% and is, therefore, ignored.38 The code used for the characteristic time is in Appendix B.

3.2.6 Simulation Methods

Simulations used the AVUS-HR approach as outlined in detail in Nellas et al.39 Very briefly, this is a Monte Carlo approach that combines aggregation-volume bias Monte

Carlo (AVBMC)40 41 42, self-adaptive umbrella sampling (US)43 and histogram reweighing

(HR)44. In addition, these simulations used the configurational-bias Monte Carlo method30

45 46 combined with the Jacobian-Gaussian algorithm47 48 for an efficient sampling of the

intramolecular geometries of n-alkanes. The cluster was defined using a Stillinger-type

49 cluster criterion with an energy cut-off of -260kB K. The transferable potentials for phase equilibria-united atom (TraPPE-UA) force field is used to describe the interactions between the model n-alkanes29 31.

FIGURE 3.2: The legend is the same for both figures. (a). Pressure and temperature profiles for condensing flows of n-pentane (n-C5), n-hexane (n-C6), and n-heptane (n-C7) at comparable inlet partial pressures of 429

Pa, 426 Pa, and 440 Pa, respectively. All expansions start at a total stagnation pressure, p0 = 30.1 kPa and the noted values of T0. For clarity, only the isentropic pressure ratio corresponding to the n-C5 mixture is shown.

For n-heptane, the strong increase in temperature near x= 5 cm is the signature of freezing. (b) The normalized nucleation rates, corresponding to the data in (a), were calculated using Classical Nucleation

Theory. Time t = 0 corresponds to the nozzle throat.

3.3 Results and discussion

3.3.1 Pressure trace experiments

Pressure trace measurements were conducted for 6 to 8 conditions for each alkane and the key results are summarized in Table 3.1. Figure 3.2(a) illustrates typical pressure and temperature profiles for expansions corresponding to n-alkane inlet partial pressures that differ by less than 3%. The onset of condensation corresponds to the location where the condensing flow profile deviates positively from the isentropic profile of the same gas mixture due to latent heat release. As expected from the trend in vapor pressure with chain length, the temperature at the onset of condensation decreases with chain length. At comparable partial pressures, heat release increases with chain length consistent with the increase in molar heat of vaporization with n. Finally, although it is not addressed in this paper, the second strong heat addition observed in the n-heptane temperature trace ~ 5 cm downstream of the throat, is due to droplet freezing50 51.

Figure 3.2(b) illustrates the corresponding normalized theoretical nucleation rates versus flow time for the three alkanes that were calculated using Classical Nucleation Theory and the temperature and partial pressure profiles determined by PTM. The nucleation pulses are reasonably symmetric with well-defined peaks, and the values of pJmax, TJmax, and SJmax reported in Table 3.1 are those corresponding to the peak in the normalized nucleation rate curves. In supersonic nozzles, nucleation is quenched by heat addition to the flow. Thus, the decrease in characteristic time with increasing chain length at constant pv0, directly reflects the more rapid heat addition for the longer chain alkanes. In particular, at the

conditions corresponding to the peak nucleation rates in Fig. 3.2(b), the rate of heat addition to the flow for n-heptane is ~ three times that for n-pentane.

Table 3.1: A summary of the pressure trace measurements results, the aerosol number densities, and the peak nucleation rates. T0 denotes the stagnation temperature, pv0 and y0 denote the initial partial pressure and initial mole fraction of the condensable, TJmax, pJmax, SJmax, and ∆푡퐽푚푎푥, denote the pressure, temperature, supersaturation, and characteristic time corresponding to the maximum nucleation rate. NZ/VV is the ratio between the density at the nucleation zone and the point of observation, N is the number density, and J is the maximum nucleation rate. All experiments used argon as the carrier gas and the stagnation pressure was p0

= 30.1kPa.

T0 pv0 푦0 TJmax pJmax 푆Jmax ∆푡Jmax NZ/VV N J

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

n-pentane

293.15 166.93 0.0055 109.37 11.30 1.69×106 14.18 1.09 2.41×1012 1.86×1017

293.15 244.91 0.0081 113.33 22.02 7.53×105 13.43 1.21 1.60×1012 1.44×1017

293.15 326.44 0.0108 118.76 30.52 1.83×105 20.75 1.26 1.02×1012 6.18×1016

293.15 428.9 0.0142 122.69 42.44 8.02×104 19.62 1.28 7.70×1011 5.02×1016

293.15 518.69 0.0172 125.44 55.36 4.88×104 17.17 1.32 4.94×1011 3.81×1016

293.15 688.49 0.0228 128.58 71.85 2.77×104 16.01 1.30 3.52×1011 2.86×1016

293.15 880.37 0.0292 131.7 99.90 1.77×104 15.94 1.32 2.76×1011 2.29×1016

n-hexane

303.15 138.26 0.0046 128.47 15.00 4.50×105 16.82 1.41 3.55×1012 2.92×1017

303.15 207.97 0.0069 132.53 22.80 2.08×105 16.35 1.43 2.08×1012 1.82×1017

303.15 279.23 0.0093 136.19 33.30 1.11×105 12.23 1.48 1.10×1012 1.33×1017

303.15 426.26 0.0141 142.99 56.13 3.33×104 11.91 1.54 6.42×1011 8.33×1016

303.15 604.19 0.0201 148.96 85.96 1.30×104 10.98 1.57 4.22×1011 6.05×1016

303.15 696.68 0.0231 150.29 100.48 1.14×104 10.64 1.57 3.12×1011 4.60×1016

n-heptane

303.15 131.9 0.0044 145.06 18.08 2.70×105 13.19 1.68 4.31×1012 5.48×1017

303.15 171.6 0.0057 149.77 25.94 1.10×105 11.73 1.74 3.35×1012 4.97×1017

303.15 250.8 0.0083 154.63 41.74 5.60×104 10.11 1.79 1.65×1012 2.91×1017

303.15 322.9 0.0107 160.09 58.10 2.30×104 9.00 1.87 1.24×1012 2.59×1017

303.15 381.4 0.0127 162.05 69.22 1.80×104 7.98 1.88 9.69×1011 2.28×1017

303.15 440.1 0.0146 165.08 83.58 1.20×104 7.82 1.91 7.92×1011 1.94×1017

303.15 554.2 0.0184 168.35 107.47 7.80×103 7.45 1.92 4.73×1011 1.22×1017

FIGURE 3.3: Change in pJmax with 1/TJmax for n-decane (C10) through n-pentane (C5). Grey and open symbols

14 are data from Ghosh et al. for the noted carrier gas, N2 or Ar, while black symbols are data from current work. The n-heptane data from Ghosh et al.14 (grey triangles) lie at higher temperature/lower pressure than the current n-heptane data consistent with the higher expansion rates used here. All data sets lie along reasonably parallel straight lines.

An interesting way to present the pJmax and TJmax values in Table 3.1, is illustrated in Fig.

3.3 In a Volmer52 plot, lines of constant nucleation rates should appear as relatively straight lines on a plot of log pJmax vs 1/TJmax. Although for the n-alkanes the measured nucleation rates increase by ~1-2 orders of magnitude as the temperature decreases over the measurement range (see Table 3.1 and Ghosh et al.14), the straight line relationship still holds for all of the data sets. With decreasing n-alkane chain length pJmax increases at fixed

TJmax or alternatively TJmax decreases at fixed pJmax, and both trends reflect the change in equilibrium vapor pressure with the size of the molecule. The difference between the two n-heptane data sets reflects the higher expansion rate of the current nozzle (T1) relative to the nozzle (C) used by Ghosh et al.14 In particular, the higher expansion rate generates a higher cooling rate and probes the metastable vapor region more deeply – i.e. faster expansions probe a region of pressure-temperature space further from the equilibrium line.

Thus, the data measured in nozzle T1 should lie above those of nozzle C – both in pressure and nucleation rate. Likewise, a carrier gas with a larger heat capacity ratio (훾) leads to a more rapid expansion in the same physical nozzle, and, thus, in nozzle C the data for n- nonane (C9) in Ar (훾 =1.67) lie above those for n-nonane in N2 (훾=1.4). As discussed in

Section 3.3.4, measuring nucleation rates in nozzles with different expansion rates is one way to obtain molecular-level information about the critical clusters that control nucleation in the supersonic flow38 53.

3.3.2 Aerosol SAXS Experiments

FIGURE 3.4: One-dimensional SAXS spectra for n-pentane measured as a function of pv0 at constant stagnation conditions p0 = 30.1kPa and T0 = 293.15 K respectively. The spectrum at pv0=880.4Pa is on the absolute intensity scale, while spectra at lower pv0 were scaled down by decreasing factors of 10. The size distribution parameters in the top right corner correspond to the best fit parameters for the pv0=880.4Pa data assuming scattering is from a polydisperse collection of spheres that follow a Schultz distribution.

Small angle x-ray scattering measurements were made 7 cm downstream of the throat where droplet formation and growth are complete. Figure 3.4 illustrates the radially averaged spectra for n-pentane droplets as pv0 increases by about a factor of 4. As in earlier

5 37 54 experiments , the point of inflection moves to lower q as pv0 increases, consistent with an increase in average droplet size as the available condensable increases. Although we only show the fit for the spectrum at 880.4 Pa, fits to the other spectra are equally good.

Table 3.3 summarizes the fit parameters as well as the values determined for N, polydispersity, 푃 = 휎/〈푟〉 and the volume fraction of droplets, 휙푆퐴푋푆, in the nozzle.

FIGURE 3.5: The variation in particle mean radius (〈푟〉), size distribution spread (σ), and number density

(N) with the partial pressure of condensable at the inlet to the nozzle. The dashed lines passing through each data are to guide the reader’s eye. Data in Nozzle C are those of Ghosh et al.14

Figure 3.5 illustrates the variation in the key size distribution parameters with pv0 for the three n-alkanes measured here. For n-heptane we also include the data of Ghosh et al.14

Since the average droplet radius increases more rapidly with pv0 than the spread, aerosol polydispersity, 휎/〈푟〉), decreases from ~0.3 to ~0.20 as pv0 increases. Although increases in 〈푟〉 with pv0 stem primarily from the commensurate increase in the volume of condensable

54 entering the nozzle, the decrease in N with increasing pv0 also pushes 〈푟〉 higher. The

14 54 decrease in N with pv0 is consistent with earlier observations and can be understood as follows: at higher pv0 the newly formed droplets can grow and release heat more quickly, thereby quenching nucleation more rapidly and reducing the number of droplets formed in the nucleation zone. This is consistent with the general trend that the characteristic time decreases with increasing pv0. The current results for n-heptane, Fig. 3.5(c), differ significantly from those of Ghosh et al.14 The higher number densities and smaller droplet sizes reflect the higher expansion rate, and therefore the higher cooling rate of the nozzle

(nozzle T1) used here relative to Ghosh et al.’s nozzle C. For some of the n-heptane and n- hexane experiments, PTMs suggest that the aerosol was frozen at the measurement location. For those conditions, we reduced the data using the liquid scattering length density but include an error bar that denotes the expected change in number density based on the difference in scattering length density between the liquid and solid states.

3.3.3 Experimental nucleation rates

Using Eq. (3.1) and the data presented in Table 3.1, we determined the experimental maximum nucleation rates for n-pentane, n-hexane, and n-heptane, and Fig. 3.6

summarizes these as a function of supersaturation for all n-alkanes investigated. Because nozzle T1 expands more rapidly than nozzle C, the nucleation rates for n-heptane are higher in nozzle T1 than in nozzle C. For all our experiments, the peak nucleation rates increase as supersaturation increases and temperature decreases. On an average, for each n-alkane, the nucleation rate changes by a factor of ~6 over an order of magnitude change in supersaturation. The degree of supersaturation required to condense n-alkanes is extremely high and reflects the relatively weak forces that govern interactions between the molecules.

FIGURE 3.6: Peak nucleation rates, Jmax, versus the supersaturation corresponding to the peak nucleation

14 rates, SJmax for n-pentane, n-hexane, and n-heptane from the current work. The heptane data of Ghosh et al. for measured in nozzle C are also included. The uncertainty in Jmax is represented by a vertical bar with a factor of 2, while the uncertainty in SJmax, primarily from temperature measurements, is within ±10%.

3.3.4 Determining the critical cluster size of n-heptane

The first nucleation theorem states that the number of molecules in the critical cluster can be determined from experimental data by calculating,55 56 57

푑(푙푛퐽) ( ) ≅ 푛∗. (3.5) 푑(푙푛푆) 푇

Within the framework of CNT or its variants, the number of monomers in the critical cluster, n*, is given by the Gibbs-Thompson equation as

2 3 ∗ 32휋푣푚휎푣푙 푛GT = 3 . (3.6) 3(푘퐵푇푙푛푆)

In our experiments, it is difficult to choose conditions that ensure the constant temperature condition in Eq. (3.5) is met for experiments conducted in different nozzles. We therefore used the approach developed in our earlier work38 53 58 where we correct the supersaturation for small temperature differences between the desired and measured values. Nucleation rates are not corrected because of the negligible difference in rates for such small changes in temperatures. Figure 3.7(a) illustrates the dependence of the n-heptane nucleation rates on supersaturation at four constant temperatures, and gives the corresponding values of n*exp, and n*GT. Further details regarding the calculation of n*exp are given in Table S3.3.

Figure 3.7(b) compares the critical cluster sizes of n-heptane derived from the first nucleation theorem to the predictions from classical nucleation theory. We also include the n* values derived from diffusion cloud chamber (TDCC) experiments of Rudek et al.59 and the MC simulations. For the latter, the temperatures are comparable to those of the current experiments, but supersaturations are lower. It is easy, however, to show analytically60 that

∗ when 푛GT for the simulations is evaluated using the physical property values consistent with the model potential, n*sim must be very close to n*GT. Moreover, as expected, the significantly higher supersaturations characteristic of our experiments yield critical cluster

sizes ~1/5 of those measured by Rudek et al.59 in the TDCC. In contrast to earlier water 53 and n-alcohol measurements 38, the measured critical cluster sizes are distinctly larger than those estimated by CNT. Furthermore, they agree quite well with the simulation results.

Very recently, Signorell and co-workers 61 investigated the nucleation and growth of n- propane and n-propane/n-ethane mixtures by sampling in the uniform flow field at the outlet of a supersonic Laval nozzle. The molecular content of the critical clusters was determined by resolving the masses of the molecular aggregates as a function of the nozzle inlet conditions via single photon ionization and mass spectrometry62. Using three supersonic nozzles characterized by different exit Mach numbers, they found average critical cluster sizes to be between 8 and 14, sizes that are close to the values derived here.

The TDCC data of Rudek et al.59, however, are in quite good agreement with the Gibbs-

Thompson predictions. Overall, the mismatch in the supersaturation dependence of the nucleation rate over large supersaturation ranges, as reflected in the critical cluster size, is not exclusive to n-heptane and has been observed for many other substances58 63 64 in particular for water. Whether this effect is real or is related to the difficulty associated with making measurements under conditions where the supersaturation dependence is steep and the critical size is large, is not fully resolved38.

FIGURE 3.7: (a) Isothermal peak nucleation rates, J, as a function of supersaturation, S. The lines through each isothermal data are power-law fits through which we determine the molecular content of the critical clusters. The values of n*exp and n*GT noted for each temperature. (b) The values of n*exp and n*sim are compared to those of n*GT. For our data, the vertical error bars correspond to the standard error of the slopes, and the horizontal error bars reflect the uncertainty in the supersaturation. The solid line corresponds to perfect agreement and the dashed lines correspond to limits.

3.4 Comparisons with theoretical predictions and simulations

3.4.1 Comparisons with Classical Nucleation Theories

To place the current results within the framework of the available literature, we first compare measured nucleation rates, Jexp, to rates predicted by three variants of the classical nucleation theory. In addition to the Becker Döring rate18 given in Eq (3.4), we also considered the Courtney variation19

1 퐽 = 퐽 (3.7) 퐶 푆 퐵퐷 and the Girshick-Chiu variant20

푒휃 퐽 = 퐽 . (3.8) 퐺퐶 푆 퐵퐷

1/3 휎 (36휋푣2 ) where 휃 = 푣푙 푚 . The latter two expressions correct the Becker Döring expression 푘퐵푇 to ensure mass balance (Courtney and GC) and return the monomer concentration for n =

1 in the underlying cluster distribution (GC).

Figure 3.8 illustrates the differences between the experimental and predicted rates as a function of normalized inverse temperatures for the current experiments (n-pentane to n- heptane) and our earlier work (n-octane and n-nonane). The disparity between the measured and predicted nucleation rates is evident in each case, as is the incorrect temperature dependence of all of the theories. Of the three approaches, JGC most closely matches the temperature dependence of the experimental observations although it generally overestimates the rates within the given temperature range. Understanding the origin of the mismatch in temperature dependence using only on experimental evidence is difficult. A complementary way to gain insight is to conduct molecular simulations.

FIGURE 3.8: Ratio of the measured and predicted nucleation rates versus reduced inverse temperature,

14 TJmax. The C5, C6, and C7 data are from the current work while the C8 and C9 data are from Ghosh et al. The straight lines through each set of data are only meant to guide the reader’s eyes.

3.4.2 Insights from molecular simulations The discrepancy between experimental nucleation rates and CNT or any of its variants, is thought to stem both from the difference in the physical properties of clusters relative to the bulk liquid phase, as well as the nonzero work of formation of the monomer inherent in JCNT. To determine how important these effects may be for the n-alkanes, we performed

MC simulations using the procedures outlined in Nellas et al.39 and found the nucleation free energy profile ∆퐺(푛) as a function of temperature and vapor phase densities over the ranges relevant to the experiments. Since the TraPPE-UA model is calibrated to match Tc, the ratio of temperature and reduced temperature are automatically the same between experiment and simulation.

Within the framework of Classical Nucleation Theory, a plot of 훿∆퐺(푛) versus

2 2 (푛3 − (푛 − 1)3) should be a straight line, where 훿∆퐺(푛) = ∆퐺(푛) − ∆퐺(푛 − 1) is the

difference in the formation free energy of clusters that differ by one monomer. When this

2 1/3 is true, a fit to the data yields the slope = (36휋⁄휌푙 ) 휎푣푙 and the y intercept, corresponding to the effective chemical potential ∆휇eff = 푘퐵푇 ln 푆, for the simulation. The size of the critical cluster is that n for which 훿∆퐺(푛∗)=0. Figure 3.9(a) illustrates typical data generated by the simulations, in this case for n-heptane at a vapor phase density, 휌v1=

2.63 × 1016 molecules/cm3 and T = 159 K. As the cluster size increases beyond n = 15, the 훿∆퐺(푛) values follow the expected straight line behavior. A fit to these data yields slope = 28.7 푘퐵T, ∆휇eff = 8.10 푘퐵푇, and n*sim = 13.7. Smaller clusters, however, have

훿∆퐺(푛) values that are distinctly lower than the straight-line extrapolation based on CNT.

Thus, it is clear that ∆퐺CNT(푛) is always larger than ∆퐺sim(푛). In either case the non-zero value corresponding to the free energy of formation of the monomer, ∆퐺(1) is subtracted from ∆퐺(푛). The behavior of ∆퐺(푛) at other vapor phase densities, 휌v2 is easily calculated using

휌 ∆퐺 (푛) = ∆퐺 (푛) − (푛 − 1)푘 푇 ln ( v2) (3.9) 휌v2 휌v1 퐵 휌v1 where the factor (n-1) ensures ∆퐺 of the monomer is zero.

Assuming that the kinetic prefactors are the same for the simulation and classical theory,

∗ the ratio Jsim/JCNT is related to the values of ∆퐺 by

퐽 Δ퐺∗ −Δ퐺∗ ln ( sim ) = CNT sim (3.10) 퐽CNT 푘퐵푇

∗ where the values required to evaluate ∆퐺CNT are derived from the fit to the simulation data.39 Over the temperature range of interest, Fig. 3.9(b) shows that on a scaled temperature plot the nucleation rate ratios for the three alkanes considered, are all very close and much closer than what we observe for the experimental data. Since both simulation and CNT are calculated ensuring ∆퐺(1) = 0, the difference in rates is primarily due to the cumulative difference in free energies of the small clusters. As temperature decreases at fixed 휌v, the critical cluster size decreases and the corrections due to smallest critical cluster sizes become more important, consistent with the increasing Jsim/JCNT ratio.

Although we attempted to simulate the longer chain alkanes, we could not produce smooth

훿∆퐺(푛) graphs at the temperatures of interest because, as illustrated in Fig. 3.10, the clusters were no longer liquid like and frequently adopted highly ordered structures. To identify the presence of a second phase transition within the cluster would require the use of an additional order parameter in the calculation for the convergence of the free energies65

66. The consistent behavior of the experimental data as a function of chain-length at these scaled temperatures (Fig. 3.8), relative to the predictions of CNT where the critical clusters are assumed to be liquid, does not provide evidence for a comparable transition in the real molecules. Whether this reflects differences in the true intermolecular potentials versus the model, or critical clusters that are warmer than the nominal temperature due to nonisothermal nucleation effects, are topics for further investigation. For example, the TraPPE-

UA model employs a single set of torsional parameters for all segments in all linear alkanes developed originally for n-butane by fitting to results obtained from the MM2 molecular mechanics calculations 67 while multiple sets of parameters were found necessary to 50

reproduce the enhanced stability of the gauche conformer observed for longer alkanes as evident from quantum mechanics calculations from butane to heptane.68 It is likely that the current model underestimates the population of the gauche defects for longer alkanes which in turn makes them more likely to form ordered structures.

16 3 FIGURE 3.9: (a) The 훿∆퐺(푛) values for a n-heptane simulation at 휌v1 = 2.62 × 10 molecules/cm and

T = 159K.The dashed red line is the fit to the simulation data for clusters containing more than 14 molecules.

(b) The ratio of nucleation rates from simulations (Jsim) and the predicted rates from classical nucleation theory (JCNT). The lines through each data set is meant to guide the reader’s eye. The simulation data exhibit almost universal behavior when plotted as a function of scaled temperature.

FIGURE 3.10: Snapshots of clusters of n-pentane through n-nonane at approximately constant Tc/TJmax ~

3.6. As the chain length increases, the clusters exhibit more order at the same scaled temperature.

3.4.3 Comparisons with nucleation rates from simulations

In order to directly compare the nucleation rates based on simulation to those of the experiments we first need to choose a prefactor for the simulations. In this paper we use the standard pre-factor from CNT and calculate the rate as

2 1/2 ∗ 휌푣 2휎푣푙 ∆퐺 퐽푠𝑖푚 = ( ) exp [− ], (3.11) 휌푙 휋휇푣 푘퐵푇

16, 17, 25, 26, 32, where 휌푙 is the liquid phase number density. We then need to scale the results

69,70 because our model compounds have different physical properties than the experimental materials. One way to do so is to follow Hale69 and others 25 , 26 and plot the rates as a

w function of the scaling parameter lnS/(Tc/T – 1) , where w is generally taken as 1.5 on the basis of scaling arguments. Empirically, Tanaka et al.25 found w = 1.3 did a better job for correlating experimental data for argon and simulations for Lennard-Jones molecules whereas Angelil et al.26 proposed w = 1.7 for water. Figures 3.11(a) and 3.11(b) compare the measured and simulated nucleation rates for temperatures between 159 and 260 K and values of w = 1.5 and 1.53, respectively. The slightly higher value of w provides better scaling for the simulation data at low S. We also conducted simulations at 150 K (see Figs.

3.14(a) and 3.14(b) in the supporting information). However, Jsim (T = 150 K) plotted as a

1.5 function of lnS/(Tc/T – 1) clearly deviates from the corresponding curves for Jsim (T =

1.5 159 – 260K) at low values of lnS/(Tc/T – 1) . This is not entirely surprising since, as illustrated in Fig. 3.10, heptane clusters at 150K (Tc/T = 3.6) appear somewhat more ordered than those of the lower alkanes, suggesting the onset of a crystalline critical cluster may occur at slightly lower T. Indeed, as illustrated in Fig. 3.15, at T = 142 K heptane clusters appear quite ordered whereas at temperatures at and above 159 K, the clusters are liquid-like. Finally, we note that the temperature range over which we have conducted the

MC simulations, ~ 100 K, is much broader than is typical, and, thus, other factors may begin to affect the scalability of Jsim.

Overall, as illustrated in Fig. 3.11, the simulations and the experimental trends are quite similar, and almost quantitative agreement is achieved when the simulated rates are reduced by ~ 3 orders of magnitude for either value of w. Although it is not obvious, the temperature dependence of the simulated rates in Fig. 3.11(a) is opposite to that of the measured rates of Rudek et al.59 and thus, increasing the value of w increases the spread in 53

the Rudek et al. data in Fig. 3.11(b). Similarly, as illustrated in Fig. 3.16, Jsim/1000 also matches the current rate measurement for n-hexane and n-pentane very well.

FIGURE 3.11: Rates from simulation agree quite well with the data if the former (solid lines) are reduced by factor of 1000 (dashed lines). (a) For the scaling exponent w = 1.5, the simulation rates do not scale perfectly at low values of of lnS/(Tc/T-1)1.5. (b) The scaling exponent of w = 1.53 does a better job of scaling the simulations but broadens the spread in the TDCC experimental data.

If we replot these data in a more standard Hale plot, Fig.3.12(a), using the same constant value of the scaling parameter Ω = 2.29 for the experiments and the simulations, the same good agreement is observed – especially if the simulated rates were reduced by a factor of

1000. Unfortunately, this approach is inconsistent in the sense that value of Ω for the simulations should differ from that of the experiments. In particular, Ω is the excess surface entropy per molecule in the cluster defined as

푣2/3휎 Ω = 푚 푣푙 (3.12) 푘퐵(푇푐−푇)

15 Earlier work found that Ωexp = 2.29, corresponding to Eq. 3.12 evaluated at T/Tc = 0.45, correlated the data well. For the simulations, the value of Ωsim is related to the slope of the

훿∆퐺(푛) curve by

slope 1 Ωsim = 1/3 . (3.13) (36π) 푘퐵(푇c−푇)

This yields values of Ωsim in the range of 2.47 < Ωsim < 2.51 depending on the temperature.

Using the correct value of Ωsim shifts the simulation curve so that it is roughly parallel to the perfect scaling line but offset by ~6 orders of magnitude as shown in Fig.3.12(b). If we interpret this offset as a difference in free energy of the critical cluster, it corresponds to roughly ∆퐺⁄푘퐵푇 = 13, i.e. the formation free energy for the simulations is consistently

27 ~13푘퐵T lower than expected. As noted in Chen et al. , the free energy of the critical clusters is not independent of the cutoff criterion and a tighter cutoff increases the free energy of the critical clusters. Thus, it may be possible to reduce this gap by several orders of magnitude by changing this factor in the simulation. In addition, the TraPPE-UA model overestimates the vapor-phase densities, which also contributes to the higher rates. Finally, we note that comparisons of this nature may be critical to improving our understanding of both natural and in silico nucleation experiments.

FIGURE 3.12: (a) In a Hale plot, simulation data follow the trend of the experimental data when Ω = Ωexp.

(b) Using Ωsim to scale Jsim yields a slope that is more consistent with scaling but introduces a substantial

3 offset. The variable C0 is related to Ω via 퐶0 = (16휋⁄3 ln 10)Ω .

3.5 Summary and Conclusions

We measured the vapor to liquid nucleation rates of n-pentane, n-hexane, and n-heptane by combining the results of PTMs and SAXS experiments. The experimental nucleation rates ranged from 2.29×1016 to 5.48×1017 cm-3s-1 at temperatures between 109 K to 168 K.

The first nucleation theorem was used to estimate that the size of n-heptane critical clusters, n*, increased from ~8 to ~12 as temperature increased from 150K to 165K. These values are larger than those previously estimated for water or the n-alcohols in supersonic nozzle experiments, and distinctly above those predicted by CNT. They are, however, comparable to values determined for n-propane condensing in supersonic flow by Signorell and co- workers using mass spectrometry61.

When comparing the measured rates to the predictions of three variants of CNT, the experimental nucleation rates for n-pentane to n-heptane are 4-7 orders of magnitude higher than JBD, 9-13 orders of magnitude higher than JC, and 1-3 orders of magnitude lower than JGC at reduced inverse temperatures ranging from 3.2 to 4.3. Despite being highly supercooled with respect to the solid, MC simulations using the TraPPE-UA model still found that the critical clusters appeared liquid-like for n-pentane through n-heptane in the temperature range of experimental interest. In contrast, n-octane and n-nonane clusters appeared well ordered. Comparing experimental nucleation rates to predictions from calculations based on MC simulation methods, yielded rates that were offset by ~ 3 orders of magnitude from the available data when rates were plotted as a function of lnS/(Tc/T –

1)w for w = 1.5 or 1.53. Furthermore, on a Hale plot the simulation rates followed the same trend as the experimental data when both used the scaling parameter derived from the experimental data. Unfortunately, doing so is inconsistent since Ωexp differs from Ωsim.

However, when the simulation data use the scaling parameter consistent with Ωsim, they follow a line reasonably parallel to perfect scaling with an offset of ~6 orders of magnitude.

Some of this offset may be due to the choice of the cut-off criterion used in the simulation.

Finally, we note that although it is both tempting and important to directly compare experiments and simulation, it is equally important to ensure that this comparison is conducted in the most consistent manner possible.

Supplemental Information

The supplemental information contains the physical property correlations used to evaluate the experimental data (Table 3.2), aerosol size distribution parameters 7 cm downstream of the physical throat (Table 3.3), experimental nucleation rates and corrected supersaturations used to determine critical cluster sizes (Table 3.4). Figure 3.13 provides additional details regarding the nozzle design. Figure 3.14 summarizes the rates from simulation and experiments as a function of scaled supersaturation over a temperature range of 110K. Figure 3.15 contains snapshots of typical n-heptane clusters as temperature

1.5 decreases. Figure 3.16 summarizes the nucleation rates as a function of lnS/(Tc/T – 1) for n-pentane and n-hexane and compares these to the simulation results.

Supplemental Information

Table 3.2: Thermophysical properties of n-pentane, n-hexane, n-heptane, n-octane, n-nonane, and argon.

71 71 71 The properties are 휇, the molecular weight ;Cpl, heat capacity of the liquid ;Tc, critical temperature ;Ttriple

72 73 point, the triple point ;Cp (T), molar heat capacity of the vapor and carrier gas ; Pe (T), equilibrium vapor

73 73 74 75 pressure ; ∆Hvap, heat of vaporization ; ρl, bulk liquid density ; ζ, surface tension .

n-pentane

μ (g mol-1) 72.15

Tc (K) 469.8

Ttriple point (K) 143.48

pc (MPa) 3.375

-1 -1 cpl (J g K ) 2.32 (at 298.15 K)

(1404.5312 /T )2  exp(−1404.5312 T ) 86.389058 +163.62772 C (T) (J mol-1 2 p (1− exp(1404.5312 T )) 2 -1 (3247.1465/T )  exp(− 3247.1465 T ) K ) +125.55904 (1− exp(− 3247.1465 T ))2

−3 −6 2 Pe (kPa) 101.325 exp((1− 309.209/T ) exp(2.73425 −1.96654410 T + 2.40840610 T ))

 1 3 2 3  1+1.177555(1− T Tc ) + 3.891572(1− T Tc ) − 5.508958(1− T Tc ) -3 0.232 ρl (g cm )  4 3  + 3.291806(1− T Tc ) 

ζ (N m-1) 0.01827 −1.089110−4 (T − 273.15)

R  exp 2.73425 −1.96654410−3T + 2.40840610−6 T 2 -1 ( ) ∆Hvap (J mol ) 309.209 + T(T − 309.209)(−1.96654410−3 + 2  2.40840610−6 T )

n-hexane

μ (g mol-1) 86.18

Tc (K) 507.9

Ttriple point (K) 177.87

pc (MPa) 3.035

-1 -1 cpl (J g K ) 2.27

(1400.5301/T )2  exp(−1400.5301 T ) 101.85997 +196.40919 C (T) (J mol-1 2 p (1− exp(−1400.5301 T )) 2 -1 (3214.2702 /T )  exp(− 3214.2702 T ) K ) +137.69426 (1− exp(− 3214.2702 T ))2

−3 −6 2 Pe (kPa) 101.325 exp((1− 341.863 T ) exp(2.79797 − 2.02208310 T + 2.28756410 T ))

0.2341+1.597561(1− T T )1 3 +1.842657(1− T T )2 3 −1.726311(1− T T ) -3 c c c ρl (g cm ) 4 3 5 3 + 0.4943082(1− T Tc ) + 0.6463138(1− T Tc ) }

ζ (N m-1) 0.02050−1.043910−4 (T − 273.15)

R  exp 2.79797 − 2.02208310−3 T + 2.28756410−6 T 2 -1 ( ) ∆Hvap (J mol )  341.863 + T(T − 341.863)(− 2.02208310−3 + 2  2.28756410−6 T )

n-heptane

μ (g mol-1) 100.204

Tc (K) 540.11

Ttriple point (K) 182.60

pc (MPa) 2.735

-1 -1 cpl (J g K ) 2.25 (at 298.15 K)

(3154.9913 T )2 exp(− 3154.9913 T ) -1 117.22475+151.73507 2 Cp(T) (J mol (1− exp(3154.9913 T )) 2 -1 (1391.9171 T ) exp(−1391.9171 T ) K ) + 227.31996 (1− exp(−1391.9171 T ))2

−3 −6 2 Pe (kPa) 101.325exp((1−371.552/T )exp(2.8647 − 2.11320410 T + 2.25099110 T ))

-3 1 3 2 3 4 3 ρl (g cm ) 0.2361+1.331593(1−T Tc ) + 3.300918(1−T Tc ) − 4.509610(1−T Tc )+ 2.765491(1−T Tc ) 

ζ (N m-1) 0.02231−1.014810−4 (T − 273.15)

R  exp 2.86470 − 2.11320410−3 T + 2.25099110−6 T 2 -1 ( ) ∆Hvap (J mol ) 371.552 + T(T − 371.552)(− 2.11320410−3 + 2  2.25099110−6 T )

argon

-1 μv (g mol ) 39.948

-1 - Cpv(T) (J g K 0.5203 1)

− . -10 gas (Å ) 1.25 10

Table 3.3: Aerosol size distribution parameters 7 cm downstream of the physical throat for the n-alkanes:

〈푟〉 is the average radius of the droplets,  is the spread in radius, N is the number density of the droplets, P is the polydispersity of the droplets, and ∅SAXS is the volume fraction of the droplets. T0 is the stagnation temperature and pv0 is the partial pressure of the condensable at the inlet to the supersonic nozzle.

T0 pv0  N

(K) (Pa) (nm) (nm) (cm-3) P= /r ∅SAXS

n-pentane

293.15 166.93 3.89 1.00 2.41×1012 0.26 7.17×10-7

293.15 244.91 5.44 1.29 1.60×1012 0.24 1.27×10-6

293.15 326.44 7.26 1.65 1.02×1012 0.23 1.89×10-6

293.15 428.90 8.68 1.88 7.70×1011 0.22 2.41×10-6

293.15 518.69 11.29 2.30 4.94×1011 0.20 3.36×10-6

293.15 688.49 13.82 2.62 3.52×1011 0.19 4.32×10-6

293.15 880.37 15.82 2.94 2.76×1011 0.19 5.06×10-6

n-hexane

138.26 3.83 1.18 3.55×1012 0.31 1.07×10-6 303.15

303.15 207.97 5.31 1.47 2.08×1012 0.28 1.62×10-6

303.15 279.23 7.47 1.95 1.10×1012 0.26 2.34×10-6

303.15 426.26 10.26  6.42×1011 0.24 3.42×10-6

303.15 604.19 12.93 2.74 4.22×1011 0.21 4.36×10-6

303.15 696.68 15.3 3.07 3.12×1011 0.20 5.25×10-6

n-heptane

131.9 3.71 1.48 4.31×1012 0.40 1.41×10-6 303.15

303.15 171.6 4.36 1.64 3.35×1012 0.38 1.71×10-6

303.15 250.8 6.49 2.04 1.65×1012 0.31 2.48×10-6

303.15 322.9 7.76 2.26 1.24×1012 0.29 3.08×10-6

303.15 381.4 8.97 2.51 9.69×1011 0.28 3.66×10-6

303.15 440.1 10.08 2.76 7.92×1011 0.27 4.20×10-6

303.15 554.2 13.01 3.02 4.73×1011 0.23 5.09×10-6

155K, 160K, and 165K.

T, K SJmax (C) SJmax (T1) Jmax (C) Jmax (T1) n*exp n*GT

150 9.3 x 104 1.2 x 105 4.5 x 1016 5.0 x 1017 8.5 4.5 155 4.5 x 104 5.4 x 104 2.6 x 1016 2.9 x 1017 9.4 4.8 160 2.2 x 104 2.5 x 104 1.4 x 1016 2.6 x 1017 12.0 5.3

165 10.5 x 103 1.2 x 104 1.1 x 1016 1.9 x 1017 12.5 5.9

Figure 3.13: (a). Side view of one of the shaped aluminum blocks, showing the dimensions in inches. (b). A view of the whole nozzle assembly from the flow entry point. The nozzle contour is described by the following equations:

0 < 푥 < 1.26" , y =0.757

1.26" < 푥 < 2.00" , y =0.40푥+0.2530

2.0" < 푥 < 2.79" , y =0.3252푥3 − 2.6725푥2 + 7.2642푥 − 5.3873

2.79" < 푥 < 7.25" , 푦 = −0.0557푥 + 1.2958

1000.

Figure 3.16: Solid lines correspond to Jsim, dashed lines to Jsim/1000. The MC simulation results agree quantitatively with the experimental data when they are reduced by a factor of 1000. (a) n-pentane (b) n- hexane. For these alkanes the scaling parameter w = 1.5 works well over the limited (20 K) temperature range of the simulations.

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CHAPTER 4

Properties of the critical clusters of the lower n-alkanes

This chapter is part of the manuscript titled “Properties of the critical clusters of the lower n-alkanes” authored by Kehinde Ogunronbi and Barbara Wyslouzil which will be submitted to The Journal of Chemical Physics. The experiments in this chapter were performed by Kehinde Ogunronbi and Barbara Wyslouzil. Analysis of experimental data was done by Kehinde Ogunronbi under the guidance of Barbara Wyslouzil.

4.1 INTRODUCTION Supersaturated vapors are a collection of molecules and loosely-bound molecular clusters in which the molecular clusters become increasingly stable as the supersaturation increases. Understanding the properties of these molecular clusters, especially for weakly bound systems, is an important first step to improving our understanding of nucleation physics. This understanding has been advanced by applying the first1, 2 and second3, 4 nucleation theorems to the measured dependence of the nucleation rate on supersaturation and temperature, respectively.

For the n-alkanes, Ford3, 4 derived the molecular content and excess energies of the critical clusters of n-heptane through n-decane from data measured in expansion 5 and diffusion chambers6, 7 where characteristic nucleation rates are 108 cm-3s-1 and 1 cm-3s-1, respectively. For these data, critical clusters were found to contain about 6 – 70 molecules, with the smallest clusters formed in the expansion chamber. These values were not compared with theory, however our prior determination of n-heptane cluster sizes using

Rudek et al.’s data7 showed that the sizes from experiment were in reasonable agreement with theory8. The excess internal energies for all of the n-alkanes studied were over- predicted (less than a factor of two) by the capillarity approximation and this discrepancy was attributed to the inappropriateness of Classical Nucleation Theory’s (CNT) use of bulk liquid phase properties to describe small molecular clusters. Vehkamäki and Ford 9 also analyzed the nucleation rate data measured by Doster et al.10 for n-octane, i-octane, and their mixtures in a pulsed expansion cloud chamber, 102 ≤ 퐽 (푐푚−3푠−1) ≤ 5 × 104. For n-octane, they found that the critical clusters contained 19 to 24 molecules with uncertainty

of ~ 5 – 10%. In their study, they found that the classical theory predicted the excess energy of the critical clusters quite well but failed to predict the size.

It is perhaps surprising that efforts to date have focused on the relatively longer

(퐶푛퐻2푛+2, 7 ≤ 푛 ≤ 10) n-alkanes, despite the importance of lower n-alkane

(퐶푛퐻2푛+2, 푛 ≤ 6) nucleation in separation processes involving raw natural gas, in understanding the role of molecular shape in vapor phase nucleation, and in developing accurate models for predicting the behavior of more complex molecules.

Studying nucleation and condensation of supersaturated n-alkane vapors in supersonic nozzles ensures that the vapor is pushed far into the metastable region, and the resulting high homogeneous nucleation rates ensure that the probability of heterogeneous nucleation is decreased. The latter also makes these data the most amenable to direct MD simulation although complete overlap is still difficult. Our recent work on the homogeneous nucleation of short-chain n-alkanes8 found a strong dependence of experimental peak nucleation rates on supersaturation and temperature, and for n-heptane we were able to estimate the critical cluster size for a limited range of conditions. Based on the success of the prior experiments, we made additional nucleation rates measurements in a significantly faster nozzle. By combining the new and old data sets, we can confirm our earlier results for n-heptane and determine both the size and excess internal energies of the critical clusters for all three lower n-alkanes – n-pentane, n-hexane and n-heptane – by applying the first and second nucleation theorems.

4.2 EXPERIMENTS AND DATA ANALYSIS 4.2.1 Chemicals

Both n-pentane and n-heptane (purity of 99%), were purchased from Sigma-Aldrich, while n-hexane (purity of 99%) was purchased from ChemSampCo. All chemicals were used without further purification. Liquid argon (Ar, purity of 99.998%) was purchased from Praxair. The thermophysical properties of the chemicals used to analyze the data are reported in Table S-1 of ref. 8.

4.2.2 Supersonic Nozzles, PTM, and SAXS

The experimental set-up, procedures, and data analysis used in this work are described in our earlier publication8, and only salient information will be highlighted here.

Determining critical cluster sizes and excess energies requires measurements of nucleation rates as a function of supersaturation at constant temperature and nucleation rates as a function of temperature at constant supersaturation, respectively. Given that nucleation and growth are spontaneous in the expanding flow of a supersonic nozzle, it is difficult to use a single supersonic nozzle to determine nucleation rates as a function of supersaturation at constant temperature. Instead, different nozzles characterized by different expansion rates yield peak nucleation rates Jmax, corresponding to different values of supersaturations SJmax at a fixed temperature TJmax, or conversely, different rates at different TJmax and fixed SJmax.

In this work we combine data from nozzles C, T1, and T3, whose effective area ratios are illustrated in Figure 4.1. The linear expansion rates, d(A/A*)/dz, of these nozzles are 0.075

cm-1, 0.147 cm-1, and 0.315 cm-1, respectively, when argon is the carrier gas. Thus, the expansion rate of nozzle T3 is 2.3 times that of nozzle T1, and 4.6 times that of nozzle C.

In a pressure trace experiment, we first characterize the effective area ratio of the nozzle by measuring the static pressure as a function of position along the centerline of the nozzle when the flow consists of pure carrier gas. We then measure the static pressure of the condensing or “wet” gas mixture and solve the diabatic flow equations using pressure and area ratio as the measured variables. This yields estimates for the temperature T, density ρ, mass fraction of condensate g, and velocity of the flow, u. Our interest is in the onset of condensation, or the point of maximum nucleation. Here boundary layer compression is 81

negligible,11 the temperatures and pressures are still very close to the isentropic values12,

13, and we do not need to solve the diabatic equations using the iterative procedures described in our earlier work14, 15, 16, 17.

We performed fixed-position SAXS measurements at the 12-ID_C beam line at the

Advanced Photon Source, Argonne National Laboratory8, 18 using 0.1 nm X-rays and a sample to detector distance of 2.0 m. The 2-D scattering data were reduced to 1-D scattering spectra and the latter were fit assuming the aerosol was a polydisperse collection of spheres that followed a Schulz size distribution. From these measurements, we determine the average particle size, , the width of the size distribution, σ, and the number density of particles, N. Using the information from both PTM and SAXS, we calculate the maximum nucleation rates in the supersonic nozzle.

All experiments in nozzle T3 started from 푝0 = 60.5 kPa with 푇0 =

293.15 K, 303.15 K, 318.15 K for pentane, hexane, and heptane, respectively.

Experiments in nozzle T1 started from 푝0 = 30.1 kPa with 푇0 = 293.15 K for pentane

8 18 and 푇0 = 303.15 K for hexane and heptane. For n-heptane, stagnation conditions in nozzles C were 푝0 = 30.2 kPa with 푇0 = 283.15 K or 318.15 K.

4.3 RESULTS AND DISCUSSION 4.3.1 Pressure Trace Measurements and characterizing the onset of condensation Typically, the expansion of a condensing mixture of carrier gas and n-alkane in a supersonic nozzle leads to increased supersaturation of the vapor mixture. This, in turn

leads to an increase in the formation rate of molecular clusters, and eventually the collapse of the supersaturated state as clusters that are large enough grow rapidly via condensation thereby depleting the vapor and adding heat to the flow. The onset of condensation, defined here as the conditions corresponding to the maximum nucleation rate, is a suitable parameter to characterize the limit of stability in an expanding flow19. Figure 4.2 summarizes these onset pressures and temperatures in nozzles C,18 T1,8 and T3 for the three n-alkanes. For each n-alkane, lower temperatures TJmax are reached at the same value of pJmax in the more rapidly expanding nozzle. For example, pJmax of n-pentane is ~ 99 Pa when TJmax = 132 K in nozzle T1 and 127 K in nozzle T3. Likewise, as summarized in

Table 4.2 of the supporting information, the position downstream of the throat at which the maximum nucleation occurs, 푥Jmax, is 3.81 cm and 1.49 cm in nozzles T1 and T3, respectively. Thus, more rapidly expanding nozzles can often explore a broader range of nucleation conditions than nozzles with slower expansion rates.

An interesting aspect of Figure 4.2 is that within each particular nozzle, the onset data form straight lines that appear to be systematically spaced with respect to carbon number. To test whether this is true, we plotted the temperatures corresponding to a constant value of pJmax and, over the limited range of carbon number, and find for a given nozzle there is a linear relationship that reflects a similar linear relationship observed for the vapor pressure.

We summarize this in Fig. 4.3 and show the consistency of the thermodynamic properties and acentric factor of these n-alkane molecules with carbon number.

Figure 4.2: Pressures and inverse temperatures corresponding to point of maximum nucleation in three nozzles characterized by different expansion rates. Grey closed symbols are data measured in nozzle T3, dark closed symbols are measured in nozzle T1, and open symbols are data measured in nozzle C. Data from nozzles T1 and C are from Ogunronbi et al.8 and Ghosh et al.18, respectively. The lines on each n-alkane data are meant to guide the reader’s eyes.

4.3.2 Particle sizes, Number densities, and Nucleation rates

Figure 4.4 summarizes the average particle sizes, size distribution spread, and number densities of the n-alkane aerosols generated in the nozzles T1 and T3. For n-heptane, the data from nozzle T3 are shown as filled solid squares where the fits were unambiguous, and as crossed squares where the uncertainty in the parameters were unacceptably large.

The difficulties associated with fitting the n-heptane data in nozzle T3 are illustrated in Fig.

4.13 of the supporting information.

Data for all n-alkanes follow similar trend in both nozzles T1 and T3. In particular, the higher number densities in nozzle T3 reflect the higher expansion rate in nozzle T3 than nozzle T1. Conversely, the average aerosol sizes in nozzle T1 are higher than in T3. This is consistent with our earlier observations8 for n-heptane in nozzles C and T1. For all of the n-hexane experiments, PTMs for all position-resolved SAXS measurement for the highest pv0 suggest that the aerosols were frozen at the measurement position. Hence, we follow the same procedure in our earlier work8 and include error bars on the number densities based on the uncertainty in the density of solid vs liquid alkane. Using these parameters, available in Table 4.3 of the supporting information, we determine the maximum nucleation rates.

Figure 4.5 summarizes the peak nucleation rates, Jmax, as a function of supersaturation corresponding to the peak nucleation rate, SJmax, derived from nozzles T3, T1, and C. The values of Jmax and parameters corresponding to the point of Jmax are available in Table 4.4 of the supporting information for nozzle T3. The higher nucleation rates in nozzle T3 for all n-alkane experiments reiterate the higher expansion rate in nozzle T3 than both nozzles

T1 and C. In addition, compared to nozzles T1, higher supersaturations and lower temperatures are reached in nozzle T3. For example, in Table 4.2 of the supporting information, at similar pv0 = 244 Pa of n-pentane, the temperatures corresponding to the point of maximum nucleation are 113 K and 107 K, in nozzles T1 and T3, respectively.

Figure 4.4: Number density, average size of droplets, and width of size distribution as a function of condensable partial pressure. Experiments in nozzles T1 and T3 started at p0 = 30.2kPa and 60.5kPa, respectively. For clarity, we omit data for n-heptane in nozzle C. The open squares correspond to the data where signals are weak and particle sizes are very small.

±10% change in supersaturation. The open symbols for n-heptane denote the conditions where the uncertainty in number density is high due to the weak scattering signals and the strong potential for coagulation to play a more significant role than normal. 88

4.3.3 Size and Excess Internal Energy of critical lower n-alkane clusters

Since the key goal of this work was to estimate the size and excess energy of the critical clusters of lower n-alkanes, extracting the properties of critical clusters directly from macroscopic measurements without having to determine the free energy of cluster formation20, 21, 22, 23 or computing growth and decay probabilities of clusters24, has proven to be an attractive one. As mentioned earlier, the first nucleation theorem, derived by

Kashchiev1, 2, 25 is written as

휕푙푛퐽 푛∗ ≅ ( ) (4.1) 휕푙푛푆 푇 where we derive the size of the critical cluster, n*, from the slope of a log-log plot of the supersaturation dependence of the nucleation rate at constant temperature. To apply this theorem to our data, we use two procedures. First, we follow the approach developed in our earlier work26 where we correct the supersaturation for the slight temperature differences between the measured and desired values. Secondly, we use equations from the fit of nucleation rates vs temperature and supersaturation over the range of overlap to estimate the nucleation theorems for all values of temperature or supersaturation in the region of overlap. To accommodate for this second procedure, we slightly modified

Equation 4.1 to determine n*. Further details on this are in the supporting information.

Shown in Table 4.1 are values of desired supersaturations, nucleation rates, experimental and theory-based critical cluster sizes of n-pentane through n-heptane. Plotted in Fig. 4.6 are isothermal supersaturation dependence of the nucleation rates for n-pentane through n- heptane. In all cases, the sizes of the critical clusters of the n-alkanes, n*, increase with 89

temperature. For n-pentane, n* increases from 1.7 to 8.7 as temperature increases from 110

K to 129 K; for n-hexane, n* increases from 3.3 to 8.0 as temperature increases from 128.8

K to 143.5 K, and for n-heptane, n* increases from 6.6 to 8.3 as temperature increases from

150 K to 155 K. The small size (n* = 1.7) of the critical cluster of n-pentane at very low temperature is not surprising as we have reported similar size for n-butanol measured in supersonic nozzles27 at 220 K. This may suggest that at such low temperature for n-pentane, where the density is very low, generation of small clusters in a collisionless free-molecular regime is pronounced28. Figure 4.7 compares the 푛∗ from nucleation theorem to the

∗ predictions from capillarity approximation, 푛퐺푇. Overall, for all n-alkanes, theory underpredicted the critical cluster sizes from experiment. This is in reasonable agreement with our prior results of the critical cluster n-heptane using two nozzles characterized by different expansion rates8.

Table 4.1: Experimental values of the critical clusters of n-pentane through n-heptane at temperatures ranging from 110 K through 155 K. At each temperature, supersaturations, SJmax, corresponding to the peak nucleation rates, Jmax, in nozzles C, T1, and T3, critical cluster sizes from experiments and capillarity approximation are shown.

n-pentane T, K SJmax (T1) Jmax (T1) SJmax (T3) Jmax (T3) n* n*GT 6 17 6 17 110 1.35 × 10 1.86 × 10 3.00 × 10 7.20 × 10 1.7 3.3 5 17 6 17 113 7.19 × 10 1.44 × 10 1.46 × 10 6.61 × 10 2.0 3.5 4 16 4 17 125 5.84 × 10 3.81 × 10 8.33 × 10 4.71 × 10 6.4 4.5 4 16 4 17 129 2.53 × 10 2.86 × 10 3.20 × 10 3.71 × 10 8.7 5.1 n-hexane T, K SJmax (T1) Jmax (T1) SJmax (T3) Jmax (T3) n* n*GT 5 17 5 18 128.8 4.01 × 10 2.92 × 10 8.03 × 10 3.02 × 10 3.3 3.7 5 17 5 18 136.5 1.09 × 10 1.33 × 10 1.73 × 10 1.60 × 10 5.1 4.1 4 16 4 17 143.5 3.33 × 10 8.33 × 10 4.27 × 10 9.42 × 10 8.0 4.8 n-heptane

T, K SJmax (T1) Jmax (T1) SJmax (T3) Jmax (T3) SJmax (C) Jmax (C) n* n*GT 150 1.15 × 105 4.97 × 1017 1.72 × 105 3.80 × 1018 9.27 × 104 4.55 × 1016 6.6 4.3 155 5.41 × 104 2.91 × 1017 7.54 × 104 2.93 × 1018 4.49 × 104 2.56 × 1016 8.3 4.6

Figure 4.6: (a). Fits to nucleation rate versus supersaturation used to derive n* for n-pentane at temperatures

(from right to left) 110 K, 113 K, 125K, and 129 K. (b). Fits used to derive n* for n-hexane at temperatures

(from right to left) 128.8 K, 136.5 K, and 143.5 K. (c). Fits for n-heptane at temperatures (from right to left)

155 K and 150 K. The vertical error bars correspond to a factor of 2, and horizontal error bars reflect the uncertainty in the supersaturation. The critical cluster sizes at different temperatures are shown in boxes.

Ex,cl. The dashed lines are from correlations between Jmax and TJmax in nozzles of different expansion rates discussed in the supporting information.

We use the slopes from Figure 4.8 to determine the excess energy of the critical cluster by applying the second nucleation theorem given by

∗ 2 휕푙푛퐽 퐸푥(푛 ) = 푘푇 ( ) + 푘푇 − 퐿 (4.2) 휕푇 푆 where k and L are the Boltzmann’s constant and latent heat of vaporization per molecule, respectively.

To determine the temperature derivative of J in Equation 4.2, we use the procedure developed by Kim et al.26 Briefly, we averaged the supersaturation of each isothermal data set and used the average value to determine the corresponding temperature from the temperature-mass flow rate correlation (plotted in Fig. 4.10 of the supporting information)

for each nozzle. Figure 4.8 illustrates the temperature dependence of n-alkane nucleation rates at constant supersaturations (ln S = 14.51, 13.84, 11.15, and 10.26 from left to right for n-pentane; ln S = 13.25, 11.83, and 10.54 from left to right for n-hexane; ln S = 11.55,

10.80, 10.06, and 9.32 from left to right for n-heptane). We use the slope from each line of constant supersaturation to evaluate Equation 4.2 and determine the excess energy, 퐸푥, of the critical clusters of each n-alkane. To the best of our knowledge, extensive data from other nucleation experiment devices for lower n-alkanes are scarce - there is only one source for n-heptane7 and none for both n-hexane and n-pentane. Since the data of Rudek et al.7 provide the supersaturation dependence of the nucleation rate at constant temperatures, we analyze the data by evaluating3

푑(ln 푆푐푟푖푡) 퐸 (푛∗)+2(퐿−푘푇) 푇 푣 = − 푥 (4.3) 푑푇 푘푇(푛∗+1)

푐푟𝑖푡 where 푆푣 is the critical supersaturation required to produce a nucleation rate 퐽 =

1푐푚−3푠−1.

To place these results within the framework of the classical nucleation theory, we compare 퐸푥 from experiments with the excess internal energy from capillarity approximation3, 4

푑휁 퐸 (푛∗) = (휁 − 푇 ) (36휋푣2 )1/3푛∗2/3 (4.4) 푥,푐푙 푑푇 푚

where 휁 is the bulk liquid surface tension and 푣푚 is the volume of a molecule.

In Figure 4.9, excess internal energy is plotted against the critical cluster size of n- alkanes. Evaluating Equation 4.4 at the two extreme temperatures for n-pentane, 110 K and

129 K, differ by only ~ 1 %. The difference is ~ 2% when Equation 4.4 is evaluated for n- hexane at 128.8 K and 143.5 K. Hence, evaluation at only one temperature is shown for each n-pentane and n-hexane. Overall, we observe an increase in excess internal energy with molecular content of the critical cluster. The trends in the data for n-pentane and n- hexane are similar – CNT overpredicts the excess internal energy for 푛∗ < 7 and underpredicts for 푛∗ > 7. However, we note here that since the excess internal energy is not the only contribution to the work of cluster formation, the over- and under-predictions of 퐸푥 by CNT do not translate to under- and over-predictions of the nucleation rates. For n-heptane, prediction by CNT underestimates the low temperature data from supersonic nozzle and overestimates the high temperature data of Rudek et al.7 In other words, all data from SSN and TDCC lie to the left and right of the CNT prediction, respectively. From

Equation 4.2, and in the context of the second nucleation theorem, points that lie to the left of the line suggest that the experimental nucleation rates have stronger dependence on temperature than theory. This is true when Equation 4.4 is evaluated at both T = 165 K and

275 K.

The dashed line is the prediction for heptane at T = 275 K. For n-heptane, current values are compared to derived values from the data of Rudek et al.7 For all n-alkanes in this work, the vertical error bars correspond to the standard error of the slopes of Fig.5 and the error bars on the number of molecules are from the standard error of the slope from the first nucleation theorem. Open symbols correspond to data from the correlations described in the text. The value of T0 is 273.15 K.

4.4 CONCLUSIONS

This work extends the previously determined critical cluster sizes of n-heptane8 to cluster formation from smaller chain n-alkane molecules in a supersonic nozzle. We measured additional nucleation rates for lower n-alkanes, in particular pentane and hexane, in a faster supersonic nozzle and applied the first and second nucleation theorems to determine the sizes and excess energies of the critical clusters of these n-alkanes. The nucleation rates in the faster nozzle (nozzle T3) are consistent with the earlier measurements of Ogunronbi et al.8 The critical clusters contained from 2 to 9 molecules 96

for n-pentane and 3 to 8 molecules for n-hexane, and using three nozzles characterized by different expansion rates, 6 to 8 molecules for n-heptane. The remarkable agreement between prior determination of n-heptane critical cluster sizes using two nozzles and current determination using three nozzles clears all doubts on the validity of the technique employed. In addition, being able to combine PTM and SAXS data between nine years apart (2010, 2018, and 2019) is impressive, and it highlights the internal consistency in our n-alkane measurements. Although we were able to derive a reasonable number of critical cluster sizes using correlations, sufficient amount of experimental data is still needed for n-pentane and n-hexane to fully rationalize our understanding of the nucleation process and provide a cornucopia of avenues for further research. Finally, significant critical cluster size gap for n-heptane exists between 15 < 푛∗ < 50, and since nucleation experiment devices only access limited range of nucleation rates, experiments yielding n* between supersonic nozzles and thermal diffusion cloud chambers are needed to fill the gap.

Supporting Information Available:

Three tables (Tables 4.2, 4.3, and 4.4) and four figures (Figures 4.10, 4.11, 4.12, and 4.13) are given in the supporting information. Table 4.2 summarizes the stagnation conditions and some values corresponding to the peak nucleation rates in nozzles T1 and T3. Table

4.3 provides data from the fits to the SAXS measurements in nozzle T3. Table 4.4 summarizes the stagnation conditions, peak nucleation rates, and all parameters corresponding to the peak nucleation rates in nozzle T3. Figure 4.10 summarizes the

correlations of TJmax and ln SJmax with mass flow rates used to determine the desired TJmax in each nozzle. Figure 4.11 summarizes the correlations of Jmax vs TJmax, Jmax vs SJmax, and

SJmax vs TJmax for n-pentane; while Figure 4.12 summarizes the correlations of Jmax vs TJmax,

Jmax vs SJmax, and SJmax vs TJmax for n-hexane. Figure 4.13 provides the 1-D SAXS data and fits for some of the measurements of n-heptane in nozzle T3.

Supporting Information

Table 4.2: Summary of the stagnation conditions, pressure pJmax, temperature TJmax, and distance downstream of the throat xJmax, corresponding to peak nucleation rates in nozzles T1 and T3.

nozzle ṁ pv0 p0 T0 pJmax TJmax xJmax (g/min) (Pa) (kPa) (K) (Pa) (K) (cm) n-pentane T1 3.15 166.93 30.1 293.15 11.30 109.61 5.81 T1 4.60 244.91 30.1 293.15 22.02 113.33 4.81 T1 6.32 326.44 30.1 293.15 30.52 118.76 4.39 T1 8.00 428.90 30.1 293.15 42.44 122.69 4.19 T1 9.97 518.69 30.1 293.15 55.36 125.44 3.89 T1 13.15 688.49 30.1 293.15 71.85 128.58 4.01 T1 16.15 880.37 30.1 293.15 99.90 131.70 3.81

T3 1.30 177.56 60.5 293.15 12.88 102.42 2.21 T3 1.80 244.25 60.5 293.15 18.29 107.19 2.01 T3 2.30 313.25 60.5 293.15 25.23 110.64 1.89 T3 2.80 380.41 60.5 293.15 32.88 113.21 1.81 T3 3.50 479.94 60.5 293.15 44.62 116.90 1.71 T3 4.50 616.72 60.5 293.15 61.85 120.80 1.61 T3 5.42 738.85 60.5 293.15 80.39 124.51 1.51 T3 6.60 903.71 60.5 293.15 99.48 126.86 1.49 T3 7.50 1031.94 60.5 293.15 117.38 128.82 1.45

n-hexane T1 3.15 138.26 30.1 303.15 15.00 128.47 3.51

T1 4.74 207.97 30.1 303.15 22.80 132.53 3.41 T1 6.32 279.23 30.1 303.15 33.30 136.19 3.19 T1 9.50 426.26 30.1 303.15 56.13 142.99 2.89 T1 13.50 604.19 30.1 303.15 85.96 148.96 2.71 T1 15.50 696.68 30.1 303.15 100.48 150.29 2.71

T3 1.58 182.8 60.5 303.15 17.49 122.46 1.59 T3 1.92 220.93 60.5 303.15 21.49 125.65 1.51 T3 2.30 264.51 60.5 303.15 27.97 129.21 1.41 T3 2.60 301.43 60.5 303.15 32.90 130.44 1.39 T3 3.10 357.72 60.5 303.15 40.40 134.15 1.31 T3 3.62 420.07 60.5 303.15 51.49 136.67 1.25 T3 4.65 541.12 60.5 303.15 68.53 140.92 1.19 T3 5.17 599.23 60.5 303.15 80.79 143.95 1.11 T3 5.70 659.76 60.5 303.15 91.97 145.30 1.09 T3 6.40 750.55 60.5 303.15 110.56 148.42 1.01

n-heptane T1 3.50 131.90 30.1 303.15 18.08 145.06 2.41 T1 4.54 171.60 30.1 303.15 25.94 149.77 2.19 T1 6.40 250.80 30.1 303.15 41.74 154.63 1.99 T1 8.50 322.90 30.1 303.15 58.10 160.09 1.81 T1 10.01 381.40 30.1 303.15 69.22 162.05 1.80 T1 11.50 440.10 30.1 303.15 83.58 165.08 1.71 T1 14.40 554.20 30.1 303.15 107.47 168.35 1.69

T3 0.96 98.10 60.5 318.15 9.17 129.61 1.53 T3 1.25 125.90 60.5 318.15 13.69 133.59 1.41 T3 1.55 158.60 60.5 318.15 17.76 138.01 1.31 T3 1.89 190.70 60.5 318.15 23.39 142.03 1.21 T3 2.10 214.90 60.5 318.15 25.43 143.06 1.21 T3 2.36 237.90 60.5 318.15 31.81 146.29 1.11 T3 2.60 266.30 60.5 318.15 35.59 147.77 1.09 T3 3.10 316.60 60.5 318.15 45.59 151.07 1.01 T3 3.60 366.20 60.5 318.15 52.95 153.02 0.99 T3 4.08 414.00 60.5 318.15 62.20 155.26 0.95

Table 4.3: Aerosol size distribution parameters 7 cm downstream of the physical throat for the n-alkanes in nozzle T3: 〈푟〉 is the average radius of the droplets,  is the spread in radius, N is the number density of the droplets, P is the polydispersity of the droplets, and ∅SAXS is the volume fraction of the droplets. T0 is the stagnation temperature and pv0 is the partial pressure of the condensable at the inlet to the supersonic nozzle.

T0 pv0  N

(K) (Pa) (nm) (nm) (cm-3) P= /r ∅SAXS

n-pentane

293.15 177.56 3.19 0.97 3.85×1012 0.302 6.77×10-7

293.15 244.25 3.75 1.28 3.20×1012 0.340 9.73×10-6

293.15 313.25 4.07 1.43 2.83×1012 0.352 1.12×10-6

293.15 380.41 4.60 1.46 2.42×1012 0.317 1.31×10-6

293.15 479.94 5.23 1.67 2.10×1012 0.320 1.67×10-6

293.15 616.72 6.37 1.89 1.52×1012 0.296 2.10×10-6

293.15 738.85 7.33 2.11 1.23×1012 0.287 2.55×10-6

293.15 903.71 8.49 2.31 9.52×1011 0.272 3.01×10-6

293.15 1031.94 9.25 2.48 8.39×1011 0.268 3.41×10-6

n-hexane

303.15 182.80 1.56 1.11 1.72×1013 0.712 8.31×10-7

303.15 220.93 2.21 1.30 1.03×1013 0.587 1.05×10-6

303.15 264.51 2.47 1.46 8.84×1012 0.592 1.29×10-6

303.15 301.43 3.20 1.53 5.27×1012 0.479 1.30×10-6

303.15 357.72 3.82  4.00×1012 0.443 1.56×10-6

100

303.15 420.07 3.97 1.84 4.00×1012 0.465 1.83×10-6

303.15 541.12 5.14 2.10 2.61×1012 0.407 2.31×10-6

303.15 599.23 5.77 2.21 2.16×1012 0.384 2.58×10-6

303.15 659.76 6.10 2.34 2.01×1012 0.383 2.84×10-6

303.15 750.55 6.82 2.47 1.68×1012 0.363 3.20×10-6

n-heptane

318.15 98.1 0.08 0.28 8.60×1014 3.236 5.63×10-7

318.15 125.9 0.09 0.31 7.53×1014 3.290 7.12×10-7

318.15 158.6 0.43 0.68 1.20×1014 1.584 8.41×10-7

318.15 190.7 0.59 0.81 9.27×1013 1.378 1.09×10-6

318.15 214.9 0.78 0.94 6.13×1013 1.198 1.16×10-6

318.15 237.9 0.85 1.00 5.73×1013 1.179 1.32×10-6

318.15 266.3 1.40 1.26 2.58×1013 0.898 1.40×10-6

318.15 316.6 2.73 1.61 7.87×1012 0.591 1.54×10-6

318.15 366.2 3.43 1.76 5.48×1012 0.514 1.78×10-6

318.15 414.0 3.54 1.87 5.48×1012 0.528 2.02×10-6

101

Table 4.4: A summary of the pressure trace measurements results, the aerosol number densities, and the peak nucleation rates in nozzle T3. T0 denotes the stagnation temperature, pv0 and y0 denote the initial partial pressure and initial mole fraction of the condensable, TJmax, pJmax, SJmax, and ∆푡퐽푚푎푥, denote the pressure, temperature, supersaturation, and characteristic time corresponding to the maximum nucleation rate. NZ/VV is the ratio between the density at the nucleation zone and the point of observation, N is the number density, and J is the maximum nucleation rate. All experiments used argon as the carrier gas and the stagnation pressure was p0 = 60.5kPa.

T0 pv0 푦0 TJmax pJmax 푆Jmax ∆푡Jmax NZ/VV N J

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

n-pentane

293.15 177.56 0.0029 102.42 12.88 2.66×107 14.16 2.04 3.85×1012 5.53×1017

293.15 244.25 0.0040 107.19 18.29 5.64×106 10.81 2.10 3.20×1012 6.23×1017

293.15 313.25 0.0051 110.64 25.23 2.19×106 8.76 2.23 2.83×1012 7.20×1017

293.15 380.41 0.0062 113.21 32.88 1.17×106 8.27 2.25 2.42×1012 6.61×1017

293.15 479.94 0.0079 116.90 44.62 4.76×105 7.84 2.40 2.10×1012 6.42×1017

293.15 616.72 0.0101 120.80 61.85 2.02×105 6.75 2.44 1.52×1012 5.48×1017

293.15 738.85 0.0121 124.51 80.39 9.13×104 6.61 2.54 1.23×1012 4.71×1017

293.15 903.71 0.0148 126.86 99.48 6.00×104 5.64 2.56 9.52×1011 4.32×1017

293.15 1031.94 0.0169 128.82 117.38 4.26×104 5.89 2.61 8.39×1011 3.71×1017

n-hexane

303.15 182.80 0.0030 122.46 17.49 3.58×106 9.83 2.46 1.72×1013 4.30×1018

303.15 220.93 0.0037 125.65 21.49 1.55×106 8.81 2.56 1.03×1013 2.99×1018

303.15 264.51 0.0044 129.21 27.97 6.69×105 7.79 2.66 8.84×1012 3.03×1018

303.15 301.43 0.0050 130.44 32.90 5.47×105 7.69 2.66 5.27×1012 1.83×1018

102

303.15 357.72 0.0059 134.15 40.40 2.33×105 7.58 2.78 4.00×1012 1.47×1018

303.15 420.07 0.0069 136.67 51.49 1.50×105 7.06 2.83 4.00×1012 1.60×1018

303.15 541.12 0.0089 140.92 68.53 6.74×104 6.95 2.91 2.61×1012 1.09×1018

303.15 599.23 0.0099 143.95 80.79 3.82×104 6.95 3.03 2.16×1012 9.42×1017

303.15 659.76 0.0109 145.30 91.97 3.17×104 6.51 3.03 2.01×1012 9.35×1017

303.15 750.55 0.0124 148.42 110.56 1.88×104 6.46 3.12 1.68×1012 8.15×1017

n-heptane

318.15 98.1 0.0016 129.61 9.17 1.54×107 11.20 2.51 8.60×1014 1.93×1020

318.15 125.9 0.0021 133.59 13.69 6.08×106 8.86 2.63 7.53×1014 2.24×1020

318.15 158.6 0.0026 138.01 17.76 1.98×106 7.90 2.74 1.20×1014 4.16×1019

318.15 190.7 0.0032 142.03 23.39 8.08×105 7.22 2.86 9.27×1013 3.67×1019

318.15 214.9 0.0036 143.06 25.43 6.58×105 7.88 2.86 6.13×1013 2.23×1019

318.15 237.9 0.0039 146.29 31.81 3.42×105 7.81 2.98 5.73×1013 2.18×1019

318.15 266.3 0.0044 147.77 35.59 2.60×105 7.24 3.01 2.58×1013 1.07×1019

318.15 316.6 151.07 45.59 1.45×105 6.43 3.10 7.87×1012 3.79×1018 0.0052

318.15 366.2 0.0061 153.02 52.95 1.05×105 5.78 3.13 5.48×1012 2.97×1018

318.15 414.0 0.0068 155.26 62.20 7.24×104 5.97 3.19 5.48×1012 2.93×1018

103

Figure 4.10: Correlations of temperature and supersaturation with mass flow rate for n-pentane, n-hexane, and n-heptane. These correlations are used to determine the adjusted TJmax corresponding to the averaged

SJmax in each nozzle.

104

105

We use these correlations to estimate the critical cluster sizes via the first nucleation theorem:

푑(푙푛퐽) (푙푛퐽) − (푙푛퐽) 푛∗ ≅ [ ] ≅ [ 퐴 퐵 ] 푑(푙푛푆) 푇 (푙푛푆)퐴 − (푙푛푆)퐵 푇

106

Where A and B correspond to conditions in nozzles characterized by different expansion rates. For example, in this work, A and B correspond to nozzles T3 and T1, respectively.

107

108

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112

CHAPTER 5

Freezing of Short Chain n-alkanes: Experiments and Molecular Dynamics Simulations

This chapter is part of the manuscript titled “Freezing of short chain n-alkanes:

Experiments and Molecular Dynamics Simulations” authored by Kehinde Ogunronbi,

Sherwin Singer, and Barbara Wyslouzil which will be submitted to the Journal of Chemical

Physics. The experiments in this chapter were performed by Kehinde Ogunronbi and

Barbara Wyslouzil. Analysis of experimental data was done by Kehinde Ogunronbi under the guidance of Barbara Wyslouzil. Molecular dynamics simulations were done by

Kehinde Ogunronbi under the guidance of Sherwin Singer.

113

5.1 INTRODUCTION

Crystallization is important in nature as it controls the formation of such diverse materials as snowflakes, diamond, and other mineral crystals, for example, stalactites and stalagmites. In the pharmaceutical industry, crystallization techniques are used in separation and purification processes,1,2,3,4 and, furthermore, the crystal polymorph formed can strongly affect a drug’s bioavailability.5 In the energy industry, the formation of crystalline hydrocarbon clathrates in natural gas pipelines at temperatures above the freezing point of water, can block transportation and increase cost 6,7. Crystallization is not guaranteed even if a liquid is supercooled or a solution is supersaturated: some compounds crystallize readily while others are more likely to form glasses, especially if cooled rapidly.

The n-alkanes are one group of compounds whose freezing behavior has been extensively investigated. These hydrocarbons are interesting because their chain like geometry is a common motif in more complex lipids, and because their high latent heat of fusion and chemical stability make them good candidates for energy storage applications8,9,10,11. Their freezing behavior is both interesting and complex.12,13,14,15

Perhaps the most well-known oddity is the jagged trend in the melt temperatures, Tm, of the low carbon number (C4< Cn < ~C12) alkanes that directly reflects the ability of even carbon number alkanes to pack more efficiently than odd alkanes16. The degree to which alkanes can be supercooled also varies with chain length. Kraack et al.14 showed that in bulk samples the degree of supercooling achieved by n-alkanes with carbon numbers 15 ≤

푛 ≤ 60, is close to zero. The even alkanes with n in the ranges 15 ≤ 푛 ≤ 20 and 40 ≤

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푛 ≤ 60 were able to supercool by up to 2 oC, but these values are still extremely small14 given that most liquid metals and molecular liquids do not begin to crystallize at an

17 appreciable rate unless the supercooling Δ푇 = 푇푚 − 푇 exceeds 0.15 to 0.25 Tm. In contrast, emulsified droplets of n-alkanes could be supercooled by up to 32 oC.

The near-zero supercooling for bulk samples with 20 < 푛 < 40 was attributed to the formation of a crystalline monolayer at the air-sample interface up to ~ 3 oC above the melt temperature. The presence of the organized surface phase then leads to surface- induced heterogeneous nucleation once the melt temperature is reached. Moreover, the finite supercooling of up to 2 oC in the case of even n-alkanes with 15 ≤ 푛 ≤ 20 and 40 ≤

푛 ≤ 60 was attributed to a mismatch between the structure of the surface phase and that of the new crystalline phase18. The large supercooling achieved in emulsified droplets suggests that the surface-induced nucleation mechanism is suppressed or nonexistent in

13 them. Finally, Ocko et al. stated since ∆Tsf is zero when n =14, the surface freezing phenomenon was restricted to n-alkanes with 푛 ≥ 14. Here, ∆Tsf is the difference between the temperature Tsf, at which surface freezing occurs and Tm, the equilibrium melting point.

In contrast, Weidinger et al.19 observed supercooling on the order of 10 K for ~ 43-

90 µm levitated n-alkane aerosol droplets (14 ≤ 푛 ≤ 17), despite the presence of a liquid/air interface. Their data also provided evidence that nucleation proceeded by both heterogeneous and homogeneous nucleation mechanisms, where heterogeneous implies that the formation of the critical nucleus is initiated by a fully developed surface monolayer, and homogeneous means the formation occurs randomly throughout the droplets. They also

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estimated that at an equal degree of supercooling, heterogeneous and homogeneous nucleation rates differed by about 5 orders of magnitude.

Using much smaller droplets, 4 ≤ 푟/nm ≤ 16, Modak et al.20,21 followed the crystallization of n-octane and n-nonane, each at single experimental condition, as well as n-decane at four different conditions. Contrary to expectations, they found spectroscopic evidence that suggests that in these rapidly cooled short chain n-alkane droplets freezing also often proceeds in 2 stages, i.e. surface freezing followed by freezing of the rest of the droplet. To support this interesting result, they conducted limited MD simulations at the united atom (UA) level and found that upon cooling an n-octane liquid droplet to below the surface melting temperature, a monolayer developed on the surface of the droplet within a few nanoseconds. After a further delay of ~ 25 ns, freezing of the droplet progressed inward on a layer-by-layer basis, i.e. via a heterogeneous nucleation mechanism.

More extensive molecular dynamics (MD) simulations have provided better understanding of the crystallization of n-alkanes by capturing the molecular details of the crystal nucleation events both in the bulk and at interfaces. Yi and Rutledge 22 used UA

MD simulations to observe homogeneous nucleation and the structure of the critical nucleus in a bulk (interface-free) crystallizing n-octane system. From the dependence of the free energy on nucleus size, they found that the critical nucleus was better quantitatively described by a cylindrical nucleus model than the spherical nucleus model. To better understand the much-debated mechanism of surface freezing, Modak et al.23 investigated

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freezing in n-octane and n-nonadecane slabs because these alkanes exhibit rather different surface freezing behavior. In particular, the longer alkane clearly develops a surface frozen layer above Tm whereas the shorter alkane is thought to surface freeze below Tm. For n- nonadecane, the simulations find that the solid-vapor surface free energy is lower than that of the vapor-liquid surface free energy, and that near Tm fluctuations within the surface layer are not much larger than those in the underlying bulk solid. These observations are consistent with the arguments of Ocko and co-workers13, 24 that surface freezing is purely a wetting phenomenon. In contrast, for n-octane, the solid-vapor surface free energy is higher than the liquid-vapor surface free energy – ruling out surface wetting as the driving force – but the fluctuations of molecules in the surface layer are far greater than those in the underlying solid. These results agrees reasonably well with the theory proposed by

Tkachenko and Rabin,25, 26 who suggested that entropic stabilization due to fluctuations of molecules in the monolayer, play an important role in surface freezing. In characterizing the formation of crystalline monolayers under highly supercooled conditions, Qiu and

Molinero 27 investigated the properties of the surface that control the interfacial orientation of three n-alkane molecules (C9, C16, and C20) and found that the orientation depends on the strength of the alkane-fluid interaction. That is, fluids that interact weakly and strongly with the alkane molecules promote perpendicular and parallel interfacial orientation of the molecules, respectively. Thus, n-alkane molecules show a strong preference to align perpendicular to a vacuum interface, and parallel to a water or metallic interface.

Despite the compelling evidence for surface freezing in n-alkanes with 8 ≤ 푛 ≤

10, a number of questions still remain. These include the value of n for which the surface- 117

templating effect ceases, or at least becomes less effective, and whether molecular level simulations can capture the trends observed in experiments. Since surface freezing in n- alkanes is driven in part by molecular shape favoring parallel chain alignment, one expects that as the chain length decreases toward the monomer, this effect should vanish. The goals of this work are, therefore, to investigate the freezing behavior of the next three shorter n- alkanes - heptane, hexane, and pentane, and conduct complementary MD simulations to better understand the role of surface freezing in crystallization of these chain-like molecules.

5.2 EXPERIMENTS AND DATA ANALYSIS

5.2.1 Chemicals and Physical properties Liquid argon, with purity of 99.998%, was purchased from Praxair. n-pentane and n-heptane, each with purity of 99%, were purchased from Sigma Aldrich and used without further purification. n-hexane, with purity > 99%, was purchased from ChemSampCo and used without further purification. The thermophysical properties of these chemicals are available in Table S-1 of the supplementary material in ref. 25.28 Pressure-volume data from the National Institute of Standards and Technology (NIST)29 were used to determine the isothermal compressibility for these materials. For n-pentane, n-hexane, and n-heptane these are 5.34 × 10−10Pa−1, 7.61 × 10−10Pa−1and 6.21 × 10−10Pa−1, respectively.

5.2.2 Particle production and characterization methods The experimental apparatus and experimental techniques used here have been described in our earlier publications,20, 28, 30, 31 and are only briefly summarized here. 118

Nanodroplets were produced by expanding argon-n-alkane gas mixtures across supersonic

Laval nozzles (nozzles T1-CaF2, T1-mica, and T3). The first two nozzles (nozzle T1 with

CaF2 windows or nozzle T1 with mica windows) are the same as those used to determine the vapor-to-liquid nucleation rates of these n-alkanes28. Both nozzles have the same linear expansion rate. A limited number of experiments were conducted using nozzle T3, a nozzle that expands roughly two times faster, thereby reaching significantly lower temperatures.

Expansions using nozzle T1 started at a stagnation pressure p0 of 30.2 kPa, and the

o o stagnation temperatures T0 were 20 C for n-pentane and 30 C for n-hexane and n-heptane.

o Expansions using nozzle T3 started at p0 of 60.4 kPa and T0 of 20 C for n-pentane.

As the gas mixture flows through the nozzle it cools and droplets form via homogeneous nucleation and growth. Upon further expansion, the droplets can freeze if temperatures are low enough. Phase transitions release heat to the flow and, consequently, the static pressure increases above that expected for an isentropic expansion. Thus, static pressure trace measurements (PTM) were used to pinpoint the location of these phase transitions.

Position resolved small angle X-ray scattering (SAXS) experiments characterized the size distributions of the evolving nanodroplets and nanoparticles. SAXS measurements were made at the Advanced Photon Source, Argonne National Lab, using the 12_ID beamline and the same instrumental settings noted in Ogunronbi et al.28 The size distribution parameters were determined by fitting the scattering spectra assuming a

Schultz distribution of polydisperse spheres and absolute calibration factors were determined as described in Manka et al.32

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Position resolved FTIR spectroscopic measurements were used to quantify the mass fraction of condensate in the vapor, liquid, and solid phases as described in detail by Modak and coworkers20, but using the modified setup developed by Park.33 To summarize briefly, in this setup two mirrors and one converging lens guide the IR beam from the side window of the spectrometer (Perkin Elmer Spectrum 100), through the supersonic nozzle perpendicular to the flow, and to an external liquid nitrogen cooled Mercury Cadmium

Telluride (MCT) detector. To determine the distribution of the condensable between the three possible phases, we assume that the total absorbance spectrum can be represented as a linear combination of the normalized absorptivities of the vapor 휖푣, liquid 휖푙, and solid phases 휖푠. These values are determined by combining data from the PTM, SAXS, and FTIR measurements as described in Modak et al.20 Once the absorptivity values are established, the absorbance spectra can be fit and the mass fraction of each phase derived from the best fit parameters.

For n-hexane experiments, we determine 휖푙 in an experiment (푝푣0 = 697 Pa) where droplets were too warm to have frozen. For n-heptane experiments, we determine 휖푙 by picking a position close to the onset of condensation at which only liquid and vapor are likely to be present. For 휖푠 we pick a position near the nozzle exit and assume the droplets are completely frozen, as confirmed by unchanging FTIR spectra.

5.2.3 Integrated Data Analysis

Flow in a supersonic nozzle with a single condensing species is described by 4 equations (momentum, energy, continuity and an equation of state) that involve 6 variables

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(pressure p, temperature T, density 휌, velocity u, area ratio A/A*, and mass fraction condensate, g). By measuring the static pressure with and without condensation, the 4 equations can be solved using p and A/A* as the known variables, and assuming that condensation does not affect boundary layer development. Characterizing the flow based solely on PTM data, however, typically underestimates both g and T because in our small nozzles heat addition can perturb the boundary layers from those that develop in the absence of condensation.

A more accurate approach is based on iteration. We start by using the estimates of temperature T and density 휌 , based on PTM alone, to calculate the mass fraction of condensate based on the SAXS measurements, gSAXS. We then solve the equations using p and gSAXS as input and obtain more accurate values of T and 휌. New values of gSAXS are then determined and the process repeats until values converge. This method is used for systems, like n-pentane in Ar carrier gas, where only condensation is observed. When droplets also freeze, a similar approach is used but now both the mass fraction of liquid condensate, gl, and the mass fraction of solid condensate, gs, based on FTIR measurements are first estimated and then refined in a similar iterative manner.

5.3 Molecular Simulations

For each n-alkane, we simulated the behavior of systems containing n-alkane chains using GROMACS and a united-atom (UA) force field model (PYS – Paul, Yoon, and

34, 35, 36 Smith) where chains consist of CH2 and CH3 beads. The bonded and non-bonded

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parameters used here are those reported by Modak et al.20 The bond-stretching potential between two beads is represented by

2 퐸푏 = 푘푏(푙 − 푙0) (5.1)

5 −1 −2 where 푘푏 = 1.46 × 10 푘퐽 푚표푙 푛푚 is the bond stretching constant, l is the bond length, 푙표 = 0.153푛푚. The bond angle bending potential between two adjacent bonds is

2 퐸휃 = 푘휃(휃 − 휃0) (5.2) where

표 푘휃 = 251.04푘퐽, 휃0 = 109.526 , and 휃 is the complement of the bond angle.

The torsional potential between three adjacent bonds is described by Ryckaert and

Bellemans as

3 푛 퐸휙 = ∑푛=0 퐶푛(푐표푠휙) (5.3)

where 휙 is the torsion angle, 퐶0 = 6.505, 퐶1 = 16.995, 퐶2 = 3.620, and 퐶3 =

−27.12 푘퐽 푚표푙−1.

Finally, the potential between beads separated by more than three bonds or beads on different chains is described by a 12-6 Lennard-Jones potential described by

휎 12 휎 6 퐸 = 4휀 [( 퐿퐽) − ( 퐿퐽) ] (5.4) 퐿퐽 퐿퐽 푟 푟

−1 where 휀퐿퐽 = 0.469 푘퐽 푚표푙 and 휎퐿퐽 = 0.401 푛푚

The simulation procedure for each n-alkane is described fully in section 5.4.3. 122

5.4 RESULTS AND DISCUSSION From the extensive PTM measurements, made as a function of n-alkane partial pressure and reported in ref 25 28, a select number of conditions were chosen to explore alkane freezing in more detail. Table 5.1 summarizes the conditions for which we characterized both droplet size distributions using position-resolved SAXS measurements and particle phase using FTIR measurements. We discuss the crystallization of n-hexane and n-heptane in Section 5.4.1, the limits to which n-pentane can be supercooled on the ~

250 microsecond timescale of our experiment without crystallizing in Section 5.4.2, and the consistency of these results with molecular simulations in Section 5.4.3.

Table 5.1: A summary of the conditions at the start of the expansion and at the initiation of the vapor-liquid phase transition. T0 denotes the stagnation temperature, 푝0 and 푝푣0 denote the stagnation pressure of the mixture and the initial partial pressure of the condensable. 푝퐽푚푎푥 and 푇퐽푚푎푥 characterize the vapor-liquid phase transition. 푇푚 is the equilibrium melting point of the n-alkanes. In all cases the flow temperature is below the melt temperature when condensation starts, although the droplet temperature can be significantly higher.

n-alkane Nozzle p0 (kPa) T0 (K) pv0 (Pa) TJmax (K) pJmax(Pa) Tm (K) 326 118.8 30.5 T1 30.2 Pentane 293.15 688 128.6 71.8 143.5 T3 60.5 1027 128.8 117.4 138 128.5 15.0 Hexane T1 30.2 303.15 279 136.2 33.3 177.8 697 150.3 100.5 172 149.8 25.9 Heptane T1 30.2 303.15 182.6 323 160.1 58.1

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5.4.1 The case for surface induced freezing in n-hexane and n-heptane

For freezing studies, n-hexane and n-heptane provide an interesting comparison because, although their boiling points differ by more than 30 K, their melt temperatures differ by only ~5 K. Fig. 5.1 summarizes the key results from the SAXS and PTM experiments. As expected, the latent heat of vaporization released to the flow as particles form and grow leads to a temperature increase above that expected for an isentropic expansion. In all cases, the temperature increase aligns well with the appearance of particles and their rapid increase in size. During the rapid growth stage the estimated droplet temperatures, determined from a simple mass and energy balance,38 lie well above the gas temperature, merging with the gas temperature as growth slows.

Both n-alkanes exhibit clear evidence for freezing at the higher partial pressure conditions investigated. In particular, there is a subtle second bump in the temperature curve – the result of the release of latent heat of fusion – that aligns reasonably well with a decrease in average particle size – a consequence of the increased density of the solid relative to the liquid. At the lower partial pressures, the second heat release is not observed, although n-hexane exhibits a distinct decrease in particle size suggestive of freezing. Since the latent heat of fusion is only about 30% of the heat of vaporization, freezing can be difficult to detect via PTM if the mass fraction of condensate is small and the flow is still expanding rapidly. Furthermore, if the particles start to freeze while they are still growing, increases in particle size will mask any shrinkage due to freezing. FTIR measurements below, Fig. 5.2, show that this is the case for the n-heptane experiment at 푝푣0 = 172 푃푎

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where the supercooling at the exit of the nozzle is ~ 64 K. The observations made here are consistent with our previous experiments with slightly longer n-alkanes – n-octane, n- nonane, and n-decane20,21.

FIGURE 5.1: Temperature profiles of the expanding flows of a mixture of carrier gas-condensable (n- hexane, and n-heptane) and the corresponding mean droplet size. The open circles represent the temperature of the droplets and the labels indicate the initial partial pressure of the condensable. In (a) there is a second bump in the n-hexane flow at ~ 6.5cm (noticeable for the high flow rate), a signature of the release of latent heat of fusion. Near this position in (b) there is a gradual decrease in the size of the droplets, signifying an increase in density and a decrease in volume of the droplets as they start to freeze. The arrows indicate where freezing starts. The same behavior is seen (c), (d) in the n-heptane flow at inlet partial pressure pv0 = 323 Pa.

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Fig. 5.2 illustrates the change in the mass fractions of the vapor gv, liquid gl, and solid gs, phases, normalized by the total mass fraction of condensable (ginf = gv + gl + gs ), for n-hexane and n-heptane derived from the integrated FTIR, PTM, and SAXS data analysis. As expected, the vapor 푔푣 drops smoothly from 1 to a finite value at the nozzle 푔푖푛푓

𝑔 exit as condensation proceed. For the liquid, 푙 starts at zero, increases and reaches a 𝑔푖푛푓 maximum, and then drops back down to zero at the nozzle exit. For the solid, 푔푠 generally 푔푖푛푓 remains at zero longer than the liquid, increasing gradually at first and then more rapidly, and finally reaching a finite value at the exit of the nozzle. In Fig. 5.2(d), however, the increase in 푔푠 is more abrupt and the rapid increase in solid occurs as vapor molecules are 푔푖푛푓 still condensing/depositing. Moreover, the liquid fraction peaks almost as soon as condensation starts, indicating the latter stages of particles growth occur by deposition rather than condensation. Overall, the range in behavior observed for n-hexane and n- heptane largely mirrors the range of behavior observed for the freezing of the longer chain n-decane droplets as the partial pressure of condensable was decreased.21

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FIGURE 5.2: Mass fractions of the condensable (vapor) and condensate (liquid and solid) as a function of position in the nozzle for (a) n-hexane at inlet partial pressure of 279 Pa (b) n-hexane at inlet partial pressure of 138 Pa (c) n-heptane at inlet partial pressure of 323 Pa and (d) n-heptane at inlet partial pressure of 171.6

Pa. In each graph, M.B is the material balance i.e. the summation of the mass fractions of the three phases at positions downstream of the nozzle throat. (The lines through the points are to guide the reader’s eye).

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To better elucidate the freezing mechanism in these short-chain alkane droplets, we

𝑔푠 determined the fraction of solid, Fs, within the droplets, where 퐹푠 = , and these values 𝑔푠+𝑔푙 are presented in Fig. 5.3. We then fit the data to a functional form that assumes a single critical nucleus induces freezing on the surface of the droplet or throughout the droplet 21.

That is, beyond the onset of solid nucleation at time 푡0, the fraction of solid droplets present in the system is given by

퐹푠(푡) = 1 − exp (−퐵(푡 − 푡0)) (5.5)

where 퐵 = 퐵푉 = 퐽푉푉 for volume-based nucleation, 퐵 = 퐵푆 = 퐽푠푆 for surface-based nucleation, 퐽푉 and 퐽푠 are the nucleation rates and 푉 and 푆 are respectively the average volume and surface area of a droplet. The fit parameters are summarized in Table 5.2.

For both n-hexane conditions and for the higher heptane condition, Figs. 5.3(a),

5.3(b) and 5.3(c), the appearance of the solid in the droplets occurs in two steps. As in earlier experiments with longer n-alkanes39,40, we interpreted the first slow step, as the signature of surface freezing. The second step, where the solid content increases rapidly corresponds to freezing of the rest of the droplet. These results suggest that, in most of these cases, freezing of supercooled liquid n-alkane nanodroplets is initiated by first aligning the molecules at the interface to form a surface frozen layer that can then catalyze freezing of the remaining liquid in the droplets.

In Fig. 5.3d, the two-step behavior is absent for the lowest flowrate of n-heptane. This is not unexpected as similar behavior was observed in our earlier experiments with n- decane at intermediate flow rate21. It is, however, surprising that the rapid freezing step 128

occurs when n-heptane is much less supercooled (~20 K) than when the second stage of freezing starts for n-hexane (~50 K) (Figure 5.2).

Surface freezing: Fs = As [1-exp (-Bs(t-t0S))] p 0 Tdrop As BS t0S JS n-alkane v (Pa) (K) (ms-1) (ms) (cm-2.s-1) 138 133.0 0.45 29.60 0.114 1.87 × 1013 n-C6 279 155.4 0.26 17.20 0.096 3.17 × 1012 n-C7 323 177.7 0.26 17.80 0.070 2.14 × 1012

Bulk freezing: Fs = [1-exp (-Bv(t-t0V))] p 0 Tdrop BV t0V Jv n-alkane v (Pa) (K) (ms-1) (ms) (cm-3.s-1) 138 121.2 77.30 0.140 7.49 × 1023 n-C6 279 126.9 70.80 0.170 8.14 × 1022 n-C7 323 162.5 68.90 0.097 4.11 × 1022

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FIGURE 5.3: Freezing kinetics of n-hexane at pv0 = 138 Pa and 279 Pa and n-heptane at pv0 = 172 Pa and

323 Pa. As the nanodroplets cool down after growth is complete, crystallization occurs via a two-step process.

The black dashed lines are the fits to the fraction of solid in the aerosol. The green dashed line is the hypothetical fraction of solid if the surface of every droplet has a fully developed solid monolayer. Plots for

(a), (b), (c), and (d) have been shifted by 0.096 ms, 0.114ms, 0.070ms, and 0.067ms, respectively, to present the analysis from all experiments on the same figure. The parameters for the black dashed lines are listed in

Table 2.

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Table 5.3: Summary of the degree of supercooling, ∆T, required for n-alkane nanodroplets from n-hexane

20,21 through n-decane to initiate “bulk” freezing. Data from n-C8, n-C9, and n-C10 are from Modak et al. pv0 is the stagnation partial pressure of the n-alkane, Tdrop and Tflow are the average temperatures of the droplets and the gas mixture, respectively, at the corresponding t0V. Tm is the equilibrium melt temperature.

n-alkane pv0 Tdrop Tflow Tm ∆T= (Tm-Tdrop) (Pa) (K) (K) (K) (K) 138 121.2 119.6 177.8 56.6 n-C6 279 126.9 127.3 177.8 50.9 n-C7 323 162.5 161.3 182.6 20.1 n-C8 209 183.1 182.1 216.4 33.3 n-C9 196 200.4 199.0 219.7 19.3 264 214.2 211.7 243.5 29.3 n-C10 322 216.1 215.9 243.5 27.4

To quantify the degree of supercooling that can be achieved in the alkane nanodroplets, we calculated the difference between the melt temperature and the temperature of the nanodroplets at the onset of rapid, bulk freezing for n-hexane through n-decane. These data are summarized in Table 5.3 and plotted in Fig. 5.4. Over this size range the data show an alternating dependence of ∆T on n, that qualitatively agrees with the odd-even variation reported by Kraack et al14, i.e. the degree of supercooling achieved by the even alkanes is higher than that achieved by the odd. Since even n-alkane droplets are more supercooled than neighboring odd n-alkanes, we would expect the supercooling of n-pentane nanodroplets to be lower than that of n-hexane, but this is clearly not the case since, as discussed below, n-pentane never froze in our experiments. Overall, our measurements show a strong dependence of the supercooling on n and the current measurements are consistent with earlier measurements in Modak et al.20, 21

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Figure 5.4: Odd-even variation of the difference between the equilibrium melting temperature and bulk freezing temperature of short-chain n-alkanes nanodroplets.

The surface-based (Js) and volume-based (Jv) nucleation rates, obtained from the fits to Equation 5.5, and available in Table 5.2, are plotted in Fig. 5.5. In addition to these rates, we also include nucleation rates corresponding to the assumption that the second nucleation occurs heterogeneously at the existing surface monolayer (Jvh). As detailed in Modak et al.20,21, the values of Jv and Jvh differ by the ratio of the characteristic volumes associated with these nucleation scenarios. To test for consistency and look for variations with chain length, we have also included data for n-octane through n-decane.20, 21 Although it is difficult to assess the uncertainty in the values of Js , Jv , and Jvh, the uncertainty stemming from the mass fraction of the condensable in the solid phase (Fig 5.2) should be small,

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since overall mass balance varies from ~ 1.0 to 1.15 and ~0.9 to 1.10 in the current work and Modak et al.20, respectively. A better measure may be to consider the repeatability of the measurements. Examining the data for hexane and decane, suggests that the uncertainty in any of these rates is less than one order of magnitude.

The trends in Fig.5.5, although admittedly rough, are quite interesting. With decreasing chain length, the surface-based nucleation rates (Js) generally decrease whereas the volume-based nucleation rates (Jv and Jvh) increase. Unlike the 5 orders of magnitude difference between Jv and Jvh measured in micron size droplets, the small size of the nanodroplets ensures that difference between these two rates is less than ~ 1 order of magnitude. The decrease in Js with decreasing n may reflect the decrease in the driving force for molecules to align with each other at the surface and, thereby, induce heterogenous nucleation of the remaining liquid. This in turn leads to a higher degree of 133

supercooling required before the “bulk” or “fast” stage of nucleation begins (Fig.5.4).

Thus, the driving force for the second stage of nucleation increases, consistent with the increased nucleation rates estimated for Jv and Jvh.

5.4.2 Limit of n-pentane crystallization In contrast to the behavior observed for n-hexane though n-decane, droplets of n- pentane did not crystallize in the time available in our experiment. Fig. 5.6 summarizes the temperature profiles and average particle sizes of n-pentane at two different inlet conditions. Temperature increase and particle growth are well aligned, but there is no evidence for a second phase transition in either the PTM or the size data, despite reaching temperatures as much as 33K below the equilibrium melting point, Tm. As summarized in

Table 5.3 and Figure 5.4, odd n-alkane droplets of slightly longer n-alkane chains always froze by the time they were supercooled by ~20K 20.

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FIGURE 5.6: Temperature profiles, (a), and size distributions, (b), as a function of position for n-pentane at inlet partial pressures of 326 Pa and 688 Pa. As expected, the onset of condensation is earlier in the higher inlet partial pressure experiment. The size distributions from both experiments suggest that growth is complete, and droplets are fully developed by the exit of the nozzle.

To confirm this interesting observation, we used vibrational spectroscopy to determine the distribution of n-pentane between the vapor and liquid phases via the integrated data

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analysis explained earlier. For both n-pentane experiments with 푝푣0 = 326 푃푎 and 푝푣0 =

688 푃푎, we determine 휖푙 by picking positions near the exit of the nozzle (z = 7.86 cm and

8.36 cm, respectively) where the droplet temperatures are 117.7 K and 137.2 K, respectively. As illustrated in Fig. 5.7(a), the spectra do not show any of the key features characteristic of n-alkane freezing, i.e. the splitting of the peaks near wavenumbers 2856

-1 -1 cm (CH2 symmetric stretch) and 2954 cm (CH3 asymmetric stretch), that are observed

20 for n-heptane and n-nonane in Fig. 5.7(b). For these two conditions the 휖푙 match very well, despite a 20 K difference in droplet temperature. We expect the spectrum of frozen n-pentane droplets should be consistent with those of the higher odd-numbered n-alkane as we have seen in earlier experiments for even-numbered n-alkanes21.

FIGURE 5.7: (a). The normalized absorptivities for n-pentane droplets at T = 117 K, and 137 K for expansions starting at pv0 = 326 Pa and 688 Pa, respectively. (b). Peak locations of normalized absorptivity of n-heptane from current experiment are consistent with those of n-nonane in Modak et al.20 Ideally, a spectrum from frozen n-pentane droplets should exhibit the same features as the spectra in (b) that were

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measured near the exit of the nozzle. For the spectra in (b), the shape and intensity of the CH3 peaks are quite close, while the intensity of the CH2 peak clearly decreases with chain length.

FIGURE 5.8: Mass fraction of vapor and liquid n-pentane, as a function of position downstream of the throat, calculated based on PTM and FTIR. Here, M.B. = (gv + gl)/ginf.

Fig. 5.8 summarizes the mass fraction and the distribution of the vapor and liquid phases as a function of position for the experiment with colder droplets. The mass balance is very good, and, thus, it is unlikely that another phase is present – even though the droplets are

~33 K supercooled at the nozzle exit.

To try and induce freezing, we conducted an additional experiment using a more rapidly expanding nozzle to supercool the droplets more drastically. Nozzle T3, described in Tanimura et al.,41 has a cooling rate of ~4 × 106K/s or roughly twice that of nozzle T1,

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and, thus, the temperature at the exit is significantly lower. Figure 5.9a illustrates the pressure and temperature profiles as an n-pentane in Ar mixture expands across nozzle T3.

For 푝푣0 = 1027 Pa, condensation starts at when Tflow ~ 128 K, i.e., about 15 K below Tm, but the Tflow quickly approaches Tm during the rapid particle growth phase. Thus, we expect that the initial droplets that form most likely start out at temperatures well above Tm, as they did in the nozzle T1 experiments. As growth slows, Tdrop should quickly approach the flow temperature and by z = 7 cm the droplets should reach ~ 84 K or ~ 60 K lower than

Tm. At this extreme condition, we took position-resolved FTIR absorbance measurement to look for evidence of a second phase transition. As illustrated in Fig. 5.9b, the raw spectrum at a supercooling of ~ 60 K matches that measured at a supercooling of 33 K when the latter is scaled by a factor of 1.8. The good match between the spectra suggests there is no difference in phase, despite a temperature difference of ~ 26 K. Furthermore, even though the signal is weak, our spectra near 1400 cm-1 more closely match those measured by Lang et al.42 and identified as liquid n-pentane nanodroplet than those identified as fully crystallized. Thus, there is no evidence for freezing of the highly cooled n-pentane.

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FIGURE 5.9: (a). Pressure and temperature profiles for a condensing flow of n-pentane in nozzle T3. The inlet conditions are noted in the legend. Although the temperature profile has not been corrected for the effect of boundary layer compression, we estimate this will only raise the exit temperature by ~ 6K. The dashed grey line is the equilibrium melting point for n-pentane, and the arrows guide the eyes to read the right axis.

(b). Absorbance spectra of n-pentane droplets characterized by different temperatures. Despite a further supercooling of ~ 26 K, the spectra do not differ from each other. The warmer spectrum was scaled by a factor of 1.8.

The stability of highly supercooled n-pentane particles against crystallization was also noted by Signorell and co-workers42 who investigated the phase behavior of propane and n-pentane aerosol particles under conditions relevant to Titan. They produced nanoparticles in a collisional cooling cell and found that n-pentane nanodroplets only started to crystallize after ~1 s at 78 K, and that crystallization required ~600 s to finish.

These times are ~ 6 orders of magnitude longer than the time available in our experiment

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at roughly the same temperature and particle size. Thus, in retrospect, we should not be surprised n-pentane did not freeze in our experiments.

One possible reason for the reluctance of n-pentane droplets to freeze under the timescale of our experiment could be explained by the greater difficulty of its molecules in overcoming the barrier to reorientation. For example, studies of the crystallization of a series of n-alkanes from n-pentane through n-hexadecane showed that the crystal-liquid surface free energies increase with decreasing chain length, and molecular configurations are expected to become more restricted as the alkane chain length decreases12.

Furthermore, if alkane freezing is highly enhanced by rapid organization of the surface frozen layer, then Molecular Dynamics simulations may be a way to rationalize at which chain length this effect vanishes, or slows significantly, in the supercooled liquid.

5.4.3. MD Simulation The goal of our MD simulations is not to directly mimic the experimental system by modeling the kinetics of freezing in nanodroplets. Rather, we want to test our hypothesis that if organization of the free surface becomes less effective as chain length decreases, then freezing of the whole droplet may not occur on the timescale of our experiment. Thus, calculations that use a slab geometry are entirely appropriate.

For each n-alkane two calculations are required. The first is to establish the equilibrium melt temperature of the UA n-alkanes and the second is to establish their surface freezing temperatures. For these short chain alkanes, the latter will be in the sub-cooled regime. If the degree of supercooling required to maintain the surface frozen layer increases rapidly

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with decreasing chain length, then this may explain the increased supercooling required to initiate freezing throughout the droplet and our interpretation of the experimental results.

5.4.3.1 Bulk Melt temperature

We follow the procedure described in the work of Morris et al.43, 44 Crystalline slabs containing more than 1700 molecules were constructed. Half of the slab was heated to high temperatures, while the other half was kept at 50 K in an NVT calculation using the v- rescale thermostat. NPT trajectories at one atmosphere were then generated at different temperatures to estimate the UA model’s melt temperature, Tm,UA, and then at 1K intervals for 10ns to refine these values. Potential energy was monitored to establish the melt temperatures for each n-alkane. Table 5.4 summarizes the simulation details and the final values of Tm,UA for each n-alkane.

Table 5.4: Details of simulation for determining the melt temperature for each n-alkane.

n-pentane n-hexane n-heptane No. of molecules in slab 2016 2880 1728 No. of atoms in slab 10080 17280 12096 Temperature for melting half of the slab (K)* 550 1100 750 Simulation time for melting half of the slab (ps) 1 1 1 Temperature of cold partition (K) 50 50 50

Simulation time for equilibration to determine Tm (ps) 10000 10000 10000 UA model melt temperature, Tm,UA (K) 130 ± 1 160 ± 1 189 ± 1 45 Experiment melt temperature, Tm (K) 143.5 177.8 182.6 * In some simulations, the half-slab was first melted under NPT conditions, using the Parrinello-Rahman pressure coupling barostat, rather than NVT conditions, but we found no difference in the final melt temperature. Fig. 5.10 summarizes the initial as well as the final configurations for the n-heptane system at temperatures between 188 K and 190 K. The solid/liquid interface moves to the

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left at 188 K (Fig. 5.10c), signifying growth of the crystal phase. The interface remains relatively constant at 189 K (Fig. 5.10d), with chains at the edge of the crystalline region starting to disorganize. However, increasing the simulation temperature by 1 K (T = 190 K in Fig. 5.10e) leads to an isotropic melt with no interface between phases.

These trends are reflected in the time evolution of the total potential energy of the system shown in Fig. 5.11a. The onset of melting should correspond to a configuration at which the potential energy is roughly constant and for this model of n-heptane, Tm,UA = 189

K. Similar simulations led to Tm,UA values of 160 ± 1 K and 130 ± 1 K for n-hexane and n-

45 pentane, respectively. These values differ from Tm of the real substances by 6 to 18 K.

As illustrated in Fig. 5.11b, the UA approach cannot capture the jagged nature of the decrease in Tm as a function of chain length although it does predict roughly the correct slope for appropriate n-alkane pairs, i.e. pentane/hexane, and heptane/octane. Given that the UA model does not treat hydrogen atoms explicitly, our estimations of Tm are in reasonable agreement with experiment.

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FIGURE 5.10: Images of n-heptane slabs at different configurations. (a). Starting configuration of a slab

(b). Final configuration of a half-melted and half-frozen slab. (c). T = 188 K, where growth of the solid phase is evident. (d). T = 189 K and (e). T = 190 K, where the whole slab is completely melted.

FIGURE 5.11: (a). Potential energy of n-heptane systems with half-melted and half-frozen partitions as a function of time at different temperatures. The potential energy at 189 K remains approximately stationary for 10 ns. Therefore, the melt temperature for n-heptane is estimated as 189 ± 1 K. (b). Melt temperatures of the real n-alkanes from experiments, Tm, and united atom model, Tm, UA.

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5.4.3.2 Surface Freezing temperature

To establish the surface freezing temperature we followed the procedure described by

Modak et al.23 In summary, we started with a crystal of n-alkane molecules. This was then melted by heating to 300 K at 1 atm and holding these conditions for 300 ps. The melt was then cooled down to a temperature slightly below the experimental melt temperature, and constant volume (NVT calculations) were run for 5 ns. Further cooling and equilibration runs were carried out at constant volume until the top and bottom layers were completely frozen. Then, half of the molecules on the top and bottom surfaces were melted by heating to 250 K while simultaneously cooling the other half of the surface to a lower temperature.

This created a system with an isotropic melt sandwiched between top and bottom surfaces, each with fully melted and frozen halves in contact along the z-direction. The final system was then thermostatted to different temperatures and the potential energies were monitored.

Table 5.5 summarizes the key simulation details and the final melt temperatures for n- pentane, n-hexane, and n-heptane.

Table 5.5: Simulation details for calculating the surface freezing temperature for each n-alkane and the values of the surface freezing temperatures.

n-pentane n-hexane n-heptane No. of molecules in slab 1680 960 960 No. of atoms in slab 8400 5760 6720 Temperature of slab with frozen top and bottom layers (K) 100 140 160 Temperature to which half of top and bottom were melted (K) 250 250 250 Temperature to which half of top and bottom were cooled (K) 70 80 140 Simulation time to melt half of the top and bottom layers (ps) 200 120 250

Simulation time for equilibration to determine Tsf (ps) 7000 5000 5000 Surface freezing temperature (K) 117 ± 1 155 ± 1 185 ± 1

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Fig. 5.12 summarizes the final configurations of an n-heptane system at different temperatures. Figure 5.12c shows that after 5ns at T = 185 K the slab configuration remains almost unchanged from the initial configuration. If the temperature changes by roughly 1 or 2 K, the surface layer either grows in, Figure 5.12(b), or vanishes, Figure 5.12(e). The difference between T = 184 K and T = 185 K was already apparent at 3 ns as illustrated in

Figure 15 of the appendix. Figures 5.12 and 5.13 shows that the orientation of all molecules in the frozen sections of the surface is nearly perpendicular to the surface.

FIGURE 5.12: Images of n-heptane slabs under different conditions (a). Initial configuration of a slab with half-melted and half-frozen surfaces. Configurations after 5 ns for (b). T = 184 K where growth of the frozen portion is evident at the top and bottom (c). T = 185 K. (d). T = 186 K, and (e). T = 187 K, the whole slab is completely melted even though the slab is supercooled.

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FIGURE 5.13: This is as discussed in relation to Fig. 5.12, except that the configurations of (a) through (e) are at 3 ns. The interface stays roughly the same at a temperature of 185 K as depicted in (c).

Monitoring changes in the potential energy of the system is a more sensitive way to establish the surface freezing temperature. Figure 5.14 summarizes the evolution of the potential energy for the configurations in Fig. 5.12. The temperature for the onset of surface freezing corresponds to a configuration at which the potential energy is roughly constant, i.e. to T = 185 K.

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FIGURE 5.14: Potential energy of n-heptane systems with half-melted and half-frozen surfaces as a function of time at different temperatures. The potential energy at 185 K stays approximately stationary for at least 3 ns. The same applies to 184 K. Therefore, taking this uncertainty into consideration, the onset of surface freezing temperature is approximately 185 ± 1 K.

The difficulty of supercooled pentane droplets to crystallize under the timescale of our experiments may be rationalized in part by, estimating and comparing the difference between the melt, Tm,UA, and surface freezing, Tsf, UA temperatures of the united atom models of n-pentane through n-octane. The degree of supercooling at the surface freezing temperature, ∆Tsf,UA = Tm,UA - Tsf,UA, of the n-alkane molecules is shown in Fig. 5.15. For the UA model used here, ∆Tsf,UA increases rather slowly from n-heptane to n-hexane with

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a much larger increase from n-hexane to n-pentane. Although the basic trend seems in rough agreement with the experiments, the change in TSF with n is relatively small, and there is little difference between ∆Tsf,UA for n-octane and n-hexane even though there is a large difference in the temperature required to induce the bulk or fast freezing stage.

Furthermore, in all of the slab simulations – especially at high supercooling – the surface layer in pentane established itself within 1 nanosecond. Thus, the simulations do not unequivocally support our hypothesis that changes in TSF correlates well with the ability to observe freezing in the n-alkane nanodroplets. More realistic simulations of nanodroplets at the all atom level may be required.

Figure 5.15: Degree of supercooling as a function of n-alkane chain length.

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5.5 CONCLUSIONS

We have studied the crystallization of short-chain n-alkanes – pentane, hexane, and heptane – using pressure trace measurements (PTM), scattering techniques (SAXS), spectroscopic methods (FTIR) and Molecular Dynamics simulations. The results from

PTM complement those from FTIR and SAXS. Temperature increases in all n-alkane experiments, inverted from PTM data, align well with the appearance of particles and their rapid increase in size. A subtle second bump in the temperature profiles for n-hexane and n-heptane – the result of the release of latent heat of fusion – aligns reasonably well with decrease in the average particle size. The average sizes of n-hexane and n-heptane droplets decrease by ~ 8% and ~ 5%, respectively, a result of the increased density of the solid relative to the liquid. Data from complementary PTM, FTIR, and SAXS experiments suggest that n-pentane droplets did not freeze under the time available in our experiments, despite the droplets reaching temperatures ~ 59 K below the melt point. These results are consistent with the earlier work of Lang et al.42 who found that n-pentane droplets required hundreds of seconds to freeze completely. Combining current data with prior measurements of n-octane through n-decane, we observe that for n-hexane through n- decane, there is a strong even-odd alternation in the degree of supercooling at the onset of the rapid stage of crystallization. With the exception of n-hexane, even alkanes required ~

30 K supercooling whereas odd n-alkanes required ~20 K supercooling.

MD simulations qualitatively verified the relative difficulty of n-pentane molecules to freeze in the sense that ∆Tsf,UA increased rapidly between n-hexane and n-pentane. The

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change was not, however, as dramatic as expected and this may be a function of the UA approach used here and our choice of interaction parameters. More sophisticated modeling may be required to understand the large increase in freezing time of n-pentane relative to the alkanes with slightly more carbon atoms.

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21. Modak, V. P.; Amaya, A. J.; Wyslouzil, B. E., Freezing of supercooled n-decane nanodroplets: from surface driven to frustrated crystallization. Physical Chemistry

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22. Yi, P.; Rutledge, G. C., Molecular simulation of crystal nucleation in n-octane melts. The Journal of chemical physics 2009, 131 (13), 134902.

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CHAPTER 6

CONCLUSIONS AND FUTURE WORK

This chapter discusses the conclusion of the different sections of my thesis and offers possible future work in this field.

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The technological importance of the vapor-liquid phase transition is observed in the separation of impurities, mainly lower alkanes, water and carbon dioxide, from raw natural gas. In the first part of this work, in Chapter 3, we investigated the vapor-liquid transitions of lower n-alkanes in a supersonic nozzle (nozzle T1) using experimental techniques and molecular simulations. For all n-alkanes, a bump in the temperature profile of the condensing flow and rapid growth of droplets, via pressure trace measurements and small angle x-ray scattering, respectively, were a signature for vapor-liquid transition. Scaled nucleation rates from experiments and predictions from Monte Carlo simulations, using the transferable potentials for phase equilibria-united atom potentials (TraPPE-UA), were offset by 3 orders of magnitude. However, when we explicitly accounted for the surface tension difference between the model and real n-alkanes, the offset increased to ~ 6 orders of magnitude, which is equivalent to a ~ 13kBT difference in formation free energy of the critical clusters. Simulations further suggested that the clusters of n-pentane through n- heptane remain liquid-like under experimental conditions, but higher alkanes, like n-octane and n-nonane, adopt more ordered structures. Also, consistent with the deficiency in the capillarity approximation, it was not surprising that experimental nucleation rates were 4 -

7 orders of magnitude higher than predictions from the classical nucleation theory. Overall, since predictions of nucleation rates from Monte Carlo simulations are 3 orders of magnitude higher than the experimental nucleation rates for n-pentane through n-heptane, we can possibly design nucleation experiments for lower n-alkanes (1 ≤ 푛 ≤ 4) based only on predictions from Monte Carlo simulations. This would serve as a promising

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scientific adventure into a complete investigation of the nucleation dynamics of the homologous series of n-alkanes.

Based on the success of prior nucleation experiment in the first work in Chapter 3, we determined the molecular content and excess internal energies of the critical clusters of n- pentane, n-hexane, and n-heptane when particles form under highly supersaturated conditions present in supersonic expansions. The difficulty of using a single nozzle (nozzle

T1) to determine nucleation rates as a function of supersaturation at constant temperature was overcome by using more than one nozzle characterized by different expansion rates.

Using a second and faster nozzle (nozzle T3), data from pressure trace measurements and small angle x-ray scattering experiment in both nozzles were combined. The higher nucleation rates in the faster nozzle (nozzle T3) are consistent with the higher expansion rate in nozzle T3 compared to those in nozzles T1 and C from Chapter 3 and Ghosh et al.1, respectively. Higher expansion rates are also consistent with lower temperatures and higher supersaturations reached in nozzle T3 compared to nozzle T1. By combining data from the first and second project, we confirmed our earlier results for n-heptane and applied the first and second nucleation theorems to determine the properties of critical clusters of n-alkanes.

For all n-alkanes, the critical clusters contained from ~ 2 to ~ 9 molecules (2 to 9 molecules for n-pentane, 3 to 8 molecules for n-hexane, and 6 to 8 molecules for n-heptane) and exhibited the expected increase in critical cluster size with decreasing supersaturation. We found remarkable consistency between the molecular content of n-heptane critical clusters from the current work and that determined using two nozzles from the work in Chapter 3.

Moreover, we observed similar trends in the critical cluster size variation of the excess 158

internal energies for n-pentane and n-hexane. In particular, the classical nucleation theory overpredicts and underpredicts the excess internal energies for n* < 7 and n* > 7, respectively. For the excess internal energy of n-heptane critical clusters, theory underpredicts low temperature data from supersonic nozzle and overpredicts the high temperature data from Rudek et al.2 Given our ability to combine data from nozzles and apply conventional approach to determine the properties of critical clusters, we can design experiments for 푛 ≤ 4 and compare our results with the work of Signorell and co- workers5,6 where they directly characterize clusters in a uniform post-supersonic nozzle flow using mass spectrometric detection and soft single-photon by vacuum ultraviolet

(VUV) light.

After nucleation and condensation, n-alkane droplets continue to get cooled and freezing of the droplets do occur if temperatures are low enough. In Chapter 5, we studied the freezing behavior of n-pentane, n-hexane, and n-heptane using three complementary experimental techniques (PTM, FTIR, and SAXS) and molecular dynamics simulation. We also tried to answer the question on the chain length of n-alkane at which freezing initiated by surface-templating ceases or at least becomes less effective on the time available in our experiments. The freezing behavior observed for n-hexane and n-heptane is consistent with earlier observations for n-octane, n-nonane, and n-decane by Modak and co-workers3,4. In particular, both n-alkanes (hexane and heptane) exhibit clear evidence for freezing – subtle second bump in the temperature curve that aligns with a decrease in the average particle size. Furthermore, we observed an odd-even variation in the supercooling of the droplets of n-hexane through n-decane before the onset of bulk nucleation. The trend suggests even 159

n-alkanes are more supercooled than their neighboring odd n-alkanes, therefore we should expect n-pentane droplets to be less supercooled than n-hexane droplets. However, this was not the case as n-pentane did not appear to freeze in our experiments despite being supercooled by ~ 60 K. This is surprising because odd n-alkane nanodroplets of slightly longer chain length freeze at supercooling of ~ 20 K. Moreover, results from experiments suggest that increase in the driving force, equivalent to the degree of supercooling, for the second stage of liquid to solid nucleation is consistent with a decrease in surface-based nucleation rates as the n-alkane chain length decreases. Using the united atom model, supercooling significantly increased for n-pentane, compared to n-hexane and n-heptane and we were able to rationalize roughly the chain length at which the surface-templating effect slowed in the supercooled liquid. Since the UA model does not treat hydrogen atoms explicitly, a better model, the all-atom model that includes hydrogen atoms, could possibly yield better results.

In addition to prior suggestions for future work, it will be interesting to investigate the freezing behavior of droplets containing binary mixtures of n-pentane through n-decane.

Since we have evidence for the freezing of droplets of n-hexane through n-decane, probing the surface freezing effect, i.e. its suppression or propagation, in such droplets will close the gap between research and implementation.

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10. Abhat, A., Low temperature latent heat thermal energy storage: heat storage materials. Solar energy 1983, 30 (4), 313-332.

11. Regin, A. F.; Solanki, S.; Saini, J., Heat transfer characteristics of thermal energy storage system using PCM capsules: a review. Renewable and Sustainable Energy Reviews

2008, 12 (9), 2438-2458.

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13. Ocko, B.; Wu, X.; Sirota, E.; Sinha, S.; Gang, O.; Deutsch, M., Surface freezing in chain molecules: Normal alkanes. Physical Review E 1997, 55 (3), 3164.

14. Kraack, H.; Sirota, E.; Deutsch, M., Measurements of homogeneous nucleation in normal-alkanes. The Journal of Chemical Physics 2000, 112 (15), 6873-6885.

15. Oliver, M.; Calvert, P., Homogeneous nucleation of n-alkanes measured by differential scanning calorimetry. Journal of Crystal Growth 1975, 30 (3), 343-351.

16. Boese, R.; Weiss, H. C.; Bläser, D., The Melting point alternation in the short‐chain n‐alkanes: single‐crystal x‐ray analyses of propane at 30 K and of n‐butane to n‐nonane at

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18. Weinstein, A.; Safran, S., Supercooling of surface modified liquids. Physical

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19. Weidinger, I.; Klein, J.; Stöckel, P.; Baumgärtel, H.; Leisner, T., Nucleation behavior of n-alkane microdroplets in an electrodynamic balance. The Journal of Physical

Chemistry B 2003, 107 (15), 3636-3643.

20. Modak, V. P.; Pathak, H.; Thayer, M.; Singer, S. J.; Wyslouzil, B. E., Experimental evidence for surface freezing in supercooled n-alkane nanodroplets. Physical Chemistry

Chemical Physics 2013, 15 (18), 6783-6795.

21. Modak, V. P.; Amaya, A. J.; Wyslouzil, B. E., Freezing of supercooled n-decane nanodroplets: from surface driven to frustrated crystallization. Physical Chemistry

Chemical Physics 2017, 19 (44), 30181-30194.

22. Yi, P.; Rutledge, G. C., Molecular simulation of crystal nucleation in n-octane melts. The Journal of chemical physics 2009, 131 (13), 134902.

23. Modak, V. P.; Thayer, M.; Wyslouzil, B.; Singer, S. J., Mechanism of surface freezing of alkanes. The Ohio State University: 2019.

24. Sirota, E.; Wu, X.; Ocko, B.; Deutsch, M., What drives the surface freezing in alkanes? Physical review letters 1997, 79 (3), 531.

25. Tkachenko, A. V.; Rabin, Y., Fluctuation-stabilized surface freezing of chain molecules. Physical review letters 1996, 76 (14), 2527.

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27. Qiu, Y.; Molinero, V., Strength of Alkane–Fluid Attraction Determines the

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28. 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

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31. Amaya, A. J.; Wyslouzil, B. E., Ice nucleation rates near∼ 225 K. The Journal of chemical physics 2018, 148 (8), 084501.

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33. Park, Y. Heterogeneous Nucleation in a Supersonic Nozzle. The Ohio State

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35. Waheed, N.; Ko, M.; Rutledge, G., Molecular simulation of crystal growth in long alkanes. Polymer 2005, 46 (20), 8689-8702.

36. Waheed, N.; Lavine, M.; Rutledge, G., Molecular simulation of crystal growth in n-eicosane. The Journal of chemical physics 2002, 116 (5), 2301-2309.

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89 (1), 1-18.

Chapter 6

1. Ghosh, D.; Bergmann, D.; Schwering, R.; Wolk, 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), 17.

2. Rudek, M. M.; Fisk, J. A.; Chakarov, V. M.; Katz, J. L., Condensation of a supersaturated vapor. XII. The homogeneous nucleation of the n‐alkanes. The Journal of chemical physics

1996, 105 (11), 4707-4713.

3. Modak, V. P.; Pathak, H.; Thayer, M.; Singer, S. J.; Wyslouzil, B. E., Experimental evidence for surface freezing in supercooled n-alkane nanodroplets. Physical Chemistry

Chemical Physics 2013, 15 (18), 6783-6795.

186

4. Modak, V. P.; Amaya, A. J.; Wyslouzil, B. E., Freezing of supercooled n-decane nanodroplets: from surface driven to frustrated crystallization. Physical Chemistry

Chemical Physics 2017, 19 (44), 30181-30194.

5. Jorge J. Ferreiro, Satrajit Chakrabarty, Bernhard Schläppi, and Ruth Signorell

Observation of propane cluster size distributions during nucleation and growth in a Laval expansion. The Journal of chemical physics 2016, 145, 211907.

6. B. Schlappi, J. H. Litman, J.J. Ferreiro, D. Stapfer, and Ruth Signorell, A pulsed uniform

Laval expansion coupled with single photon ionization and mass spectrometric detection for the study of large molecular aggregates - Physical Chemistry Chemical Physics 2015,

17, 25761-25771.

187

APPENDIX A Fortran Inversion Code to determine flow properties using p and g as input ccc this program version includes the ability to update the latent heat as a c function of temperature for h2o and d2o based on clausius-clapeyron c approximations to liquid-vapor equations. !chhfeb2001 ccc this version of the program calculates a "wet" isentrope based on the c measured dry isentrope and corrected for the differences in gamma. it c also starts the wet condensing flow integration on the desired data c point rather than on the wet isentrope to avoid any extraneous extra c shifts. ccc smoothes all of the good density data first, then integrates from an c initial value using finer integration grid (up to 5x) ccc modified to take in pressure data instead of density data ....jul97, jlc ccc note: stein used to smooth the integrated values as well... may consider c doing this for rough data... not yet implemented but easy to do.... bew ccc this version has been modified for Nozzle H on train B with Velmex (PP) ccc RTD probe is calibrated and temperature calibration factor is included ccc Now nu.dat has "tempcal" and this program reads in the value and does c temperature calibration as "to(i)=to(i)+tempcal"...... jun02, PP ccc fc=g *wi/(w10+w20) was replaced by fc=g/(w10+w20) March04 Shinobu ccc Inert gas is a mixture of N2 and CH4 March04 Shinobu ccc tisd=pp0d(i)**c0*t0 was replaced by pp0d(i)**c3*t0 July05 Shinobu ccc Function fk has been corrected 3/31/2007 Shinobu, Hartawan ccc See Vol.6, p9 and Vol. 9, p68 ccc tempcal is not used from cal07. 10/17/2007 Shinobu ccc Gas constant was set to 8.3145. 10/18/2007 Shinobu ccc Changing for Nonane-D2O, new Properties for Nonane 17/06/2009 ccc Changed for Heptane, 06/17/2015 Kehinde implicit real*8(a-h,o-z) 188

real*8 fcon real*8 msq,msqw,mssq,dumvar real*8 rg, pi, avog real*8 dotm,dotncal,pc10,pc20,zc10 real*8 p0, t0, tempcal real*8 xstart, xthroat, xnum real*8 tt0(2000),fc(2000),g(2000),u(2000), *rr0(2000),pp0(2000),pp0d(2000) real*8 tt0_is(2000),t_is(2000),rr0_is(2000),pp0_is(2000) real*8 t_is_s(2000),pp0_is_s(2000) real*8 aratio(2000),wg(2000),t(2000),tisd(2000) real*8 xd(2000),xw(2000),x(2000) real*8 dry(2000),dryf(2000),sdry(2000) real*8 wet(2000),swet(2000),wetf(2000) real*8 po(400),p(400),deltapo(400),deltap(400),to(400) real*8 deltadry(2000),deltadryf(2000),deltato(2000) real*8 deltawet(2000),deltawetf(2000),dtemp(2000,20) real*8 m_1,mssq_is,m_0,m_2 real*8 mdry(2000) real*8 t_is_up(2000) c c ccccccccccccccccccccccccccccccc ! Shinobu ccccccccc real*8 mssq_TDL,cp_TDL,cpr_TDL real*8 fc_TDL(2000),tt0_TDL(2000),u_TDL(2000),rr0_TDL(2000) real*8 g_TDL(2000),g_TDL2(2000),g_TDL3(2000),sol_TDL(2000) real*8 sol_TDL2(2000),sol_TDL3(2000),fsol_TDL(2000) real*8 x_TDL(2000),t_TDL(2000),ar_TDL(2000) real*8 cpv_TDL,cpc_TDL,y1_TDL,y2_TDL,y3_TDL,cpsol_TDL cccccccccccccccccccccccccccccccccccccccccccccccccccccccc character*30 dryfil,wetfil,a character*8 specie(2)

189

character*60 progname c character*4 title(3,2) common /xval/ xs(2000) *------nomenclature c dhc,fdhc(zc10,t(i)) latent heat of condensable vapor c pc10,pc20 condensable vapor pressure (read in 2*Torr, works in dyne/cm^2) c t(i) Temperature of inert in Kelvin c zc10 Initial molar fraction of condensable vapor1 (zc10+zc20=1) *------nomenclature progname='nuetodd2o_irCH4MFC_DryCp2Up_cal07' open(5,file='nu_CH4_MFC_heptane_argon.dat',status='old') ! March04 Shinobu open(10,file='4pp.out',status='unknown') open(11,file='wilson.out',status='unknown') open(9,file='new4pp.out',status='unknown') open(13,file='dtemp.out',status='unknown') open(7,file='upstream.out',status='unknown') write(*,*)'break' read(*,*)dumvar call echo pi=3.14159d0 rg=8.3145d7 avog=6.022d23 c read two condensable species read(5,41,end=50)specie c print 1006, specie 1006 format (2a8) 41 format(2a8) c read stagnation conditions-temp, pressure, partial pressure of c condensable--pressures are in mm of hg--note t0 and p0 are calculated from data files later. read(5,*)tempcal !PP02 !RTD probe calibration added

190

write(*,*)'tempcal = ',tempcal,' (not used)' c convert pressures to dyn/cm**2 pconv = 760.d0/1.01325d6 c read molecular weights of carrier (1), condensable (2,3) and CH4 (4) read(5,*)wmN,wm2,wm3,wm4 ! March04 Shinobu c read specific heats of gases read(5,*)cpN,cp2,cp3,cp4 ! March04 Shinobu c read correction factor for MKS flowmeter read(5,*)demf c read latent heat of fusion units in J/g read(5,*)fdhd c read starting value and the number of points in the output read(5,*) xstart2, ilast2 c read the integration end points (may be different than ifrst, ilast) c the number of integrations attempts, and the number of integration c sub-intervals. c istart >= 2, ifin < ilast, ni=1 (for useless roop, do k=1,ni) read(5,*)istart2, ifin2, ni, nint c read name of dry pressure data file read(5,1)dryfil 1 format(a30) c read smoothing parameters: m-order, n-number of points read(5,*)md,nd c read x values and all of the dry data p0, p(x), and the associated standard deviations. c correct the pressures i.e. baratron calibrations and pressure loss due to mesh. open(unit=4,file=dryfil,status='old') c read total number of values in dry pressure data file read(4,1)a read(4,*)idend dotncal=0.d0

191

p0dry=0.d0 c t0dry=273.15d0 +tempcal t0dry=0.0d0 do i=1,idend read(4,*)xd(i),po(i),deltapo(i),p(i),deltap(i),to(i),deltato(i), * dummy1,dummy2,flowmain,deltaflowmain,flowsub,deltaflowsub, * pthroat,deltathroat,pexit,deltapexit, * dummy3,dummy4 ! xd(i) in 0.01mm, po in 2*torr to(i)=to(i)*0.99375d0+0.352d0 ! Shinobu cal08 Vol.11, p.33 (same as cal 13) ccc Following equation includes the effect of poloss ccc po(i)=po(i)*1.00895d0-0.00995d0 !Yensil cal16 c p0dry=p0dry+po(i) t0dry=t0dry+to(i) end do p0dry=p0dry/idend t0dry=t0dry/idend+273.15d0 c do i=1,idend p(i)=0.0374041+1.00004*p(i) !Shinobu Cal 2014 dry(i)=p(i)/p0dry deltadry(i)=dry(i) * *((deltap(i)/p(i))**2.0+(deltapo(i)/p0dry)**2.0)**0.5 c write(*,*) i, xd(i), dry(i), deltadry(i) !debug dotncal=dotncal+(flowmain+flowsub)*demf/idend/22.41d0 c p0dry=p0dry+po(i) c t0dry=t0dry+to(i) end do close(unit=4)

192

write(*,*)p0dry c p0dry=p0dry/idend c t0dry=t0dry/idend+273.15d0 c read the number of wet data sets and flow rate of CH4 read(5,*)ndata, dotCH4 wm0dry=(dotncal*wmN+dotCH4*wm4)/(dotncal+dotCH4) ! March04 Shinobu w40dry=dotCH4*wm4/(dotncal*wmN+dotCH4*wm4) y30dry=dotCH4/(dotncal+dotCH4) cp0dry=( dotncal*wmN*cpN+dotCH4*wm4*fcp4(t0dry,p0dry,y30dry) ) * / (dotncal*wmN+dotCH4*wm4) c write(*,*)'wm0dry,w40dry,y30dry,cp0dry=', c & wm0dry,w40dry,y30dry,cp0dry write(11,1302)dryfil do kd = 1,ndata read(5,*)ntype,entry1,entry2,entry3 ! March04 Shinobu if(ntype.eq.0)then !pressure input (torr) pc10=entry1 pc20=entry2 tCH4=entry3 ! March04 Shinobu else if(ntype.eq.1)then !massflow and weight fraction input dotm=entry1 wfc10=entry2 wfc20=1.0d0-wfc10 c write(7,*)wfc10 tCH4=entry3 ! March04 Shinobu else write(*,*)'need to specify pressure (0)' write(*,*)'or mass flow with first weight fraction input(1)' stop end if

193

c read name of wet pressure data file read(5,1)wetfil open(unit=4,file=wetfil,status='old') c read total number of values in wet pressure data file read(4,1)a read(4,*)idenw c read x values and all of the wet data p0, p(x), and the associated standard deviations. c correct the pressures i.e. baratron calibrations and pressure loss due to mesh. tN=0.d0 t0set=0.d0 c do i=1,idenw read(4,*)xw(i),po(i),deltapo(i),p(i),deltap(i),to(i),deltato(i), * dummy1,dummy2,flowmain,deltaflowmain,flowsub,deltaflowsub, * pthroat,deltathroat,pexit,deltapexit, * dummy3,dummy4 !po in 2*torr ccc Following equation includes the effect of poloss ccc po(i)=po(i)*1.00895d0-0.00995d0 ! Yensil Cal2016 c t0set=t0set+to(i)/idenw to(i)=to(i)*0.99375d0+0.352d0 ! Shinobu cal08 Vol.11, p.33 p(i)=0.0374041+1.00004*p(i) ! Shinobu Cal2014 wet(i)=p(i)/po(i) deltawet(i)=wet(i) * *((deltap(i)/p(i))**2.0+(deltapo(i)/po(i))**2.0)**0.5 c write(*,*) xw(i), wet(i), deltawet(i) !debug tN=tN+(flowmain+flowsub)*demf/idenw/22.41d0 enddo c cccccccccccccccccccccccccccccc ! Shinobu cccc read(4,*) iden_TDL

194

write(*,*)iden_TDL do i=1,iden_TDL read(4,*) x_TDL(i),g_TDL2(i),sol_TDL2(i) enddo ccccccccccccccccccccccccccccccccccccccccccccc close(unit=4) c figure out the average stagnant pressure and temperature p0=0.0 t0=0.0 do i=1,idenw p0=p0+po(i) t0=t0+to(i) enddo p0=p0/idenw t0=t0/idenw+273.15 devp0=0.0 do i=1,idenw devp0=devp0+(po(i)-p0)**2.0 enddo devp0=(devp0/(idenw-1))**0.5 write(*,*) 'average p0 is ', p0,'torr' write(*,*) 'p0 std dev is ', devp0,'torr' write(*,*) 'average t0 is ',t0,'k' allflux=tN+tCH4+dotm*wfc10/wm2+dotm*wfc20/wm3 wm1=(tN*wmN+tCH4*wm4)/(tN+tCH4) w40=tCH4*wm4/(tN*wmN+tCH4*wm4+dotm) y30=tCH4/allflux c !chh99 c now figure out pcondensable from calibration and average properties if(ntype.eq.1)then !now calculate pcondensable

195

pc10=p0*dotm*wfc10/wm2/allflux ! June05 Shinobu write(*,*)'pc10= ',pc10,' torr' pc20=p0*dotm*wfc20/wm3/allflux write(*,*)'pc20= ',pc20,' torr' endif c convert pressures to dyn/cm**2 p0=p0/pconv pct0=pc10+pc20 pc10=pc10/pconv pc20=pc20/pconv if((pc10+pc20).lt.1.d-18) then zc10=0.d0 else zc10=pc10/(pc10+pc20) !chh22.02.01 endif y10=pc10/p0 y20=pc20/p0 c calculate stagnation gas mass density and condensable monomer mass c density (g/cm**3) c w2,w3 are mass fraction of condensable vapor in gas wmav=(wm1*(p0-pc10-pc20)+wm2*pc10+wm3*pc20)/p0 w20=wm2*pc10/p0/wmav w30=wm3*pc20/p0/wmav wi=1.d0-w20-w30 wN0=wi-w40 c gw17-2-00 assuming vapor condenses at constant composition let's define c a fictitious mean condensable vapor molecular weight wmc if((pc10+pc20).lt.1.d-18) then wmc=0.d0 else

196

wmc=(wm2*pc10+wm3*pc20)/(pc10+pc20) endif c also let's save the inital average molecular weight wmav0=wmav cp0= wN0*cpN+w40*fcp4(t0,p0,y30) & +w20*fcp2(t0,p0,y10)+w30*fcp3(t0,p0,y20) gamma=cp0dry/(cp0dry-rg*1.d-7/wm0dry) !n2 gamma gamma0=cp0/(cp0-rg*1.d-7/wmav) !initial mixture gamma rhog0=p0/rg/t0*wmav write(*,*)'wmav',' w20',' w30',' wi',' cp0',' gamma0' !chh061098 write(*,*)wmav, w20,w30,wi,cp0,gamma0 c calculate various exponents and constants involving gamma eai = 2.d0*(gamma-1.d0)/(gamma+1.d0) eai0 = 2.d0*(gamma0-1.d0)/(gamma0+1.d0) ep = -gamma/(gamma-1.d0) ep0 = -gamma0/(gamma0-1.d0) erho = -1.d0/(gamma0-1.d0) emrho = gamma-1.d0 emrho0 = gamma0-1.d0 eam2 = (gamma+1.d0)/(gamma-1.d0) eam20 = (gamma0+1.d0)/(gamma0-1.d0) c1 = 2.d0/(gamma-1.d0) c10 = 2.d0/(gamma0-1.d0) c2 = (gamma0+1.d0)/2.d0 c0 = (gamma0-1.d0)/gamma0 c3 = (gamma-1.d0)/gamma c figure out where the throat is for the dry data c first figure out the value of pstar/p0=pstp0 pstp0 = (1.d0+ 1.0d0/c1)**ep tstt0=pstp0**c3

197

*********** Values at throat under Dry condition, Shinobu ************* stepm=100.0 nstepm=100 dm_1=1.d0/stepm m_1=0.d0+dm_1 t_0=t0dry p_0=p0dry t_1=t_0 p_1=p_0 g_1=gamma cp_1=cp0dry do i=2,nstepm fk_1=( dgdt(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_1-1.d0)*m_1*m_1 ) dg_1=-2.d0*fk_1*m_1*(g_1-1.d0)*2.d0*dm_1/ & (1.d0+fk_1*m_1*m_1*(g_1-1.d0)/g_1) dt_1=-t_1/(2.d0+(g_1-1.d0)*m_1*m_1)* & ( 2.d0*(g_1-1.d0)*m_1*2.d0*dm_1 + & m_1*m_1*(g_1-1.d0)/g_1*dg_1 ) dp_1=p_1/t_1*g_1/(g_1-1.d0)*dt_1 dr_1=r_1/t_1/(g_1-1.d0)*dt_1 t_2=t_0+dt_1 p_2=p_0+dp_1 r_2=r_0+dr_1 cp_2=(1.d0-w40dry)*cpN+w40dry*fcp4(t_2,p_2,y30dry) g_2=cp_2/(cp_2-rg*1.d-7/wm0dry) t_0=t_1 p_0=p_1

198

t_1=t_2 p_1=p_2 cp_1=cp_2 g_1=g_2 m_1=m_1+dm_1 c write(*,*)'m_1,cp_1,g_1',m_1,cp_1,g_1 write(*,*)'dg_1, dt_1, dp_1, ',dt_1,dp_1 c write(*,*)'p_1/p0, t_1/t0',p_1/p0,t_1/t0,i enddo tstcpdry=t_1 pstcpdry=p_1 write(*,*) ' pstp0, tstt0 (for constant Cp) =', pstp0,tstt0 write(*,*) 'pstcpdry/p0dry, tstcpdry/t0dry = ', & pstcpdry/p0dry, tstcpdry/t0dry *********************************************************************** pstp0dry=pstcpdry/p0dry do i=1,idend c write(*,*) i, xd(i), dry(i) !debug if((dry(i).gt. pstp0dry).and.(dry(i+1).le. pstp0dry))then c write(*,*) 'true' !debug xthroat=( pstp0dry-dry(i))/(dry(i+1)-dry(i))*(xd(i+1)-xd(i)) & +xd(i) go to 5001 endif enddo 5001 continue write(*,*) 'dry throat of ',dryfil,' is at ',xthroat c now shift all the x and scale so that x(i) is in units of cm. c find number of unused points before xstart !chh110698 do i=1,idend

199

x(i)=(xd(i)-xthroat)/1000.0 ! in units of cm !Shinobu for Velmex on Train B enddo c now do linear interpolation to get fixed x intervals ************************************************ ixstart=1 xstart= int(x(1)*10.d0)/10.d0 ilast=ilast2+int( (xstart2-xstart)/0.1+0.1 ) write(*,*) 'xstart= ',xstart ************************************************ c save steps in inner loop by beginning interp. where left off lasti=ixstart !chh110698 do j=1,ilast xs(j)=xstart+(j-1)*0.1 !in intervals of 1 mm do i=lasti,idend c write(*,*)i, x(i) !chh110698 if((x(i).le.xs(j)).and.(x(i+1).gt.xs(j)))then dryf(j)=dry(i)+(xs(j)-x(i))*(dry(i+1)-dry(i))/(x(i+1)-x(i)) deltadryf(j)=deltadry(i) lasti=i !chh110698 goto 5 endif write(*,*)xs(j), dryf(j) enddo write(*,*) 'can not interpolate for point', j 5 continue enddo do j=1,(nd-1)/2 k0=1 i=j call smooth(md,nd,i,k0,sval,dryf)

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sdry(i)=sval k0=ilast-nd+1 i=ilast+1-j call smooth(md,nd,i,k0,sval,dryf) sdry(i)=sval enddo c next do points in good data range do i=3,(ilast-2) k0=i-(nd-1)/2 call smooth(md,nd,i,k0,sval,dryf) sdry(i)=sval enddo 1201 format(f10.4,f10.4,g14.4,f10.4,f10.4,g14.4) pstp0w= (1.d0+ 1.0d0/c10)**ep0 *********** Values at throat under Wet condition, Shinobu ************* stepm=250.0 nstepm=250 dm_1=1.d0/stepm m_1=0.d0+dm_1 t_0=t0 p_0=p0 r_0=rhog0 t_1=t_0 p_1=p_0 r_1=r_0 g_1=gamma0 cp_1=cp0 do i=2,nstepm fk_1=( dgdt(t_1,p_1,y10,y20,y30,w20,w30,w40,wmav0) & *t_1 + p_1*g_1/(g_1-1.d0)*

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& dgdp(t_1,p_1,y10,y20,y30,w20,w30,w40,wmav0) )/ & ( 2.d0+(g_1-1.d0)*m_1*m_1 ) dg_1=-2.d0*fk_1*m_1*(g_1-1.d0)*2.d0*dm_1/ & (1.d0+fk_1*m_1*m_1*(g_1-1.d0)/g_1) dt_1=-t_1/(2.d0+(g_1-1.d0)*m_1*m_1)* & ( 2.d0*(g_1-1.d0)*m_1*2.d0*dm_1 + & m_1*m_1*(g_1-1.d0)/g_1*dg_1 ) dp_1=p_1/t_1*g_1/(g_1-1.d0)*dt_1 dr_1=r_1/t_1/(g_1-1.d0)*dt_1 t_2=t_0+dt_1 p_2=p_0+dp_1 r_2=r_0+dr_1 cp_2= wN0*cpN+w40*fcp4(t_2,p_2,y30) & +w20*fcp2(t_2,p_2,y10)+w30*fcp3(t_2,p_2,y20) g_2=cp_2/(cp_2-rg*1.d-7/wmav) t_0=t_1 p_0=p_1 r_0=r_1 t_1=t_2 p_1=p_2 r_1=r_2 cp_1=cp_2 g_1=g_2 m_1=m_1+dm_1 enddo gammam=g_1 tstarcp=t_1 pstarcp=p_1 rstarcp=r_1 ustarcp=dsqrt(gammam*rg*tstarcp/wmav)

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write(*,*)'p*/p0, t*/t0, r*/r0 (constant Cp) ', & pstp0w,1.d0/c2,c2**erho write(*,*)'p_1/p0, t_1/t0, r_1/r0 ',p_1/p0,t_1/t0,r_1/rhog0 write(*,*) 'gamma0, gammam',gamma0,gammam *********************************************************************** do i=1,idenw c write(*,*) i, xw(i), wet(i) !debug if((wet(i).gt.pstarcp/p0).and.(wet(i+1).le.pstarcp/p0))then c write(*,*) 'true' !debug xthroat=xw(i)+(pstarcp/p0-wet(i))/ & (wet(i+1)-wet(i))*(xw(i+1)-xw(i)) go to 5002 endif enddo 5002 continue write(*,*) 'wet throat of ',wetfil,' is at ',xthroat c now shift all the x and scale so that x(i) is in units of cm. c find the number of unused points before xstart. !chh110698 ixstart=0 !chh110698 do i=1,idenw x(i)=(xw(i)-xthroat)/1000.0 ! in units of cm if(x(i).le.xstart)ixstart=i !chh110698 enddo c write(*,*) 'ixstart= ',ixstart !chh110698 write(*,*) 'throat shifted' !debug c now do linear interpolation to get fixed x intervals lasti=ixstart !chh110698 do j=1,ilast c xs values have already been assigned in dry data analysis c write(*,*) xs(j) !debug

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do i=lasti,idenw !chh110698 if((x(i).le.xs(j)).and.(x(i+1).gt.xs(j)))then wetf(j)=wet(i)+(xs(j)-x(i))*(wet(i+1)-wet(i))/(x(i+1)-x(i)) deltawetf(j)=deltawet(i) lasti=i !chh110698 goto 6 endif enddo write(*,*) 'can not interpolate for point', j 6 continue enddo c cccccccccccccccccccccccccccccccccc ! Shinobu lasti=1 do j=1,ilast c write(*,*) xs(j) !debug do i=lasti,iden_TDL if((x_TDL(i).le.xs(j)).and.(x_TDL(i+1).gt.xs(j))) then g_TDL3(j)=g_TDL2(i)+(xs(j)-x_TDL(i))* & (g_TDL2(i+1)-g_TDL2(i))/(x_TDL(i+1)-x_TDL(i)) sol_TDL3(j)=sol_TDL2(i)+(xs(j)-x_TDL(i))* & (sol_TDL2(i+1)-sol_TDL2(i))/(x_TDL(i+1)-x_TDL(i)) lasti=i !chh110698 goto 62 endif enddo write(*,*) 'can not interpolate for point', j 62 continue enddo ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c we now have an array wetf(j) at fixed xs(j) intervals. now put

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c through smoothing routine. write(*,*) 'put through smoothing' c c smooth wet pressure values c first do points at ends of good data range do j=1,(nd-1)/2 k0=1 i=j call smooth(md,nd,i,k0,sval,wetf) swet(i)=sval k0=ilast-nd+1 i=ilast+1-j call smooth(md,nd,i,k0,sval,wetf) swet(i)=sval enddo c next do points in good data range do i=3,(ilast-2) k0=i-(nd-1)/2 call smooth(md,nd,i,k0,sval,wetf) swet(i)=sval enddo write(*,*) 'finished interpolating points' c use finer integration step size than measured point spacing c generate interior points by linear interpolation c nint is the number of subintervals between each pair of original x values write(*,*) 'nint= ', nint c calculated the finer grid, interpolating on the wet condensing and c wet isentrope data write(*,*) 'calculate the finer grid' **********************************************

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ifin=ifin2+int( (xstart2-xstart)/0.1+0.1 ) istart0=1 ********************************************** npts=ifin-istart0+1 nnpts=(npts-1)*nint+1 jinit=nnpts+2*nint+istart0-1 do i=ifin+1,istart0,-1 delx=xs(i)-xs(i-1) delprd = sdry(i)-sdry(i-1) c delprwi = sweti(i)-sweti(i-1) delprw = swet(i)-swet(i-1) delg=g_TDL3(i) - g_TDL3(i-1) ! Shinobu delsol=sol_TDL3(i) - sol_TDL3(i-1) jinit=jinit-nint jp=0 write(*,*)jinit,jinit-nint+1 do j=jinit,jinit-nint+1,-1 fint=1.d0*dfloat(jp)/(1.d0*nint) xs(j)=xs(i)-delx*fint write(*,*)fint, xs(j) if(dabs(xs(j)).LT.1.d-4) ithroat=j pp0d(j) = sdry(i)-delprd*fint c pp0i(j) = sweti(i)-delprwi*fint pp0(j) = swet(i)-delprw *fint g_TDL(j)=g_TDL3(i)-delg*fint ! Shinobu sol_TDL(j)=sol_TDL3(i)-delsol*fint jp=jp+1 enddo enddo ifin1=istart0+nnpts-1

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***************************************************************** istart= istart0+ & int( (xstart2-xstart)/0.1+istart2-istart0+0.1 )*nint write(*,*)'ithroat =',ithroat ***************************************************************** *** Pressure and temperature upstream of the integration region **** t_0=tstarcp p_0=pstarcp cp_0= wN0*cpN+w40*fcp4(t_0,p_0,y30) & +w20*fcp2(t_0,p_0,y10)+w30*fcp3(t_0,p_0,y20) g_0=cp_0/(cp_0-rg*1.d-7/wmav) dp=( pp0(ithroat+1)-pp0(ithroat-1) )*p0 dt=t_0/p_0*(g_0-1.d0)/g_0*dp t_is_up(ithroat)=tstarcp t_is_up(ithroat+1)=tstarcp+dt/2.d0 t_is_up(ithroat-1)=tstarcp-dt/2.d0 t_1=t_is_up(ithroat+1) p_1=pp0(ithroat+1)*p0 do i=ithroat+2,istart cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav) dp=( pp0(i)-pp0(i-2) )*p0 dt=t_1/p_1*(g_1-1.d0)/g_1*dp t_is_up(i)=t_is_up(i-2)+dt t_1=t_is_up(i) p_1=pp0(i)*p0 enddo t_1=t_is_up(ithroat-1) p_1=pp0(ithroat-1)*p0

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do i=ithroat-2,istart0,-1 cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav) dp=( pp0(i)-pp0(i+2) )*p0 dt=t_1/p_1*(g_1-1.d0)/g_1*dp t_is_up(i)=t_is_up(i+2)+dt t_1=t_is_up(i) p_1=pp0(i)*p0 enddo write(7,700) 700 format(' x(cm) pp0 p(Torr) Tisw(K)') do i=istart0,istart write(7,710) xs(i),pp0(i),pp0(i)*p0*pconv,t_is_up(i) 710 format(f7.2,f8.4,2f10.2) enddo close(unit=7) *********************************************************************** do k = 1,ni c need to calculate at istart-1 so adjust if istart=1 c since there is no good data avaiable before 1 write(*,*) 'start' write(*,5000) istart 5000 format(3(I3,2x)) if(istart.eq.1)istart=istart+1 ***********Values at the start point of integration for Dry, Shinobu ********** stepp=100.0 nstepp=100 dp_1=(pp0d(istart-1)*p0dry-pstcpdry)/stepp t_0=tstcpdry

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p_0=pstcpdry cp_0=(1.d0-w40dry)*cpN+w40dry*fcp4(t_0,p_0,y30dry) g_0=cp_0/(cp_0-rg*1.d-7/wm0dry) m_0=1.d0 fk_0=( dgdt(t_0,p_0,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_0 + p_0*g_0/(g_0-1.d0)* & dgdp(t_0,p_0,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_0-1.d0)*m_0*m_0 ) a_0=1.d0 dt_0=t_0/p_0*(g_0-1.d0)/g_0*dp_1 dm_0= -(2.d0+(g_0-1.d0)*m_0**2) & *( g_0+fk_0*(m_0**2)*(g_0-1.d0) ) & /(2.d0*(g_0**2)*m_0)/p_0*dp_1 da_0= -a_0*(m_0**2-1.d0)/(g_0*m_0**2)/p_0*dp_1 t_1=t_0+dt_0 m_1=m_0+dm_0 p_1=p_0+dp_1 a_1=a_0+da_0 cp_1=(1.d0-w40dry)*cpN+w40dry*fcp4(t_1,p_1,y30dry) g_1=cp_1/(cp_1-rg*1.d-7/wm0dry) fk_1=( dgdt(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_1-1.d0)*m_1*m_1 ) do i=2, nstepp dt_1=t_1/p_1*(g_1-1.d0)/g_1*2.d0*dp_1 dm_1= -(2.d0+(g_1-1.d0)*m_1**2) & *( g_1+fk_1*(m_1**2)*(g_1-1.d0) ) & /(2.d0*(g_1**2)*m_1)/p_1*2.d0*dp_1 da_1= -a_1*(m_1**2-1.d0)/(g_1*m_1**2)/p_1*2.d0*dp_1

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t_2=t_0+dt_1 m_2=m_0+dm_1 a_2=a_0+da_1 p_2=p_0+2.d0*dp_1 cp_2=(1.d0-w40dry)*cpN+w40dry*fcp4(t_2,p_2,y30dry) g_2=cp_2/(cp_2-rg*1.d-7/wm0dry) fk_2=( dgdt(t_2,p_2,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_2 + p_2*g_2/(g_2-1.d0)* & dgdp(t_2,p_2,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_2-1.d0)*m_2*m_2 ) t_0=t_1 m_0=m_1 a_0=a_1 p_0=p_1 t_1=t_2 m_1=m_2 a_1=a_2 g_1=g_2 fk_1=fk_2 p_1=p_2 c write(*,*)'dt_1,dm_1,da_1 =',dt_1,dm_1,da_1 enddo tisd(istart-1)=t_1 aratio(istart-1)=a_1 ar_TDL(istart-1)=a_1 !harshad mdry(istart-1)=m_1 dp_2=dp_1+(pp0d(istart)-pp0d(istart-1))*p0dry dt_1=t_1/p_1*(g_1-1.d0)/g_1*dp_2 dm_1= -(2.d0+(g_1-1.d0)*m_1**2) & *( g_1+fk_1*(m_1**2)*(g_1-1.d0) )

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& /(2.d0*(g_1**2)*m_1)/p_1*dp_2 da_1= -a_1*(m_1**2-1.d0)/(g_1*m_1**2)/p_1*dp_2 t_2=t_0+dt_1 m_2=m_0+dm_1 a_2=a_0+da_1 p_2=p_0+dp_2 tisd(istart)=t_2 mdry(istart)=m_2 aratio(istart)=a_2 ar_TDL(istart)=a_2 !harshad ********************************************************************************* c note! start the wet condensing flow integration on the desired data c point (i.e. on the wet curve data) rather than on the wet isentrope c to avoid any extraneous extra shifts/offsets in t etc. ***********Values at the start point of integration for Wet, Shinobu ********** stepp=100.0 nstepp=100 dp_1=(pp0(istart-1)*p0-pstarcp)/stepp t_0=tstarcp p_0=pstarcp r_0=rstarcp g_0=gammam dt_0=t_0/p_0*(g_0-1.d0)/g_0*dp_1 dr_0=r_0/p_0/g_0*dp_1 t_1=t_0+dt_0 r_1=r_0+dr_0 p_1=p_0+dp_1 cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav)

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do i=2,nstepp dt_1=t_1/p_1*(g_1-1.d0)/g_1*2.d0*dp_1 dr_1=r_1/p_1/g_1*2.d0*dp_1 t_2=t_0+dt_1 r_2=r_0+dr_1 p_2=p_0+2.d0*dp_1 cp_2= wN0*cpN+w40*fcp4(t_2,p_2,y30) & +w20*fcp2(t_2,p_2,y10)+w30*fcp3(t_2,p_2,y20) g_2=cp_2/(cp_2-rg*1.d-7/wmav) t_0=t_1 r_0=r_1 p_0=p_1 t_1=t_2 r_1=r_2 cp_1=cp_2 g_1=g_2 p_1=p_2 enddo tt0_is(istart-1)=t_1/t0 rr0_is(istart-1)=r_1/rhog0 pp0_is(istart-1)=p_1/p0 msqw = c10*((1.d0/pp0(istart-1))**c0-1.d0) write(*,*)'pp0(istart-1), tt0, rr0', & pp0(istart-1),1.d0/(1.d0+msqw/c10),(1.d0+msqw/c10)**erho write(*,*)'p_1/p0,t_1/t0,r_1/rhog0', & pp0_is(istart-1),tt0_is(istart-1),rr0_is(istart-1) dp_2=dp_1+(pp0(istart)-pp0(istart-1))*p0 dt_1=t_1/p_1*(g_1-1.d0)/g_1*dp_2 dr_1=r_1/p_1/g_1*dp_2 t_2=t_0+dt_1

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r_2=r_0+dr_1 tt0_is(istart)=t_2/t0 rr0_is(istart)=r_2/rhog0 pp0_is(istart)=(p_0+dp_2)/p0 msqw = c10*((1.d0/pp0( istart))**c0-1.d0) write(*,*)'pp0(istart), tt0, rr0 ', & pp0(istart),1.d0/(1.d0+msqw/c10),(1.d0+msqw/c10)**erho write(*,*)'p_2/p0,t_2/t0,r_2/rhog0', & pp0_is(istart),tt0_is(istart),rr0_is(istart) tt0(istart-1)=tt0_is(istart-1) tt0_TDL(istart-1)=tt0_is(istart-1) !harshad rr0(istart-1)=rr0_is(istart-1) rr0_TDL(istart-1)=rr0_is(istart-1) !harshad tt0(istart)=tt0_is(istart) tt0_TDL(istart)=tt0_is(istart) !harshad rr0(istart)=rr0_is(istart) rr0_TDL(istart)=rr0_is(istart) !harshad *********************************************************************** g(istart)=0.d0 g(istart-1)=0.d0 fc(istart)=0.d0 !fraction condensed fc_TDL(istart)=0.d0 !harshad fsol_TDL(istart)=0.d0 write(10,1024) progname 1024 format('Program: ',a60) write(10,1011)p0*pconv,devp0,t0-273.15, t0set, rhog0*1.0d3 !4pp plots write(10,1010) dotm,specie(1),wfc10 !4pp plots 1010 format('Weight flux of condensable =',f6.2, & ' g/min , Fraction of ',a, '=',f7.3) !4pp plots 1011 format('p0= ',f6.2,'+/-',f4.2,' Torr T0=',f6.2,

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& ' C (set T0=',f6.2,') rho0=', e11.4,' kg/m3') !4pp plots write(10,1012)pc10*pconv,specie(1),pc20*pconv, +specie(2) !4pp plots 1012 format('@subtitle "',2(f7.4,'torr ',a),'"') !4pp plots write(10,1013) allflux, (dotncal+dotCH4) !4pp plots 1013 format( 'Total mole flux =', & f6.3,' mol/min (',f6.3,' in Dry )' ) !4pp plots write(10,1019) tCH4, dotCH4 !4pp plots 1019 format( ' CH4 mole flux =', & f6.3,' mol/min (',f6.3,' in Dry )' ) !4pp plots write(10,1014)wetfil,dryfil !4pp plots 1014 format('@subtitle \"',a30,'with dry trace ',a30,'\"') !4pp plots write(10,1015) !4pp plots 1015 format(' x(cm) u(m/s) T(K) p/p0 Tis p/p0_is', &' MoleFract. g g/g_inf A/A* r/r0', &' Tisd p/p0_isd') !4pp plots

write(9,1037) !4pp plots in new output file 1037 format(' x(cm) u_PTM T_PTM g_PTM T_is Tptm-Tis P/po_is', &' P/po g/g_inf sol/g_inf T_tdl A/A*_PTM A/A*_TDL gginf_TDL', &' r/r0_TDL r/r0 g_TDL u_TDL') write(*,*) 'start integration' write(*,5000)istart,ifin1 do i=istart,ifin1 c calculate local value of effective area ratio, aratio c msq is local mach number squared, mssq = (u/u*)^2 *********** Integration of the isentropic curve for Dry, Shinobu ************** dp_dry=( pp0d(i+1)-pp0d(i-1) )*p0dry/2.d0 p_dry=pp0d(i)*p0dry cp_dry=(1.d0-w40dry)*cpN + w40dry*fcp4(tisd(i),p_dry,y30dry)

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g_dry=cp_dry/(cp_dry-rg*1.d-7/wm0dry) fk_dry=( dgdt(tisd(i),p_dry,0.d0,0.d0,y30dry, & 0.d0,0.d0,w40dry,wm0dry) & *tisd(i) + p_dry*g_dry/(g_dry-1.d0)* & dgdp(tisd(i),p_dry,0.d0,0.d0,y30dry, & 0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_dry-1.d0)*mdry(i)**2 ) dt_dry=tisd(i)/p_dry*(g_dry-1.d0)/g_dry*dp_dry dm_dry= -( 2.d0+(g_dry-1.d0)*mdry(i)**2 ) & *( g_dry+fk_dry*(mdry(i)**2)*(g_dry-1.d0) ) & /(2.d0*(g_dry**2)*mdry(i))/p_dry*dp_dry da_dry= -aratio(i)*(mdry(i)**2-1.d0)/(g_dry*mdry(i)**2)/ & p_dry*dp_dry tisd(i+1)=tisd(i-1)+2.d0*dt_dry mdry(i+1)=mdry(i-1)+2.d0*dm_dry aratio(i+1)=aratio(i-1)+2.d0*da_dry dar=dlog(aratio(i+1)/aratio(i-1))/2.d0 c write(*,*) 'Integration of the isentropic curve for Dry OK' ********************************************************************************* *********** Integration of the isentropic curve for Wet, Shinobu ************** dp=(pp0(i+1)-pp0(i-1))/2.d0 t_is(i)=tt0_is(i)*t0 u_is=ustarcp*rstarcp/rhog0/rr0_is(i)/aratio(i) mssq_is=(u_is/ustarcp)**2 cp_is= wN0*cpN+w40*fcp4(t_is(i),pp0_is(i)*p0,y30) & +w20*fcp2(t_is(i),pp0_is(i)*p0,y10) & +w30*fcp3(t_is(i),pp0_is(i)*p0,y20) cpr_is=cp_is/cp0 hpara_is=1.d0-(gamma0-1.d0)/gamma0/cpr_is tempA=(1.d0-t_is(i)/tstarcp/gammam/mssq_is)/rr0_is(i)

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tempG=t0/tstarcp/gammam/mssq_is tempJ=rr0_is(i)*tt0_is(i)/ & (hpara_is-t_is(i)/tstarcp/gammam/mssq_is) dtt0_is=(tt0_is(i)-tempA*tempJ)*dar dpp0_is=-tempJ*dar drr0_is=-(rr0_is(i)+tempG*tempJ)*dar tt0_is(i+1)=tt0_is(i-1)+2.d0*dtt0_is pp0_is(i+1)=pp0_is(i-1)+2.d0*dpp0_is rr0_is(i+1)=rr0_is(i-1)+2.d0*drr0_is c write(*,*) 'Integration of the isentropic curve for Wet OK' *************** Smoothing, Shinobu ************************************ t_is_s(i)=t0*( tt0_is(i-1)+2.d0*tt0_is(i)+tt0_is(i+1) )/4.d0 pp0_is_s(i)=( pp0_is(i-1)+2.d0*pp0_is(i)+pp0_is(i+1) )/4.d0 c t_is_s(i)=t0*tt0_is(i) c pp0_is_s(i)=pp0_is(i) *********************************************************************** *********************************************************************** t(i)=tt0(i)*t0 t_TDL(i)=tt0_TDL(i)*t0 !harshad u(i)=ustarcp*rstarcp/rhog0/rr0(i)/aratio(i) u_TDL(i)=ustarcp*rstarcp/rhog0/rr0_TDL(i)/ar_TDL(i) ! harshad mssq=(u(i)/ustarcp)**2 mssq_TDL=(u_TDL(i)/ustarcp)**2 fcon=dotm * (wfc10/wm2 + (1-wfc10)/wm3) if((pc10+pc20).lt.1.d-18) then y1=0.d0 y2=0.d0 else y1=(1-fc(i))*fcon/(allflux-fc(i)*fcon)*y10/(y10+y20) y2=(1-fc(i))*fcon/(allflux-fc(i)*fcon)*y20/(y10+y20)

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endif y3=y30*allflux/(allflux-fc(i)*fcon) c write(*,*)'(y1+y2+y3+tN/allflux*y3/y30), (wN0+w20+w30+w40)', c & y1+y2+y3+tN/allflux*y3/y30, wN0+w20+w30+w40 c harshad-y values required only in case of clustering y1=0 y1_TDL=0 y2=0 y2_TDL=0 y3=0 y3_TDL=0 c gw2-17-00 update specific heat if((pc10+pc20).lt.1.d-18) then cpv=0.d0 cpc=0.d0 else cpv=( w20*fcp2(t(i),pp0(i)*p0,y1) + & w30*fcp3(t(i),pp0(i)*p0,y2) )/(w20+w30) cpc=( w20*fcpl2(t(i))+w30*fcpl3(t(i)) )/(w20+w30) endif cp= wN0*cpN+w40*fcp4(t(i),pp0(i)*p0,y3) & +(w20+w30-g(i))*cpv+g(i)*cpc cpr=cp/cp0 c c gw2-17-00 update specific heat for TDL if((pc10+pc20).lt.1.d-18) then cpv_TDL=0.d0 cpc_TDL=0.d0 cpsol_TDL=0.d0 else cpv_TDL=( w20*fcp2(t_TDL(i),pp0(i)*p0,y1_TDL) +

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& w30*fcp3(t_TDL(i),pp0(i)*p0,y2_TDL) )/(w20+w30) cpc_TDL=( w20*fcpl2(t_TDL(i))+w30*fcpl3(t_TDL(i)) )/(w20+w30) cpsol_TDL=( w20*fcpsol2(t_TDL(i)) + & w30*fcpsol3(t_TDL(i)))/(w20+w30) endif cp_TDL= wN0*cpN+w40*fcp4(t_TDL(i),pp0(i)*p0,y3_TDL) & +(w20+w30-g_TDL(i))*cpv_TDL+g_TDL(i)*cpc_TDL cpr_TDL=cp_TDL/cp0 c gw2-17-00 update "mu/(1-g)" = wmu, and related factors if((pc10+pc20).lt.1.d-18) then wmu=wm1 wg(i)=0.d0 else wmu=wm1*wmc/(wi*wmc+(w20+w30-g(i))*wm1) wg(i)=wmu/(wmc) endif wmuu0=wmu/wmav0 c c gw2-17-00 update "mu/(1-g)" = wmu_TDL, and related factors if((pc10+pc20).lt.1.d-18) then wmu_TDL=wm1 wg_TDL=0.d0 else wmu_TDL=wm1*wmc/(wi*wmc+(w20+w30-g_TDL(i))*wm1) wg_TDL=wmu_TDL/(wmc) endif wmuu0_TDL=wmu_TDL/wmav0 hpara=wmuu0-(gamma0-1.d0)/gamma0/cpr dr=dp/tstarcp*t0/gammam/mssq-rr0(i)*dar dgp=(hpara-t(i)/tstarcp/mssq/gammam)/rr0(i)*dp+tt0(i)*dar if((pc10+pc20).lt.1.d-18) then

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dg=0.d0 else dg=dgp*cp*t0/(fdhc(wfc10,t(i))-cp*t(i)*wg(i)) ! Shinobu endif c gw2-17-00 update dtt0 dtt0=(wmuu0-t(i)/tstarcp/gammam/mssq)/rr0(i)*dp+ & tt0(i)*(dar+wg(i)*dg) tt0(i+1)=tt0(i-1)+2.0d0*dtt0 rr0(i+1)=rr0(i-1)+2.0d0*dr g(i+1)=g(i-1)+2.0d0*dg if((w20+w30).gt.0.0)then fc(i+1)=g(i+1)/(w20+w30) ! March04 Shinobu else fc(i+1)=0.0 end if c c write(*,*) 'Integration of the Wet trace OK' cc ***********************correction for the code when g is input******** ap_TDL=wmuu0_TDL*gamma0-(gamma0-1.d0)/cpr_TDL ! Shinobu h_TDL=ap_TDL/gamma0 ! Shinobu cc ccccccccccccccccccc ! Shinobu cccccccccccccccccccccccc dg_TDL=(g_TDL(i+1)-g_TDL(i-1))/2.0d0 dsol_TDL=(sol_TDL(i+1)-sol_TDL(i-1))/2.0d0 tempA=(wmuu0_TDL-t_TDL(i)/tstarcp/gammam/mssq_TDL)/rr0_TDL(i) tempB=tt0_TDL(i) tempC=tt0_TDL(i)*wg_TDL tempF=fdhc(wfc10,t_TDL(i))/cp_TDL/t0-tt0_TDL(i)*wg_TDL tempD=(h_TDL - t_TDL(i)/tstarcp/gammam/mssq_TDL)/rr0_TDL(i)/tempF tempE=tt0_TDL(i)/tempF tempS=fdhd/cp_TDL/t0

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dlar_TDL=dg_TDL/tempE-tempD/tempE*dp+tempS/tt0_TDL(i)*dsol_TDL dtt0_TDL=tempA*dp+tempB*dlar_TDL+tempC*dg_TDL dr_TDL=dp/tstarcp*t0/gammam/mssq_TDL-rr0_TDL(i)*dlar_TDL ar_TDL(i+1)=ar_TDL(i-1)*dexp(2.0d0*dlar_TDL) rr0_TDL(i+1)=rr0_TDL(i-1)+2.0d0*dr_TDL tt0_TDL(i+1)=tt0_TDL(i-1)+2.0d0*dtt0_TDL wm_TDL=wmav0*dotncal/ & (dotncal-(dotncal-tN-tCH4)*fc_TDL(i)) ! Molecular weight for c c sound velocity ga_TDL=cp_TDL/(cp_TDL-8.3145/wm_TDL) ! Specific heat ratio for c sound velocity a_TDL=(ga_TDL*8.3145*t_TDL(i)/wm_TDL*1000.0)**0.5 if((w20+w30).gt.0.0)then fc_TDL(i+1)=g_TDL(i+1)/(w20+w30) ! Shinobu fsol_TDL(i+1) = sol_TDL(i+1)/(w20+w30) else fc_TDL(i+1)=0.0 fsol_TDL(i+1)=0.0 end if write(9,1105)xs(i),u(i)/100,t(i),g(i),t_is_s(i),t(i)-t_is_s(i), & pp0_is_s(i),pp0(i),fc(i), fsol_TDL(i), & t_TDL(i),aratio(i),ar_TDL(i),fc_TDL(i),rr0_TDL(i), & rr0(i),g_TDL(i),u_TDL(i)/100 1105 format(3f8.2,f8.4,2f8.2,4f8.4,f8.2, & 5f8.4,f8.4,f8.2) cc ***********************correction for the code when g is input over******** write(10,1020)xs(i),u(i)/100,t(i),pp0(i),t_is_s(i),pp0_is_s(i), ! June05 Shinobu * (1-fc(i))*fcon/(allflux-fc(i)*fcon),g(i),fc(i),aratio(i), * rr0(i),tisd(i),pp0d(i) 1020 format(f8.3,f10.2,f8.2,f8.4,f8.2,f8.4,2e13.4,f8.4,f8.4,f8.4,

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& f8.2,f8.4) !4pp plots 1000 format(e12.3,e12.4,f8.2,e12.3,f8.2,f7.3,2e12.4,e12.4) 1100 format(i5,e12.3,5e12.4) 1110 format(e12.3,e13.5) enddo write(*,*)'start search' c now search for the onset conditions using both t(i)-t_is_s(i) and t(i)-tisd do i = istart,ifin1-1 dtemp(i,kd) = t(i)-t_is_s(i) dt1 = t(i) - t_is_s(i) dt2 = t(i+1) -t_is_s(i+1) if(dt1.le.0.98d0.and.dt2.gt.0.98d0)then xon = xs(i)+(0.98-dt1)/(dt2-dt1)*(xs(i+1)-xs(i)) pp0on = pp0(i)+(0.98-dt1)/(dt2-dt1)*(pp0(i+1)-pp0(i)) pp0ion = pp0_is_s(i)+(0.98-dt1)/(dt2-dt1)*(pp0_is_s(i+1)-pp0(i)) ton = t(i)+(0.98-dt1)/(dt2-dt1)*(t(i+1)-t(i)) tiswon = t_is_s(i)+(0.98-dt1)/(dt2-dt1)*(t_is_s(i+1)-t_is_s(i)) else endif enddo write(*,*)'using the t-t_is_s = 0.98 k' write(*,1300)xon,pp0on*pct0,ton, & pp0ion*pct0,tiswon 1300 format('onset occurs at x =',f8.4,f8.4,f7.2,f8.4,f7.2) write(11,1301)t0,p0*pconv,pct0,ton,pp0on*pct0, & pp0on*pc10*pconv,pp0on*pc20*pconv,wetfil,xon 1301 format(f8.2,f8.2,f8.4,f8.2,f8.4,f8.4,f8.4,2x,a13,f7.1) 1302 format('@\"t0 p0 pct ton pon p1on p2on', & 6x,a13,'\"') xon=0.0

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pp0on=0.0 ton=0.0 do i = istart,ifin1-1 dt1 = t(i) - tisd(i) dt2 = t(i+1) - tisd(i+1) if(dt1.le.0.98d0.and.dt2.gt.0.98d0)then xon = xs(i)+(0.98-dt1)/(dt2-dt1)*(xs(i+1)-xs(i)) pp0on = pp0(i)+(0.98-dt1)/(dt2-dt1)*(pp0(i+1)-pp0(i)) pp0don = pp0d(i)+(0.98-dt1)/(dt2-dt1)*(pp0d(i+1)-pp0(i)) ton = t(i)+(0.98-dt1)/(dt2-dt1)*(t(i+1)-t(i)) tisdon = tisd(i)+(0.98-dt1)/(dt2-dt1)*(tisd(i+1)-tisd(i)) else endif enddo write(*,*)'using the criterion t-tisd = 0.98 k' write(*,1300)xon,pp0on*(pc10+pc20)*pconv,ton, & pp0don*(pc10+pc20)*pconv,tisdon write(*,*) 'finished integration' enddo enddo c now write out the dtemp files to dtemp.out do i = istart,ifin1 write(13,1313)xs(i),(dtemp(i,j),j=1,ndata) enddo 1313 format(f8.4,20(f8.2)) 50 stop end c subroutine smooth(m,n,k,k0,sval,y) c this subroutine produces smoothed values of a tabulated function y

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c based on technique described in ralston, "a first course in num. anal." c y values do not have to be equally spaced, but x values must be supplied c regardless of the spacing c c m - order of the highest polynomial used in smoothing c n - number of y points in interval over which smoothing is performed c k - point whose smoothed value is desired c k0 - first point in set of n c sval - smoothed value returned to calling program c real*8 p(-2:5,1:200),b(0:5),omega(0:5),gamma(0:5),beta(-1:5) *,alpha(0:5),y(200),sval,x common /xval/ x(1000) beta(-1)=0. beta(0)=0. gamma(0)=n omega(0)=0. alpha(1)=0. do i=k0,(n+k0-1) omega(0)=omega(0)+y(i) alpha(1)=alpha(1)+x(i) p(-2,i)=0. p(-1,i)=0. p(0,i)=1. enddo b(0)=omega(0)/gamma(0) alpha(1)=alpha(1)/gamma(0) sval=b(0) do j=1,m gamma(j)=0.

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omega(j)=0. alpha(j+1)=0. do i=k0,(n+k0-1) p(j,i)=(x(i)-alpha(j))*p(j-1,i) - beta(j-1)*p(j-2,i) gamma(j)=gamma(j)+p(j,i)*p(j,i) alpha(j+1)=alpha(j+1)+x(i)*p(j,i)*p(j,i) omega(j)=omega(j)+y(i)*p(j,i) enddo alpha(j+1)=alpha(j+1)/gamma(j) beta(j)=gamma(j)/gamma(j-1) b(j)=omega(j)/gamma(j) sval=sval+b(j)*p(j,k) enddo return end subroutine echo character*100 a write(9,3) 15 read(5,1,end=99)a write(9,2)a goto 15 99 continue rewind 5 return 1 format(a100) 2 format(1x,a100) 3 format(1h1,20x,'input file',//) end *------chh22.02.01---* *

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* real*8 function fdhc(wfc10,tk) double precision zc10,wfc10,tk,rg *------general nomenclature-* c rg universal gas constant in units of c tk temperature of vapor condensing in kelvin c zc10 molar fraction of condensable 1 in vapor (zc10+zc20=1.0) *------condensable nomenclature-* c a2h2o - a4h2o h2o vapor pressure constants, wagner correlation c a1d2o - a6d2o d2o vapor pressure constants, c mwd2o d2o molecular weight c z d2o intermediate variable *------* double precision a1d2o,a2d2o,a3d2o,a4d2o,a5d2o,a6d2o,z,mwd2o !d2o pve constants double precision bbu,cbu,mwetod,dhcetod

rg=8.3145d0 *------* *-d2o clausius-clapeyron relation applied to equilibrium vapor pressure *-d2o valid for temperature range of 275-823K *-d2o hill, mcmillan, and lee, j. phys chem ref data, vol 11, no.1, p1-14 (1982) a1d2o= -7.81583d0 a2d2o= 17.6012d0 a3d2o=-18.1747d0 a4d2o= -3.92488d0 a5d2o= 4.19174d0 a6d2o=643.89d0 mwd2o=20.03d0 z=1-tk/a6d2o d2oa=a1d2o*z+a2d2o*z**1.9+a3d2o*z**2+a4d2o*z**5.5+a5d2o*z**10.

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d2ob=a1d2o+1.9d0*a2d2o*z**0.9+2.d0*a3d2o*z+5.5d0*a4d2o*z**4.5 &+10.d0*a5d2o*z**9. dhcd2o=-rg*(a6d2o*d2oa+tk*d2ob)/mwd2o *------* *-BuOH clausius-clapeyron relation applied to equilibrium vapor pressure *-BuOH valid for temperature range of 243.2-303.2 K * And T. Schmeling and R. Strey, Ber. Bunsenges. Phys. Chem., vol 87, p871-874 (1983) c Use corrected equation 20 of Ruzicka and Majer J physical chem ref data 23, 1994 p 1-39 c Note the T is missing from the a1 term!Original units are J/mol a0 = 2.86470d0 a1 = -2.113204d-3 a2 = 2.250991d-6 Tb = 371.552 mwetod=100.204d0 Term1 = rg*exp(a0 + a1*tk + a2*tk*tk) Term2 = Tb + tk*(tk - Tb)*(a1+2.0d0*a2*tk) ccc dhcetod= Hvap von heptane!!! dhcetod = (Term1*Term2)/mwetod *------* fdhc = (wfc10*dhcetod)+(1.d0-wfc10)*dhcd2o c write(28,*)'dhc debug: fdhc= ',fdhc,' wfc10= ',wfc10,' tk= ',tk,' K' !debug dhc return end *------* ***************** Functions for Cp ********************************** real*8 function fcp2(tk,p,y1) double precision tk,p,y1 double precision mw,a0,a1,a2 c** Cp of BuOH ****** p: Total static pressure

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****** y1: Mole fraction of condensable 1 in vapor phase c [J/g*K] *----Cp heptane--Ruzicka & Majer J.Phys.Chem.Ref.Data.Vol.23.No.1,1994 ------***NEU machen * taken from Bures et al, Chem Eng Sci, 36, 529-537 (1981) c------Einheit J/molK------A = 117.22475d0 B1 = 151.73507d0 B2 = 227.31996d0 C1 = 3154.9913d0 C2 = 1391.9171d0 Term1 = (C1/tk)*(C1/tk)*dexp(-C1/tk)/(1.0d0 - dexp(-C1/tk))**2 Term2 = (C2/tk)*(C2/tk)*dexp(-C2/tk)/(1.0d0 - dexp(-C2/tk))**2 cpheptane = A + B1*Term1 + B2*Term2 mw=100.204d0 fcp2 = cpheptane/mw return end C Functions for Cp solid (J/g-K) real*8 function fcpsol3(tk) fcpsol3= ((-23900+1940.8*tk-10.48*tk**2+0.02719*tk**3)) & /100204 return end real*8 function fcpsol2(tk) fcpsol3= ((-23900+1940.8*tk-10.48*tk**2+0.02719*tk**3)) & /100204 return end c unit for the code should be J/g*K real*8 function fcp3(tk,p,y2) 227

double precision tk,p,y2 double precision rg,mw,a0,a1,a2,a3 c** Cp of D2O: Fitted to the data in JCP 22, 2051 (1954) at T= 160-340 K rg=8.3145 mw=20.03 a0=4.1712 a1=-2.2388d-3 a2= 8.6096d-6 a3=-5.6304d-9 fcp3=rg*(a0+a1*tk+a2*tk**2+a3*tk**3)/mw c fcp3=1.710d0 return end real*8 function fcp4(tk,p,y3) double precision tk,p,y3 double precision rg,mw,a0,a1,a2,a3 c** Cp of CH4: Fitted to the data in J. Chem. Eng. Data 8, 547 (1963) at T= 160-340 K rg=8.3145 mw=16.04 a0=4.337 a1=-3.7677d-3 a2= 9.1107d-6 a3= 1.0178d-9 fcp4=rg*(a0+a1*tk+a2*tk**2+a3*tk**3)/mw c fcp4=2.226d0 return end ************** Temperature derivative of Cp **************** real*8 function fdcp2dt(tk,p,y1) double precision tk,p,y1,mw,a1,a2,c1,c2

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c----Cp Nonane--Ruzicka & Majer J.Phys.Chem.Ref.Data.Vol.23.No.1,1994 ------***NEU machen * taken from Bures et al, Chem Eng Sci, 36, 529-537 (1981) c------Einheit J/molK------A = 117.22475d0 B1 = 151.73507d0 B2 = 227.31996d0 C1 = 3154.9913d0 C2 = 1391.9171d0 mwheptane=100.204d0 ccc Näherung für cp/dt tk1 = tk+ 0.01d0 tk2 = tk - 0.01d0 Term1 = (C1/tk1)*(C1/tk1)*dexp(-C1/tk1)/ & (1.0d0 - dexp(-c1/tk1))**2 Term2 = (C2/tk1)*(C2/tk1)*dexp(-C2/tk1)/ & (1.0d0 - dexp(-c2/tk1))**2 cpheptane1 = (A + B1*Term1 + B2*Term2)/mwheptane Term1 = (C1/tk2)*(C1/tk2)*dexp(-C1/tk2)/ & (1.0d0 - dexp(-c1/tk2))**2 Term2 = (C2/tk2)*(C2/tk2)*dexp(-C2/tk2)/ & (1.0d0 - dexp(-c2/tk2))**2 cpheptane2 = (A + B1*Term1 + B2*Term2)/mwheptane fdcp2dt = (cpheptane1-cpheptane2)/(0.02d0) return end real*8 function fdcp3dt(tk,p,y2) double precision tk,p,y2 double precision rg,mw,a1,a2,a3 c** Cp of D2O: Fitted to the data in JCP 22, 2051 (1954) at T= 160-340 K rg=8.3145 229

mw=20.03 a1=-2.2388d-3 a2= 8.6096d-6 a3=-5.6304d-9 fdcp3dt=rg*(a1+2.d0*a2*tk+3.d0*a3*tk**2)/mw c fdcp3dt=0.0 return end real*8 function fdcp4dt(tk,p,y3) double precision tk,p,y3 double precision rg,mw,a1,a2,a3 c** Cp of CH4: Fitted to the data in J. Chem. Eng. Data 8, 547 (1963) at T= 160-340 K rg=8.3145 mw=16.04 a1=-3.7677d-3 a2= 9.1107d-6 a3= 1.0178d-9 fdcp4dt=rg*(a1+2.d0*a2*tk+3.d0*a3*tk**2)/mw c fdcp4dt=0.0 return end *************** Pressure derivative of Cp ********************* real*8 function fdcp2dp(tk,p,y1) double precision tk,p,y1 fdcp2dp=0.d0 return end real*8 function fdcp3dp(tk,p,y2) double precision tk,p,y2 fdcp3dp=0.d0

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return end real*8 function fdcp4dp(tk,p,y3) double precision tk,p,y3 fdcp4dp=0.d0 return end ***************** Functions for Cpl ********************************** c** Cpl of Heptane real*8 function fcpl2(tk) double precision tk fcpl2=2.250 return end c** Cpl of D2O real*8 function fcpl3(tk) double precision tk fcpl3=4.205d0 return end ******** Temperature derivative of gamma of gas mixture *** real*8 function dgdt(tk,p,y1,y2,y3,w20,w30,w40,wmav0) double precision tk,p,y1,y2,y3,w20,w30,w40,wmav0 double precision rg,cpN,cp,gamma,dcpdt rg=8.3145 cpN=0.5203 cp= (1.d0-w20-w30-w40)*cpN+w40*fcp4(tk,p,y3) & +w20*fcp2(tk,p,y1)+w30*fcp3(tk,p,y2) gamma=cp/(cp-rg/wmav0) dcpdt= w20*fdcp2dt(tk,p,y1)+w30*fdcp3dt(tk,p,y2)

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& +w40*fdcp4dt(tk,p,y3) dgdt=gamma*(1.d0-gamma)/cp*dcpdt return end ******** Pressure derivative of gamma of gas mixture ****** real*8 function dgdp(tk,p,y1,y2,y3,w20,w30,w40,wmav0) double precision tk,p,y1,y2,y3,w20,w30,w40,wmav0 double precision rg,cpN,cp,gamma,dcpdp rg=8.3145 cpN=0.5203 cp= (1.d0-w20-w30-w40)*cpN+w40*fcp4(tk,p,y3) & +w20*fcp2(tk,p,y1)+w30*fcp3(tk,p,y2) gamma=cp/(cp-rg/wmav0) dcpdp= w20*fdcp2dp(tk,p,y1)+w30*fdcp3dp(tk,p,y2) & +w40*fdcp4dp(tk,p,y3) dgdp=gamma*(1.d0-gamma)/cp*dcpdp return end

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APPENDIX B Fortran Code for determining Characteristic time c This Code calculates the characteristic time for the corresponding c peak nucleation rates. The basic principal is c deltat=int.(J(S,T)dt)/Jmax c In order to run this code the 4pp.txt file is required. c David Ghosh 02 Sep. 2005 c this code is for Propanol c Alex Manka 30.10.2007 c Dirk Bergmann changes for n-nonane 11/15/2007 c Dirk Änderung der Phys. Properties 4/09/2008 für Nonane c Change for D20 18/06/2009 c Change for H2O 1/22/2015 c Change for Hexane 05/19/2015 *------definition ------implicit real*8(a-h,o-z) real*8 x(1200),u(1200),T(1200),pp0(1200),fc(1200) real*8 umean(1200),timestep(1200),pc(1200),S(1200) real*8 JBD(1200),A(1200),B(1200),sig(1200),d(1200),v(1200) real*8 intJBDu(1200),intJBDo(1200),intJBD(1200),rhorat(1200) real*8 Na,k,pi,M,locMax,globMax,time(1200),rateint(1200) real*8 chartime(1200) real*8 d1, d2, d4, d5, pe character*30 dryfil,wetfil character*4 a0 character*18 a1 character*29 a2 character*11 a3 character*17 a4 character*13 a5 233

character*50 a6,a7,a10,a11 character*100 a8,a9 character*9 specie character*26 a26 *------open files ------open(5,file='nu_CH4_MFC_hexane_argon.dat',status='old') open(7,file='4pp_9_50_corrected.txt',status='old') open(8,file='chartime.txt',status='unknown') open(9,file='results.txt',status='unknown') open(11,file='screen.txt',status='unknown') *------*------Read from nuc_CH4_MFC_Cp.for ------c read two condensable species read(5,1012)specie write(11,*)'Specie is ',specie write(9,*)'Specie is ',specie write(9,*) ' file : ',wetfil write(9,*)' flow rate = ',dotm,' g/min' 1012 format(a8) read(5,1004)a10 c read molecular weights of carrier (1) and condensable (2,3) read(5,*)wm1,wm2,wm3 write(11,*)'The molecular weight of the carrier gas is ',wm1 write(11,*)'The molecular weight of the 1. condensable is ',wm2 write(11,*)'The molecular weight of the 2. condensable is ',wm3 write(9,*)'The molecular weight of the carrier gas is ',wm1 write(9,*)'The molecular weight of the 1. condensable is ',wm2 c read specific heats read(5,*)cp1 ! March04 David write(11,*)'The cp of the carrier gas is ',cp1

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c read latent heat, and specific heat of condensate read(5,1004)a11 c im Code von 2007 fehlte conv. Faktor, Wurde nachträglich eingefügt! c von Dirk 22/09/2008 read(5,*)convMKS write(11,*) 'convMKS is' , convMKS c read starting value and the number of points in the output read(5,*) xstart, ilast write(11,*)'xstart is ', xstart write(11,*)'ilast is ', ilast c read the integration end points (may be different than ifrst, ilast) c the number of integrations attempts, and the number of integration c sub-intervals read(5,*)istart, ifin ,ni, nint write(11,*)'istart is ', istart write(11,*)'ifin is ', ifin write(11,*)'ni is ', ni write(11,*)'nint is ',nint c read name of dry pressure data file read(5,1)dryfil 1 format(a30) c read smoothing parameters: m-order, n-number of points read(5,*)md,nd write(11,*)'md is ',md write(11,*)'nd is ',nd c read the number of wet data sets read(5,*)ndata write(11,*)'ndata is ', ndata *------Title line for output files ------write(8,*)'File mdot xrmax Trmax pcrmax Srmax

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& JBDmax charactime xvv rhorat' *------number of lines in file ------c Caculating the number of lines in a file. The nomenclautur of the c variables is taken from the final data inversion CODE npts=ifin-istart+1 nnpts=(npts-1)*nint + 1 ifin1=istart + nnpts - 1 write(11,*)'ifin1 is ',ifin1 *------constants ------c Avogadro-Konstante [1/mol]: Na=6.02205e23 c Boltzmann-Konstante [J/K]: k=1.38066e-23 pi=3.141593 M= 0.08618 *------input from 4pp.out ------do j=1,ndata c read title columne read(7,1011)a9 read(7,1000)a0,p0,a1,t0,a26,rho0 c convert p0 from torr to pascal c convert t0 from Celcius to Kelvin p0 = 133.322 * p0 t0 = t0 + 273.15 write(11,*)'T0 = ', t0 ,' Kelvin' write(11,*)'p0 = ', p0 ,' Pascal' write(11,*)'rho0 = ',rho0, 'kg/m3' write(9,*)'T0 = ', t0 ,' Kelvin' write(9,*)'p0 = ', p0 ,' Pascal' write(9,*)'rho0 = ',rho0, 'kg/m3'

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read(7,1001)a2,dotm write(11,*)' dotm = ', dotm ,' g/min ' write(9,*)' dotm = ', dotm ,' g/min ' read(7,1002)a3,pc10 c convert pc10 from torr to pascal pc10 = pc10 * 133.322 write(*,*)'pc10 = ', pc10 ,' Pascal' write(9,*)'pc10 = ', pc10 ,' Pascal' read(7,1003)a4,allflux,a5,dotncal write(11,*)'allflux(wettrace) = ', allflux write(11,*)'dotncal(drytrace) = ', dotncal write(9,*)'allflux(wettrace) = ', allflux read(7,1004)a6 read(7,1005)a7 read(7,1010)a8 1000 format(a4,f6.2,a18,f5.2,a26,e10.4) 1001 format(a29,f6.2) 1002 format(a11,f7.4) 1003 format(a17,f6.3,a13,f6.3) 1004 format(a50) 1005 format(a50) 1010 format(a100) 1011 format(a100) *------Data input from nu_CH4_MFC.dat ------read(5,*)dummy,dotm,wfc10 write(11,*)'The massflowrate of the condensable is ',dotm, & ' gmin-1' write(9,*)'The massflowrate of the condensable is ',dotm, & ' gmin-1' read(5,1)wetfil

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write(11,51)'Wetfile is ',wetfil write(9,51)'Wetfile is ',wetfil 51 format(a13) w=(dotm/wm2)/(allflux) write(9,*)'mole fraction w is ',w write(11,*)'mole fraction w is ',w write(*,*)'pc10 using w=', w*p0 *------Data output file ------write(9,*)' ' write(9,*)'x(i) T(i) pc(i) S(i) JBD(i) time rhorat' write(11,*)'x [cm] u[m/s] T [K] JBD [cm-3s-1] rateint' *------Now starting DATA analysis ------c read numeric input from 4pp.out, calculate the partial pressure, c supersaturation, time etc do i=1,ifin1-1 read(7,*)x(i),u(i),T(i),pp0(i),d5,d6,d7,d8,fc(i),dummy, & rhorat(i) pc(i) = pc10*pp0(i)*(1.0-fc(i)) c The equlibrium vapor pressure for n-Nonane is from: c Dissertation David Ghosh 2007 c Hung , Krasnopoler Katz (1989) c ------NEUE PHYS PROPERTIEs------c Neue Phys Properteis von Rudek et al. J.Chem. Phys.,Vol.105, No. 11 (1996) c geändert am 4.09.2008 von Dirk c pe=(128.77889-9467.401/T(i)-17.56832*log(T(i))+0.0152556*T(i))*1000 ------! alte von Dave c pe=133.3224*exp(pe) ------!alte von Dave c------Hexane Geleichgweichtsdampfdruck! in Pa------vap1=2.79797-2.022083d-3*T(i)+2.287564d-6*T(i)**2.0d0 pe=101.325*exp((1.0d0-341.863/T(i))*exp(vap1))*1000.0d0 c------vapor pressure H2O------1/22/2015 238

c d1 = 77.34491296 c d2 = 7235.42 c d4 = 8.2d0 c d5 = 0.0057113 c pe = exp(d1-d2/(T(i))-d4*dlog(T(i))+d5*T(i) ) !in Pa S(i) = pc(i)/pe if(S(i).le.2.0)S(i)= 2.0d0 enddo c calculate the flow time from the throat (t=0 at throat) time(1) = x(1)/u(1)/100.d0 timestep(1) = x(1)/u(1)/100.d0 umean(1)=u(1) do i=2,ifin1-1 umean(i)=(u(i)+u(i-1))/2. timestep(i)=(x(i)-x(i-1))/umean(i)/100.d0 time(i) = time(i-1)+(x(i)-x(i-1))/umean(i)/100.d0 enddo c do i=1,ifin1-1 c write(11,*)i,x(i),umean(i),timestep(i) c enddo *------Calculate the Nucleation rate *------do i=1,ifin1-1 c Temperaturumrechnung in °C: tt=T(i)-273.15 aa1=1.597561*(1.0d0-T(i)/507.9)**(0.333d0) aa2=1.842657*(1.0d0-T(i)/507.9)**(0.666d0) aa3=-1.726311*(1.0d0-T(i)/507.9) aa4=0.4943082*(1.0d0-T(i)/507.9)**(1.333d0) aa5=0.6463138*(1.0d0-T(i)/507.9)**(1.666d0)

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d(i)=234*(1.0d0+aa1+aa2+aa3+aa4+aa5) c------surface tension of Hexane (N/m)------sig(i)=0.02050-1.0439d-4*tt c Molekülvolumen von Nonan [m^3]: v(i)=M/(Na*d(i)) c Keimbildungsrate nach Becker und Döring [1/(s*cm^3)]: A(i)=v(i)*((pc(i)/(k*T(i)))**2.0)*((2.0*sig(i)*Na/(pi*M))**0.5) B(i)=16.0*pi*(v(i)**2.0)*(sig(i)**3.0)/ & (3.0*(k**3.0)*(T(i)**3.0)*(log(S(i))**2.0)) JBD(i)=A(i)*(1.0e-6)*exp(-1.0*B(i)) if(JBD(i).lt.1)JBD(i)=1.d0 write(11,*)i,S(i),JBD(i) enddo *------JBD max ------c search for the first nucleation rate peak in the nozzle, calculate the rate c integral, c until the rate has dropped by 6 orders of magnitude and the integral is no c longer changing significantly ratemax = JBD(1) rateint(1)= 0.0 test=0.0 do i=2,ifin1-1 if(JBD(i).ge.ratemax)then ratemax = JBD(i) xrmax = x(i) srmax = S(i) Trmax = T(i) pcrmax=pc(i) imax = i endif

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rateint(i) = rateint(i-1) + 0.5*(JBD(i)+JBD(i-1))*timestep(i) chartime(i)= rateint(i)/ratemax if((JBD(i)/ratemax).lt.1.0e-3)goto 3000 Write(11,*)i,x(i),xrmax,JBD(i),ratemax,srmax,chartime(i) write(9,1021)x(i),T(i),pc(i),S(i),JBD(i), & chartime(i),rhorat(i) 1021 format(f8.2,f10.2,f9.3,f10.3,g12.4,e12.4,f9.4) enddo 3000 continue c find the ratio of rhoNZ/rhoVV assume VV is 6.5 cm downstream of throat rhoNZ_rhoVV=rhorat(imax)/rhorat(641) write(8,1020)wetfil,dotm,xrmax,Trmax,pcrmax,Srmax,ratemax, & chartime(i),x(641),rhoNZ_rhoVV enddo 1020 format(a13,f6.2,f8.2,f10.2,f9.3,f10.3g12.4,e12.4,f7.2,f9.4) end

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APPENDIX C Mathematica Code to produce a n-alkane crystal Initial configuration of solid phase set-up SetDirectory[NotebookDirectory[]] /Users/kehindeemekaogunronbi/Desktop/AlkaneSimulations/Heptane/FRESH/HeptaneSolid "/Users/kehindeemekaogunronbi/Desktop/AlkaneSimulations/Hexane/NEW/HexaneSolid" /Users/kehindeemekaogunronbi/Desktop/AlkaneSimulations/Hexane/NEW/HexaneSolid resname={" METH"," ETH "," PROP"," BUT "," PENT"," HEX "," HEP "," OCT "}; nCarbon=7; (* number of carbon atoms *) targetL=5; (* approximate box side length, in nm *) nnDist=0.470; (* nearest neighbor distance within layers *) a={nnDist,0,0}; b={nnDist Cos[Pi/3],nnDist Sin[Pi/3],0}; c = {0,0,N[1.233 (nCarbon/8)]}; Na0=Round[targetL/a[[1]] ]; If[OddQ[Na0],Na0+=1]; Nb0=Round[targetL/b[[2]]]; If[OddQ[nB0],Nb0+=1]; Nc0=Round[targetL/c[[3]] ]; Print[" suggested simulation cell size"]; Print["Na:", Na0," Nb:",Nb0," Nc:",Nc0]; Na: 12 Nb: 12 Nc: 5 suggested simulation cell size Na: 12 Nb: 12 Nc: 5 suggested simulation cell size Na: 12 Nb: 12 Nc: 5 suggested simulation cell size Na: 12 Nb: 12 Nc: 5 suggested simulation cell size Na: 12 Nb: 12 Nc: 5 suggested simulation cell size " suggested simulation cell size Na: 12 Nb: 12 Nc: 6 Na=12; Nb=24; Nc=6; construct initial configuration (* a={.468,0,0}; b={.468 Cos[Pi/3],.468 Sin[Pi/3],0}; c = {0,0,1.233}; *) (* To make a system of 480 alkanes, I originally used Na=10, nB=12, nC=4 *) (* To make a long sysem of 960 alkaes, nB was doubled: Na=10, nB=24, nC=4 *) th2=ArcCos[-1/3]/2; offset=a/2+b/2+.05c; l=.153; rho=l Cos[th2];

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t1a={ rho Cos[Pi/6], rho Sin[Pi/6],l Sin[th2]}; t1b={-rho Cos[Pi/6],-rho Sin[Pi/6],l Sin[th2]}; t2a={-rho Cos[Pi/6], rho Sin[Pi/6],l Sin[th2]}; t2b={ rho Cos[Pi/6],-rho Sin[Pi/6],l Sin[th2]}; imol=0; iatom=0; atomname={" CAA"," CAB"," CAC"," CAD"," CAE"," CAF"," CAG"," CAH"," CAI"," CAJ"}; stream=OpenWrite["chains"<>ToString[nCarbon]<>".gro"]; WriteString[stream,"initial alkane config\n"]; WriteString[stream,ToString[Na Nb Nc nCarbon],"\n"]; Do[ r0=offset+na a+(b-a/2)nb+If[OddQ[nb],a/2,0]+nc c; If[EvenQ[nb],ta=t1a;tb=t1b, ta=t2a;tb=t2b]; ia=0;ib=0;imol+=1; Do[ iatom+=1; t=r0+ia ta + ib tb; If[OddQ[i],ia+=1,ib+=1];

WriteString[stream,IntegerString[imol,10,5],resname[[nCarbon]],atomname[[i]],IntegerString[iatom,1 0,5], PaddedForm[t[[1]],{6,3}], PaddedForm[t[[2]],{6,3}], PaddedForm[t[[3]],{6,3}], "\n"],{i,nCarbon}]; (* Print[na," ",nb," ",nc]; *)

,{nc,0,Nc-1},{nb,0,Nb-1},{na,0,Na-1}]; WriteString[stream, PaddedForm[Na a[[1]],{12,6}], PaddedForm[Nb b[[2]],{12,6}], PaddedForm[Nc c[[3]],{12,6}] (*, PaddedForm[Na a[[2]],{12,6}], PaddedForm[Na a[[3]],{12,6}], PaddedForm[Nb b[[1]],{12,6}], PaddedForm[Nb b[[3]],{12,6}], PaddedForm[Nc c[[2]],{12,6}], PaddedForm[Nc c[[3]],{12,6}]] *)]; WriteString[stream,"\n"]; Close[stream];

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APPENDIX D Fortran Code to divide top and bottom frozen layers into halves implicit none integer nMolMax,nAlkane,nAtomMax parameter (nMolMax=1200,nAlkane=6,nAtomMax=nMolMax*nAlkane) character*40 title integer i,j,k,nAtom,nMol, iRes,iAtom,FreezeGroup(nAtomMax), > nFreezeGroup,MeltGroup(nAtomMax),nMeltGroup,iSF, > i1,i2 double precision r(3,nMolMax,nAtomMax),L(3),t,zSF1,zSF2 character*5 resname(nMolMax,nAtomMax),atomname(nMolMax,nAtomMax) c------c chains with z coordinate of carbon iSF < zSF1 or > zSF2 are c counted as part of surface frozen layer. data zSF1,zSF2 /1.23d0,3.63d0/ data iSF /4/ c------c compile: gfortran -O2 MeltGroup.f -o MeltGroup c compile: ifort -O2 MeltGroup.f -o MeltGroup c usage: ./MeltGroup < hexane960-02_120K.gro > melt.dat read(*,*) title read(*,*) nAtom nMol=nAtom/nAlkane do i=1,nMol do j=1,nAlkane read(*,'(i5,2a5,i5,3f8.3,3f8.4)') iRes,resname(i,j), > atomname(i,j),iAtom,(r(k,i,j),k=1,3) end do end do read(*,*) (L(k),k=1,3) 244

c nFreezeGroup=0 nMeltGroup=0 t=L(2)/2.d0 do i=1,nMol if( (r(3,i,iSF).lt.zSF1.or.r(3,i,iSF).gt.zSF2) .and. > r(2,i,iSF).lt.t) then do j=1,nAlkane nFreezeGroup=nFreezeGroup+1 FreezeGroup(nFreezeGroup)=(i-1)*nAlkane+j end do else do j=1,nAlkane nMeltGroup=nMeltGroup+1 MeltGroup(nMeltGroup)=(i-1)*nAlkane+j end do end if end do c write(*,*) 'FreezeGroup has ',nFreezeGroup,' members' write(*,*) 'MeltGroup has', nMeltGroup,' members' write(*,*) 'Freeze + Melt groups has ',(nFreezeGroup+nMeltGroup), > ' members' write(*,*) 'nAtom in input gro file ', nAtom c open(unit=1,file='IndexFileAdditions.ndx') k=nFreezeGroup/6 if(k*6.ne.nFreezeGroup) k=k+1 write(1,*) '[ FreezeGroup ] ' do j=1,k

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i1=(j-1)*6 + 1 i2=Min(i1+5,nFreezeGroup) write(1,'(6i10)') (FreezeGroup(i),i=i1,i2) end do c k=nMeltGroup/6 if(k*6.ne.nMeltGroup) k=k+1 write(1,*) '[ MeltGroup ] ' do j=1,k i1=(j-1)*6 + 1 i2=Min(i1+5,nMeltGroup) write(1,'(6i10)') (MeltGroup(i),i=i1,i2) end do close(1) c open(unit=1,file='Freeze.gro') write(1,*) 'melt group members' write(1,*) nFreezeGroup do iAtom=1,nFreezeGroup i=(FreezeGroup(iAtom)-1)/nAlkane i=i+1 j=FreezeGroup(iAtom)-(i-1)*nAlkane write(1,'(i5,2a5,i5,3f8.3,3f8.4)') i, > resname(i,j), > atomname(i,j),FreezeGroup(iAtom),(r(k,i,j),k=1,3) end do write(1,*) (L(k),k=1,3) close(1) c open(unit=1,file='Melt.gro')

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write(1,*) 'melt group members' write(1,*) nMeltGroup do iAtom=1,nMeltGroup i=(MeltGroup(iAtom)-1)/nAlkane i=i+1 j=MeltGroup(iAtom)-(i-1)*nAlkane write(1,'(i5,2a5,i5,3f8.3,3f8.4)') i, > resname(i,j), > atomname(i,j),MeltGroup(iAtom),(r(k,i,j),k=1,3) end do write(1,*) (L(k),k=1,3) close(1) c stop end

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APPENDIX E Fortran Code that divides the slab into halves in the y-direction implicit none integer nMolMax,nAlkane,nAtomMax parameter (nMolMax=2900,nAlkane=6,nAtomMax=nMolMax*nAlkane) character*40 title integer i,j,k,nAtom,nMol, iRes,iAtom,FreezeGroup(nAtomMax), > nFreezeGroup,MeltGroup(nAtomMax),nMeltGroup,iSF, > i1,i2 double precision r(3,nMolMax,nAtomMax),L(3),t,zSF1,zSF2 character*5 resname(nMolMax,nAtomMax),atomname(nMolMax,nAtomMax) c------c chains with z coordinate of carbon iSF < zSF1 or > zSF2 are c counted as part of surface frozen layer. c data zSF1,zSF2 /1.35d0,3.40d0/ data iSF /4.89/ c------c compile: gfortran -O2 MeltGroup.f -o MeltGroup c compile: ifort -O2 MeltGroup.f -o MeltGroup c usage: ./MeltGroup < hexane960-02_90K.gro > melt.dat c read(*,*) title read(*,*) nAtom nMol=nAtom/nAlkane do i=1,nMol do j=1,nAlkane read(*,'(i5,2a5,i5,3f8.3,3f8.4)') iRes,resname(i,j), > atomname(i,j),iAtom,(r(k,i,j),k=1,3) end do end do 248

read(*,*) (L(k),k=1,3) c nFreezeGroup=0 nMeltGroup=0 t=L(2)/2.d0 do i=1,nMol if( r(2,i,iSF).lt.t) then do j=1,nAlkane nFreezeGroup=nFreezeGroup+1 FreezeGroup(nFreezeGroup)=(i-1)*nAlkane+j end do else do j=1,nAlkane nMeltGroup=nMeltGroup+1 MeltGroup(nMeltGroup)=(i-1)*nAlkane+j end do end if end do c write(*,*) 'FreezeGroup has ',nFreezeGroup,' members' write(*,*) 'MeltGroup has', nMeltGroup,' members' write(*,*) 'Freeze + Melt groups has ',(nFreezeGroup+nMeltGroup), > ' members' write(*,*) 'nAtom in input gro file ', nAtom c open(unit=1,file='IndexFileAdditions.ndx') k=nFreezeGroup/6 if(k*6.ne.nFreezeGroup) k=k+1 write(1,*) '[ FreezeGroup ] ' do j=1,k

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APPENDIX F GROMACS Topology file for n-pentane ; This file was generated by PRODRG version 071121.0636 ; PRODRG written/copyrighted by Daan van Aalten ; and Alexander Schuettelkopf ; Questions/comments to [email protected] ; When using this software in a publication, cite: ; A. W. Schuettelkopf and D. M. F. van Aalten (2004). ; PRODRG - a tool for high-throughput crystallography ; of protein-ligand complexes. ; Acta Crystallogr. D60, 1355--1363. [ defaults ] ; nbfunc comb-rule gen-pairs fudgeLJ fudgeQQ 1 2 no 1.0 1.0 [ atomtypes ] ;name at.num mass charge ptype sigma eps CH2 6 14.02700 0.000 A 0.401 0.469 CH3 6 15.03500 0.000 A 0.401 0.469 ;[ bondtypes ] ;[ pairtypes ] ;[ angletypes ] ;[ dihedraltypes ] [ nonbond_params ] ; i j func sigma eps CH3 CH3 1 0.401 0.469 CH3 CH2 1 0.401 0.469 CH2 CH2 1 0.401 0.469 [ moleculetype ] ; Name nrexcl PEN 3 252

[ atoms ] ; nr type resnr resid atom cgnr charge mass 1 CH3 1 PEN CAA 0 0.000 15.0350 2 CH2 1 PEN CAB 1 0.000 14.0270 3 CH2 1 PEN CAC 2 0.000 14.0270 4 CH2 1 PEN CAD 3 0.000 14.0270 5 CH3 1 PEN CAE 4 0.000 15.0350 [ bonds ] ; ai aj fu c0, c1, ... 1 2 1 0.153 292000.0 ; CAA CAB 2 3 1 0.153 292000.0 ; CAB CAC 3 4 1 0.153 292000.0 ; CAC CAD 4 5 1 0.153 292000.0 ; CAD CAE [ pairs ] ; ai aj fu sigma eps 1 4 1 0.401 0.00 ; CAA CAD 2 5 1 0.401 0.00 ; CAB CAE [ angles ] ; ai aj ak fu theta0 k_theta 1 2 3 1 109.526 502.08 ; CAA CAB CAC 2 3 4 1 109.526 502.08 ; CAB CAC CAD 3 4 5 1 109.526 502.08 ; CAC CAD CAE [ exclusions ] 1 2 3 4 2 1 3 4 5 3 1 2 4 5 4 1 2 3 5 5 2 3 4 [ dihedrals ] ; ai aj ak al fu c0, c1, c2, c3, c4, c5

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4 3 2 1 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAD CAC CAB CAA 5 4 3 2 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAE CAD CAC CAB ; system name [ system ] pentane solid or liquid [ molecules ] PEN 2016

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APPENDIX G GROMACS Topology file for n-hexane ; This file was generated by PRODRG version 071121.0636 ; PRODRG written/copyrighted by Daan van Aalten ; and Alexander Schuettelkopf ; Questions/comments to [email protected] ; When using this software in a publication, cite: ; A. W. Schuettelkopf and D. M. F. van Aalten (2004). ; PRODRG - a tool for high-throughput crystallography ; of protein-ligand complexes. ; Acta Crystallogr. D60, 1355--1363. [ defaults ] ; nbfunc comb-rule gen-pairs fudgeLJ fudgeQQ 1 2 no 1.0 1.0 [ atomtypes ] ;name at.num mass charge ptype sigma eps CH2 6 14.02700 0.000 A 0.401 0.469 CH3 6 15.03500 0.000 A 0.401 0.469 ;[ bondtypes ] ;[ pairtypes ] ;[ angletypes ] ;[ dihedraltypes ] [ nonbond_params ] ; i j func sigma eps CH3 CH3 1 0.401 0.469 CH3 CH2 1 0.401 0.469 CH2 CH2 1 0.401 0.469 [ moleculetype ] ; Name nrexcl HEX 3 255

[ atoms ] ; nr type resnr resid atom cgnr charge mass 1 CH3 1 HEX CAA 0 0.000 15.0350 2 CH2 1 HEX CAB 1 0.000 14.0270 3 CH2 1 HEX CAC 2 0.000 14.0270 4 CH2 1 HEX CAD 3 0.000 14.0270 5 CH2 1 HEX CAE 4 0.000 14.0270 6 CH3 1 HEX CAF 5 0.000 15.0350 [ bonds ] ; ai aj fu c0, c1, ... 1 2 1 0.153 292000.0 ; CAA CAB 2 3 1 0.153 292000.0 ; CAB CAC 3 4 1 0.153 292000.0 ; CAC CAD 4 5 1 0.153 292000.0 ; CAD CAE 5 6 1 0.153 292000.0 ; CAE CAF [ pairs ] ; ai aj fu sigma eps 1 4 1 0.401 0.00 ; CAA CAD 2 5 1 0.401 0.00 ; CAB CAE 3 6 1 0.401 0.00 ; CAC CAF [ angles ] ; ai aj ak fu theta0 k_theta 1 2 3 1 109.526 502.08 ; CAA CAB CAC 2 3 4 1 109.526 502.08 ; CAB CAC CAD 3 4 5 1 109.526 502.08 ; CAC CAD CAE 4 5 6 1 109.526 502.08 ; CAD CAE CAF [ exclusions ] 1 2 3 4 2 1 3 4 5 3 1 2 4 5 6

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4 1 2 3 5 6 5 2 3 4 6 6 3 4 5 [ dihedrals ] ; ai aj ak al fu c0, c1, c2, c3, c4, c5 4 3 2 1 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAD CAC CAB CAA 5 4 3 2 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAE CAD CAC CAB 6 5 4 3 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAF CAE CAD CAC ; system name [ system ] hexane solid or liquid [ molecules ] HEX 2880

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APPENDIX H GROMACS Topology file for n-heptane ; This file was generated by PRODRG version 071121.0636 ; PRODRG written/copyrighted by Daan van Aalten ; and Alexander Schuettelkopf ; Questions/comments to [email protected] ; When using this software in a publication, cite: ; A. W. Schuettelkopf and D. M. F. van Aalten (2004). ; PRODRG - a tool for high-throughput crystallography ; of protein-ligand complexes. ; Acta Crystallogr. D60, 1355--1363. [ defaults ] ; nbfunc comb-rule gen-pairs fudgeLJ fudgeQQ 1 2 no 1.0 1.0 [ atomtypes ] ;name at.num mass charge ptype sigma eps CH2 6 14.02700 0.000 A 0.401 0.469 CH3 6 15.03500 0.000 A 0.401 0.469 ;[ bondtypes ] ;[ pairtypes ] ;[ angletypes ] ;[ dihedraltypes ] [ nonbond_params ] ; i j func sigma eps CH3 CH3 1 0.401 0.469 CH3 CH2 1 0.401 0.469 CH2 CH2 1 0.401 0.469 [ moleculetype ] ; Name nrexcl HEP 3 258

[ atoms ] ; nr type resnr resid atom cgnr charge mass 1 CH3 1 HEP CAA 0 0.000 15.0350 2 CH2 1 HEP CAB 1 0.000 14.0270 3 CH2 1 HEP CAC 2 0.000 14.0270 4 CH2 1 HEP CAD 3 0.000 14.0270 5 CH2 1 HEP CAE 4 0.000 14.0270 6 CH2 1 HEP CAF 5 0.000 14.0270 7 CH3 1 HEP CAG 6 0.000 15.0350 [ bonds ] ; ai aj fu c0, c1, ... 1 2 1 0.153 292000.0 ; CAA CAB 2 3 1 0.153 292000.0 ; CAB CAC 3 4 1 0.153 292000.0 ; CAC CAD 4 5 1 0.153 292000.0 ; CAD CAE 5 6 1 0.153 292000.0 ; CAE CAF 6 7 1 0.153 292000.0 ; CAF CAG [ pairs ] ; ai aj fu sigma eps 1 4 1 0.401 0.00 ; CAA CAD 2 5 1 0.401 0.00 ; CAB CAE 3 6 1 0.401 0.00 ; CAC CAF 4 7 1 0.401 0.00 ; CAD CAG [ angles ] ; ai aj ak fu theta0 k_theta 1 2 3 1 109.526 502.08 ; CAA CAB CAC 2 3 4 1 109.526 502.08 ; CAB CAC CAD 3 4 5 1 109.526 502.08 ; CAC CAD CAE 4 5 6 1 109.526 502.08 ; CAD CAE CAF 5 6 7 1 109.526 502.08 ; CAE CAF CAG

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[ exclusions ] 1 2 3 4 2 1 3 4 5 3 1 2 4 5 6 4 1 2 3 5 6 7 5 2 3 4 6 7 6 3 4 5 7 7 4 5 6 [ dihedrals ] ; ai aj ak al fu c0, c1, c2, c3, c4, c5 4 3 2 1 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAD CAC CAB CAA 5 4 3 2 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAE CAD CAC CAB 6 5 4 3 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAF CAE CAD CAC 7 6 5 4 3 6.505 16.995 3.62 -27.12 0.00 0.00 ; dih CAG CAF CAE CAD ; system name [ system ] heptane solid or liquid [ molecules ] HEP 1728

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