NUCLEATION THEORY USING EQUATIONS OF STATE by ABDALLA A. OBEIDAT ATHESIS Presented to the Faculty of the Graduate School of the UNIVERSITY OF MISSOURI-ROLLA in Partial FulfillmentoftheRequirementsfortheDegree DOCTOR OF PHILOSOPHY in PHYSICS 2003 Gerald Wilemski, Advisor Barbara N. Hale Jerry L. Peacher Paul E. Parris Daniel Forciniti ii ABSTRACT Various equations of state (EOS) have been used with the most general Gibbsian form (P form) of classical nucleation theory (CNT) to see if any improvement − could be realized in predicted rates for vapor-to-liquid nucleation. The standard or S form of CNT relies on the assumption of an incompressible liquid droplet. With − the use of realistic EOSs, this assumption is no longer needed. The P form results − for water and heavy water were made using the highly accurate IAPWS 95 EOS − and the CREOS.TheP form successfully predicted the temperature (T ) and − supersaturation (S) dependence of the nucleation rate, although the absolute value was in error by roughly a factor of 100. The results for methanol and ethanol using alessaccurateCPHB EOS showed little improvement over the S form results. − Gradient theory (GT ), a form of density functional theory (DFT), was applied to water and alcohols using the CPHB EOS. The water results showed an improved T dependence, but the S dependence was slightly poorer compared to the S form − of CNT. The methanol and ethanol results were improved by several orders of magnitude in the predicted rates. GT and P form CNT were also found to be in − good agreement with a single high T molecular dynamics rate for TIP4P water. The P formof binary nucleation theory was studied for a fictitious water-ethanol − system whose properties were generated from DFT and a mean-field EOS for a hard sphere Yukawa fluid. The P form was not successful in removing the unphysical − behavior predicted by binary CNT in its simplest form. The DFT results were greatly superior to all forms of classical theory. iii ABSTRACT Various equations of state (EOS) have been used with the most general Gibbsian form (P form) of classical nucleation theory (CNT) to see if any improvement − could be realized in predicted rates for vapor-to-liquid nucleation. The standard or S form of CNT relies on the assumption of an incompressible liquid droplet. With − the use of realistic EOSs, this assumption is no longer needed. The P form results − for water and heavy water were made using the highly accurate IAPWS 95 EOS − and the CREOS.TheP form successfully predicted the temperature (T ) and − supersaturation (S) dependence of the nucleation rate, although the absolute value was in error by roughly a factor of 100. The results for methanol and ethanol using alessaccurateCPHB EOS showed little improvement over the S form results. − Gradient theory (GT ), a form of density functional theory (DFT), was applied to water and alcohols using the CPHB EOS. The water results showed an improved T dependence, but the S dependence was slightly poorer compared to the S form − of CNT. The methanol and ethanol results were improved by several orders of magnitude in the predicted rates. GT and P form CNT were also found to be in − good agreement with a single high T molecular dynamics rate for TIP4P water. The P formof binary nucleation theory was studied for a fictitious water-ethanol − system whose properties were generated from DFT and a mean-field EOS for a hard sphere Yukawa fluid. The P form was not successful in removing the unphysical − behavior predicted by binary CNT in its simplest form. The DFT results were greatly superior to all forms of classical theory. iv ACKNOWLEDGEMENTS I would like to express my gratitude and appreciation to my research advisor, Dr. Gerald Wilemski, for his constant support and patience through the time I spent at UMR. Without his guidance and motivation this work would have never been done. I would like also to thank the members of my Ph.D. committee Dr. B. Hale, D. P. Parris, Dr. J. Peacher, and Dr. D. Forciniti for their help and support. I also would like to thank Dr. J-S. Li for his support and suggestions during my thesis work. I am very thankful to my parents, without their endless love and support, I would not have been neither in UMR nor in this life. I would like also to thank my roommates and friends, Eyad, Malik, Ahmad, Abdul, Vikas, and Sabrina, who made my life much easier while staying in the US. This dissertation is dedicated to the wonderful lady Enas. This work was supported by the Engineering Physics Program of the Division of Materials Sciences and Engineering, Basic Energy Sciences, U. S. Department of Energy. v TABLE OF CONTENTS Page ABSTRACT.................................... iii ACKNOWLEDGEMENTS ............................ iv LIST OF ILLUSTRATIONS . ........................... viii LISTOFTABLES................................. xii SECTION...................................... 1. INTRODUCTION......................... 1 1.1. NUCLEATION PHENOMENOLOGY AND BASIC THEORY 1 1.2.BRIEFOVERVIEWOFBINARYNUCLEATION..... 8 1.3.MOTIVATION......................... 10 2. EQUATION OF STATE APPROACH FOR CLASSICAL NUCLEATIONTHEORY..................... 12 2.1.THEORY............................ 12 2.1.1.WorkofFormation.................... 12 2.1.2.Gibbs’sReferenceState................. 15 2.3.1.NumberofMoleculesinCriticalNucleus........ 17 3. EQUATIONS OF STATE FOR UNARY SYSTEMS . ...... 19 3.1.WATER............................. 19 3.1.1.IAPWS-95........................ 19 3.1.2.CrossOverEquationofState(CREOS-01)...... 20 3.1.3. JefferyandAustinEOS(JA—EOS).......... 21 3.1.4.CubicPerturbedHardBody(CPHB)......... 22 vi 3.1.5.Peng-Robinson(PR).................. 23 3.2.HEAVYWATER:CREOS-02................. 24 3.3. METHANOL AND ETHANOL: CPHB ........... 24 4. RESULTS OF EOS APPROACH FOR UNARY SYSTEMS . 25 4.1.WATER............................. 25 4.2.HEAVYWATER........................ 29 4.3.DISCUSSIONOFWATERRESULTS............ 32 5. GRADIENTTHEORYOFUNARYNUCLEATION...... 36 5.1.BACKGROUND........................ 36 5.2.THEORY............................ 37 6. RESULTS OF GRADIENT THEORY FOR UNARY NUCLEATION........................... 40 6.1.WATERANDTIP4PWATER................ 40 6.1.1. Planar and Droplet Density ProfilesfromGT..... 40 6.1.2.WaterNucleationRates................. 44 6.1.3.TIP4PWaterNucleation................ 47 6.2.COMPARISONOFTHEWATEREOS........... 48 6.3. RESULTS FOR METHANOL AND ETHANOL ...... 52 7. BINARYNUCLEATIONTHEORY................ 55 7.1CLASSICALNUCLEATIONTHEORY........... 55 7.2DENSITYFUNCTIONALTHEORY(DFT)........ 58 7.3SURFACETENSIONANDREVERSIBLEWORK..... 59 vii 7.4DFTFORHARDSPHERE-YUKAWAFLUID....... 60 8. PROPERTIES OF THE MODEL BINARY HARD-SPHERE YUKAWA(HSY)FLUID...................... 61 8.1EQUATIONOFSTATE.................... 61 8.2FITTEDPROPERTYVALUES............... 66 9. RESULTSOFTHEHSYBINARYFLUID............ 67 9.1.CRITICALACTIVITIESATCONSTANTW*....... 67 9.2. NUMBER OF MOLECULES IN CRITICAL DROPLET . 69 10.CONCLUSIONS.......................... 71 APPENDICES................................... A.IMPORTANTTHERMODYNAMICRELATIONS........ 73 B.DETAILSOFVARIOUSEQUATIONSOFSTATE....... 77 C. PHYSICAL PROPERTIES OF WATER AND HEAVY WATER 89 D. SURFACE TENSION AND WORK OF FORMATION IN DFT 91 BIBLIOGRAPHY................................. 95 VITA........................................ 100 viii LIST OF ILLUSTRATIONS Figures Page 1.1. Schematic pressure — volume phase diagram for a pure substance. The green solid line is a true isotherm whose intersections (e) with the bin- odal dome give the equilibrium pressure and volumes of the coexising vapor-liquid phases. Binodal: solid heavy curve; spinodal: red dashed curve.................................... 1 1.2. The contributions of the surface and volume terms of the free energy of formation versus the cluster size. The free energy of formation has amaximumatthecriticalsize...................... 4 1.3. Experimental data for water from Ref.[18] illustrating the inadequate temperature dependence predicted by the classical Becker-Doering the- ory[4], labeled S-form in the figure.................... 6 2.1. Same as Figure 1.2 but the free energy of formation is plotted as a functionoftheradiusofthecluster.................... 13 2.2. The concept of the reference liquid state using a pressure-density isotherm for a pure fluid. The full circles represent the equilibrium vapor-liquid states, while the diamonds mark the metastable vapor phase and the referenceliquidphase........................... 17 4.1. The work of formation for water droplets using the IAPWS-95 EOS withthethreeformsofCNTatT=240,250,and260K........ 25 4.2. Comparison of the experimental rates of Woelk and Strey (open circles) for water with two versions of CNT based on the IAPWS-95 EOS; P- formandS-form.............................. 26 4.3. Comparison of the experimental rates of Woelk and Strey (open circles) for water down to T=220 K with two versions of CNT based on the CREOS-01andwiththescaledmodel.................. 27 4.4. The number of water molecules in the critical cluster as predicted by the nucleation theorem and the P-form calculations. The dashed-line showsthefullagreementwiththeGibbs-Thomsonequation...... 28 ix 4.5. The experimantal rates of heavy water by Woelk and Strey down to T=220 K with the predictions of the P-form of the CREOS-02. 30 4.6. The P-form results using CREOS-02 at high S compared with two different sets of supersonic nozzle experiments. The scaled model and the empirical function also shown at T=237.5, 230, 222, 215, and 208.8 K from