NUCLEATION THEORY USING EQUATIONS of STATE By

Home , Nucleation

NUCLEATION THEORY USING EQUATIONS OF STATE

ABDALLA A. OBEIDAT

ATHESIS

Presented to the Faculty of the Graduate School of the

UNIVERSITY OF MISSOURI-ROLLA

in Partial FulﬁllmentoftheRequirementsfortheDegree

DOCTOR OF PHILOSOPHY

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

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 binodal 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 theory[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 left to right...... 31

4.7.AsinFigure4.4butforheavywater...... 32

4.8. The temperature-density isobars of water using the IAPWS-95 EOS and the CREOS-01 compared to experimental data of Kell and Whal- ley[75]andPetitetetal.[81]...... 33

4.9. The work of formation of water at T=240, 250, and 260 K predicted bytheIAPWS-95andCREOS-01...... 34

4.10. Isothermal compressibility of liquid water at 10 MPa and 190 MPa calculated from the ﬁt of Kanno and Angell[83]...... 35

6.1. The thickness of ﬂat water interfaces at diﬀerent T using GT, MD simulations[90],andexperimentaldata[91]...... 40

6.2. Density proﬁles of water droplets predicted by CPHB at diﬀerent T, forasupersaturationratioof5...... 42

6.3.SameasFigure6.2butatS=20...... 42

6.4. Density proﬁles of water droplets at T=350 K for diﬀerent values of thesupersaturationratiousingtheCPHBEOS...... 43

6.5.SameasFigure6.4butusingtheJA-EOS...... 44 x

6.6. Nucleation rate predictions of the CPHB using the P-form and the GT comparedtoexperimentaldataofWoelkandStrey[18]...... 45

6.7. The ratio of the GT work of formation to that of the P-form of CNT asafunctionofsupersaturationratioat260K...... 46

6.8. The number of water molecules in the critical cluster as predicted by the nucleation theorem and the GT calculations. The dashed line rep- resentsfullagreementwithGibbs-Thomsonequation...... 47

6.9. Nucleation rates for GT and two forms of CNT at T=350 K using diﬀerent EOSs, as shown in the ﬁgure, compared with the MD rate for TIP4P water and the result of the P-form of CNT using CREOS-01. . 48

6.10. The predictions of diﬀerent EOSs for the equilibrium liquid density of water at diﬀerent T compared to the experimental data generated usingtheIAPWS-95...... 49

6.11. Density of liquid water using the CPHB EOS (stars) at diﬀerent P (0.1, 50, 100, 150, 190 MPa) compared to the experimental data calculated usingtheIAPWS-95(circles)...... 49

6.12. The predictions of diﬀerent EOSs for the equilibrium vapor pressure at diﬀerent T compared to the experimental data calculated by using theIAPWS-95...... 50

6.13. Same as for Figure 6.12 except for the equilibrium vapor density. . . . 50

6.14. Experimental nucleation rates of methanol compared to the predictions ofGTandtheP-formofCNTwiththeCPHBEOS...... 52

6.15.AsinFigure6.14butforethanol...... 53

6.16. Liquid ethanol density vs. P at diﬀerent temperatures using the CPHB EOS(opensymbols)andexperimentaldata(solidsymbols)...... 53

6.17. Experimental nucleation rates of ethanol compared to calculated rates using the S-form and the P-form of CNT with the CPHB EOS and the P-form of CNT using ﬁttedexperimentaldensitydata[94]...... 54 xi

8.1. The total and partial equilibrium vapor pressures of the HSY model ﬂuidatT=260Kversusmixturecomposition,x...... 62

8.2.P-xphasediagramofthebinaryHSYmodelsystem...... 63

8.3. Surface tension for the pseudo water-ethanol system and measured valuesforwater-ethanolversusethanolmolefraction,x...... 64

8.4. Variation of the partial molecular volume of p-water with composition. 65

8.5.SameasFigure8.4.butforp-ethanol...... 65

9.1. Critical activities of p-water (1) and p-ethanol (2) needed to produce aconstantworkofformationof40kT...... 67

9.2. The number of molecules of each component of the critical droplet as a function of the p-water activity using version 1 and version 2 of the CNT,aswellastheDFT...... 69

9.3. The number of molecules of each component of the critical droplet as a function of the p-water activity using version 3 of the CNT and the DFT...... 70

9.4. The number of molecules of each component of the critical droplet as a function of the p-water activity using versions 1, 2, and 3 of the CNT. 70

A.1. Schematic depiction of a spherical critical nucleus in a metastable gas phase...... 76 xii

LIST OF TABLES

Tables Page

B.1. The coeﬃcientsvaluesoftheidealgaspart...... 78

B.2. The coeﬃcientsandparametersoftheresidualpart...... 79

B.3. The other coeﬃcientsandparametersoftheresidualpart...... 81

B.4. The coeﬃcientsoftheCREOSequationofstate...... 83

B.5. The coeﬃcientsofCREOS-01andCREOS-02EOSs...... 84

B.6. The coeﬃcientsandparametersoftheJA-EOS...... 85

B.7. The C parameters for water, ethanol, and methanol of the CPHB EOS. 87

B.8. The parameters of the CPHB EOS used for water, ethanol, and methanol...... 88 1. INTRODUCTION

1.1 NUCLEATION PHENOMENOLOGY AND BASIC THEORY This thesis is primarily concerned with the accuracy of theories of vapor-to-liquid nucleation based on equations of state of real ﬂuids. Nucleation refers to the kinetic processes involved in the initiation of ﬁrst order phase transitions in nonequilibrium systems. Two phase equilibrium states for a pure substance, e.g. vapor and liquid, occur at unique pairs of temperature T and pressure P . For two-phase vapor- liquid equilibrium, the pressure is referred to as the equilibrium vapor pressure Pve.

Now, if the actual pressure of the vapor Pv is larger than the equilibrium vapor pressure, the vapor is said to be supersaturated. This new state of the vapor is either metastable or unstable. These two types of states are distinguished by their location with respect to the mean-ﬁeldspinodal,whichisillustratedinFigure1.1.

c riti c a l p P oi nt

T 6 T 5

e e l T =T t 4 c b a S a T t 3 t a e b e P aS t e l e T e 2 Unstable M T 1 0 0 V V V le ve

Figure 1.1. Schematic pressure — volume phase diagram for a pure substance. The green solid line is a true isotherm whose intersections (e) with the binodal dome give the equilibrium pressure and volumes of the coexising vapor-liquid phases. Binodal: solid heavy curve; spinodal: red dashed curve.

This figure shows a P -V phase diagram for a pure fluid with several isotherms 2 based on a van der Waals equation of state (EOS). The heavy dome-shaped curve denotes the binodal line, the locus of two-phase, vapor-liquid equilibrium states, which ends at the critical point. The true isotherms are flat in the two-phase region inside the dome. All mean-field EOSs are similar in that they artificially treat the fluid as a homogeneous phase with a continuously varying density inside the two-phase region. This unphysical oversimplification generates the "van der Waals loops" shown by the isotherms. The spinodal boundary separates mechanically stable regions (metastable states for which (∂P/∂V )T 0) from mechanically unstable ≤ regions (for which (∂P/∂V )T > 0) as determined by the slope of the isotherms of the mean-field EOS. If the supersaturated vapor is in contact with its own liquid, it will condense until the vapor again reaches saturation. If a container of volume V contains only pure vapor, at a suitably large supersaturation value S = Pv/Pve, droplets will start to form at an appreciable rate as a result of collisions among vapor molecules. This process of forming a droplet is known as homogeneous nucleation. If impurities are also present in the container, the supersaturated vapor will first condense on them in a process referred to as het- erogeneous nucleation. Since nucleation plays a key role in many fields ranging from atmospheric applications to materials science, the study of nucleation has a long his- tory and is currently receiving much attention stimulated by the development of new experimental and theoretical techniques to measure and predict nucleation rates. The first comprehensive treatment of the thermodynamics of the nucleation process is due to Gibbs[1]. Gibbs showed that the reversible work required to form a nucleus of the new phase consists of two terms: a bulk (volumetric) term and a surface term. Later, in 1926 Volmer and Weber[2] developed the first nucleation rate expression, by arguing that the nucleation rate should be proportional to the frequency of collisions between condensable vapor molecules and small droplets (critical clusters) of the new phase of a size, the critical size, that just permits spon- taneous growth. A more detailed kinetic approach for the process of vapor-to-liquid nucleation was subsequently developed by Farkas[3]. The theory of Volmer, We- ber, and Farkas was extended a few years later by Becker and Döring[4], Frenkel[5], Zeldovich[6], and is now known as classical nucleation theory (CNT). 3

The basic kinetic mechanism assumed by these authors was that small clusters grow and decay by the absorption or emission of single molecules. In this theory the clusters are treated as spherical droplets. As in Gibbs’s treatise, the work of formation of a critical droplet consists of volumetric and surface contributions, but in the absence of knowledge of the microscopic cluster properties, bulk thermodynamic properties are used to evaluate the two contributions. Gibbs’s result for W is

W = Aγ Vl(Pl Pv) ,(1) − − 2 4π 3 W =4πr γ r (Pl Pv) , (2) − 3 − and it strictly applies to just the droplet of critical size. The surface term Aγ represents the free energy needed to create a surface. This term always opposes droplet formation. The volume term -Vl(Pl Pv) represents the stabilizing free − energy obtained by forming a fragment of new phase. The negative sign before the volume term ensures that new phase formation ultimately lowers the free energy of the system. In developing the kinetics of nucleation, it is necessary to know the free energy of formation of droplets of noncritical size. The simplest approximation is to assume that Gibbs’s result for W applies to all sizes and to rewrite it in terms of n, the number of molecules in the droplet. In terms of the spherical liquid droplet model, the surface area and volume are straightforward to rewrite since r n1/3 ∝ for this model. It is customary to assume that the droplet is an incompressible liquid and to replace the pressure difference by the difference in chemical potential between the old and new phases at the same pressure Pv, as explained in more detail later. It is generally a very good approximation to treat the vapor phase as ideal, so that the chemical potential difference can then be expressed in terms of the supersaturation ratio S. One other approximation is needed: the surface tension of a planar interface is used to evaluate the surface term because the surface tension of microscopic droplets is unknown. With these simplifications the free energy of formation of a cluster of n molecules is

∆G γ A 2/3 = ∞ n ln S = θn n ln S ,(3) kT kT − − 4 where γ is the surface tension of a planar gas-liquid interface, A is the surface ∞ area of the cluster, which is estimated from the liquid number density ρl,andθ = 1/3 2/3 (36π) γ ρ−l /kT. The dependence of ∆G on n is illustrated in Figure 1.2. ∞

surface term Free Energy of Formation Free Energy 0 n 0 n*

volume term

Figure 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 a maximum at the critical size .

Asseenintheﬁgure, ∆G has a maximum at the value n = n∗,knownasthe critical size. If the cluster size n is less than n∗, the surface term is dominant. As a result, the cluster has a higher tendency to shrink, i.e., to reduce its free energy, than to grow, i.e., to increase its free energy. On the other hand if n>n∗,the volume term is dominant, and the cluster has a higher tendency to grow than to shrink. From the extremum condition, [d∆G/dn] =0, one obtains the simple n∗ relation for the critical size n∗,

2θ (n )1/3 = , (4) ∗ 3lnS which is equivalent to the Kelvin equation for the critical radius r∗ 2γ r∗ = ∞ .(5) ρlkT ln S 5

The barrier height is equal to the work of formation of the critical droplet, ∆G∗ =

W ∗. Volmer and Weber[2] in 1926 argued that the nucleation rate depends exponentially on the work of formation of the droplet. The nucleation rate expression, which derives from the work of Becker-Döring[4], Frenkel[5], Zeldovich[6], is often referred to as Becker-Döring theory. The expression is given by Abraham[7], for example, as

W ∗ JCL = J0 exp , (6) − kT µ ¶ with the pre-exponential factor

2 2γ Pv J = ∞ v , (7) 0 πm l kT r µ ¶ where m is the mass of a condensible vapor molecule, vl(= 1/ρl) is the molecular volume, Pv is the vapor pressure, and the barrier height of nucleation is

16πγ3 W ∗ = ∞ 2 . (8) 3(kTρl ln S) As seen from the nucleation expression, all the inputs to the theory are experimental quantities which makes the theory easy and popular to use. For many years, it was impossible to measure nucleation rates accurately. Instead, what was usually determined experimentally was the critical supersaturation at which nucleation was observable at a signiﬁcant rate, whose value was typically not known to better than one or two orders of magnitude. (One can see from Eqs.(6) and (8) that J depends sensitively on S, but that S is rather insensitive to changes in J.) These critical supersaturation measurements were generally in good agreement with the predictions of CNT for many substances. Over the past twenty years, the development of new experimental techniques with improved precision has allowed the accurate measurement of nucleation rates for many substances[8—16]. Comparison of these results with the predictions of CNT has shown that the theory is usually in error, giving rates that are too low at low temperatures and too high at high temperatures[10, 17, 18], as illustrated in Figure 1.3. 6

1010 260 K 250 K H O 2 Expt S-form

9 10 240 K ) -1 230 K s -3 108

J (cm 107 T=220 K

106 5 10152025 S

Figure 1.3. Experimental data for water from Ref.[18] illustrating the inadequate temperature dependence predicted by the classical Becker- Doering theory[4], labeled S-form in the ﬁgure.

Due to limits of CNT, there has been much eﬀorttoimprovetheclassicalmodel, but most of the newer models take the form of correction factors to CNT. [19, 20]. In addition, the improvements of these models are often substance speciﬁc, which limits their applicability. One of the most successful and most general treatments of the temperature dependence of nucleation rates is the so-called scaled model of Hale[21, 22] introduced in 1986. The scaled model is based on CNT, and it yields a universal dependence of nucleation rate on Tc/T 1.Thetwoparametersof − this model are the nearly universal constant Ω, which is interpreted as the excess 26 3 1 surface entropy per molecule, and the constant rate prefactor J0( 10 cm− s− ). ≈ The value of Ω for nonpolar substances is around 2.2, whereas for polar materials it is about 1.5. For later use, and as an example, Ω is 1.476 for heavy water and 1.470 for water. The model works well for many substances for which the CNT fails. In the scaled model, the nucleation rate is given by the expression, 7

3 16π 3 Tc 2 J = J0 exp Ω 1 /(ln S) . (9) − 3 T − Ã µ ¶ ! The most fundamental approach to improving CNT is through the development of microscopic theories. The goal of any microscopic theory is to avoid the overly simplistic use of bulk thermodynamic properties in evaluating the free energy of cluster formation. There are several different types of microscopic approaches, which will only be mentioned here. Molecular dynamics (MD) computer simulations have been used to explore properties of small molecular clusters, e.g. by Gubbins and coworkers[23, 24] and Tarek and Klein[25], and to directly simulate nucleation, as in the work of Rao and Berne[26], Yasuoka and Matsumoto[28], and ten Wolde and Frenkel[29]. Monte Carlo (MC) computer simulations have also been used ex- tensively to calculate free energies of cluster formation, e.g. by Lee et al.[27, 30], Hale et al.[31, 32], and Oh and Zeng[33], and to examine the subcritical cluster size distribution directly[34]. Hybrid approaches like those of Weakliem and Reiss[35], Schaff et al.[36] that combine MC or MD simulation results with analytical theory have also been developed. A brief review by Reiss[37] discusses other approaches by many other authors not mentioned here. Another important approach known as the density functional theory (DFT)[38— 40] will be discussed later in detail. To briefly summarize here, in DFT the free energy of the nonuniform system, F [ρ(r)], is written as a functional of the local density ρ(r) at each position r in the fluid. The presence of the nucleus renders the fluid inhomogeneous. The inhomogeneity is characterized by the density that varies continuously from its value at the center of the nucleus to its value in the metastable mother phase far from the nucleus. The properties of the critical nucleus are determined by finding the density profile that minimizes the nonuniform fluid’s free energy. Cahn and Hilliard[40] were the first who developed a type of DFT for nucleation theory. They proposed the Helmholtz potential to be

c 2 F [ρ (r)] = dr f0 [ρ (r)] + [ ρ (r)] ,(10) 2 ∇ Z ³ ´ 8

where f0 is the Helmholtz free energy density of the homogenous fluid of density ρ, ρ is the gradient of the density, and c is the so-called influence parameter related ∇ to the intermolecular potential. Because the Helmholtz potential above depends on the gradient of the density, minimizing Eq.(10) one obtains a differential Euler-Lagrange equation. This theory is called gradient theory (GT ), or square gradient theory, because of the form of the free energy functional. The first GT was actually devised many years earlier by van der Waals[41] to describe the structure of planar interfaces. To apply the GT ,one needs a well-behaved EOS. It should have the form of a cubic equation, similar in spirit to the van der Waals EOS, that describes the system as a single homogeneous phase whose density varies continuously throughout the two-phase region. A more general form of DFT was developed and applied to nucleation theory by Oxtoby and coworkers[42, 43]. It is a molecular theory that explicitly uses an intermolecular potential. The theory uses a hard sphere fluid as a reference state and treats the attractive intermolecular potential as a perturbation. The theory is developed in terms of the grand potential Ω, which is written in the perturbation theory as the following functional of the density,

Ω [ρ (r)] = dr (fh [ρ (r)] µρ (r)) + drdr0φ ( r r0 )ρ(r)ρ(r0) ,(11) − att | − | Z ZZ Here, fh is the Helmholtz free energy density of the hard sphere ﬂuid, φatt is the long-range attractive part of the potential, and µ is the chemical potential. The simpler GT canbederivedfromthemoregeneralDFT by expanding the density in a Taylor series and retaining only the leading nonzero terms. Minimization of Eq.(11) generally leads to an integral Euler-Lagrange equation, which must be solved for the density proﬁle of the nonuniform system.

1.2 BRIEF OVERVIEW OF BINARY NUCLEATION Many of the above considerations apply as well to homogeneous nucleation of binary systems, commonly referred to as binary nucleation, but there is a major difference as well. In binary nucleation, the initial metastable phase and the final phase are two component systems. Thus, the kinetics of nucleation involves the 9 formation of clusters of the new phase that generally contain both components. To apply CNT to binary systems, the most important quantity needed to predict nucleation rates is the composition of a critical nucleus. If the surface tension is known as a function of the composition and if ∆P , the difference in pressure inside and outside the droplet, is known, then the critical radius can be calculated using the Laplace formula, ∆P =2γ/r∗. Assuming the droplet is incompressible, the

Gibbs-Thomson equations can be derived: ∆µ = 2γvi/r. The diﬀerences in the i − chemical potential between liquid and vapor phases are represented by ∆µi,whilevi is the molecular volume of component i in the liquid. From the two Gibbs-Thomson equations, one can determine the composition and the critical radius of the droplet. In 1950, Reiss[44] proposed a theory based on kinetic and thermodynamic argu- ments showing that the binary nucleation rate is determined by the passage over a saddle point in the two-dimensional droplet size space. Later, Doyle[45] used this theory to study the sulfuric acid-water system, but the Gibbs-Thomson equations he found contained a term involving the compositional derivative of the surface tension. Because these terms were small for the sulfuric acid-water system, they had essentially no eﬀect on the calculated critical cluster compositions. When Doyle’s equations were subsequently applied to strongly surface active systems, such as ethanol-water or acetone-water, these terms became very important for water-rich cluster compositions. As a result, the theoretically calculated vapor compositions needed to produce experimentally observed nucleation rates were many orders of magnitude lower in the concentration of the surface active component than the experimental concentrations. Renninger, Hiller, and Bone[46] argued that Doyle’s treatment of the Gibbs-Thomson equations was inconsistent. Wilemski[47] proposed a revised classical theory in which the Gibbs surface adsorption equation was used to cancel the derivative of the surface tension, thus permitting the conventional Gibbs-Thomson equations to be recovered. It is interesting to note that the conventional Gibbs-Thomson equations had been used in the original, early work on binary nucleation by Flood[48] and by Döring and Neumann[49], but had then been forgotten. 10

The predictions of either version of CNT for ideal binary mixtures are fairly reasonable, but for mixtures with a component that strongly segregates on the droplet surface, e.g. alcohol-water or acetone-water mixtures, problems arise. Doyle’s version of CNT predicts unrealistic results, as just noted, while the revised binary CNT gives rise to unphysical behavior that violates the nucleation theorem[50] for binary systems. In an important step to resolving these difficulties, Laaksonen[51, 52] proposed a so-called explicit cluster model to study water-alcohol systems. The model makes realistic predictions for the vapor concentrations while predicting physical behavior for the nucleus composition, in accord with the nucleation theorem. Compared to unary nucleation, less work on microscopic theories of binary nucleation has been performed. Zeng and Oxtoby[43] extended the DFT to treat binary nucleation for Lennard-Jones mixtures. Talanquer and Oxtoby have used the GT to study highly nonideal binary systems with parametrized hard-sphere—van der Waals EOS[53]. Napari and Laaksonen have recently performed DFT calculations for a site-site interaction model that simulates systems with a highly surface active component.[54] Hale and Kathmann have performed Monte Carlo simulations to calculate the free energies of formation of sulfuric acid-water clusters[55]. 1.3 MOTIVATION The principal goal of this thesis is to test a form of classical nucleation theory closest in spirit to the original pioneering work of Gibbs. The usual forms of CNT are well-known to provide a poor quantitative description of the temperature dependence of measured nucleation rates, although the predicted dependence on supersaturation is generally quite satisfactory. To explore this, Gibbs’s original formula was used to calculate nucleation rates for several different substances: water and heavy water, methanol, ethanol. Significant improvement in the predicted temperature dependence of the nucleation rate was realized only for water and heavy water. This appears to be due to the extraordinary isothermal compressibility of these two substances at the low temperatures where nucleation rates are generally measured. The other materials studied are much less compressible at low temperatures, and the customary approximation of an incompressible fluid, universally used in the classical theory, is valid for these substances. The implementation of 11

Gibbs’s original formula requires the use of an accurate equation of state for the fluid properties. In the case of water, two different EOS were used, but each accurately treated the anomalously high compressibility of fluid water. With various equations of state available, it was possible to test a nonclassical theory of nucleation known as gradient theory. A second goal of this thesis is to determine whether or not the predicted temperature dependence of the nucleation rate would be improved by this simplest form of density functional theory. Reasonably good results were found for water using a so-called CPHB EOS, but the gradient theory results for , methanol, and ethanol were only slightly improved compared to the predictions of classical theory. Finally, the application of Gibbs’s original formula to binary nucleation was explored. The goal of this aspect of the work is to see whether certain unphysical aspects of classical binary nucleation theory could be alleviated by using a more exact formulation of the theory. A key difficulty in carrying out this phase of the research was that for the most interesting binary systems, such as the ethanol-water system, there are no accurate EOSs in the temperature range of interest. To sur- mount this difficulty, a model system was devised with properties resembling those of the ethanol-water system. The EOS for the model system consists of a binary hard sphere fluid contribution plus an attractive term of the van der Waals form. The bulk surface tension was computed as a function of mixture composition using density functional theory for a planar interface. To facilitate the DFT calculations, attractive potentials of the Yukawa form were employed. The results showed that Gibbs’s original formula, with the bulk surface tension, also suffered from the same unphysical behavior as simpler forms of the classical binary theory. 12

2. EQUATION OF STATE APPROACH FOR CLASSICAL NUCLEATION THEORY

2.1 THEORY In this chapter, three diﬀerent versions of classical nucleation theory (CNT) are explored to study nucleation rates of water and heavy water. For two of these versions,anovelapproachbasedondiﬀerent equations of state is used to calculate the work of formation of a critical droplet, W ∗, which is then used to evaluate the nucleation rate. The theoretical predictions are compared with the experimental rates of water and heavy water[18]. The theoretical results are also compared with the predictions of the scaled model of Hale[21]. The number of molecules in a critical cluster are compared with the experimental data using the nucleation theorem[50]. 2.1.1 Work of Formation.ConsideravolumeV containing N molecules of vapor at a chemical potential µv and pressure Pv. The Helmholtz free energy of this vapor is

Fi = Nµ PvV. (12) v −

Upon forming a droplet with n molecules, if we ignore the very small changes in µv and Pv,theﬁnal Helmholtz free energy of the system is

Ff =(N n)µ + nµ (V Vl)Pv VlPl + Aγ , (13) − v l − − − where µl is the chemical potential of a molecule at the internal pressure Pl of the droplet, Vl isthevolumeofthedroplet,A is its surface area, and γ is the surface tension. The diﬀerence in the free energy between the initial and ﬁnal systems is

∆F = Ff Fi = n(µ µ ) (Pl Pv)Vl + Aγ . (14) − l − v − − (It should be noted that Eqs.(13) and (14) are not quite rigorous since they fail to include the surface excess number of molecules[50]. As shown in Appendix A, the ﬁnal results below are, nevertheless, correct.)

This free energy diﬀerence has a maximum at a speciﬁcradius,r∗,whenµl = µv, 13

Max 4π 3 2 ∆F = (Pl Pv) r∗ +4πr∗ γ ,(15) − − 3 which represents the free energy barrier required to be overcome to form a spherical critical droplet of radius r∗. Using results of Appendix A, Eq.(14) can be approximated as the following sum of a surface and a volume term,

4πr3 ∆µ ∆F = +4πr2γ, (16) − 3 vl where ∆µ is the diﬀerence in chemical potential between the initial metastable phase and the ﬁnal stable phase and vl is the molecular volume of the stable phase. Figure 2.1 schematically shows the dependence of this free energy as a function of the droplet radius.

surface term

MAX ∆F Free Energy of Formation Free Energy 0 r 0 r*

volume term

Figure 2.1. Same as Figure 1.2 but the free energy of formation is plotted as a function of the radius of the cluster.

Applying the Laplace equation, which governs the pressure drop across a curved interface, speciﬁcally Pl Pv =2γ/r∗(see Appendix A), we then obtain −

16π γ3 W ∗ = 2 ,(17) 3 (Pl Pv) − 14

where W ∗ is the minimum free energy, i.e., the reversible work, required to form a critical droplet of radius r∗. To apply the above formula, which is known as Gibbs formula, one has to know the exact surface tension at that radius and the pressure inside the droplet. Lack- ing knowledge of the exact surface tension, the ﬁrst approximation is to use the experimental surface tension of a ﬂat interface, i.e., set γ = γ to obtain ∞

16π γ3 W ∗ = ∞ 2 . (18) 3 (Pl Pv) − We call this equation the P form. − Usually the pressure inside the droplet is found approximately by making the assumption that the droplet is incompressible. In this case, we can replace ∆P =

Pl Pv with (µ µ (Pv))/vl, which follows from the thermodynamic identity − v − l

∆µ = µ µ (Pv)=µ µ (Pv)= vldP, (19) v − l l − l ZPv when the molecular volume vl is assumed to be constant and the condition of unstable equilibrium between the critical droplet and the metastable vapor is used. Note that this deﬁnition of ∆µ is identical to Kashchiev’s[56]. Equation (18) then becomes

3 2 16π γ vl W = ∞ . (20) ∗ 3 (∆µ)2 We call this equation the µ form. This form is most useful if the chemical potential − diﬀerence can be found from an equation of state. However, ∆µ is more commonly found using simpler, but approximate thermodynamic relations. If we assume the supersaturated and saturated vapors are ideal gases and that the droplet is a tiny piece of incompressible bulk liquid, then it is easily shown (e.g. in Appendix A) that

∆µ = kT ln S vl(Pv Pve) , (21) − − where k is the Boltzmann constant, T is the absolute temperature, and S is the 15 supersaturation value deﬁned as the ratio of the actual vapor pressure to the equilibrium vapor pressure Pve, i.e., S = Pv/Pve. It is customary to neglect the Pv term, which is almost always extremely small. Equation (20) then reduces to the most familiar form used in CNT,

3 2 16π γ vl W = ∞ . (22) ∗ 3 (kT ln S)2 For simplicity we call this equation the S form. − Applying the ﬁrst two forms of W ∗ requires knowledge of the actual pressure and chemical potential inside the droplet. Usually this information is unavailable, and for this reason experimentalists compare their results with rates predicted using the S form because the supersaturation ratio is readily determined from the − experimental data.

A less approximate way to evaluate the P form of W ∗ involves calculating the − internal pressure Pl using the equation

kT ln S = vldP , (23)

PZve which follows from Eq.(19) and the conditions for stable and unstable equilibrium,

µl(Pve)=µv(Pve) ,(24)

µl = µv(Pv) , (25) along with the ideal gas approximation,

µ (Pv) µ (Pve)=kT ln S. (26) v − v The integral on the right-hand-side of Eq.(23) can be evaluated quite accurately if the liquid density or, equivalently, the molecular volume is known as a function of pressure. If the pressure dependence of the density is not available from direct measurements, it may be calculated using the measured liquid isothermal compressibility, preferably as a function of pressure. 2.1.2 Gibbs’s Reference State. A more comprehensive approach for calculating the pressure and chemical potential diﬀerences needed in the P and − 16

µ forms of W ∗ involves using a complete equation of state (EOS).Acom- − plete EOS consists of a functional representation, either analytical or tabular, of the Helmholtz free energy F of the substance as a function of density and temperature. From the Helmholtz free energy, the pressure and the chemical potential are readily calculated from standard thermodynamic identities. Thus, F contains all the information needed to calculate the work formation of a droplet using the ﬁrst two forms.

The calculation of the internal pressure Pl from an EOS follows Gibbs’s[1] original reasoning[56—58]. Upon forming a droplet within a homogeneous fluidwithuniform chemical potential and temperature, the droplet may be so small that its internal state may not be homogeneous even at the center of the drop. The meaning of the internal pressure and density of the droplet is then obscured, and these values are difficult to determine. To overcome this difficulty, Gibbs introduced the concept of the reference state as the thermodynamic state of a bulk phase whose internal pressure Pref and density ρref are determined by the same conditions that exist for the new phase and the mother phase, i.e., by assuming that the temperature andthechemicalpotentialarethesameeverywhereinthenonuniformsystem. In mathematical terms, the pressure inside the droplet is calculated such that the chemical potentials are equal in both the metastable vapor and reference liquid phases

µv(ρv)=µl(ρref ) ,(27) where ρv is the density of the supersaturated vapor and ρref is the density of the reference liquid state. As a practical matter, one always calculates differences in chemical potential, and because Eq.(27) involves phase densities that generally differ by many orders of magnitude it is convenient to rewrite this equation as an equality of chemical potential differences measured from the common equilibrium state, for which

P(ρve)=P(ρle) ,(28)

µv(ρve)=µl(ρle) ,(29) where ρve and ρle are the equilibrium vapor and liquid densities, respectively. After 17 subtracting the equilibrium value of µ from both sides of Eq.(27), we obtain

µ (ρ ) µ (ρ )=µ (ρ ) µ (ρ ) . (30) v v − v ve l ref − l le

The chemical potentials are calculated from µ =(∂f/∂ρ)T ,wheref is the appropriate Helmholtz free energy density for the EOS.Onceρref has been found by solving Eq.(30), the reference pressure Pref is straightforward to calculate from the EOS. Figure 2.2 shows the concept of the reference state.

(ρ ,P ) ref ref P =pressure of metastable region v P =pressure at which µ(ρ ) = µ (ρ ) ref l ref v v

P (ρ ,P ) v v P eq

Figure 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 reference liquid phase.

Once W ∗ has been evaluated, the nucleation rate can be calculated using Eq.(6). Comparisons of the calculated rates with experimental values will be made in later sections for various substances. 2.1.3 Number of Molecules in the Critical Nucleus. In addition to the nucleation rate, another physical quantity of interest is the size of the critical nucleus, which is experimentally determinable from measured nucleation rates using 18 the nucleation theorem in the approximate form[50, 59],

∂ ln J n∗ . (31) ≈ ∂ ln S

The experimentally determined values of n∗ can be compared with the theoretical values based on the diﬀerent forms of W ∗ using the rigorous form of the nucleation theorem[56] ∂W∗ = ∆n∗/ (1 ρ /ρ ) . (32) ∂∆µ − − v l For the formation of liquid droplets in a dilute vapor, Eq.(32) reduces to

∂W∗ = n∗ . (33) ∂∆µ −

The critical number n∗ can also be computed from more classical considerations. 3 Since the volume of a spherical critical nucleus is V ∗ =4πr∗ /3, one can calculate the number of molecules in the nucleus from the relation n∗vl = Vl. Applying the

Gibbs-Thomson or Kelvin equation, Eq.(5), for r∗,oneﬁnds

2 3 32πvl γ n = ∞ , (34) ∗ 3(kT ln S)3 which is equivalent to Eq.(4). To implement the approach outlined above, there is clearly a need for a satisfactory EOS. There are many possible candidates in the literature. Not all of these are suitable for use in the EOS approach because they are not suﬃciently accurate. Curiously, these less accurate EOSs are actually the only ones suitable for the gradient theory calculations presented later. For completeness all of the EOS used in this thesis are presented in the next chapter. 19

3. EQUATIONS OF STATE FOR UNARY SYSTEMS

3.1 WATER Five EOSsforwaterwereusedinthedifferentphasesofthisthesiswork. 3.1.1 IAPWS 95.ThisEOS was published by the International Associa- − tion for the Properties of Water and Steam [60, 61]. It is an analytical equation based on a multiparameter fit of all the experimental data available at temperatures above 234 K. It is very accurate and therefore highly suitable for use in the EOS approach, but only for T 234 K. This limitation strictly applies to the low T— ≥ low P vapor-liquid equilibrium states. Liquid densities at high P and low T are in good agreement with the few experimental data available. The low T —lowP vapor behavior also is reasonable. This EOS has one other significant drawback. It fails to provide a continuous representation of single phase fluid states in the metastable and unstable regions of the phase diagram, and is, therefore, unsuitable for use in gradient theory calculations. In the IAPWS 95 EOS,thespecific Helmholtz free energy f is represented in − dimensionless form as φ = f/RT ,andφ is separated into an ideal part, φ0 and a residual part φr,i.e,

f = φ = φ0(δ, τ)+φr(δ,τ) ,(35) RT 3 where δ = ρ/ρc, τ = Tc/T with Tc =647.096 K, ρc =322kg/m ,andR = 0.46151805 kJ/(kg K).Thesubscriptc designates a value at the critical point. We also have the following deﬁnitions

8 0 0 0 0 0 0 γi τ φ =ln(δ)+n1 + n2τ + n3 ln(τ)+ ni ln(1 e− ), (36) i=4 − X

7 51 54 c r di ti di ti δ i di ti 2 2 φ = niδ τ + niδ τ e− + niδ τ exp αi(δ εi) βi(τ γi) i=1 i=8 i=52 − − − − X56 X X ¡ ¢ bi + ni∆ δΨ, (37) i=55 X 20 with

2 2 ai ∆ = θ + Bi[(δ 1) ] , (38) −

1 2 2β θ =(1 τ)+Ai[(δ 1) ] i , (39) − −

2 2 Ψ =exp Ci(δ 1) Di(τ 1) . (40) − − − − All the values of coeﬃcients and¡ parameters of φ0 and φr arelistedinAppendixB.¢ 3.1.2. Cross Over Equation of State (CREOS 01).MostEOSs − are attempts to improve the van der Waals EOS to give better representations of the properties of real systems, but these equations generally fail to reproduce the singular behavior observed at the critical point. This failure stimulated a search for a new type of EOS that could describe classical mean-ﬁeld behavior far away from the critical region and smoothly cross over to the singular behavior near the critical point. New equations with this capability have been developed by Kiselev and Ely for water, which they termed CREOS 01[62]. Since the − concept behind the crossover EOS is to get the right behavior near a critical point, to make this equation work at low temperatures, the scenario of a second critical point at low temperature[67] was exploited by Kiselev and Ely[62]. Even though the CREOS 01 equation is a cubic equation, it describes only the liquid states of − the system. Since it does not provide any representation of the vapor states, it is unsuitable for use in GT calculations for vapor-to-liquid nucleation. In the CREOS equation, the Helmholtz free energy of the system is cast in terms of Landau theory[63] as

ρ A(T,ρ)=∆A(τ,∆η)+ µo(T)+Ao(T) , (41) ρc where A is the dimensionless Helmholtz free energy, A = ρA/ρcRTc,andwhere

τ = T/Tc 1, ∆η = ∆ρ = ρ/ρ 1,andµ (T),andAo(T ) are analytical functions − c − o of T . 21

5 2 α α 2 ∆i ∆i ∆A(r, θ)=kr − R (q) aψ0(θ )+ cir R (q)ψi(θ) , (42) Ã i=1 ! X f τ = r(1 b2θ2) , (43) −

β β+ 1 ∆ρ = kr R− 2 (q)+d1τ. (44) All the coefficients and parameters of CREOS 01 are given in Appendix B. − 3.1.3 Jeffery and Austin EOS (JA EOS).Jeffery and Austin[64] have − developed an analytical equation of state to describe water. It has several interesting properties, but also an important drawback. Similar to the CREOS equation, it predicts a low temperature critical point associated with two metastable phases of supercooled water. It also provides a continuous description of single-phase states in the two phase region, similar to the van der Waals and other cubic EOSs. However, since it does not accurately predict the low temperature vapor-liquid binodal line, it is suitable for quantitative use in gradient theory calculations only for a small range of higher temperatures. The JA EOS consists of three parts. The first part, developed by Song and − Mason[65, 66], is a generalized van der Waals EOS of the specificform,

PSM a 1 =1+ α b∗ ρ + αρ 1 . (45) ρRT − − RT 1 λbρ − ³ ´ 2 2· − ¸ 6 2 Here, a is the van der Waals constant, (a =27R Tc /64Pc =0.5542 Pa m mol− ), λ is a constant equal to 0.3241,b∗ and α are related to the Boyle volume, vb, through

α =2.145vb,b∗ =1.0823vb,andb is a function of temperature given by

b(T) T T =0.2exp b3( + b4) b1 exp b5 + b2 ,(46) v − T − T b µ b ¶ µ b ¶ where Tb is the Boyle temperature. The second part of the JA EOS incorporates the effects of hydrogen bonding. − This effect was first treated approximately by Poole et al.[67]. Jeffery and Austin modified the results of Poole et al. to get this part of the Helmholtz free energy as

HB FHB = fRT ln Ω0 + ΩHB exp (1 f)RT ln(Ω0 + ΩHB) ,(47) − − RT − − ³ ³ ´´ 22

where Ω0, and ΩHB are the numbers of conﬁgurations of weak hydrogen bonds with energy 0 and of strong hydrogen bonds, respectively. These are given by

Ω0 =exp( S0/R), and ΩHB =exp( SHB/R),whereS0 and SHB are the entropies − − of formation of a mole of weak and strong hydrogen bonds respectively, and HB is the hydrogen bond energy. In this term f is a function of temperature and density through the following relation

8 1+C1 T f = exp C2 ,(48) exp((ρ ρ )/σ)+C1 − Tf − HB Ã µ ¶ ! where σ is the width of the region where hydrogen bonds are able to form, ρHB is the hydrogen bond density, C1 and C2 are constants and Tf is the normal freezing temperature. The ﬁnal part of the equation is called the vapor correction term, and it reads

2 Pcorr = I1ρ RT , (49) where the correction function I1 is given by

I1 =(α B)ξ(T )φ(ρ) . (50) − The auxiliary functions ξ and φ are deﬁned as

2 Tc 6 (T κTc) + A2 ξ(T)=A1 A5 exp( ) − ,(51) − T T 2 µ ¶ C 6.7 ρ exp A4 ρc φ(ρ)= µ ³ ´3.2¶ ,(52) ρ 1+A3 ρc ³ ´ where A1, A2,A3,A4,κ are constants, B is the second virial coeﬃcient, deﬁned as

B = α b∗ a/RT, Tc and ρ are the critical temperature and density. All of the − − c constants are evaluated in Appendix B. The total pressure for the JA EOS is − given by

P = PSM +2PHB + Pcorr .

3.1.4 Cubic Perturbed Hard Body (CPHB).ThisEOS was developed by Chen et al[68, 69] to study the vapor-liquid equilibrium of nonpolar and polar 23

fluids. It employs a generalized hard sphere EOS to treat the hard-core repulsions between molecules and uses a simple, modified van der Waals term to treat the effects of attractive forces. The hard-body compressibility factor of Walsh and Gubbins[70] is used. Walsh and Gubbins modified the well-known Carnahan-Starling[71] EOS, which is based on hard sphere simulation data. The Walsh and Gubbins modification covers all the shapes of the molecules from a single sphere to a chainlike molecule through the use of a nonspherical factor, α. The compressibility factor of Walsh and Gubbins was simplified to the form

v + k b Zrep = 1 ,(53) v k2b − where b is the hard core molar volume, v is the molar volume, and k1,k2 are given in Appendix B. After adding an empirical attractive term to Eq.(53), the CPHB EOS reads as

RT (1 + k b/v) a P = 1 .(54) v k2b − v(v + c) − The parameters a, b,andc are determined from the critical properties of the fluid. Details of the CPHB EOS are given in Appendix B. 3.1.5 Peng-Robinson (PR).VanderWaalsintroducedthefirst mean-field theory to study phase behavior in real systems. His EOS was a qualitative break- through in understanding, but it lacks quantitative accuracy, particularly with respect to the vapor-liquid equilibrium states. Peng and Robinson[72] proposed several modifications to overcome these shortcomings for nonpolar fluids. The PR equation gives good results in describing nonpolar fluidbehavior,butitismoder- ately successful for polar fluids as well. The PR equation takes the form

RT a(T ) P = . (55) v b − v(v + b)+b(v b) − − At the critical point, the following relations are satisﬁed: 24

2 2 R Tc a(Tc)=.45724 , (56) Pc RTc b(Tc)=.0778 , (57) Pc Zc =0.307 . (58)

At any other T , Peng and Robinson assumed

a(T )=α(Tr,ω)a(Tc) ,

b(T )=b(Tc) , (59)

where Tr = T/Tc,

1 1 2 α 2 =1+k(1 Tr ) , (60) − and k is a substance-speciﬁc constant. This constant was correlated to the acentric factor, ω, and the result was:

k =0.37464 + 1.54266ω 0.26992ω2 . (61) − 3.2 HEAVY WATER: CREOS 02 − CREOS 02 has the same functional form as CREOS 01, but with different − − parameter values[73]. The parameters and coefficients of this equation are given in Appendix B. 3.3 METHANOL AND ETHANOL: CPHB The original CPHB equation of state was developed for non-polar fluids. An extension has been made to cover many polar fluids including alcohols. This equation is very sensitive to the parameter values. To use this equation successfully, very careful attention should be paid to the values of the critical properties, i.e., they should be the same as used by Chen et al.[74] Refer to Appendix B for the parameters. 25

4. RESULTS OF EOS APPROACH FOR UNARY SYSTEMS

4.1 WATER Before applying the different equations of state to calculate nucleation rates, differences in the critical work of formation, W ∗, for the various forms of CNT were examined. Figure 4.1 shows W ∗ of water droplets using the IAPWS 95[60] at − T =240, 250,and260K. As can be seen from the graph, the results for the µ form − and for the S form are close to each other at low S and start to deviate slightly − at high S. The maximum deviation is of order kT, which will give a difference in nucleation rates of only a factor of three and is, thus, inconsequential. It is clear from this figure that the P form gives significantly different results. The W ∗ − for the P form is much lower than for the other forms. Since the nucleation − rate depends exponentially on ( W ∗), higher nucleation rates will result for the − P form. AnimportantpointtonoteisthatthegapbetweentheP form − − and other versions grows as T decreases, so the predicted temperature dependence should also be greatly improved.

48 T=260 K H O 240 K 2 µ-form 44 250 K S-form P-form 40

36 W*/kT 32

67891011121314 S

Figure 4.1. The work of formation for water droplets using the IAPWS-95 EOS with the three forms of CNT at T=240, 250, and 260 K. 26

The nucleation rates of water using the IAPWS 95 EOS[60] and applying the − different versions of CNT are shown in Figure 4.2. Since the calculated nucleation rates using the P form are higher, they were divided by 200 for the figure. The − figure shows that the P form performed excellently regarding both the temper- − ature dependence and the supersaturation dependence. Because the predictions of S form and µ form are so close to each other, only the results of the S form − − − are plotted.

1010 260 K H O 250 K T=240 K 2

109 -1 8 s 10

-3

7 J cm 10 S-form Expt P-form/200 106 67891011121314 S

Figure 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-form and S-form.

The other EOS used to describe water at low temperature is the CREOS 01. − Because it fails to describe the vapor states of the ﬂuid, the CREOS 01 was − used only for the liquid states, while the JA EOS was used for the vapor, in the − following way. To calculate the equilibrium vapor density, ρve, and liquid density,

ρle, one solves, respectively, the two equations,

exp Pve (T )=PJA(ρve) , (62) 27

exp Pve (T )=PCREOS 01(ρle) , (63) − where Pve is the experimental equilibrium vapor pressure[18]. Then, to ﬁnd ρref the JA EOS and the CREOS 01 were combined in the following equation − −

µJA(ρv) µJA(ρve)=µCREOS 01(ρref ) µCREOS 01(ρle) . (64) − − − − The rationale for this procedure is that the JA EOS is expected to be accurate − for densities and chemical potential diﬀerences of vapor states, while the same thing is true of the CREOS 01 for the liquid states. − With CREOS 01[62] results can be calculated over a wider range of tempera- − tures down to T =220K, as shown in Figure 4.3. The P form results are again − divided by the factor of 200.Theﬁgure also shows the predictions of the scaled model[31].

1010 260 K 250 K Expt H O S-form 2 Scaled P-form/200 109 ) 240 K -1

s 230 K -3 108

J (cm T=220 K 107

106 5 10152025 S

Figure 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-01 and with the scaled model. 28

Both the P form and the scaled model results describe the data well. The − classical Becker-Döring result, based on the S form gives a clearly inferior account − of the temperature dependence. From the experimental rates and the nucleation theorem, the number of molecules in the critical droplet, n∗, can be determined. Figure 4.4 shows the experimental values[18] and the values derived from the P form of W ∗ versus the predictions − of the Gibbs-Thomson formula, Eq.(34), at the diﬀerent temperatures. Only the

CREOS 01 EOS was used to calculate n∗ using the formula − 32πγ3 n∗ = ∞ 3 ρref , (65) 3(Pref Pve) − which is readily found from Eqs.(18) and (32). The experimental data were found by Wölk and Strey[18] using the equation ∂ ln J n∗ = 2 . (66) ∂ ln S −

50 H O 2 40 Gibbs-Thomsom Expt 30 P-form * n 20

0 0 1020304050 n* Gibbs-Thomson

Figure 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 shows the full agreement with the Gibbs-Thomson equation. 29

The calculated n∗ values using the P form of the CNT show excellent agree- − ment with the measured ones. This result is not unexpected since the P form of − the CNT gives the right T and S dependence, and since n∗ is essentially equal to the derivative of ln J with ln S. 4.2 HEAVY WATER

The only EOS valid at low T to describe D2O is the CREOS 02[73]. As − for CREOS 01, this equation also describes only liquid states, and there is no − other EOS to describe the vapor states. Consequently, to evaluate the chemical potential of the metastable vapor, the assumption that the vapor is ideal has been used, i.e., µ(ρ ) µ(ρ )=kT ln S. To calculate the equilibrium liquid density, ρ , v − ve le the experimental equilibrium vapor pressure[18], Pve(T), has been equated with the CREOS 02 pressure at the equilibrium liquid density −

Pve(T )=PCREOS 02(ρle) . (67) −

To ﬁnd ρref the ideal vapor assumption was used to obtain

kT ln S = µCREOS 02(ρref ) µCREOS 02(ρle) . (68) − − −

The reference pressure is then obtained as Pref = PCREOS 02(ρref ) after the solution − to Eq.(68) is found. Figure 4.5 shows the rates, divided by a factor of 100, predicted by the P form − using the CREOS 02[73] equation. The results show good agreement with the − experimental T and S dependence. 30

260 K D O Expt 250 K 2 109 P-form/100 240 K S-form 230 K

8 -1 10 s

-3

107 J cm T=220 K

106 5 10152025 S

Figure 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.

All the aforementioned experimental data has been taken by Wölk and Strey[18] using a pulse chamber. Other interesting experimental data have been taken by Heath[76, 77], Khan[78], and Kim[79] using a supersonic nozzle technique. This technique yields a very high nucleation rate at high supersaturation values. The results predicted by the P form with CREOS 02 have been compared with − − both the scaled model and an empirical function by Wölk et al[80]. The empirical function was developed by ﬁtting all the nucleation rate data of Wölk and Strey at low S. Figure 4.6 shows all the results. 31

D O Khan et al 2 1018 Kim et al P-form Empirical scaled model -1 s -3

1017 J cm

1016 20 40 60 80 100 120 140 160 S

Figure 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 Kfromlefttoright.

From the above ﬁgure, we notice that the scaled model gives very good results at these high supersaturation values, while the P form results based on CREOS 02 − − lie within an order of magnitude of the measured values, but do not reproduce the T dependencequiteaswellasforthelowS pulse chamber data. The following graph (Figure 4.7) shows the number of molecules in the critical droplet calculated from the experimental data[18] and the P form of W ∗ using − the nucleation theorem plotted versus the number of molecules predicted by using the Gibbs-Thomson formula at the diﬀerent temperatures. 32

50 D O 2 40 Gibbs-Thomsom Expt 30 P-form

* n 20

0 0 1020304050 n* Gibbs-Thomson

Figure 4.7. As in Figure 4.4 but for heavy water.

As for ordinary water, n∗ calculated from the P form of the CNT is in excellent − agreement with the measured values. Again, since the P form of the CNT − reproduces the experimental T and S dependence of J and since n∗ is essentially the slope of the lnJ —ln S curve, this good agreement is not surprising. 4.3 DISCUSSION OF WATER RESULTS The results show a clear advantage of using the P form over the other versions. − Note that the µ and S forms, which were based on the assumption of liquid − − incompressibility, give poor results when compared with the experimental data. This is strong evidence for the invalidity of the assumption that liquid water is incompressible. Figure 4.8 shows the liquid density as a function of temperature at different pressures as calculated from IAPWS 95 and CREOS 01,which − − are excellent agreement with each other and with experiment[75, 81] over wide ranges of pressure and temperature. From this figure, one can see that at all temperatures the density of liquid water depends strongly on the pressure. This means that liquid water is very compressible, especially at the lower temperatures. Also note that at a pressure between 190 and 300 MPa, the densities predicted 33 by CREOS 01 and IAPWS 95 equations start to differ qualitatively. The − − CREOS 01 equation predicts that at the higher pressures the well-known density − maximum of water no longer occurs. This is in accord with the experimental density measurements of Petitet, Tufeu, and Le Neindre[81] that show no density maximum for P 200 MPa down to T =251.15 K. The disappearance of the density ≥ maximum is also consistent with the observation that water’s viscosity decreases and its diffusivity increases with increasing pressure up to a pressure of about 200 MPa. At higher pressures, these anomalies in water’s transport coefficients vanish, and water behaves more normally with further increases in pressure[82]. In contrast, the IAPWS 95 equation continues to predict this feature. This suggests that − nucleation rates calculated using the IAPWS 95 equation at low T would differ, − perhaps substantially, from those found here using CREOS 01.Thisconjecture − awaits a means of using the IAPWS 95 equation at low T before it can be tested. − It should be noted that for T 240 K, there is essentially no difference between ≥ the W ∗(P form) predictions of these two EOSs, as can be seen in Figure 4.9. −

300 H O Expt (Petitet) 2 280 Expt (Kell) IAPWS-95 CREOS-01 260 500 MPa 0.1 MPa 240 T (K) 220 200 50 400 200 100 300 150 190 180 800 900 1000 1100 1200 1300 3 ρ (kg/m )

Figure 4.8. The temperature-density isobars of water using the IAPWS- 95 EOS and the CREOS-01 compared to experimental data of Kell and Whalley[75] and Petitet et al.[81]. 34

40 260 K CREOS 38 250 K IAPWS-95 T=240 K 36

/(kT) 32 P ∆ 30 W

26 6 7 8 9 10 11 12 13 14 S

Figure 4.9. The work of formation of water at T=240, 250, and 260 K predicted by the IAPWS-95 and CREOS-01.

Figure 4.10 shows the isothermal compressibility as a function of temperature at 10 MPa (the differences in the isothermal compressibility between 1 atm and 10 MPa are small) and at 190 MPa, calculated using the fit of Kanno and Angell[83]. From this figure, it is clear that the isothermal compressibility decreases sharply when the pressure is increased to values typical of critical nuclei. It should be kept in mind that the reference pressure for critical droplets can reach very high values, up to 400 MPa or higher, and so the high pressure behavior of the EOS is of considerable importance in calculating nucleation rates using the P form of CNT. − 35

1000 )

-1 800

10 MPA (Pa 600 12 *10 t 400 K 190 MPA

200 230 240 250 260 270 280 290 T (K)

Figure 4.10. Isothermal compressibility of liquid water at 10 MPa and . 190 MPa calculated from the ﬁt of Kanno and Angell[83]

One last point concerns a purely practical matter. In Chapter 2, an alternative to using a full EOS to do the P form calculations was noted. This method − was tested using accurate ﬁts for the liquid density as a function of pressure and employing Eq.(23). Results essentially identical to those shown here were obtained. 36

5. GRADIENT THEORY OF UNARY NUCLEATION

5.1 BACKGROUND Classical nucleation theory is the most frequently used theory to explain the nucleation process. In its basic form, the theory depends on experimental measurements as the inputs, which makes it convenient to use. With what is known as the capillarity approximation, the theory treats the droplet as if it has a sharp interface with a surface tension equal to that of flat equilibrium interface. Furthermore, the droplet is usually assumed to be incompressible with a density equal to the bulk liquid density at low pressure. In reality, the interface between the gas and liquid phases is never as sharp as envisioned in the model, and it is often quite diffuse. The density of the droplet is not spatially uniform. For this reason many different theoretical approaches have been adopted to avoid the approximations of CNT,in particular the assumptions of a homogeneous droplet with a sharp interface. Among the different approaches to realistically treat the inhomogeneity of the droplet is the density functional theory (DFT). The DFT is a rigorous statistical mechanical approach in which the free energy of the system is expressed as a functional of the density profile of the entire system. Because DFT involves the use of realistic intermolecular potentials, it is considerably more difficult to use than the CNT. Moreover, these potentials are often quite complicated and not so easy to develop. Gradient theory (GT ) is a methodologically less demanding, but more approximate relative of DFT.WhereasDFT depends on the intermolecular potential as its main ingredient, the GT , instead, requires a well-behaved mean-field EOS. In this chapter, GT and the P form of CNT are applied to study the − nucleation of water, methanol, and ethanol using the CPHB EOS. The results are compared to experimental measurements and to the predictions of the S form − of CNT. Using the CPHB EOS and the JA EOS,theGT and the P form − − of CNT are also applied to study the nucleation of TIP4P water at T =350K, which was first simulated by Yasuoka and Matsumoto[84] using molecular dynamics techniques. 37

5.2 THEORY Neglecting all the external ﬁelds, the GT expression for the Helmholtz free energy, F,ofapurecomponentis

c 2 F = f0(ρ)+ ( ρ) dV ,(69) 2 ∇ Z ³ ´ where f0 is the Helmholtz free energy density of the homogenous fluid of density ρ, c is the so-called influence parameter, which is related to the intermolecular potential, and ρ is the gradient of the density. The analysis of GT is greatly simplified by ∇ noting[85] that c is only a weak function of density. Then the influence parameter can be assumed to be constant at constant T . This parameter characterizes the inhomogeneity of the fluid. The density distribution that makes F an extremum is determined by the Euler-Lagrange equation,

µ = µ c 2ρ, (70) 0 − ∇ where µ is the constant chemical potential of the inhomogeneous ﬂuid, and µ0 is the chemical potential of the homogeneous ﬂuid at density ρ. For a planar interface, we have a one-dimensional problem, and the above equation can be written as

d2ρ 1 1 ∂ = (µ µ)= (f0 ρµ) .(71) dx2 c 0 − c ∂ρ − 1 Let ω(ρ)=f0 ρµ, multiply the above equation by dρ/dx,andintegratefrom − 2 to . Then apply the boundary conditions, i.e., ρ( ) ρ and ρ( ) −∞ ∞ ∞ −→ ve −∞ −→ ρle,whereρve and ρle are the equilibrium vapor and liquid densities, respectively. The above equation then reduces to

c dρ = dx , (72) 2 − r ∆ω(ρ) where ∆ω(ρ)=ω(ρ) ω(ρe),ρe is thep equilibrium density of either bulk phase, and − the negative sign indicates that the bulk liquid is located at . To apply the above −∞ equation to a real system, one needs to know the value of the inﬂuence parameter. 38

This can be established by using the experimental surface tension of the planar interface, γ ,tocalculatec from the GT expression for the surface tension[86, 87] ∞

ρle

γ = 2c[ω(ρ) ω(ρve)]dρ .(73) ∞ − ρZ ve p The solution to Eq.(72) is the density profile of a flat interface. Equation (72) can also be used to determine the thickness of the interfacial region by integrating between fixed density limits. A common definition uses ρ =0.1ρl +0.9ρv and − ρ+ =0.1ρv +0.9ρl as the lower and upper limits, respectively, so that the "10—90" interfacial thickness is defined as

ρ+ c dρ t = . (74) 2 ∆ω(ρ) r Zρ − p The primary use for the flat density profile is as the initially guessed profile to solve Eq.(70) for the radial profile at some initial supersaturation value, S ' 1. Assuming the droplet is spherical, Eq.(70) can be written as

d2ρ 2 dρ 1 + = (µ µ) , (75) dr2 r dr c 0 − with the boundary conditions

dρ 0 as r 0 and ρ ρ as r , (76) dr −→ −→ −→ b −→ ∞ where ρb is the density of the initially uniform metastable phase. Solving Eq.(75) under the conditions of Eq.(76), enables one to determine the density profilesofdropletsatdifferentvaluesofS. The numerical procedure used tosolveEq.(75)isbasedonaniterativefinite difference scheme very similar to the one already described by Li and Wilemski[88]. In one scheme S is slowly increased, and the profile from the previous value of S is used as an initial guess. The true profile is obtained by iterating until a predetermined convergence criterion is satis- fied at each point. A significant limitation of this scheme is that, particularly at low temperature, S can be increased by only very small amounts to ensure convergence. 39

An alternative scheme is to apply the foregoing scheme at some relatively high temperature, where convergence is fast. From the high temperature proﬁles generated out to an appropriately large value of S,saySM , one lowers the temperature (by 1

K at high T and by 0.5 K at low T)attheconstantSM value thereby generating a complete set of profiles over the entire temperature range of interest. Each high S profile corresponds to a small droplet size, and this profile serves as the initial guess for the next higher or lower value of S at the given temperature. The most difficult aspect of these calculations is finding converged profiles when the droplet size is relatively large, i.e., for S values not much larger than 1. This calculation is easier at high temperatures where the profiles are much more diffuse, as are the profiles at high S at any T.ConvergedprofilesarefoundmoreeasilyathighS values. With the density profile, all other thermodynamic properties of the droplet can be calculated: Of particular interest are the excess number of molecules in the droplet and the work of formation of the droplet. The reversible work is defined as the difference in the free energy of the system containing the droplet and that of the homogeneous system:

0 W ∗ = [f(ρ) f0(ρ )]dV , (77) − Z 2 0 0 0 where f(ρ)=f0(ρ)+(c/2)( ρ) ,andf0(ρ )=ρ µ P is the Helmholtz free ∇ 0 − energy density of the uniform gas phase. This result is easily shown[88] to be equivalent to the original result of Cahn and Hilliard[40]

c W GT = [∆ω + ρ 2]dV . (78) 2 |∇ | Z The excess number of the molecules in the droplet can be calculated using the following deﬁnition

∞ ∆n =4π [ρ(r) ρ ]r2dr , (79) − b Z0 which is quite general, except for being restricted to spherical droplets. 40

6. RESULTS OF GRADIENT THEORY FOR UNARY NUCLEATION

6.1 WATER AND TIP4P WAT E R 6.1.1 Planar and Droplet Density Profiles from GT . To get a feel for how well these EOSs can describe interfacial properties, the CPHB EOS and the JA EOS were first used with GT to calculate the thickness of planar liquid- − vapor interfaces using Eq.(74) at different temperatures. Among others, Alejandre, Tildesley, and Chapela (AT C)[90] have simulated these interfaces using molecular dynamics (MD)withtheSPC/E potential and full Ewald summation. The ATC simulations provided excellent estimates for the surface tension of water. For this reason, their results were chosen for comparison with the present GT results. From their simulation results, they determined the "10—90" thickness of the interface at various temperatures. Figure 6.1 shows the comparison of the GT results, the MD simulation results, and a few experimental values determined from ellipsometry data[91]. The GT calculations were done using the experimental surface tension to evaluate the influence parameter c.

11 Expt 10 GT/CPHB GT/JA 9 MD-SPC/E

) 7 o

6 t (A 5

3 300 350 400 450 500 550 T (K)

Figure 6.1. The thickness of ﬂat water interfaces at diﬀerent T using GT, MD simulations[90], and experimental data[91]. 41

The results clearly depend on the EOS used, but roughly follow the trend of both the simulations and the experiments. Although the GT results are closer to the experimental values than are the simulation results, absolute agreement is lacking. The larger experimental thicknesses have been attributed to the effects of capillary waves, which are not present in either the GT calculations or the MD simulations. The lack of agreement between the GT results and the simulation results is not surprising since neither EOS precisely represents the model fluid based on the SPC/E potential. Turning now to the droplet calculations, Figures 6.2 and 6.3 show how the density profiles for the CPHB EOS change with temperature. The influence parameter used in these calculations was evaluated from the experimental surface tension at each temperature. Results are given at S =5and 20, respectively, representative low and high supersaturation values. In Figure 6.2, for low S at low T ,thedroplets have a distinct core where the density is essentially constant. As T increases, the extent of the uniform core shrinks steadily until the droplet is practically all interface at 350 K. The steady decrease in size with increasing T can be understood classically in terms of the Gibbs-Thomson (or Kelvin) and Laplace equations. As T increases at fixed S, the increase in the chemical potential difference between the equilibrium and metastable phases is given by ∆µ = kT ln S. Then, the thermodynamic relation, (∂µ/∂P)T =1/ρ, shows that the chemical potential in the droplet can be increased by raising the droplet’s internal pressure, which is governed by the Laplace equation, ∆P =2γ/r. Since the surface tension decreases with increasing T, r must necessarily decrease to provide the required pressure increase. Simi- lar behavior is seen at S =20, but now the droplets are quite diffuseevenatthe lowest value of T . The increase in diffuseness of the droplets at higher S is a conse- quence of increased proximity to the spinodal. In mean-field theory, at the spinodal the nucleus and the mother phase are indistinguishable. Thus, as the spinodal is approached, the nucleus becomes more diffuse with a decreasing core density. 42

S=5

0.06 T(K) 220 0.05 ) 250 3 280 0.04 320 350

0.03

(mol/cm 0.02 ρ

0.01

0.00 0.0 0.5 1.0 1.5 r (nm)

Figure.6.2.Densityproﬁles of water droplets predicted by CPHB at diﬀerent T, for a supersaturation ratio of 5

S=20

0.06 T(K) 220 0.05 250 280

) 320 3 0.04 350

0.03

0.02 (mol/cm

ρ 0.01

0.00 0.0 0.5 1.0 1.5 r (nm)

Figure.6.3.SameasFigure6.2butatS=20 43

Figures 6.4 and 6.5 show density profiles for the CPHB and JA EOS, − respectively, at T =350K for several supersaturation ratios. In contrast to the preceding results, these calculations were made with the influence parameter evaluated from the TIP4P surface tension so they could be used for comparison with the MD results for water nucleation[84]. The JA EOS produces smaller − droplets with higher density cores, whereas the CPHB predicts somewhat larger droplets with less dense cores. In each case, these high T profiles are quite diffuse, i.e., the the flat region in the droplet core is very small. The droplets are practically all interface.

density profiles of CPHB water at T=350K 0.06

0.05 S=5 0.04 S=10 ) 3 S=15 0.03 S=20

0.02 (mol/cm

ρ 0.01

0.00

0.0 0.2 0.4 0.6 r (nm)

Figure 6.4. Density proﬁles of water droplets at T=350 K for diﬀerent values of the supersaturation ratio using the CPHB EOS. 44

density profiles of JA water at T=350K 0.06

0.05 S=5 )

3 0.04 S=10 S=15 0.03 S=20

0.02 (mol/cm ρ 0.01

0.00

0.0 0.2 0.4 0.6 r (nm)

Figure 6.5. Same as Figure 6.4 but using the JA-EOS.

6.1.2 Water Nucleation Rates.BoththeGT and the P form of the CNT − have been used with CPHB EOS to predict nucleation rates of water. The results were compared with the experimental measurements of Wölk and Strey[18]. Figure 6.6 shows the calculated nucleation rates compared with the experimental data. The results of GT have been multiplied by a factor of 100 while those of CNT were divided by 100. Asseenintheﬁgure, GT gives a better temperature dependence for theratethandoestheP formof the CNT.Also,GT gives the right S dependence − at low supersaturation ratios at any T, but it starts to deviate as S increases. It is also seen that the P form of CNT gives the right S dependence, but the predicted − temperature dependence is hardly diﬀerent from that of the usual S form of CNT, − shown in Figure 1.3, for example. This behavior is not surprising because, as shown below, the CPHB EOS does not reproduce the density of supercooled liquid water very accurately. Recall that CREOS-01 and IAPWS-95 were successful because they accurately treated the anomalous compressibility of supercooled liquid water. 45

1010 260K 250K EXP (Wolk-Strey) 240K GT*100 109 P-form/100

) 230K -1 108 s -3

T=220K 107 J (cm 106 H O 2 105 5 10152025 S

Figure. 6.6. Nucleation rate predictions of the CPHB using the P-form and the GT compared to experimental data of Woelk and Strey[18].

It is also curious that the GT rates are roughly a factor of 104 times smaller than the classical values. Typically, DFT or GT rates are higher than classical rates because the nonclassical work vanishes as the spinodal is approached while the classical W does not. Hence at supersaturations close enough to the spinodal nonclassical rates are higher. But far away from the spinodal, this behavior is reversed: nonclassical W’s are larger than classical values, and the rates are correspondingly lower. This behavior is implicit in the recent work of Koga and Zeng[92]. Another way of expressing their results is shown in the following ﬁgure. The Figure shows that the free energy of formation using the GT is bigger than the CNT result for S / 69.6, and smaller at larger S.At220 K,themaximumshiftstoS =21.5 and the point of equality now lies at S>100. This behavior indicates that the spinodal boundary for the CPHB EOS must be very far away from the region of S relevant to the nucleation experiments. It should be noted that the spinodal boundary is dependent on the speciﬁc EOS used and is not unique. 46

1.25 T=260 K (CPHB) 1.20

1.15 P ∆ 1.10 /W

1.05 GT

W 1.00

0.95

0.90

020406080100 S

Figure 6.7. The ratio of the GT work of formation to that of the P-form of CNT as a function of supersaturation ratio at 260 K.

Figure 6.8 shows the excess number of molecules found from GT with the CPHB EOS using Eq.(79) and the experimental number of molecules using Eq.(66) from the Wölk and Strey data. Although the GT predicts the right T dependence of the rates, its predicted S dependence is not as good as that of the P-form. From the nucleation theorem, one can see that the bigger the isothermal slope of ln J vs. ln S is, the higher the number of molecules is predicted. From Figure 6.6 it is clear that the slope of nucleation rates against S is bigger for GT than for the experimental results. This behavior is reflected in the results shown in Figure 6.8. The deficiency of GT in predicting the S dependence is not unexpected. Li and Wilemski found similarly poor behavior for GT in their study of a hard sphere Yukawa fluid[88]. 47

50 H O 2 40 Expt GT-CPHB 30 Gibbs-Thomson * n 20

0 0 1020304050 n* Gibbs-Thomson

Figure 6.8. The number of water molecules in the critical cluster as predicted by the nucleation theorem and the GT calculations. The dashed line represents full agreement with Gibbs-Thomson equation.

6.1.3 TIP4P Water Nucleation. Yasuoka and Matsumoto (YM)[84] investigated the homogeneous nucleation of water using the molecular dynamics technique and the TIP4P water potential at T =350K. They obtained one simulated nucleation 26 3 1 rate with a value of 9.62 10 cm− s− at a supersaturation ratio of 7.3.Itisof × interest to see how this value compares with other theoretical estimates. Thus, GT was used to calculate water nucleation rates with both the CPHB EOS and the JA EOS. Rate calculations have also been made with the P form of CNT − − using the CPHB EOS,theJA EOS,thePeng-RobinsonEOS,andCREOS 01 − − 28 3 EOS. The last EOS used only at S =7.3 and yielded the rate 1.83 10 cm− × 1 s− . In all of these calculations, the value of the surface tension reported by YM for the TIP4P potential, γ =39erg/cm2, was used as needed. The calculated results are compared with the molecular dynamics simulation result in Figure 6.9. From the ﬁgure one can see that there is no advantage in using the GT over the P form of CNT or vice versa. The CPHB predicts higher rates than either − the JA EOS or the Peng-Robinson (PR) EOS. Since, none of the three EOSs − 48 was developed speciﬁcally for TIP4P water, the results are strictly not comparable; nevertheless, it is satisfying to see that the CPHB, PR and JA EOS results − bracket the simulated result of the molecular dynamics.

TIP4P-water at T=350K 1029 CREOS-01 CPHB EOS 1028

) Peng-Robinson EOS -1

27 s 10 -3 MD S-Form P-Form 26 JA EOS GT J (cm 10 MD CREOS-01

5678910 S

Figure 6.9. Nucleation rates for GT and two forms of CNT at T=350 K using diﬀerent EOSs, as shown in the ﬁgure, compared with the MD rate for TIP4P water and the result of the P-form of CNT using CREOS-01.

6.2 COMPARISON OF THE WATER EOS To better understand the reasons for the success or failure of the preceding nucleation rate calculations, it is interesting to see to what degree these four diﬀer- ent models of water agree or disagree in predicting the actual properties of water. 4 For TIP4P , the calculated saturated vapor and liquid densities are 4.66 10− × g/cm3 and 0.9356 g/cm3, respectively, and the pressure at vapor-liquid equilibrium is 0.0753 MPa. Figures 6.10, 6.11, 6.12, and 6.13 compare the various equations with the single TIP4P datum. Note that the results of the IAPWS 95 EOS − may be regarded as the experimental values since this equation describes real water to high precision. 49

1.00

0.95 ) 3 0.90

0.85 CPHB IAPWS-95 (g/cm JA le PR ρ 0.80 MD

0.75 200 250 300 350 400 450 T (K)

Figure 6.10. The predictions of diﬀerent EOSs for the equilibrium liquid density of water at diﬀerent T compared to the experimental data generated using the IAPWS-95.

P=0.1 MPa 190 MPa 300

280

260 T (K)

240

220 960 1000 1040 1080 3 ρ (kg/m )

Figure 6.11. Density of liquid water using the CPHB EOS (stars) at diﬀerent P (0.1, 50, 100, 150, 190 MPa) compared to the experimental data calculated using the IAPWS-95 (circles) 50

1.0x105 IAPWS-95 4 CPHB 8.0x10 JA PR 6.0x104 MD

4.0x104 P (Pa) 2.0x104

0.0 275 300 325 350 375 T (K)

Figure 6.12. The predictions of diﬀerent EOSs for the equilibrium vapor pressure at diﬀerent T compared to the experimental data calculated by using the IAPWS-95

-3 5.0x10 10-6 IAPWS-95 CPHB

-7 JA 10 4.0x10-3 PR MD 10-8

) -3 3 3.0x10 10-9

10-10 2.0x10-3

(g/cm 10-11 220 230 240 250 260 ve ρ 1.0x10-3

0.0 200 250 300 350 400 450 T (K)

Figure 6.13. Same as for Figure 6.12 except for the equilibrium vapor density. 51

Starting with the PR EOS, one can see that this equation predicts fairly accu- − rately the equilibrium vapor pressure and vapor density, but it is severely deficient in predicting the equilibrium liquid density. One shouldn’t be too critical of the PR EOS, because as noted earlier, this equation was developed to predict the − properties of non-polar fluids. For this reason, the PR EOS was not used with − the GT to predict nucleation rates of water. The JA EOS gives generally poor predictions of all properties on the binodal − except for the equilibrium liquid density. It is worth noting that the JA EOS is − capable of accurate predictions of the equilibrium vapor pressure if the either the correct equilibrium vapor or liquid density is supplied independently. It is when the simultaneous calculation of the vapor and liquid binodal densities is attempted that the JA EOS fails. The JA EOS binodal vapor densities are many orders of − − magnitude too low (note particularly the inset of Figure 6.13), and their use would lead to a gross overestimate of the extent of the metastable vapor phase. For this reason, we avoided using this equation with GT to calculate the nucleation rates of water at low T. The CPHB equation shows good agreement with the experimental values of the equilibrium properties over most of the temperature range considered, but it fails to show the liquid density maximum at T 4 ◦C. Moreover, it incorrectly predicts ' a monotonically increasing density with decreasing T at low P . It shows the oppo- site behavior at high P and is generally in poor agreement with the experimental values. These major flawsdisfavortheuseofthisequationatlowT. Despite these flaws, this equation still displays the appropriate mean-field behavior needed for GT calculations, and its quantitative predictions of other water properties are generally acceptable. As a result, it was used for both the low T and high T rate calculations. In conclusion, it should be noted that none of the equations provides a particularly accurate description of real water. This failing is clearly shared by TIP4P water as the figures also show. 52

6.3 RESULTS FOR METHANOL AND ETHANOL Both the GT and the P form of the CNT have been used with the CPHB − EOS to predict nucleation rates of methanol and ethanol. In each case the calculated rates exceed the experimental values by many orders of magnitude. In the case of methanol, Figure 6.14 shows that the GT results are better than the results of the P form of the CNT by about a factor of 300 when compared with the − experimental results of Strey, Wagner, and Schmeling[93]. Similar trends are seen for the ethanol results in Figure 6.15, but here the improvement is by a factor of 500. In each ﬁgure, the scaling factors that reduce the calculated rates were chosen to force rough agreement at the higher temperature.

Methanol 109 T=272 K T=257 K

108 ) -1

s 7

10 -3

106 10 J (cm J /10 P J Expt 7 J /3*10 GT 105 2.2 2.4 2.6 2.8 3.0 3.2 S

Figure 6.14. Experimental nucleation rates of methanol compared to the predictions of GT and the P-form of CNT with the CPHB EOS.

It should be noted that neither theory reproduces the experimental temperature dependence, and the reason might be related to the inaccurate predictions of the CPHB EOS for the density at diﬀerent temperatures and high pressures. Figure 6.16 shows such predictions for the CPHB EOS as a function of temperature compared to experimental data[94] at diﬀerent pressures. 53

Ethanol 1010 T=286 K

9 T=293 K 10 )

-1 8

s 10 -3

107 Jexpt J (cm J /5*107 6 P 10 J /105 GT

2.4 2.5 2.6 2.7 2.8 2.9 3.0 S

Figure 6.15. As in Figure 6.14 but for ethanol.

Ethanol 0.88 T=273 K T=283 K 0.86 T=293 K T=303 K

) T=313 K 3 0.84

0.82 (g/cm

ρ 0.80

0.78 -5 0 5 10 15 20 25 30 35 40 P (MPa)

Figure 6.16. Liquid ethanol density vs. P at diﬀerent temperatures using the CPHB EOS (open symbols) and experimental data (solid symbols). 54

The experimental results suggest that the incompressible droplet approximation might not be so unreasonable for ethanol, although the density profiles for ethanol (and methanol) droplets under these conditions look qualitatively similar to those of water droplets. The glaring inconsistency between the CPHB and experimental ethanol densities suggests that the P form calculations based on fits to the − experimental density and Eq.(23) would not be in good agreement with the CPHB results. This is confirmed in Figure 6.17 which compares the two sets of P form − results. It is not clear why the S form results, which assume an incompressible − droplet, have the same T dependence as the CPHB P form results. It is most − likely accidental agreement in the small T range examined.

1010 Ethanol T=293 K T=286 K

109

) 108 -1 s -3

107

P-form/5*107 (CPHB)

J (cm 6 10 S-form/5*106 P-form/107 (fitting) Jexpt 105 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 S

Figure 6.17. Experimental nucleation rates of ethanol compared to calculated rates using the S-form and the P-form of CNT with the CPHB EOS and the P-form of CNT using ﬁtted experimental density data[94].

In contrast, for methanol the P form results based on ﬁts to the experimental − density and Eq.(23) were identical to those calculated using the CPHB EOS.This was expected because liquid methanol densities given by the CPHB EOS as a function of pressure agree well with the experimental values[95]. 55

7. BINARY NUCLEATION THEORY

7.1 CLASSICAL NUCLEATION THEORY Reiss generalized the theory of binary nucleation by including a kinetic mechanism that allows for cluster growth and decay in the two-dimensional space compris- ing the number of molecules, n1 and n2, of each species in the cluster[44]. The major diﬀerence between unary and binary nucleation is that in the latter the critical cluster is distinguished not only by its size, but also by its composition (xi = ni/( ni)). For the Gibbs free energy of a binary cluster, Reiss used the capillarity approxima-P tion and found the following approximate result,

2 ∆G = n1∆µ1 + n2∆µ2 +4πr γ, (80) where ∆µi is the chemical potential diﬀerence for a molecule of species i in the vapor and liquid phases at the same pressure. The free energy surface represented by this ∆G contains a saddle point (sometimes more than one), which functions much as a pass through a ridge of mountains. That is, the saddle point is the lowest point on a free energy ridge that separates small, unstable clusters from larger, stable (actually, growing) fragments of the new phase. The critical droplet is located at the saddle point, i.e., where the ﬁrst derivatives of ∆G with respect to n1 and n2 are zero. In this case, Doyle obtained the following forms for the Gibbs-Thomson equations[45]

2γv 3xv dγ ∆µ + 1 m =0, (81a) 1 r − r dx 2γv2 3(1 x)vm dγ ∆µ + + − =0, (81b) 2 r r dx whichmustbesolvedforx and r. Here, x isthemole(ornumber)fractionofthe second component of the ﬂuid, and vm is the mean molecular volume. The mean molecular volume is related to the partial molecular volume, vi,ofeachcomponent through the relation vm =(1 x)v1 + xv2. Note that, in general, all of the − thermodynamic properties of binary mixtures depend on the composition of the mixture. 56

Under the assumption that the liquid is incompressible, the exact thermodynamic relation of the pressure and the chemical potential, ∂µ i = v , i =1, 2 (82) ∂P i µ ¶T,ni readily integrates to the result

∆µ = vi(Pl Pv) . (83) i − Combining Eqs.(81) and (83) with the Laplace equation for the pressure diﬀerence, one can show that vmdγ/dx =0must be true[96]. Since, in general the surface tension depends on composition, this result must be invalid, and it suggests that Doyle’s versions of the binary Gibbs-Thomson equations are wrong. Following another line of reasoning stimulated by the work of Renninger, Hiller, and Bone[46], Wilemski[97] proposed a revised thermodynamic cluster model that led to the classical Gibbs-Thomson equations, in which the surface tension derivatives are missing

2γv ∆µ + 1 =0 , (84a) 1 r 2γv ∆µ + 2 =0 .(84b) 2 r These two equations can be combined to obtain the following relation

∆µ ∆µ 1 = 2 ,(85) v1 v2 whose solution yields the so-called bulk, or interior, composition of the cluster. All of the properties of the critical nucleus can be evaluated using this composition[47]. For many binary systems the composition dependence of the surface tension is very weak and dγ/dx 0. These two different methods for finding the critical ≈ nucleus composition then produce very similar results. However, for systems whose surface tension varies strongly with x, such as water-alcohol systems, the two methods give very different results, and each approach suffers from a serious deficiency, as described in Section 1.2. A third approach is possible if one evaluates the critical composition and critical work of formation using Gibbs’s fundamental conditions of (unstable) equilibrium. 57

Assume the metastable binary system is at a total vapor pressure Pv and temperature T , with a vapor mole fraction yi. Then the properties of the liquid reference phase that represents the critical nucleus are determined by solving the following equations simultaneously

µiv(T,Pv,yi)=µil(T,Pl,xi) , (86) where µiv and µil are the vapor and liquid chemical potentials of component i,and yi and xi are, respectively, the vapor and liquid mole fractions of component i.As- suming that the surface tension of the droplet is the same as that of a macroscopic liquid mixture with composition xi, the radius of the critical droplet can be determined from the Laplace equation (see Appendix A.2) and the pressure diﬀerence

Pl Pv. − As in unary nucleation, if an EOS (introduced in the next chapter) is known for the system, one can apply Eq.(86) to determine the internal reference pressure Pl. The free energy of formation at the critical radius can then be evaluated from the relation 3 16πγ∗ ∆G∗ = W ∗ = 2 , (87) 3(Pl Pv) − which is formally identical to the unary result. We call this version 1. Applying Eqs.(84) at the critical radius, it can be shown easily that

2γ∗vm∗ r∗ = , (88) −(1 x )∆µ + x ∆µ − ∗ 1 ∗ 1 where denotes a property evaluated at the critical nucleus composition. The ∗ critical free energy of formation, then follows from Eq.(80) as

16πγ 3v 2 W = ∗ m∗ . (89) ∗ 3((1 x )∆µ + x ∆µ )2 − ∗ 1 ∗ 2

We will call this version 2 when x∗is determined using the classical Gibbs-Thomson equations of Wilemski’s revised model, and we call it version 3 when Doyle’s form of the Gibbs-Thomson equations, Eqs.(81) are used.

In experiments, the main independent variable is known as the vapor activity, ai, which is deﬁned as the ratio of the partial vapor pressure of component i, Piv,to 58

0 0 the equilibrium vapor pressure of its pure liquid, Pi , i.e., ai = Piv/Pi . One can also eq eq deﬁne the supersaturation ratio of component i as Si(x)=Piv/Pi (x),wherePi (x) is the equilibrium partial vapor pressure of component i over the binary solution of composition x.

If the vapor is assumed to be ideal, one can write ∆µi = kT ln(ai/ail)=kT ln Si, eq 0 where ail = Pi (x)/Pi is the liquid activity of component i.Thentheworkof formation can be written as

3 2 16πγ∗ vm∗ W ∗ = 2 , (90) 3(kT ln S∗) with ln S∗ =(1 x∗)lnS1(x)+x∗ ln S2(x). Thisformisverypopularbecauseitis − expressed in terms of conveniently measured properties. In Chapter 9, results of these three classical versions are compared for a model binary fluid whose properties resemble those of an ethanol-water mixture. The model fluid approach was used because there are no accurate EOSs for the most interesting binary systems that show surface segregation or enrichment. Since the results are strictly not comparable with experiment, density functional theory was used to calculate the most important thermodynamic property of the model fluid, namely, its surface tension. The other properties were obtained from the mean field EOS for the system. A binary hard sphere—Yukawa mixture was used as the basis for the model fluid. 7.2 DENSITY FUNCTIONAL THEORY (DFT) Zeng and Oxtoby[43] developed an approximate density functional theory for binary nucleation in which the functional for the Helmholtz free energy takes the form 1 2 F [ρ (r),ρ (r)] = drfh [ρ (r),ρ (r)] + drdr0φ ( r r0 )ρ (r)ρ (r0) . 1 2 1 2 2 ij | − | i j Z i,j=1 ZZ X (91) where fh is the Helmholtz free energy density of a uniform hard sphere mixture and

φij is the perturbed attractive part of the potential. The grand potential, Ω,ofthe system is 2

Ω [ρ1(r),ρ2(r)] = F [ρ1(r),ρ2(r)] µi drρi(r) , (92) − i=1 X Z 59

th where µi is the (constant) chemical potential of the i component in the system. The equilibrium droplet density proﬁles can be generated by applying the conditions

∂Ω/∂ρi =0,or∂F/∂ρi = µi, and solving the resulting Euler-Lagrange equations that read 2

µi = µih [ρ1(r),ρ2(r)] + dr0φij( r r0 )ρj(r0) , (93) j=1 | − | X Z where µih is the hard sphere chemical potential. The free energy density of the homogeneous ﬂuidcanbederivedfromEq.(91)by taking the limit of uniform densities. This yields 1 2 f(ρ ,ρ )=f (ρ ,ρ ) α ρ ρ , (94) 1 2 h 1 2 2 ij i j − i,j=1 X where

αij = drφ (r) .(95) − ij Z The pressure and the chemical potential of the homogeneous ﬂuid are then readily derived from Eq.(94), and they are 1 2 P = P (ρ ,ρ ) α ρ ρ , (96) h 1 2 2 ij i j − i,j=1 X and 2

µi0 = µih(ρ1,ρ2) αijρj ,(97) − j=1 X where Ph is the pressure of the binary hard sphere mixture. 7.3 SURFACE TENSION AND REVERSIBLE WORK An expression for the surface tension of a planar interface can be obtained by using the deﬁnition of the grand potential, i.e., Ω = PV + γA,whereV is the − system volume and A is the area of the interface. If we substitute Eq.(97) into Eq.(92), it is shown in Appendix D that 1 1 γ = ρ (x)[µ (ρ1,ρ2) µ ]+ ρ (x)[µ (ρ ,ρ ) µ ]+P Ph dx . (98) 2 1 1h − 1 2 2 2h 1 2 − 2 − Z ½ ¾ Another important relation is the work of formation of a spherical droplet, which is derived in Appendix D as

1 1 2 Wrev =4π ρ (r)[µ (ρ1,ρ2) µ ]+ ρ (r)[µ (ρ ,ρ ) µ ]+P Ph r dr . 2 1 1h − 1 2 2 2h 1 2 − 2 − Z ½ ¾ (99) 60

7.4 DFT FOR THE HARD SPHERE—YUKAWA FLUID In this thesis, the model ﬂuid is a binary hard sphere—Yukawa mixture. The Yukawa potential, 3 αijλ exp( λr) φ = − , (100) ij − 4πλr hasbeenchosenastheattractivepartofthepotential. Thereisamajoradvantage of using this particular potential that arisesfromitsstatusastheGreen’sfunctionof the Helmholtz equation. By acting with 2 on the coupled integral Euler-Lagrange ∇ equations, Eq.(93), we transform them into two coupled diﬀerential equations,

d2µ 2 dµ 2 ih + ih = λ2 µ (ρ ,ρ ) µ α ρ , (101) dr2 r dr ih 1 2 i ij j " − − j=1 # X that are much simpler to solve numerically.

Further simpliﬁcations are possible. For the particular choice, α12 = √α11α22, the so-called Bertholet mixing rule, one can combine the equations for the chemical potentials, Eq.(93), in a single formula, µ (ρ ,ρ ) µ µ (ρ ,ρ ) µ 1h 1 2 − 1 = 2h 1 2 − 2 . (102) √α11 √α22 With this expression, Eq.(98) for the surface tension simpliﬁes to

ρil 1 2 1/2 dµih γ = (µih(ρ1,ρ2) µ1) 2αii(Ph P) dρi , i =1or 2 , λαii − − − dρ Zρiv µ i ¶ £ ¤ (103) where ρiv and ρil are the equilibrium vapor and liquid densities of component i. Note that this equation does not require the actual density proﬁle in contrast to the earlier Eq.(98), although one should be careful about choosing which component to integrate over. The smart choice is to pick the component whose density varies monotonically through the interface[99]. A derivation of Eq.(103) is given in Appendix D.

Since Eq.(102) defines ρ1 as a function of ρ2 everywhere in the system, only one of the differential Euler-Lagrange equations, Eq.(101), needs to be solved. This further reduces the amount of computational work required. Please note that all the above equations will reduce to those of a unary fluid by setting either ρ1 or ρ2 to zero. 61

8. PROPERTIES OF THE MODEL BINARY HARD-SPHERE YUKAWA (HSY) FLUID

8.1 EQUATION OF STATE In order to approximate the properties of real ﬂuids using the hard sphere—

Yukawa (HSY ) ﬂuid, one has to evaluate the hard core diameters σi and the

αij parameters to produce the right surface tensions and vapor pressures. This can be achieved by choosing carefully the critical temperatures of each component, 3 this choice is implemented using the scaled αii parameters, αii = αii/(kTσi )=

11.1016Tci/T ,whereTci is the critical temperature of component i. This choice of e e αii has been used previously[42, 100]. All the parameters in this chapter were scaled using the following rules:

3 2 3 µi = µi/kT , P = Pσ1/kT , γ = γσ1/kT , ρi = ρiσ1 (104) 3 3 f = fσ /kT, σ = σ2/σ1, vi = vi/σ . (105) e 1 e e 1 e

All the lengthse are scaled with respect to σe1. To optimize the HSY ﬂuid properties to resemble those of the water—ethanol system, the following parameters have been estimated to produce the right surface tension of the pure components: λ =0.709 1 ◦ − ◦ ◦ A , σ1 =3A, σ2 =4A, Tc1 =610K, Tc2 = 544 K,where1 and 2 refer to water and ethanol, respectively. Accordingtotheabovescalingrules,theEOS , i.e., Eq.(96) appears in dimensionless variables as 1 2 P = P (ρ , ρ ) α ρ ρ , (106) h 1 2 2 ij i j − i,j=1 X where e e e e e e e 6 ξ 3ξ ξ + ξ3 2ξ3 P = 0 + 1 2 2 + 2 , (107) h π 1 ξ (1 ξ )2 (1 ξ )3 · − 3 − 3 − 3 ¸ i with ξi = π (ρ1 + ρ2(eσ2/σ1) ) /6. The chemical potential of each component can be derived from the free energy density using µi = ∂f/∂ρi,where e e 1 2 f = f α ρ ρ , (108) h 2 ij i j − i,j=1 X e e e e e 62 and the Helmholtz free energy density of a hard sphere mixture, 3 3 ξ2 3ξ1ξ2 ξ2 fh = ρ ln ρ + ρ ln ρ + 1 ln(1 ξ )+ + , (109) 1 1 2 2 ξ ξ2 − − 3 ξ (1 ξ ) ξ ξ (1 ξ )2 µ 0 3 ¶ 0 − 3 0 3 − 3 ise givene bye the binarye e Carnahan-Starling equation of Mansoori et al.[101] At a given temperature T and liquid composition x, the coexisting vapor liquid densities are determined by solving the following equations simultaneously

µ1v(ρ1,ρ2)=µ1l(ρ1,ρ2) , (110)

µ2v(ρ1,ρ2)=µ2l(ρ1,ρ2) , (111)

Pv(ρ1,ρ2)=Pl(ρ1,ρ2) , (112)

x2 = ρ2/(ρ1 + ρ2) . (113)

After solving the above equations, one can produce the whole equilibrium phase diagram. The physical properties shown in Figures 8.1-8.5 were also generated, because they are needed to calculate the work of formation using CNT.The components of the model ﬂuid are referred to as p-water and p-ethanol, where "p" stands for pseudo, because of their resemblance to real water and ethanol.

T=260 K 4.0x10-4

3.0x10-4

/kT -4

3 2.0x10

1 σ P

-4 P 1.0x10 ve P1 P2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 x

Figure 8.1. The total and partial equilibrium vapor pressures of the HSY model ﬂuid at T=260 K versus mixture composition, x. 63

Figure 8.1 shows how the equilibrium partial vapor pressures vary with composition, and it also shows the total vapor pressure. Although the absolute magnitudes of the pure vapor pressures are too high by factors of 120 and 70 for water and ethanol, respectively, the qualitative behavior is quite similar to that of the water— ethanol system. Also, the ratio of the calculated equilibrium vapor pressures of the pure components (p-water to p-ethanol) is 0.64 compared to 0.495 for the real system. Figure 8.2 shows the vapor-liquid equilibrium phase diagram as a function of the p-ethanol composition. The azetropic composition is realistic for water-alcohol systems.

-4 T=260 K 4.0x10 liquid

3.5x10-4 /kT 3 1 -4

σ vapor

3.0x10 ve P

2.5x10-4

2.0x10-4 0.0 0.2 0.4 0.6 0.8 1.0 x

Figure 8.2. P-x phase diagram of the binary HSY model system.

We have seen earlier that the composition dependence of the bulk surface tension is a key ingredient in classical binary nucleation theory. Figure 8.3 shows the variation of the calculated surface tension using Eq.(103) with the p-ethanol composition compared to the measured surface tension by Viisanen et al.[102]. The calculated values (in mN/m) of the pure components, 74.42 for p-water and 25.08 for p-ethanol, are close to the experimental values, 77.45 and 25.04, and the trend 64

captures the desired behavior, the steep decline at small x2, quite nicely.

T=260 80 DFT 70 Expt

(mN/m) 40 γ

20 0.0 0.2 0.4 0.6 0.8 1.0 x

Figure 8.3. Surface tension for the pseudo water-ethanol system and measured values for water-ethanol versus ethanol mole fraction, x.

Figures 8.4 and 8.5 show the variation of the partial molecular volumes with the p-ethanol composition. The partial molecular volumes have been evaluated at the usual constant pressure, 1 atm, using the deﬁnitions (∂V/∂n ) = v ,and i P,nj i

ρi = ni/V ,whereni is the number of molecules of component i and V = n1v1 +n2v2 is the total volume. The following expression was derived from Eq.(106):

∂Ph/∂ρi (α1iρ1 + αi2ρ2) vi = − . (114) 2(P Ph)+ρ ∂Ph/∂ρ + ρ ∂Ph/∂ρ − 1 1 2 2 The partial molecular volumes show virtually no dependence on composition, in contrast with the behavior of real water-alcohol systems. 65

1.070 T=260 K

1.065

1.060 3 1 σ /

1 1.055 v

1.050

1.045

0.0 0.2 0.4 0.6 0.8 1.0 x

Figure 8.4. Variation of the partial molecular volume of p-water with composition.

T=260 K 2.67

2.66

3 2.65 1 σ /

2 2.64 v

2.63

2.62

2.61 0.0 0.2 0.4 0.6 0.8 1.0 x

Figure 8.5. Same as Figure 8.4. but for p-ethanol. 66

8.2 FITTED PROPERTY VALUES In order to calculate the work of formation at the critical cluster using version 1 from Eq.(87), one needs the use of the EOS and the value of the surface tension at the critical composition x∗.AnEOS is not used to predict the work of formation using version 2,andversion 3. In these versions, one solves either the Eqs.(84) for version 2 or Eqs.(81) for version 3.Inordertofind x∗ and to evaluate W ∗ in the traditional manner of binary CNT, i.e., without using a full EOS, one needs eq eq to have all the physical properties, γ,v,P1 ,P2 , as functions of composition. ln γ was correlated by fitting ln γ to a polynomial of seventh order as a function of e 7x/(1+6x). The molecular volume, v =(1 x)v1 +xv2 was fitted such that both v1 − eq eq and v2 are polynomials of fourth order in x, while P1 ,P2 are written as a function of the activity coefficients γi as

eq 0 Pi = Pi xiγi . (115)

In this form the γi are chosen to follow the van Laar equations: A ln γ = , (116) 1 (1 + A(1 x)/Bx)2 − B ln γ = . (117) 2 (1 + Bx/A(1 x))2 − For the pseudo water-ethanol system, the ﬁtting parameters A and B are 1.0481 and 2.68438, respectively. The corresponding experimental values at 260 K are 0.92 and 1.47 for water-ethanol[102] and 1.313 and 2.3652 for water-propanol[103]. The

ﬁtted equations of v1, v2,andγ are in dimensionless units

2 3 4 v1 =1.06717 0.00263x 0.08125x +0.09417x 0.03174x − − − 2 3 4 v2 =2.61886 + 0.19638x 0.29593x +0.21051x 0.05972x e − − ln γ =1.40523 3.71918y +15.09805y2 54.43205y3 +114.48491y4 e − − 128.96306y5 +74.21279y6 17.8y9 (118) e − − where y =7x/(1 + 6x). 67

9. RESULTS OF THE HSY BINARY FLUID

9.1 CRITICAL ACTIVITIES AT CONSTANT W∗ In this thesis, we compare the various theoretical results in the form of a critical vapor activity plot at T =260K for a constant value of W ∗, W ∗/kT =40.This 7 3 1 valuewillproduceanucleationrateoftheorder10 cm− s− .Figure9.1shows how a2 varies with a1for the diﬀerent versions of binary CNT and also includes the DFT results.

4.0 * DFT 3.5 W /kT=40 Version 1 T=260 K Version 2 3.0 Version 3 2.5

2 2.0

a 1.5

1.0

0.5

0.0 01234567 a 1

Figure 9.1. Critical activities of p-water (1) and p-ethanol (2) needed to produce a constant work of formation of 40 kT.

It is well known that for binary systems with a highly surface active component, the critical activity curves calculated using the CNT (version 2) exhibit unphysical behavior[47, 104, 105]. This version predicts that at the same activity a2,different activities of a1 will produce the same value of the work of formation. Such behavior has been produced again in the pseudo water-ethanol system. Version1 also predicts the same behavior as the standard version 2 calculations, but it lowers the magnitude of the discrepancy slightly. The close correspondence of version 1 and version 2 is not surprising, because the classical Kelvin equations, Eqs.(84), are 68 easily derived from the fundamental Gibbs condition, Eq.(86), using the assumption of an incompressible fluid. The predictions of both version 1 and version 2 show a contradiction with the nucleation theorem[106], which states in mathematical form that the work of formation has to have a negative slope with increasing activity of either component. As shown earlier in the thesis, the P form,orwhatwecalledit − here the version 1 is an exact formula except for the approximation of the droplet surface tension as that of the flat interface. The major reason for the failure of version 1 and version 2 is probably the use of the bulk surface tension which forces the highly curved droplet surface to implicitly assume the composition of a flat interface. Models such as the explicit cluster model[107] that relax this requirement, but otherwise use the same ingredients as the classical CNT, produce physically realistic behavior in reasonable agreement with experiment. A second contributing factor might refer to nature of the EOS, in that this equation probably does not realistically capture the isothermal compressibility behavior of a real alcohol-water mixture. The behavior of version 3 is quite similar to that predicted for real water—alcohol systems. While no unphysical behavior is predicted, the p-ethanol activities rapidly drop to unrealistically low values as the p-water activity increases to rather modest values. It is interesting that the drop-off region for version 3 mirrors the overshoot region of version 1 and 2. The figure also shows several points generated by using DFT to predict the work of formation. It is quite obvious that the DFT improves the results significantly. It is not unexpected that the results of the DFT willdoabetterajobovertheCNT regarding many aspects as the temperature dependence (which is not shown here) and activity dependence. Also note that the results of the DFT show systematic agreement with the nucleation theorem, which is another of the major advantages of using the DFT. Similar results have been found previously by Napari and Laaksonen using DFT with models based on site-site Lennard-Jones potentials[54, 108]. 69

9.2 NUMBER OF MOLECULES IN THE CRITICAL DROPLET Because it gives information about the composition and size of the critical nucleus, a very interesting piece of information to evaluate is the number of molecules of each component in the critical droplet. Figures 9.2 and 9.3 show how the numbers of molecules vary as a function of the p-water activity. Figure 9.2 shows the predictions of version 1 and version 2 of CNT, which are in very close agreement, and also shows the DFT calculations. Figure 9.3 shows the predictions of version 3 with the predictions of the DFT. Figure 9.4 compares the results for versions 1 and 2, which are essentially identical, with those of version 3 in the region a1 < 1, where the critical activity curves for the differentversionsdonotdeviategreatly.Itshouldbenotedthattheresults are not directly comparable because version 3 gives the total numbers of each type of molecule in the droplet, whereas, versions 1 and 2 give only the numbers in the bulk core of the droplet. For this highly surface enriched system, one would expect differences between the total numbers and the core numbers. Thus, the differences between the different versions are not too surprising.

100 version 1 version 2 80 DFT

60 n 1 n

20 n 2 0 0123456 a 1

Figure 9.2. The number of molecules of each component of the critical droplet as a function of the p-water activity using version 1 and version 2 of the CNT,as well as the DFT. 70

T=260 K 100

60 n (version 3) n * 1 1 n (version 3) n 2 40 DFT

20 n 2 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 a 1

Figure 9.3. The number of molecules of each component of the critical droplet as a function of the p-water activity using version 3 of the CNT and the DFT.

T=260 K

100 n 2

version 1 60 * version 2

n version 3 40

20 n 1 0

01 a 1

Figure 9.4. The number of molecules of each component of the critical droplet as a function of the p-water activity using versions 1, 2, and 3 of the CNT. 71

10. CONCLUSIONS

Generally, in assessing how well the CNT predicts the experimental results for unary systems, the P-form was ignored in all previous studies due to lack of suitable EOSs. Instead, the assumption of the incompressibility of the liquid droplet was made to simplify the analysis. In this thesis, the P-form of CNT has been studied on several different substances for the first time. This version of CNT proved fairly successful inpredicting the nucleation rates of water and heavy water using accurate EOSs. Compared to the more approximate versions of CNT, the P-form shows major improvements regarding the T dependence, as well as the S dependence, of the rates for water and heavy water. The highly compressible nature of supercooled water is clearly the most important factor in understanding water behavior, and this feature is accurately described in the EOSs that gave the most successful predictions. On the basis of the water results, the P-form of CNT was expected to predict better results than the other CNT forms for the nucleation of alcohols. Unfortu- nately, this expectation was not realized. The disappointing results for ethanol and methanolmightbeduetotheinaccurateCPHBEOSusedtodescribealcohols, or because of inaccurate experimental density measurements, or simply because alcohols are much less compressible than water or heavy water. Water nucleation has been also studied for the firsttimeusinggradienttheory with the CPHB EOS. The results of this theory show an improvement over the standard CNT regarding the T dependence. However, these GT results also show a somewhat poorer prediction for the S behavior at high values of S. Also, it was found that the predicted rates for GT are lower than the predictions of the CNT, contrary to what is usually found for a nonclassical theory. This behavior is understandable in terms of the spinodal location predicted by the EOS: if the relevant supersaturations are far away from the spinodal, the nonclassical work of formation is actually larger than the classical value. The GT, with the CPHB EOS, also gave improved results compared to the CNT forms for the nucleation rates of ethanol and methanol. A final area of study concerned the novel application of the P-form of CNT to binary nucleation theory. The goal of this study was to see if using an EOS 72 could eliminate nonphysical behavior found to arise in one of the standard versions of the theory. To carry out this study, a simple model of a highly surface active system, with properties similar to ethanol-water or propanol-water mixtures, was devised and analyzed using DFT. The properties of this model fluid were employed in conventional calculations with the different version of classical binary theory. The P-form of the binary CNT failed to significantly improve the predicted critical activity curve. The other (Doyle) version gave unrealistic results similar to those found for alcohol-water systems. The DFT results were both realistic and physically sound, and they obviously constitute a significant improvement over classical binary nucleation theory. 73

APPENDIX A IMPORTANT THERMODYNAMIC RELATIONS 74

APPENDIX A IMPORTANT THERMODYNAMIC RELATIONS

A.1. CHEMICAL POTENTIAL DIFFERENCE IN THE IDEAL GAS LIMIT To obtain Eq.(21) in Sec. 2.1, start from the general thermodynamic relation

∂µ l = v , (119) ∂P l where vl is the partial molecular volume. Then apply this relation to an incompressible liquid droplet for which vl is independent of pressure, and integrate Eq.(119) over the pressure range Pve to Pv to get

µ (Pv)=µ (Pve)+vl(Pv Pve) . (120) l l − The deﬁnition of ∆µ is

∆µ = µ (Pl) µ (Pv) , (121) l − l and with the substitution of Eq.(120) this becomes

∆µ = µ (Pl) µ (Pve) vl(Pv Pve) , (122) l − l − −

At unstable equilibrium, we have µl(Pl)=µv(Pv), and at bulk two-phase equilibrium, we have µl(Pve)=µv(Pve). With these two identities, Eq.(122) can be written as

∆µ = µ (Pv) µ (Pve) vl(Pv Pve) . (123) v − v − −

In the ideal gas limit µ (Pv) µ (Pve)=kT ln S ,then∆µ ﬁnally reduces to v − v

∆µ = kT ln S vl(Pv Pve) . (124) − −

The term vl(Pv Pve) is negligible except at extremely high supersaturation values − S 106,whicharevirtually unattainable. ≥ A.2. THE LAPLACE EQUATION Assuming a container contains a gas of total volume V , and total number of molecules N. Assume also a spherical droplet has been formed of volume Vl with 75

radius r with number of molecules nl. The gas will have a chemical potential µv, volume Vv = V Vl, and number of molecules nv = N nl ns,wherens is the − − − surface excess number of molecules. For simplicity, the Gibbs surface of tension dividing surface will be adopted[109]. The droplet molecules will have a chemical potential µl. The total Helmholtz free energy of the system containing the droplet is

F = Pv(V Vl) PlVl + γA +(N nl ns)µ + nlµ + nsµ , (125) − − − − − v l s at equilibrium, dF =0,then

Pv(dV dVl) PldVl + γdA + µ (dN dnl dns)+µ dnl + µ dns − − − v − − l s

(V Vl)dPv VldPl + Adγ +(N nl ns)dµ + nldµ + nsdµ =0 (126) − − − − − v l s Using the constraints of constant total volume and total number of molecules, i.e., dV = dN =0, and employing the constant T Gibbs-Duhem identities (VvdPv = nvdµ , VldPl = nldµ , Adγ = nsdµ ), the above equation becomes v l − s

(Pv Pl)dVl + γdA +(µ µ )dnl +(µ µ )dns =0. (127) − v − l v − s

To maintain equilibrium for an arbitrary virtual variations dnl and dns, the equilibrium condition, µv = µl = µs,mustbesatisﬁed. This leaves the remaining equation

(Pv Pl)dVl + γdA =0. (128) −

2 Assuming the droplet is spherical, we have dVl =4πr dr and dA =8πrdr,thenthe above equation reduces to 2γ Pl Pv = , (129) − r which is known as the Laplace equation. A.3. THE WORK OF FORMATION Subtract Eq.(12) from Eq.(125) to obtain

∆F = (Pl Pv)Vl + γA + nl(µ µ )+ns(µ µ ) , (130) − − l − v s − v and apply the equilibrium condition, µv = µl = µs,toobtain

Max ∆F = (Pl Pv)Vl + γA , (131) − − 76 which equals Eq.(15) for a spherical droplet.

gas A, µ ,n s s

Liquid r

v,µ ,n l l l

V-V, µ , N-n-n l v l s

0 0

Figure A.1. Schematic depiction of a spherical critical nucleus in a metastable gas phase.

A.4. THE GIBBS-THOMSON EQUATION For a quick route to the Gibbs-Thomson equation, note that if the droplet is incompressible then Eq.(119) integrates to

∆µ = µ (Pl) µ (Pv)= vldP =(Pl Pv)vl . (132) l − l − ZPv

As we showed in Appendix A.1.,

∆µ = kT ln S (133)

to a very good approximation. Then if ∆P = Pl Pv is replaced with Laplace’s − equation, Eq.(129), we get

2γv ∆µ = kT ln S = l , (134) r which is known as Gibbs-Thomson or Kelvin equation. More general derivations of Laplace’s formula and Gibbs-Thomson equation are available[109, 110]. 77

APPENDIX B DETAILS OF VARIOUS EQUATIONS OF STATE 78

APPENDIX B DETAILS OF VARIOUS EQUATIONS OF STATE

B.1. IAPWS-95 The equation formulated by Wagner and Pruss[61] was adopted by The Interna- tional Association for Properties of Water and Steam, who released it in 1996. Here, it will be referred to as IAPWS 95. The equation represents all the thermody- − namic properties for water over the range of available experimental data down to T =234K.Moreinformationaboutthisequationisavailableatwww.IAPWS.org.. Table B.1 gives all the numerical values of the coeﬃcients in the ideal gas part, while TablesB.2andB.3showallthecoeﬃcients and parameters of the residual part.

Table B.1. The coeﬃcients values of the ideal gas part.

0 0 0 0 ini γi ini γi 1 8.32044648201 0.050.97315 3.53734222 − 26.6832105268 0.061.27950 7.74073708 33.00632 0.070.96956 9.24437796 40.012436 1.28728967 8 0.24873 27.5075105 79

Table B.2. The coeﬃcients and parameters of the residual part.

ici di ti ni 1 1 0 1 0.5 0.12533547935523 10− − × 2 0 1 0.875 0.78957634722828 101 × 3 0 1 1 0.87803203303561 101 − × 4 0 2 0.5 0.31802509345418 5 0 2 0.75 0.26145533859358 − 2 6 0 3 0.375 0.78199751687981 10− − × 2 7 0 4 1 0.88089493102134 10− × 8 1 1 4 0.66856572307965 − 9 1 1 6 0.20433810950965 4 10 1 1 12 0.66212605039687 10− − × 11 1 2 1 0.19232721156002 − 12 1 2 5 0.25709043003438 − 13 1 3 4 0.16074868486251 1 14 1 4 2 0.40092828925807 10− − × 6 15 1 4 13 0.39343422603254 10− × 5 16 1 5 9 0.75941377088144 10− − × 3 17 1 7 3 0.56250979351888 10− × 4 18 1 9 4 0.15608652257135 10− − × 8 19 1 10 11 0.11537996422951 10− × 6 20 1 11 4 0.36582165144204 10− × 11 21 1 13 13 0.13251180074668 10− − × 9 22 1 15 1 0.62639586912454 10− − × 23 2 1 7 0.10793600908932 − 1 24 2 2 1 0.17611491008752 10− × 25 2 2 9 0.22132295167546 80

Table B.2. continued

ici di ti ni 26 2 2 10 0.40247669763528 − 27 2 3 10 0.58083399985759 2 28 2 4 3 0.49969146990806 10− × 1 29 2 4 7 0.31358700712549 10− − × 30 2 4 10 0.74315929710341 − 31 2 5 10 0.47807329915480 1 32 2 6 6 0.20527940895948 10− × 33 2 6 10 0.13636435110343 − 1 34 2 7 10 0.14180634400617 10− × 2 35 2 9 1 0.83326504880713 10− × 1 36 2 9 2 0.29052336009585 10− − × 1 37 2 9 3 0.38615085574206 10− × 1 38 2 9 4 0.20393486513704 10− − × 2 39 2 9 8 0.16554050063734 10− − × 2 40 2 10 6 0.19955571979541 10− × 3 41 2 10 9 0.15870308324157 10− × 4 42 2 12 8 0.16388568342530 10− − × 1 43 3 3 16 0.43613615723811 10− × 1 44 3 4 22 0.34994005463765 10− × 1 45 3 4 23 0.76788197844621 10− − × 1 46 3 5 23 0.22446277332006 10− × 4 47 4 14 10 0.62689710414685 10− − × 9 48 6 3 50 0.55711118565645 10− − × 49 6 6 44 0.19905718354408 − 50 6 6 46 0.31777497330738 51 6 6 50 0.11841182425981 − 81

Table B.3. The other coeﬃcients and parameters of the residual part.

ici di ti ni αi βi γi εi 52 0.03 0 0.31306260323435 102 20 150 1.21 1 − × 53 0.03 1 0.31546140237781 102 20 150 1.21 1 × 54 0.03 4 0.25213154341695 104 20 250 1.25 1 − × iai bi Bi ni Ci Di Ai βi 55 3.50.85 0.2 0.14874640856724 28 700 0.32 0.3 − 56 3.50.95 0.20.31806110878444 32 800 0.32 0.3

Some important relations

∂f P = ρ2 , (135) ∂ρ µ ¶T P (δ,τ) =1+δφr , (136) ρRT δ where δ = ρ/ρc and

r 7 51 ∂φ c r di 1 ti δ i di ti ci φ = = nidiδ − τ + nie− δ τ (di ciδ ) δ ∂δ − · ¸τ i=1 i=8 54 X X £ ¤ 2 2 d di ti αi(δ εi) βi(τ γi) i + niδ τ e− − − − 2αi (δ εi) δ − − i=52 · ¸ X56 ∂δ∆bi ψ + n , (137) i ∂δ i=55 X · ¸ with

2 2 ai ∆ = θ + Bi (δ 1) , (138) − 1 2 2β θ =(1 τ)+£ Ai (δ ¤ 1) i , (139) − − 2 2 Ci(δ 1) Di(τ 1) ψ = e− − − £ − .¤ (140) 82

B.2. CREOS-01, -02

The parameters β, α, and ∆i in the CREOS 01, 02 are the universal criti- − − 2 cal exponents, b is a universal constant parameter, the scaled functions ψi(θ) are universal analytical functions of the parametric variable θ,asdeﬁnedinSection3.2.

The other parameters, k, d1 ,a,and ci are characteristic parameters of the system of interest. The universal functions, ψi,aregivenby

2 1 b 1 2γ 1 2 2 1 2β 2 2 2 Ψ0(θ)= 2β − +2β − (1 b θ ) − (1 b θ ) , 2b4 2 α γ(1 α) − − α − · − − ¸ 1 γ + ∆1 2 2 Ψ1(θ)= 2 (1 2β)b θ , 2b (1 α + ∆1) 2 α + ∆1 − − · − ¸· − ¸ 1 γ + ∆2 2 2 Ψ2(θ)= 2 (1 2β)b θ , 2b (1 α + ∆2) 2 α + ∆2 − − · − ¸· − ¸ 1 2 2 4 4 Ψ3(θ)= θ 3 2(e0 β)b θ + e1(1 2β)b θ , 3 − − − 1 2£ 3 2 2 ¤ Ψ4(θ)= b θ 1 e2(1 2β)b θ , 3 − − 1 2 3 £ 2 2¤ Ψ5(θ)= b θ 1 e4(1 2β)b θ . (141) 3 − − £ ¤ The crossover function, R(q),isdeﬁned in the following expression

q2 2 R(q)= 1+ , (142) 1+q µ ¶ where the crossover variable, q, is related to the parameter r through

q = √rq , (143) with

τ = r(1 b2θ2) , (144) − where τ = T/Tc 1. The µ (T ), Ao(T) are analytical functions of temperature and − o are given by 83

3 j µo(T )= mjτ , (145) j=1 X 3 j Ao(T )= Zc + Ajτ , (146) − j=1 X where Zc is the critical compressibility given by Pc/ρcRTc . Table B.4. shows all the universal constants, while Table B.5. shows the system dependent parameters for H2O and D2O.

Table B.4. The coeﬃcients of the CREOS equation of state. α =0.11 β =0.325 γ =2 α 2β =1.24 − − 2 γ 2β b = γ(1− 2β) = 1.359 − ∼ ∆1 = ∆1 =0.51

∆2 = ∆2 =2∆1 =1.02 f ∆3 = ∆4 = γ + β 1=0.565 f − ∆5 =1.19 1 ∆3 = ∆4 = ∆3 =0.065 − 2 1 ∆5 = ∆5 =0.69 f f − 2 e0 =2γ +3β 1=2.455 f − e1 =(e0 β)(2e0 3) / (e0 5β) = 4.9 − − − ∼ e2 =(e0 3β) / (e0 5β) = 1.773 − − ∼ e3 =2 α + ∆5 =3.08 − e4 =(e3 3β) / (e3 5β) = 1.446 − − ∼ 84

Table B.5. The coeﬃcients of CREOS-01 and CREOS-02 EOSs. a b Parameter H2O D2O k 0.372389 0.319254

d1 0.171848 0.200011 a 192.657 211.969

c1 86.1386 379.046

c2 2116.6 265.477 − − c3 180.877 407.576

c4 298.053 458.119 − − g 9.71434 13.9632

A1 0.873229 0.712249 − − A2 173.177 215.237

A3 32.5782 50.5706

m1 0.88195 1.91944

m2 110.191 155.576 − − m3 10.2527 14.7238 − −

The data in the second column have been taken from Kiselev and Ely[62], while column three has been taken from the same authors[73]. Note that in the second paper[73], there are misprints in the signs of c1, C2,andc3 . The critical parameters of the second critical point in the supercooled water are: 3 Tc =188K, Pc =230.MP a, ρc =1100Kg.m− , while for supercooled heavy water 3 they are: Tc =195K, Pc =230MPa, ρc =1220Kg.m− . 85

B.3. JA-EOS RefertoSec.3.3forthedeﬁning equations.

Table B.6. The coeﬃcients and parameters of the JA-EOS. first term second term third term

Pc =220.5 MPa

Tc =647.3 K

Tb = 1408.4 K Tf =273.15K ρ =1./56 g/cm3 3 3 c vb =41.782 cm /mol ρHB =0.8447g/cm κ =0.836575 λ =0.3241 C1 =0.714024 A1 = 12.1637 α =2.145 vb C2 =0.18 − 5 × A2 =0.228358 10 b∗ =1.0823 vb σ =0.168695ρHB × × A3 =13.3273 b1 =0.250810 S0 = 61.468 J/mol.K − A4 = 0.0610028 b2 =0.998586 SHB = 5.1278 J/mol.K − − A5 =1.87317 b3 =21.4 HB = 11490 KJ/mol. − b4 =0.0445238

b5 =1.01603 86

B.4. CPHB EOS

Chen et al.[69] applied the Walsh-Gubbins EOS and simpliﬁed it to v + k b Zrep = 1 , (147) v k2b − where b is the hard core volume, and k1, k2 were correlated to the nonspherical factor α as

k1 =4.8319α 1.5515 , (148) − 1.3683 k2 =1.8177 .1778α− . (149) − Note that when α =1, then Eq.(147) gives numerical results equivalent to the Carnahan-Starling expression[69]. After adding an empirical attractive term to Eq.(147), the CPHB EOS is

RT (1 + k b/v) a P = 1 , (150) v k2b − v(v + c) − where a, b,andc are calculated from the conditions deﬁning the critical point ∂P =0, (151) ∂v µ ¶c ∂2P =0, (152) ∂v2 µ ¶c Pcvc = Zc . (153) RTc From these conditions, a, b,andc are deﬁned as

2 2 ΩacR Tc a = F1(T ) , (154) Pc ΩbcRTc b = F2(T ) , (155) Pc Ω RT c = cc c , (156) Pc where Ωac, Ωbc, Ωcc are determined through the following equations

Ωcc =1+k2Ωbc 3ζc , (157) 3 − ζc k1ΩbcΩcc Ωac = − . (158) k2Ωbc 87

2 2 2 2 3 k2Ωbc3+(2k1k2+2k 3k ζ )Ω +(k1+k2 3k1ζ 3k2ζ +3k2ζ )Ωbc ζ =0. (159) 2 − 2 c bc − c− c c − c

The last term of Eq.(159) in the original paper has been mistyped as 3ζc instead 3 of ζc . On solving this equation the lowest positive root is used. Thenonsphericalfactorisgivenby

2 3 4 3 5 α =1.0003 0.2719M +3.731M 1.0827M +0.1144M 4.1276 10− M , (160) − − − × where M = mwω/39.948 , mw is the molecular weight.

The parameters ζc, F1,andF2 are calculated from the following equations

2 F1 = 1+C1 1 TR + C2 (1 TR) , (161) − − h ³ ´ i 2 3 2 p2/3 2/3 2/3 F2 = 1+C3 1 T + C4 1 T + C5 1 T , (162) − R − R − R · ³ ´ ³ ´ ³ ´ ¸ where C2 and C5 depend on the ﬂuid properties through

2 Tc Tc C2 = 1.4671 + 3.6889 2.0005 +5.2614 ωZc , (163) − αT − αT µ b ¶ µ b ¶ ω 3.063 p 1.667 C5 =7.9885 4.3604e +1.4554mwω 21.395α(ζ Zc) 4.0692Zcα (164). − − c − −

The parameter ζc was correlated through

4 Tb ζc =0.2974 + 0.1123ω 0.9585ωZc +7.7731 10− mwω , (165) − × Tc µ ¶ where Tb, Tc, TR(= T/Tc) are the boiling, critical and reduced temperatures, respectively. Equation (153) deﬁnes Zc, the critical compressibility factor. The C1, C3, and C4 coeﬃcients are given in the following table for water, methanol, and ethanol.

Table B.7. The C parameters for water, ethanol, and methanol of the CPHB EOS.

Substance C1 C3 C4 Water 0.28111 2.18987 -2.03823 Methanol -1.73089 4.55337 -7.42214 Ethanol -1.20047 7.36221 -13.2154

Note that in the table in the original paper, C3,andC4 were mistyped as C2,and

C3. 88

The following table contains diﬀerent properties of water, ethanol, and methanol. As mentioned in Chapter 3, it is important to use the same critical temperature as Chen et al. to get reasonable results.

Table B.8. The parameters of the CPHB EOS used for water, ethanol, and methanol.

Substance Tc(K) Tb(K) Pc(MPa) mw(g/mol) ω Water 647.3 373.2 220.5 18.015 .348 Methanol 512.6 337.8 80.972 32.042 0.559 Ethanol 512.93 351.443 61.37 46.069 0.6436 89

APPENDIX C PHYSICAL PROPERTIES OF WATER AND HEAVY WATER 90

APPENDIX C PHYSICAL PROPERTIES OF WATER AND HEAVY WATER

C.1 WATER

Correlations for the surface tension (mN/m) and equilibrium liquid density (g/cm3)are γ =93.6635 + 0.009133T 0.000275T 2 , (166) − 0.33 ρle =0.08 tanh x +0.7415tr +0.32 , (167) where tr =(Tc T)/Tc is the reduced temperature, and x =(T 225)/46.2,andTc − − (647.15 K) is the critical temperature of water.

The experimental equilibrium vapor pressure (Pa)ofwater[75],Pve(T ) can be evaluated from

P exp(T)=exp(77.34491 7235.42465/T 8.2lnT + .0057113T) . (168) ve − − C.2 HEAVY WATER Correlations for the surface tension and equilibrium liquid density are

γ =93.6635 + 0.009133T 0 0.000275T 02 , (169) − 0.33 ρle =0.09 tanh x +0.84tr +0.338 , (170) where T 0 =1.022T, tr =(Tc T)/Tc is the reduced temperature, with x =(T − − 231)/51.5,andTc (643.89K) is the critical temperature of heavy water. Please note that in the original paper of Wölk and Strey[18], a misprint is found in the formula of the surface tension for heavy water. The experimental equilibrium vapor pressure[75], of heavy water can be evaluated from T P exp(T )=P exp c (α τ + α τ 1.9 + α τ 2 + α τ 5.5 + α τ 10) , (171) ve c T 1 2 3 4 5 µ ¶ where

α1 = 7.81583,α2 =17.6012,α3 = 18.1747,α4 = 3.92488,α5 =4.19174,Tc = − − − 643.89,Pc =21.66MPa, and τ =1 T/Tc. − 91

APPENDIX D SURFACE TENSION AND WORK OF FORMATION IN DFT 92

APPENDIX D SURFACE TENSION AND WORK OF FORMATION IN DFT

D.1 SURFACE TENSION Start with the thermodynamic deﬁnition of the grand potential for a two-phase system at pressure P ,withvolumeV , and interfacial area A,

Ω = PV + γA . (172) − From Eq.(172) with Eqs.(92) and (91), the Aγ(= Ω + PV) term can be expressed as

1 Aγ = fh(ρ ,ρ )dr+ drdr0φ ( r r0 )ρ (r)ρ (r0) µ ρ (r)dr+ Pdr . 1 2 2 ij | − | i j − i i Z X Z X Z Z(173)

The term involving the attractive potential φij can be eliminated with the help of Eq.(93). The resulting equation can then be simpliﬁed by noting that the density varies solely in the x direction, perpendicular to the planar interface. The resulting expression for the surface tension, then reads as

1 γ = fh(ρ ,ρ )dx + ρ (µ µ )dx µ ρ (x)dx + Pdx. (174) 1 2 2 i i − ih − i i Z X Z X Z Z The fh term can be eliminated using the deﬁnition

fh = ρ µ Ph . (175) i ih − X Then, we obtain the somewhat simpler result,

1 1 γ = ρ (µ µ )+ ρ (µ µ )+(P Ph) dx . (176) 2 1 1h − 1 2 2 2h − 2 − Z ½ ¾ Now, for our HSY model ﬂuid, writing Eq.(93) for the ﬂat interface we obtain

2 d µ1h 2 = λ (µ (ρ ,ρ ) µ α11ρ α12ρ ) , (177) dx2 1h 1 2 − 1 − 1 − 2 and 2 d µ2h 2 = λ (µ (ρ ,ρ ) µ α21ρ α22ρ ) . (178) dx2 2h 1 2 − 2 − 1 − 2 93

Equations (177) and (178) can be simpliﬁed further by using the so-called Bertholet mixing rule (α12 = √α11α22). Note that by using this mixing rule, the pressure EOS, Eq.(96) in Chapter 7, can also be simpliﬁed to

1 2 P = Ph (√α11ρ + √α22ρ ) , (179) − 2 1 2 which will be used to replace the terms α11ρ1 + α12ρ2 or α21ρ1 + α22ρ2 in Eqs.(177) and (178).

Now multiply Eq.(177) by dµ1h/dx and Eq.(178) by dµ2h/dx and note that 1 d dµ 2 dµ d2µ ih = ih ih . (180) 2 dx dx dx dx2 µ ¶ Equations (177) and (178) can then be written as

2 1 d dµ1h 2 dµ1h = λ (µ (ρ ,ρ ) µ α11ρ α12ρ ) , (181) 2 dx dx 1h 1 2 − 1 − 1 − 2 dx µ ¶ and 2 1 d dµ2h 2 dµ2h = λ (µ (ρ ,ρ ) µ α21ρ α22ρ ) . (182) 2 dx dx 2h 1 2 − 2 − 1 − 2 dx µ ¶ With the help of the Gibbs-Duhem identity

dPh = ρidµih , (183) X Eq.(179), and the diﬀerential of the Eq.(102),

√α11dµ2h = √α22dµ1h , (184) it is easily seen that after integrating both sides of Eq.(181), we obtain

2 dµ1h 2 2 = λ (µ µ ) 2α11 (Ph P ) . (185) dx 1h − 1 − − µ ¶ £ ¤ Now substitute Eq.(102) into Eq.(176), and use Eq.(177) to further simplify this equation. Then the expression for the surface tension can ﬁnally be written as

1 dµ 2 1 dµ 2 γ = 1h dx = 2h dx , (187) λ2α dx λ2α dx 11 Z µ ¶ 22 Z µ ¶ 94 with the use of Eq.(185), or the similar pair of equations with i =2. These speciﬁc forms still require detailed knowledge of the structure of the interface to complete their evaluation. To avoid this, Eq.(187) can be transformed by changing the independent variable from x to one of the densities by noting that

dµ 2 dµ dµ dµ 1h dx = 1h dµ = 1h 1 dρ . (188) dx dx 1 dx dρ 1 µ ¶ µ ¶ µ ¶ 1 The ﬁnal formula may then be cast in terms of either component 1 or 2 as

ρil 1 2 1/2 dµih γ = (µih(ρ1,ρ2) µi) 2αii(Ph P ) dρi , i =1or 2 . λαii − − − dρ Zρiv µ i ¶ £ ¤ (189) D.2 WORK OF FORMATION Thereversibleworkisdefined as the difference in the grand potential between the initial uniform system and the final system containing a droplet,

Wrev = Ω (ρ ,ρ ) Ω0(ρ ,ρ ) , (190) 1 2 − 1b 2b where ρ1b, ρ2b are the densities of the uniform system. For the initial system, we have Ω0(ρ ,ρ )= PV. For the nonuniform system, we have 1b 2b − 1 Ω = dr fh(ρ ,ρ ) µ ρ + ρ (µ µ ) , (191) 1 2 − i i 2 i i − ih Z ½ X X ¾ where, as usual,

fh = ρ µ Ph . (192) i ih − Substitute Eq.(192) into Eq.(191), andX assume the droplet is spherical, then Eq.(190) becomes

1 1 2 Wrev =4π ρ (r)[µ (ρ1,ρ2) µ ]+ ρ (r)[µ (ρ ,ρ ) µ ]+P Ph r dr . 2 1 1h − 1 2 2 2h 1 2 − 2 − Z ½ ¾(193) 95

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VITA

MynameisAbdallaAhmadObeidat;IwasbornonDecember4,1970inYoubla, Jordan. I graduated from Koforsoom High School in 1988. I received my Bachelor degree in Physics with minor in Mathematics in May of 1992 from Yarmouk Uni- versity. In August of 1992, I began my graduate work in the same university, and I received my Masters in the ﬁeld of Magnetism from Physics department by October of 1994. In January of 1995, I start teaching undergraduate labs at Jordan University of Science and Technology till the end of 1997. By August of 1998, I started working toward my Doctor of Philosophy degree in Physics at University of Missouri-Rolla. I worked one year as a teaching assistant, while I was working as a research assistant for Dr. Wilemski that involves Unary and Binary Nucleation. In December of 2003, I was awarded a Doctor of Philosophy degree in Physics.