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Nucleation and condensation in gas-vapor mixtures of alkanes

and water

Citation for published version (APA):

Peeters, P. (2002). Nucleation and condensation in gas-vapor mixtures of alkanes and water. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR559332

DOI:

10.6100/IR559332

Document status and date: Published: 01/01/2002

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Nucleation and Condensation

in

Gas-Vapor Mixtures

of

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Omslagontwerp: Paul Verspaget Druk: Universiteitsdrukkerij, TUE

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Peeters, Paul

Nucleation and Condensation in Gas-Vapor Mixtures of Alkanes and Water / by Paul Peeters.

Eindhoven : Technische Universiteit Eindhoven, 2002. -Proefschrift. - ISBN 90-386-2039-x

NUR 910

Trefw.: condensatie / druppelvorming / gasdynamica / aardgas.

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Nucleation and Condensation

in

Gas-Vapor Mixtures

of

Alkanes and Water

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr. R.A. van Santen, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op

donderdag 31 oktober 2002 om 16.00 uur

door

Paul Peeters

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prof.dr.ir. M.E.H. van Dongen en

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A clever person solves a problem. A wise person avoids it.

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Contents

1 Introduction 1

1.1 Nucleation and growth rate . . . 1

1.2 Motivation of this research . . . 3

1.3 Thesis overview . . . 5

References . . . 6

2 Phase equilibrium 9 2.1 Thermodynamics of phase equilibrium . . . 9

2.1.1 Fugacity . . . 10

2.2 Equations of state . . . 11

2.2.1 CPA . . . 12

2.3 Mixtures of methane, n-nonane, and/or water . . . 14

2.3.1 Pure components . . . 14

2.3.2 Mixtures . . . 15

2.3.3 Liquid versus vapor fraction . . . 16

References . . . 25

3 Nucleation 27 3.1 Cluster distribution . . . 27

3.2 Steady nucleation rate . . . 30

3.3 Dilute vapor in a high pressure carrier gas . . . 33

3.4 Heterogeneous nucleation . . . 35

3.4.1 Wetting and contact angles . . . 36

3.5 Mixtures of methane, n-nonane, and/or water . . . 36

3.5.1 Supersaturation ratio . . . 36

3.5.2 Ternary nucleation . . . 37

3.6 Nucleation theorem . . . 40

References . . . 41

4 Droplet growth 43 4.1 Homogeneous droplet model . . . 44

4.1.1 Continuum region . . . 45

4.1.2 Knudsen layer . . . 47 vii

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4.1.3 Complete set of equations . . . 50

4.2 Layered droplet model . . . 51

4.2.1 Molar fluxes in liquid layer . . . 51

4.2.2 Complete set of equations . . . 52

4.3 Mixtures of methane, n-nonane and/or water . . . 53

4.3.1 Binary mixtures . . . 53

4.3.2 Ternary mixtures . . . 54

References . . . 55

5 Wave tube experiments 57 5.1 Nucleation pulse method . . . 57

5.2 Pulse-expansion wave tube . . . 58

5.2.1 Pressure profile . . . 60

5.2.2 Bursting of the diaphragm . . . 62

5.2.3 Thermodynamic state . . . 63

5.3 Droplet detection . . . 64

5.3.1 Light scattering by dielectric particles . . . 65

5.3.2 Scattering intensity . . . 67

5.3.3 Light extinction . . . 69

5.3.4 Layered droplets . . . 69

5.4 Mixture preparation . . . 69

5.4.1 Saturation section . . . 71

5.4.2 Flushing through the HPS . . . 74

5.5 Experimental procedure . . . 75

References . . . 76

6 Experimental results and discussion 79 6.1 Nucleation . . . 79

6.1.1 Binary mixtures . . . 79

6.1.2 Ternary mixtures . . . 83

6.2 Droplet growth . . . 86

6.2.1 Droplet growth rates . . . 87

6.2.2 Droplet growth model . . . 92

References . . . 99

7 Nucleation of ice 101 7.1 Introduction . . . 101

7.2 Nucleation . . . 101

7.3 Surface energy of ice . . . 102

7.4 Experiment . . . 103

7.5 Results and discussion . . . 105

7.6 Conclusions . . . 112

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Contents ix

8 Conclusions and recommendations 115

A Physical properties 119

References . . . 122

B Energy of cluster formation 125

C Droplet growth; derivation of equations 127

C.1 Incoming mass flux . . . 127 C.2 Energy flux . . . 128 C.3 Liquid layer . . . 129

D Tables of experimental data 131

E Droplet growth curves 135

Summary 145

Samenvatting 147

Nawoord 149

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Chapter 1

Introduction

In this thesis vapor to liquid nucleation and subsequent droplet growth is studied. Nu-cleation and droplet growth are processes that bring a system that is not at a state of thermodynamic equilibrium (i.e. is in a supersaturated state), to a new equilibrium, by means of the formation of a new phase. Homogeneous nucleation refers to the formation of small stable clusters of vapor molecules, and the nucleation rate is the rate at which these homogeneous condensation nuclei are formed. We will first give a short description of the physical processes that determine the nucleation rate and the subsequent droplet growth rate. Then, the choice of the system studied, and the conditions at which it is studied will be motivated. In the final section an overview of this thesis is given.

1.1

Nucleation and growth rate

The process of nucleation is a statistical process. It involves the formation of clusters of molecules of a new phase. These clusters are formed at subsaturated as well as supersatu-rated conditions. When the parent phase is not supersatusupersatu-rated, these cluster are unstable and therefore disappear again. When the system is supersaturated, the clusters of the new phase can become stable if they have a certain minimum size. This can be explained as follows. When a system is supersaturated it can lower its energy by forming the new phase. This decrease of energy is proportional to the volume of the new phase. However, the formation of the new phase also involves the formation of an interface between the old (parent) phase and the new phase, which increases the total energy in proportion to the area of this interface. Hence, the difference in the energy of a system with and without a cluster of the new phase has a maximum as a function of the cluster size. This maxi-mum forms an energy barrier for the formation of the new stable phase. The maximaxi-mum height of this energy barrier is called the energy of (cluster) formation, and it depends on the degree of supersaturation of the system. The clusters of size corresponding to this maximum are called the critical clusters. The further away from equilibrium the system is, the lower the energy barrier will be, and the smaller the critical clusters will be. The probability of obtaining a critical cluster is proportional to the Boltzmann factor with the

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Figure 1.1: Calculations for water vapor to liquid nucleation at 1 bar and 298 K, using the classical nucleation theory. Left: Energy of cluster formation as a function of the cluster size n, for different constant values of the supersaturation S. Right: Nucleation rate J as a function of the supersaturation S.

energy of formation in the exponent. This was first formulated by Volmer and Weber in 1926 [1]. Later, the rate at which critical clusters are formed was included by Becker and D¨oring [2]. The description of the nucleation rate, as formulated by Volmer and Weber, and Becker and D¨oring, is now known as the classical nucleation theory. This theory was extended by Reiss [3] to describe two-component nucleation, and over the years, many modifications to the theory have been made. An up to date discussion of the theory is given by Kashchiev [4]. As an example, figure (1.1) shows the energy barrier W/kBT for

the nucleation of water vapor to liquid water as a function of the number of molecules in the cluster, for different values of the supersaturation. Also shown is the corresponding nucleation rate J as a function of the supersaturation S. The curves are calculated using the classical nucleation theory with the capillarity approximation, at a pressure of 1 bar and temperature of 298 K. As can be seen in figure (1.1), the nucleation rate strongly depends on the supersaturation.

It is noteworthy that all nucleation theories can be related to the principle of determin-ing the height of the energy barrier, and the rate at which clusters can cross it. Theories differ in the means of determining the height of this energy barrier and the rate factor. In the capillarity approximation an analytical expression for the height of the energy barrier is found based on the assumption that the microscopic clusters can be described using macro-scopic properties. The height of the barrier can also be obtained from a density functional approach [5], or by a direct molecular simulation using Monte Carlo techniques [6–8]. In this work we will confine ourselves to the description of the capillarity approximation. Rate factors are mostly obtained using kinetic considerations.

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1.2 Motivation of this research 3 two regimes. One regime is found when the size of the cluster is (still) much smaller than the mean free path of the molecules. The growth rate is then accurately described by kinetics. When the new cluster has grown much larger than the mean free path of the molecules, the growth of the cluster is determined by relations from continuum theory. Of course, when clusters grow from very small to larger sizes, they generally grow from the kinetic to the continuum regime. Therefore, a model that can describe the growth in both regimes, including the transition from one to the other regime, is needed. For this purpose, the flux-matching method is applied [9–11], which is based on the following idea. A cluster of the new phase is always surrounded by a layer of finite thickness, in which the transport of mass and energy are described by kinetic relations. The thickness of the layer is of the order of the molecular mean free path. Outside this layer, the transport of mass and energy are described by continuum relations. During (quasi-)steady growth the fluxes have to be continuous across the interface between the continuum region and kinetic region. By applying the continuity of the fluxes across the interface a complete set of equations can be obtained describing the growth of the cluster. In the limit of kinetic growth the kinetic layer stretches out to infinity, while in the limit of continuum growth the thickness of the kinetic layer approaches zero.

1.2

Motivation of this research

In this work the nucleation behavior and subsequent droplet growth of supersaturated n-nonane and supersaturated water vapor in methane are studied. The idea of studying this particular system originates from new developments in natural gas industry. Recently, the controlled generation of nucleation and droplet growth in newly developed gas/vapor separators has been shown to be possible [12]. In figure (1.2) such a new separator is shown schematically. The natural gas is contained at high pressure beneath the earth surface in large cavities or porous rock formations, called gas deposits. The main component of the natural gas is methane. Besides methane, it often contains water vapor and many different hydrocarbon vapors. Before the natural gas is delivered to the customers, a large part of the vapors needs to be removed from the gas. With the newly developed gas/vapor separator this is achieved in the following way. First, the gas is accelerated to a supersonic speed by the nozzle. Due to the isentropic acceleration the temperature and pressure of the natural gas will drop, making the natural gas mixture supersaturated. Nucleation will take place and droplets will start to grow. Subsequently, a vortex is induced in the gas flow by means of a vortex generator, which is placed in the tube behind the nozzle. The droplets in the flow will be swirled to the outside of the tube. As a result the core of the flow through the tube will become dry, while the outside of the flow will contain most of the vapor components. The outer layer of the flow is then separated from the core of the flow, leaving only the dry gas in the main flow. Although the principles on which the apparatus operates are quite simple, the actual operation is not. If the droplets are too small they will just follow the streamlines of the flow, and many of them will not be forced to the outside. If they are too large, their inertia will be too large for them to be swirled

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i n l e t n o z z l e w i n g d r y g a s w e t g a s w e t g a s

Figure 1.2: Schematic view of a newly developed gas/vapor separator, designed for the natural gas industry.

to the outside. Therefore, the successful operation of the apparatus depends on a delicate balance between the flow speed, the strength of the vortex, the nucleation rate, the growth rate, and so on, which are all related to each other. In order to increase the performance of the apparatus, a good understanding of all the individual processes is needed. Two of these processes are the nucleation and the subsequent droplet growth, which are the subject of this study.

When looking at the nucleation and growth rate in natural gas, two characteristic aspects are evident. These are that the nucleation and growth take place at high pressures, and that it involves the nucleation and growth of many different (vapor) components. At the Technische Universiteit Eindhoven nucleation and growth rates of droplets have been studied using an expansion chamber [13], and a pulse-expansion wave tube. The latter is basically a modified shock tube [14]. Both devices are well suited for performing measurements at high pressure (up to 50 bar). The pulse-expansion wave tube has the advantage that the nucleation stage and the growth stage are separated in time, making the analysis of the data much simpler. Over the past 10 years several systems have been studied, amongst others were systems of n-octane in methane [15], n-nonane in methane [13, 15,16], and samples of actual dry (i.e. without water) natural gas [17]. Except for the case in which samples of natural gas were used, so far, all the studies were confined to systems of a single vapor in a carrier gas. To study multi-component nucleation and droplet growth in a systematic way, it is desirable that gas mixtures containing more than one vapor component can be prepared in a controlled manner. To achieve this, the existing method of mixture preparation has been altered in such a way that gas/vapor mixtures containing two vapor components can be prepared in a controllable way. The components chosen in this work are methane, water, and n-nonane. The choice of methane is evident, this is the main component of natural gas. The vapor component water is chosen because it

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1.3 Thesis overview 5 is often abundantly present in natural gas reservoirs, and because it is a polar molecule, unlike the hydro-carbon vapors. As the third component n-nonane is chosen, as being a typical hydro-carbon vapor. As can be seen from the composition analysis of the sample of natural gas, given in reference [17], most vapor components are different kinds of (non-polar) alkanes, like n-nonane. Furthermore, the vapor/liquid phase envelope describing phase equilibrium in dry natural gas is very similar to the phase envelope of mixtures of n-nonane in methane, as was pointed out by Muitjens [13].

1.3

Thesis overview

In chapter 2 a short overview of the thermodynamics of phase equilibria will be given. Calculations of phase equilibria can be performed using an equation of state. An appro-priate equation of state was chosen for the specific system of interest. In chapter 2 we will show that the CPA equation of state is well suited to describe the vapor/liquid equilibria in mixtures of methane, n-nonane, and water.

In chapter 3 the description of the classical nucleation theory, using the capillarity approximation, will be given. The influence of the carrier gas on the nucleation of a single vapor component will be discussed. Then, we will distinguish between homogeneous nucleation and heterogeneous nucleation. Homogeneous nucleation is the nucleation of a new phase in the parent phase, while heterogeneous nucleation is the nucleation of a new phase onto a substrate or ’host’ particle. The possible occurrence of heterogeneous nucleation in mixtures of supersaturated n-nonane and water in methane will be discussed. In the last section of chapter 3 we will describe the well-known nucleation theorem. With this theorem, information about the composition of critical clusters (i.e. clusters on top of the energy barrier) can be obtained from experimental nucleation rate data.

Chapter 4 is devoted to the description of droplet growth. The growth description is based on the flux-matching model. First, the droplet growth description will be given for homogeneous multi-component droplets in a real (not-inert) carrier gas. Then, this model will be extended to describe the growth of multi-component droplets, that consist of two different liquids. The second liquid is assumed to form a layer around the core, which consists of the first liquid.

In chapter 5 the experimental setup is described. In the first part the nucleation pulse method is highlighted. It will then be explained how the nucleation pulse method can be applied using a shock tube. The droplets in our setup are detected by means of an optical setup. By measuring both the intensity of the light that is scattered at an angle of 90◦, and the light intensity that is transmitted through the cloud of droplets, their

radius and number density can be determined. Attention will be given to the method of mixture preparation. When performing high-pressure nucleation experiments the vapor fractions are of the order 10−4 to 10−5. In order to obtain accurate nucleation rate data

as a function of the supersaturation, the values of these small vapor fractions have to be accurate within a few percent. This makes the mixture preparation a very challenging aspect of the experimental procedure.

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In chapter 6 the experimental results are given. The nucleation rates of n-nonane in methane and that of water in methane will be compared with the predictions of the classical nucleation theory. Then the nucleation rates of supersaturated n-nonane and supersaturated water in methane are presented. Likewise, the droplet growth rates of the binary systems will be discussed first, followed by a discussion of the droplet growth rates in the ternary systems. In the last part of this chapter the experimentally obtained growth rates will be compared to the predictions of the growth model given in chapter 4.

Chapter 7 is on the nucleation behavior of supersaturated water vapor in helium. The reason for studying this system is the following. For nucleation of supersaturated water in methane or helium at the conditions studied in this thesis, liquid water is not the most stable new phase. In helium the most stable phase is ice, while in methane hydrate formation is also possible. It is generally assumed that liquid water will form before any ice is formed. In this chapter this is investigated experimentally. Once this has been achieved for water in helium, for which only liquid water and solid water are possible as new phases, the research can be extended to nucleation of water in methane, in which hydrate formation is also possible.

The final chapter, chapter 8, contains the general conclusions of this work, together with some recommendations for future experiments.

References

[1] M. Volmer and A. Weber, Z. Phys. Chem. 119, 277 (1926). [2] R. Becker and W. D¨oring, Ann. Phys. 5, 719 (1935).

[3] H. Reiss, J. Chem. Phys. 18, 840 (1950).

[4] D. Kashchiev, Nucleation; Basic Theory with Applications, Butterworth-Heinemann, Oxford, 2000.

[5] D.W. Oxtoby and R. Evans, J. Chem. Phys. 89, 7521 (1988). [6] C.L. Weakliem and H. Reiss, J. Chem. Phys. 101, 2398 (1994). [7] I. Kusaka and D. W. Oxtoby, J. Chem. Phys. 110, 5249 (1998). [8] P.R. ten Wolde and D.J. Frenkel, J. Chem. Phys. 109, 9919 (1998). [9] N.A. Fuchs, Phys. Z. Sowjet 6, 224 (1934).

[10] N. Fukuta and L.A. Walter, J. Atmos. Sci. 27, 1160 (1956). [11] J.B. Young, Int. J. Heat Mass Transfer 36, 2941 (1993).

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1.3 References 7 [13] M.J.E.H. Muitjens, Homogeneous condensation in a vapour/gas mixture at high pres-sures in an expansion cloud chamber, PhD thesis, Eindhoven University of Technology, 1996, ISBN 90-386-0199-9.

[14] K.N.H. Looijmans, P.C. Kriesels, and M.E.H. van Dongen, Exp. Fluids 15, 61 (1993). [15] K.N.H. Looijmans, Homogeneous nucleation and droplet growth in the coexistence region of n-alkane/methane mixtures at high pressures, PhD thesis, Eindhoven Uni-versity of Technology, 1995.

[16] C.C.M. Luijten, P. Peeters, and M.E.H. van Dongen, J. Chem. Phys. 111, 8535 (1999).

[17] C.C.M. Luijten, R.G.P. van Hooy, J.W.F. Janssen, and M.E.H. van Dongen, J. Chem. Phys. 109, 3553 (1998).

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Chapter 2

Phase equilibrium

Nucleation and condensation are non-equilibrium processes. These are processes that bring a system that is out of equilibrium to a new equilibrium. Therefore, it is appropriate that a few words are spent on the description of thermodynamic equilibrium, and the methods that are used to calculate equilibrium states. In the first part of this chapter some principles of thermodynamic equilibrium are given. In the second part of this chapter an equation of state is given, used to carry out calculations of phase equilibria for the systems of interest to us. This is described in the third part of this chapter. Throughout this chapter, and the following ones, the letter y will be used to indicate molar vapor fractions, and the letter x will be used to indicate molar liquid fractions.

2.1

Thermodynamics of phase equilibrium

In this section the principles of thermodynamic equilibrium will be highlighted. For the derivations and proof of the statements the reader is referred to standard textbooks on thermodynamics, e.g. [1]. A combination of the first and second law of thermodynamics results in the following expression for a reversible process in an open system that can exchange energy, volume, and mass with its environment:

dU = T dS − pdV +

m

X

i=1

µidni, (2.1)

where m is the total number of components. The internal energy of the system U is a function of the entropy S, the volume V , and the molar quantities ni, all of which are

extensive parameters. The temperature T , the pressure p, and the chemical potential µi

are intensive. By introducing the Gibbs energy, which is defined as

G ≡ U + pV − T S, (2.2)

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the change in energy of a closed system can equally well be described by the change in Gibbs energy, dG = −SdT + V dp + m X i=1 µidni. (2.3)

Applying this equation to an isothermal and isobaric open homogeneous system, it follows that the Gibbs energy has to be linear in ni, therefore

G =

m

X

i=1

µini. (2.4)

The total differential of G is therefore also equal to: dG = m X i=1 µidni+ m X i=1 nidµi. (2.5)

Subtracting this result from equation (2.3) results in the well known Gibbs-Duhem equation SdT − V dp +

m

X

i=1

nidµi = 0, (2.6)

that relates all the intensive parameters to each other. It also shows that in a system consisting of m components, m + 1 of the m + 2 variables are independent.

When a closed system consists of several phases it can be thought to exist of ˜p open homogeneous phases that coexist, where ˜p is the number of phases present. The closed het-erogeneous system is in equilibrium when the temperature, pressure, and chemical potential of each component are uniform throughout the entire heterogeneous system. Therefore, there are (˜p − 1)(m + 2) equilibrium relations between the ˜p homogeneous phases in the heterogeneous system. Since each phase has m + 1 degrees of freedom, the number of degrees of freedom of the closed heterogeneous system is

F = ˜p(m + 1) − (˜p − 1)(m + 2) = m − ˜p + 2 ≥ 1. (2.7) This is the famous phase rule first formulated by Gibbs.

2.1.1

Fugacity

For practical applications the equilibrium condition of equal chemical potential for a com-ponent in each phase is often converted into another equivalent condition. This is the condition that the fugacity of each component has to be equal in each phase. The fugacity fi of component i is defined as µi− µ0i = RT ln µ fi f0 i ¶ , (2.8)

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2.2 Equations of state 11 where the superscript 0 denotes some arbitrary reference state. For an ideal gas mixture the fugacity fi is equal to the partial pressure yip (yi is the molar vapor fraction). From

this the fugacity coefficient is introduced φi =

fi

yip

, (2.9)

which is equal to one for an ideal gas. The value of the coefficient indicates the deviation from the ideal gas behavior. We will now derive an expression for the equilibrium between a gas phase and a liquid phase, from which several simplifying assumptions will be made in section 2.3.3. Throughout this work the superscripts G and L will be used for gas phase and liquid phase properties, respectively. For the liquid phase one can write in a similar way µLi = µLi,pure+ RT lnµ xiφ L i p fL i,pure ¶ = µLi,pure+ RT ln(γixi), (2.10)

where the activity coefficient γ = φp/fref has been introduced, which is common for the

description of liquid equilibria. The pure component chemical potential at pressure p and temperature T can be rewritten as

µLi,pure= µ0i + RT ln à φL i,purepsat,i f0 i ! + Z p psat,i viLdp. (2.11)

Here, the Gibbs-Duhem equation in the form dµi = vidp has been used, and vi is the

partial molar volume. We now turn to the chemical potential of the gas phase, for which we write µGi = µ0i + RT lnµ yiφ G i p f0 i ¶ . (2.12)

By applying the equilibrium condition µL

i = µGi the following equation is obtained:

ln(yiφGi p) = ln(φGi,purepsat,i) + ln(xiγi) +

Z p

psat,i

vL i

RTdp (2.13)

The pure component saturated liquid fugacity coefficient φL

i,pure has been replaced by the

pure component saturated gas fugacity coefficient φG

i,pure, which can be done since it is an

equilibrium property.

2.2

Equations of state

In order to calculate equilibria an expression is needed that relates the thermodynamic properties. An example of such an expression is p = p(T, V, ni), where the pressure is

expressed as a function of the temperature, volume, and the molar quantities of all the components present. Such an expression defines the thermodynamic state of a system

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at given thermodynamic properties, and is called an equation of state (eos). Once the equation of state of a system is known, all its (equilibrium) properties can be calculated from the thermodynamic relations. Finding the proper eos for a given system is the real problem. An important category of eos is formed by the cubic eos1 like the Peng-Robinson

eos and the Redlich-Kwong-Soave eos [2, 3]. These equations have two or more adjustable parameters which can be fitted to experimental data. For non-polar substances and mix-tures thereof these eos often give good results [1, 2]. However, for substances that have the ability to form strong associating bonding interactions between molecules, like hydrogen bonding, predictions are poor. In the last ten to fifteen years much progress has been made in the statistical theory of associating fluids. The idea behind this theory is that the Helmholtz free energy of a fluid can be split into several contributions. Each contri-bution covers a specific kind of interaction within the molecules or between the molecules. Examples of eos resulting from this are SAFT (Statistical Associating Fluid Theory) [4–6] and APACT (Associated Perturbed Anisotropic Chain Theory) [7]. These eos give much insight into how the physics on a molecular level determines the macroscopic behavior of a fluid. Once the parameters of these eos have been fitted to experimental data, they are able to predict the phase behavior of highly non-ideal systems. Moreover, for systems for which no experimental data are available, parameters can often be extrapolated from sim-ilar systems. The disadvantage of these eos is their complexity, even for non-associating fluid mixtures. It has therefore been proposed to combine the association term from the statistical associating fluid theory with a cubic equation of state. In short these models are called CPA, which stands for cubic plus association. In a number of papers by Tassios et al. [8–13] the Redlich-Kwong-Soave (RKS) eos in combination with the association term of SAFT has proven to give good descriptions of the phase behavior of mixtures of alkanes, alcohols and water. The predictions are comparable to the predictions from the original SAFT model by Huang and Radosz [11,13]. The price of this gain in simplicity is of course the loss of physical insight. However, if the main goal is to have an accurate description of the phase behavior of a system, and experimental data to which parameters can be fitted are available, then CPA is to be preferred above equations like SAFT. For an extensive review on equations of state the reader is referred to [14].

2.2.1

CPA

As mentioned above, the CPA eos by Tassios et al. [8–13] combines the relative simple cubic RKS eos with the powerful association term of the SAFT model. In terms of compressibility factors the eos can be written as

ZCP A = ZRKS + Zassoc. (2.14)

The compressibility factor ZRKS is equal to

ZRKS =

v v − b −

α

RT (v − b). (2.15)

1The term ”cubic” implies an eos which, if expanded, would contain volume terms raised to either the

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2.2 Equations of state 13 The parameter b is a constant while the parameter α is temperature dependent, and is given by

α = a(1 + c(1 −pTr))2, (2.16)

where Tr is the reduced temperature defined as Tr = T /Tc, with Tc the critical temperature.

This gives a total of three independent fitting parameters for ZRKS, namely a, b and c. For

mixtures the classical van der Waals one-fluid rules are applied:

α =X i X j xixjαij, (2.17) and b =X i X j xixjbij, (2.18) with αij = √αiαj(1 − kij), (2.19) and bij = bi+ bj 2 (1 − lij). (2.20)

The parameters kij and lij are used to fit the equation to experimental (phase equilibrium)

data of the mixture. They should have a value close to zero. The compressibility factor due to association is given by

Zassoc = X i xi X j ρj X Aj ·µ 1 XAj − 1 2 ¶ ∂XAj ∂ρi ¸ , (2.21)

where ρi is the molar density of component i. The last summation is a summation over

all possible association sites (Aj, Bj, ...) of component j. The fraction of molecules of

component j that is not bonded at site Aj of component j is given by

XAj = Ã 1 + NAρ X i X Bi xiXBi∆AjBi !−1 , (2.22)

where NAis the Avogadro constant, and ∆AjBi is the association strength parameter given

by ∆AjBi = g ij · expµ ² AjBi RT ¶ − 1 ¸ bijβAjBi. (2.23)

Here ²AjBi and βAjBi are the association energy and the interaction volume of site A on

molecule j with site B on molecule i, respectively. The radial distribution function gij is

taken from Boublik [15] and Mansoori et al. [16]. It is given by gij = 1 1 − ξ3 + 3ξ2 (1 − ξ3)2 didj di+ dj + 2ξ 2 2 (1 − ξ3)3 µ didj di+ dj ¶2 , (2.24)

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with ξk = π 6NAρ X i xidki. (2.25)

The molecular diameter d is in this case related to the molar volume parameter via bi =

2πNAd3i

3 . (2.26)

For pure self-associating components there are a total of five parameters that need to be fitted to pure component data. These are a, b, and c for the RKS eos and ² and β for the association part. For mixtures containing only one associating component no cross association is possible and only two extra mixture parameters are needed. These are kij

and lij. Very often it is sufficient to use only one mixture parameter kij, and set lij to zero.

2.3

Mixtures of methane, n-nonane, and/or water

In this section we will use the CPA eos, discussed in the previous section, to calculate phase equilibria in mixtures of methane, n-nonane and water within a range of pressures and temperatures. In order to do so the parameters of the eos need to be determined. The pure component parameters are determined by fitting the eos to saturated vapor pressures and liquid densities. The mixture parameters are determined by fitting the eos to liquid equilibrium fractions and, when possible, to vapor equilibrium fractions. The fitting procedure was performed using the program PE2000 by Pfohl et al. [17], which is available on the internet.

2.3.1

Pure components

The pure component parameters of methane have been determined using the saturated vapor pressure and liquid density data of NIST [18], available via the NIST webbook. Data were taken in the temperature range of 95 K to 185 K, at 10 K intervals. For n-nonane the saturated vapor pressure and liquid density were taken from Hung et al. [19], in the temperature range of 220 K to 320 K, in 10 K intervals. Finally, for water the saturated vapor pressure was taken from Vargaftik [20] and the liquid density was taken from Pruppacher and Klett [21]. These sources include data of supercooled water. Hence, the parameters of the eos are obtained by fitting in the temperature range of 220 K to 320 K, in 10 K intervals, making the eos valid for calculations with supercooled water, without having to extrapolate. In the case of water, the term Zassoc becomes finite, and has to be

included in the fitting procedure. In order to do this, the number of association sites on a water molecule has to be determined. In figure (2.1) the water molecule is shown. Due to the covalent bonding between the hydrogen and oxygen atom, the hydrogen atom is electrically positive on the outside. Furthermore, the side of the oxygen atom not bonded at the hydrogen atoms, has two electrically negative sites, due to the two free electron pairs. In literature the water molecule has therefore been described by either a three or four site

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2.3 Mixtures of methane, n-nonane, and/or water 15 model, depending on wether the two negative sites are taken as a single negative site, or are accounted for separately. However, in a number of papers it has been shown that appointing 4 sites gives somewhat better results in the overal prediction of the eos [11, 22]. We will therefore adapt the 4-site model for water. The two sites of equal sign are completely equal, and only interactions between a positive and a negative site are allowed. Therefore, only a single kind of association interaction is possible, being the interaction between a positive and a negative site.

The pure component parameters are calculated by minimizing the objective function, which is defined as obj = 1 2   1 n n X i µ pexp,i− pcalc,i pexp,i ¶2 + 1 n n X i à ρL exp,i− ρLcalc,i ρL exp,i !2 . (2.27)

The results of the minimization are shown in table 2.1. After having obtained the pure a (bar l2/mol) b (l/mol) c (-) ²AB/k

B (K) βAB (-) obj (%)

methane 2.287 0.02859 0.4486 2.10

n-nonane 38.93 0.1564 1.128 1.17

water 0.7622 0.01524 2.055 1721 0.1353 1.54

Table 2.1: Pure component parameters for the CPA model.

component parameters, the mixing parameters of the Van der Waals mixing rules have to be obtained.

2.3.2

Mixtures

In this section the binary interaction parameters of the CPA eos are determined for the mixtures methane/n-nonane, methane/water, and n-nonane/water. Inclusion of the inter-action parameter lij did not result in an improvement of the performance of the eos, and

it was therefore set to zero. The interaction parameter kij is obtained by fitting the eos to

experimentally obtained equilibrium fractions.

+

-

-+

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In the case of methane/n-nonane the eos is fitted to the equilibrium liquid composition only. The data from reference [23] were used. No experimental equilibrium vapor com-position data were found in the temperature range of interest. However, as was shown by Voutsas et al. [10], the binary interaction is much more sensitive to liquid equilibrium fractions than to vapor equilibrium fractions. This can easily be understood, since in the liquid phase mixture specific molecular interactions dominate, while in the vapor phase ideal gas behavior dominates. The parameter kmn was fitted for several temperatures,

which resulted in a temperature dependent parameter. The result is shown in figure (2.2). In figure (2.3) an xp-diagram is shown for mixtures of methane and n-nonane at different temperatures. The calculations have been performed using the temperature-dependent parameter kmn(T ).

For the methane/water system both liquid and vapor equilibrium data were available. The vapor equilibria are taken from Rigby and Prausnitz [24]. The liquid equilibria are taken from an analytical fit through a large compilation of experimental data [25]. Points are taken at the same temperatures as the vapor equilibria. To demonstrate the importance of the liquid equilibria, the parameter kmw is now obtained by fitting the eos either to

the liquid equilibrium composition, the vapor equilibrium composition, or the liquid and vapor equilibrium compositions. The results are shown in figure (2.4). The results clearly indicate the importance of liquid phase equilibria. Not including the liquid equilibria results in a non-linear temperature dependence for the parameter kmw, while fitting to only

liquid equilibria gives practically the same results as fitting to vapor and liquid equilibria. In figure (2.5) the vapor liquid equilibria of methane and water are shown for different temperatures, together with experimental data. The kmw value is taken from the fitted

function, shown in fig. (2.4).

The determination of the binary interaction parameter kij for mixtures of n-nonane and

water is more difficult, since experimental data are scarce. Moreover, for cases for which comparison of experimental liquid/liquid equilibrium data is possible, the data differ [26]. In figure (2.6) the fitted values of knw are shown, together with a proposed temperature

dependency. For mixtures of n-nonane and water there are two different equilibrium liquid phases. One liquid phase is rich in n-nonane, and the other is rich in water. In figure (2.7) these two equilibrium liquid phases are shown in an xT-diagram. The predictions of the CPA eos are obtained using the proposed temperature dependent values of knw. The

agreement of the CPA eos with the experimental data is reasonable. When using this temperature dependent knw a qualitative correct temperature dependence of the n-nonane

solubility in water is obtained (lower curve), while this is not possible when using a constant binary interaction parameter knw [13].

2.3.3

Liquid versus vapor fraction

Equation (2.13) will now be used to introduce some simplifying expressions for a ternary mixture consisting of methane (m), n-nonane (n), and water (w). These expressions will later be used in the description of nucleation and droplet growth. We will consider condi-tions where methane is supercritical while n-nonane and water are subcritical.

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2.3 Mixtures of methane, n-nonane, and/or water 17

Figure 2.2: The binary interaction parameter kmn for methane/n-nonane mixtures as a

function of temperature.

Figure 2.3: xp-diagram of methane/n-nonane mixtures. Results of the CPA eos at several temperatures, together with experimental data [23].

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Figure 2.4: The binary interaction parameter kmw for methane/water mixtures as a

function of temperature, when fitted to either equilibrium liquid fractions, vapor fractions, or liquid and vapor fractions.

Figure 2.5: px-diagram of methane/water mixtures. Results of the CPA eos at several temperatures, together with experimental data [24, 25].

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2.3 Mixtures of methane, n-nonane, and/or water 19

Figure 2.6: The binary interaction parameter kmw for n-nonane/water mixtures as a

function of temperature.

Figure 2.7: xT-diagram of n-nonane/water mixtures. Results of the CPA eos for proposed temperature dependent knw(T ), together with experimental data.

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Phase equilibrium

In the limiting case where the molar fraction of one of the vapor components is zero we have a binary system. At vapor-liquid equilibrium this system has two independent variables which can be freely chosen. By choosing p and T the fractions in both phases are fixed. Equation (2.13) can now be rewritten to give

yieq = feij

psat,i

p , (2.28)

where feij is the enhancement factor of component i in j, which is a function of the pressure

and temperature only. It has been fitted to the data obtained by the CPA eos, using the following function ln(feij) = Bij µ p psat,i − 1 ¶ + Cij µ p psat,i − 1 ¶2 + Dij µ p psat,i − 1 ¶3 . (2.29)

The parameters Bij, Cij, and Dij have been determined in the temperature range between

230 K and 300 K, in 10 K intervals. Subsequently, a temperature-dependent polynomial function was fitted through the parameters. The results are shown in figures (2.8) and (2.9). For the ternary system at three-phase equilibrium there are two different liquids. One is rich in n-nonane and the other is rich in water. Here, we will consider the system which is rich in n-nonane. At three-phase equilibrium the three component system has also two independent variables. Therefore, the water vapor fraction can be written as

yw = Kw,nm(p, T )

psat,w

p , (2.30)

where Kw,nmis again a function of the pressure and temperature only. The subscript stands

for water (solute) in n-nonane and methane. The value of Kw,nm has been determined at

the three-phase equilibria in the temperature range of 220 K to 300 K, in intervals of 10 K. At each temperature, the pressure was increased from 1 bar to 50 bar, in 10 bar intervals. The values of Kw,nm have been fitted to a function similar as equation (2.29).

The temperature-dependent parameters are shown in figure (2.10).

The coefficients of the temperature dependent parameters are listed in table (2.2). Note that in each case the saturated vapor pressure is equal to the saturated vapor pressure of the vapor component of interest, while the critical temperature is taken equal to the critical temperature of methane for each case.

Ternary two phase equilibrium

Next, we consider (two-phase) vapor-liquid equilibria in the ternary system. If the liquid phase is rich in n-nonane it is denoted with the superscript Ln. From equation (2.13) it

then follows that yw = xLwn γLn w (p, T, xLwn)φw,pure(T ) φG w(p, T, yw(p, T, xLwn)) exp à Z p psat,w(T ) vLn w RTdp ! psat,w(T ) p . (2.31)

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2.3 Mixtures of methane, n-nonane, and/or water 21

Figure 2.8: Temperature-dependent parameters for the enhancement factor for saturated n-nonane vapor in methane.

Figure 2.9: Temperature-dependent parameters for the enhancement factor for saturated water vapor in methane.

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a0 a1 a2 a3 a4 a5 ln(Bnm) -121.78 191.35 -113.24 23.737 - -ln(Cnm) 30.855 -225.24 219.24 -61.704 - -ln(−Dnm) 1794.2 -5510.6 6032.2 -2874.9 507.40 -ln(Bwm) -98.794 148.37 -85.582 17.528 - -ln(Cwm) -2110.0 6438.0 -7521.5 3922.6 -767.24 -ln(Dwm) 33568.33 -131491.1 204218.5 -157517.4 60371.55 -9200.210 ln(Bw,nm) -53.439 48.803 -13.058 - - -ln(Cw,nm) -115.25 112.45 -33.252 - - -ln(Dw,nm) - - -

-Table 2.2: Temperature-dependent coefficients of the factors feij and Kw,nm, given by

equation (2.29). The temperature dependency is given by the polynomial ln(Xij) = a0 +

a1(T /Tc,m) + a2(T /Tc,m)2+ a3(T /Tc,m)3+ a4(T /Tc,m)4+ a5(T /Tc,m)5.

The vapor-liquid ternary system has three degrees of freedom and is therefore completely determined for given p, T , and xw. Furthermore, for small fractions of water in liquid

n-nonane (xLn

w ¿ 1) the activity coefficient γwLn and partial molar liquid volume vLwn are

independent of the liquid water fraction. Likewise, for small vapor fractions of water (i.e. high (partial) methane pressure), the fugacity coefficient of water φV

w can be assumed to

be independent of the water vapor fraction. These simplifying assumptions result in yw = xLwn Kw,nm(p, T ) xLn,eqw psat,w p , (2.32) where xLn,eq

w is the equilibrium water fraction at three-phase equilibrium in the liquid which

is rich in n-nonane. The expression is closely related to the equation of Krichevsky and Kasarnovsky [27, 28], that gives the solubility of a gas in a liquid at elevated pressures.

Now, we want to relate the liquid water fraction to its vapor fraction in the case the liquid is rich in water. Similar to equation (2.31), we then have

yw = xLww γLw w (p, T, xLww)φw,pure(T ) φG w(p, T, yw(p, T, xLww)) exp à Z p psat,w(T ) vLw w RTdp ! psat,w(T ) p . (2.33)

In this case xw is close to unity. It is now again assumed that the water activity coefficient

γLw

w is independent of the liquid water fraction. This assumption is based on the Lewis

fugacity rule [1] which reads

fi = xifi,pure. (2.34)

The Lewis fugacity rule holds for fractions xi close to unity. For an ideal gas this rule

reduces to Raoult’s law. Then, assuming a fraction-independent partial liquid volume and a small vapor fraction (i.e., high (partial) methane pressure), one can write

yw = xLww

fewm

xeqwm

psat,w

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2.3 Mixtures of methane, n-nonane, and/or water 23 where xeq

wm is the equilibrium liquid water fraction for the binary methane/water system

at given temperature and pressure. Similarly, for n-nonane the result is yn = xLnn

fenm

xeqnm

psat,n

p . (2.36)

What remains to be determined are the equilibrium liquid compositions. For the binary vapor-liquid equilibria, the methane solubility in either water or n-nonane has been calcu-lated as a function of temperature and pressure, using again the CPA eos. The temperature was varied between 220 K and 300 K, in 10 K intervals, while at each temperature, the pressure was varied between 1 bar and 100 bar, in steps of 1 bar. The three-phase equi-librium water fraction in liquid n-nonane was calculated at 10 K intervals between 220 K and 300 K. At each temperature the pressure was increased from 1 bar to 50 bar in 10 bar intervals. At each temperature the equilibrium solubility was fitted to the following poly-nomial: xeqij = aij + bij µ p psat,j − 1 ¶ + cij µ p psat,j − 1 ¶2 . (2.37)

Subsequently, the parameters aij, bij and cij were again fitted to temperature dependent

polynomials. The results are shown in figures (2.11) to (2.13) and table (2.3).

b0 b1 b2 ln(amn) - - -ln(bmn) -59.095 51.856 -13.297 ln(−cmn) -119.91 105.12 -27.112 ln(amw) - - -ln(bmw) -39.114 23.588 -4.8709 ln(−cmw) -92.900 70.780 -16.931 ln(aw,nm) -31.88 25.10 -5.913 ln(−bw,nm) -87.22 72.74 -17.94 ln(cw,nm) - -

-Table 2.3: Fraction xeqij, given by equation (2.37). The temperature dependent coefficients are given by the polynomial ln(xij) = b0+ b1T /Tc,m+ b2(T /Tc,m)2.

The advantage of having these functions is that the relations between the fractions in the different phases can now be determined by the variables pressure and temperature, which is convenient in the experiments.

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Figure 2.11: Temperature-dependent parameters for the solubility of methane in n-nonane.

Figure 2.12: Temperature-dependent parameters for the solubility of methane in water.

Figure 2.13: Temperature-dependent parameters for the solubility of water in liquid n-nonane.

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2.3 References 25

References

[1] J.M. Prausnitz, R.N. Lichtenthaler, and E. Gomes de Azevedo, Molecular Thermody-namics of Fluid-phase Equilibria, Prentice-Hall, New Jersey, 2nd edition, 1986. [2] H. Orbey and S.I. Sandler, Modeling Vapor-Liquid Equilibria; Cubic Equations of

State and Their Mixing Rules, Cambridge University Press, Cambridge, 1998.

[3] R.C. Reid, J.M. Prausnitz, and B.E. Poling, The Properties of Gases and Liquids, McGraw–Hill Book Company, New York, 1987.

[4] W.G. Chapman, K. E. Gubbins, G. Jackson, and M. Radosz, Ind. Eng. Chem. Res. 29, 1709 (1990).

[5] S.H. Huang and M. Radosz, Ind. Eng. Chem. Res. 29, 2284 (1990). [6] S.H. Huang and M. Radosz, Ind. Eng. Chem. Res. 30, 1994 (1991).

[7] G.D. Ikonomou and M.D. Donohou, Fluid Phase Equilibria 39, 129 (1988).

[8] G.M. Kontogeorgis, E.C. Voutsas, I.V. Yakoumis, and D.P. Tassios, Ind. Eng. Chem. Res. 35, 4310 (1996).

[9] I.V. Yakoumis, G.M. Kontogeorgis, E.C. Voutsas, and D.P. Tassios, Fluid Phase Equilibria 130, 31 (1997).

[10] E.C. Voutsas, G.M. Kontogeorgis, I.V. Yakoumis, and D.P. Tassios, Fluid Phase Equilibria 132, 61 (1997).

[11] I.V. Yakoumis, G.M. Kontogeorgis, E.C. Voutsas, E.M. Hendriks, and D.P. Tassios, Ind. Eng. Chem. Res. 37, 4175 (1998).

[12] E.C. Voutsas, I.V. Yakoumis, and D.P. Tassios, Fluid Phase Equilibria 158, 151 (1999).

[13] E.C. Voutsas, G.C. Boulougouris, I.G. Economou, and D.P. Tassios, Ind. Eng. Chem. Res. 39, 797 (2000).

[14] S.I. Sandler, Models for Thermodynamic and Phase Equilibria Calculations, Marcel-Dekker, New York, 1994.

[15] Y. Boublik, J. Chem. Phys. 53, 471 (1970).

[16] G.A. Mansoori, N.F. Carnahan, K.E. Starling, and T.W. Leland, Jr., J. Chem. Phys. 54, 1523 (1971).

[17] O. Pfohl, S. Petkov, and G. Brunner, Usage of PE - A program to Calculate Phase Equilibria, Herbert Utz Verslag, M 1st edition.

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[18] P.J. Linstrom and W.G. Mallard (eds.), NIST Chemistry WebBook, NIST Standard REference Database Number 69, 2001, http://webbook.nist.gov.

[19] C.-H. Hung, M.J. Krasnopoler, and J.L. Katz, J. Chem. Phys. 90, 1856 (1989). [20] N.B. Vargaftik, Tables on the thermophysical properties of liquids and gases 2nd

edition, Wiley, New York, 1975.

[21] H.R. Pruppacher and J.D. Klett, Microphysics of Clouds and Precipitation, Reidel, Dordrecht, Holland, 1978.

[22] T. Kraska and K.E. Gubbins, Ind. Eng. Chem. Res. 35, 4727 (1996). [23] L.M. Shipman and J.P. Kohn, J. Chem. Eng. Data 11, 176 (1966). [24] M. Rigby and J.M. Prausnitz, J. Phys. Chem. 72, 330 (1968).

[25] A.S. Kertes (ed.), Solubility Data Series, 1989, Volume 38; Hydrocarbons with Water and Seawater, Part II: Hydrocarbons C8 to C36.

[26] A.S. Kertes (ed.), Solubility Data Series, 1987, Volume 27/28; Methane. [27] I.R. Krichevski and J.S. Kasarnovski, J. Am. Chem. Soc. 57, 2168 (1935). [28] B.F. Dodge and R.H. Newton, Ind. Eng. Chem. Res. 29, 718 (1937).

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Chapter 3

Nucleation

Nucleation is the process of phase transition by heterogeneous density fluctuations (i.e. cluster formation). As seen in the previous chapter, the number of phases present in equi-librium depends on the thermodynamic conditions. When these conditions are changed, the number of equilibrium phases can also change. The driving force behind phase change is the difference in chemical potential of the old and the new equilibrium phase. Phase transformation by cluster formation is a statistical process in which the difference in chem-ical potential appears in the exponent of the Boltzmann factor. This will be the subject of section 3.1. The subject of section 3.2 will be the rate at which the clusters grow, which is a kinetic process. Up to this point the description will be in terms of single-component nucleation. From that basis the effects of a high pressure carrier gas on homogeneous va-por phase nucleation will be introduced in section 3.3, while in section 3.4 heterogeneous nucleation will be treated. Then, section 3.5 will be on the specific case of nucleation in supersaturated gas mixtures of methane, n-nonane and water. Finally, the last section of this chapter will be on the nucleation theorem. This theorem provides a way to get infor-mation about the (micro-scale) nucleation clusters from experimental nucleation rate data. The properties of the nucleation clusters play an important role in all nucleation theories. Therefore, the nucleation theorem gives a direct link between theory and experiment. The treatment of nucleation in this chapter will be all but exhaustive. For a complete treatment on nucleation the reader is referred to the book by Kashchiev [1], which also contains many references to specific subjects.

3.1

Cluster distribution

Consider an isolated system at pressure p and temperature T which contains n molecules, and the same system which contains a single cluster containing nL bulk molecules, nS

sur-face molecules and nG gas molecules with corresponding chemical potential. The difference

in energy of the two states of the same system is then (see appendix B)

W = nGµG+ nLµL+ nSµS+ φ + Vclus(p − pclus) − nµG. (3.1) 27

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where Vclusis the total volume of the cluster and φ is the energy contribution of the surface

of the cluster. The energy difference W is called the energy of formation of a cluster. When it is assumed that the surface of the cluster is in thermodynamic equilibrium with the bulk of the cluster, then µS = µL= µclus and the surface molecules nS can be lumped together

with the bulk molecules nL, giving nclus = nL + nS. Furthermore, µG = µI, since the

pressure and temperature are the same. Noticing that n = nG + nS + nL the above

equation can then be simplified to give

W = nclus(µclus− µG) − Vclus(pclus− p) + φ. (3.2) The chemical potential of the cluster is equal to the chemical potential of the bulk con-densed phase at the same temperature, but at a higher pressure. This higher pressure is due to the Laplace pressure in the cluster. Using the Gibbs-Duhem equation the chemical potential of the cluster can be expressed as

µclus(pclus) = µL(p) + Z pclus

p

vclusdp0. (3.3)

Combining this equation with equation (3.2) results in W = −nclus∆µ +

Z pclus

p

Vclusdp0

− Vclus(pclus− p) + φ. (3.4)

Here, the saturation ∆µ is introduced. It has a dominant role in the description of nucle-ation processes, and it is defined as

∆µ = µG− µL. (3.5)

Note that µGis the chemical potential at the supersaturated state, while µLis the chemical

potential at equilibrium, at the same pressure and temperature. For incompressible con-densed phases the integral in equation (3.4) can easily be calculated to give Vclus(pclus−p),

exactly cancelling the third term on the right-hand side of equation (3.4). What remains to be determined is an expression for the surface energy φ(nclus), which depends on the size

of the cluster. Using the capillarity approximation for homogeneous nucleation, φ(nclus) is

expressed as

φ = a0σ(nclus)2/3, (3.6)

where σ is the size-independent surface tension. For spherical clusters with molar density ρclus, a

0 is given by

a0 = (36π)1/3(NAρclus)−2/3. (3.7)

We can now use this result to determine the equilibrium density of clusters of size nclus.

This equilibrium density C(nclus) can be expressed as

C(nclus) = C1exp µ −W (n clus) − W (1) kBT ¶ , (3.8)

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3.1 Cluster distribution 29

Figure 3.1: Cluster density distribution of water at 273.15 K in case of subsaturation (∆µ/(kBT ) = −2), saturation (∆µ/(kBT ) = 0), and supersaturation (∆µ/(kBT ) = 2.

where kB is the Boltzmann constant, C1 is the density of monomers at the given

thermo-dynamic conditions, W (nclus) is the energy needed to form a cluster of size nclus, and W (1)

is the energy of formation of one monomer. By substituting equations (3.4) and (3.6) into equation (3.8) the following density distribution of (incompressible) clusters is obtained:

C(nclus) = C1exp µ ∆µ(nclus − 1) kBT − a0σ((nclus)2/3− 1) kBT ¶ (3.9) In figure (3.1) the cluster density distribution of water at 273.15 K is shown when the system is subsaturated (∆µ/(kBT ) = −2), saturated (∆µ/(kBT ) = 0), and supersaturated

(∆µ/(kBT ) = 2. In the case of subsaturation the cluster density drops quickly with

in-creasing cluster size. This holds up to the case of saturation. In these cases the cluster distributions are real equilibrium distributions. Although the monomer is the energeti-cally most stable configuration, larger clusters do exist due to statistical fluctuations in the (sub)saturated phase. When the parent phase becomes supersaturated a minimum ap-pears in the cluster distribution. This minimum corresponds to a maximum in the energy of formation of a cluster. In order to undergo the transition from the parent phase to the new more stable phase an energy barrier has to be crossed. For homogeneous nucleation from the vapor phase this barrier is entirely due to the formation of an interface between the old and new phase. What is the physical significance of this meta-stable cluster distri-bution? Clearly, the ascending part (large nclus) of the cluster density does not represent

the actual cluster distribution, since any cluster that has crossed the barrier will keep growing, continuously decreasing its energy. The cluster distribution in the ascending part is therefore not determined by statistical fluctuations. However, the descending part in the

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n c l u s- 1 n c l u s n c l u s+ 1 e ( n c l u s) f ( n c l u s- 1 ) e ( n c l u s+ 1 ) f ( n c l u s) J ( n c l u s - 1 ) J ( n c l u s )

Figure 3.2: Schematic picture of the nucleation process according to the Szilard scheme. e and f are the detachment and attachment rates of molecules, respectively. J(nclus) is

the net rate of change of clusters of size nclus to size nclus+ 1.

cluster distribution (small nclus) is still determined by statistical fluctuations. And these

clusters play an essential role in the description of nucleation, since the nucleation rate is largely dependent on the concentration of these subcritical clusters.

3.2

Steady nucleation rate

In this section an expression will be given for the steady nucleation rate J, when the nucleation process occurs by the Szilard scheme. This means that clusters can only change size by adding or removing a single monomer, and the clusters do not interact with each other. This process is schematically shown in figure (3.2). In case of steady-state nucleation J is finite and constant. Becker and D¨oring [2] were the first to give a complete description of the steady nucleation rate, using this method. The steady nucleation rate is given by

J = f (nclus) ˆC(nclus) − e(nclus+ 1) ˆC(nclus+ 1), (3.10) where f (nclus) is the attachment rate of monomers onto a cluster of size nclus, e(nclus+ 1)

is the evaporation rate of monomers from clusters of size nclus + 1, and ˆC(nclus) is the

actual cluster distribution. For nucleation from the vapor phase, the attachment rate of monomers can be defined as the impingement rate of monomers times the area of the cluster of size nclus times a sticking probability. Usually, the sticking probability is taken

equal to unity, rendering the following expression for the attachment rate: f (nclus) = a0(nclus)2/3 µ NA 2πM kBT ¶1/2 p, (3.11)

where M is the molar mass of the molecules, and the impingement rate is obtained from the Maxwell velocity distribution for an ideal gas. Historically, there are two different ways to obtain an expression for the evaporation rate. The first method is called the method by constraint equilibrium. In this case a detailed balance is applied to the meta-stable phase, as if it were a true equilibrium state. This results in

e(nclus+ 1) = f (nclus) C(n

clus)

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3.2 Steady nucleation rate 31 where C(n) is given by equation (3.8). Substituting this expression for e(nclus+ 1) into

equation (3.10) and dividing by f (nclus)C(nclus) gives

J f (nclus)C(nclus) = ˆ C(nclus) C(nclus) − ˆ C(nclus+ 1) C(nclus+ 1). (3.13)

Taking the sum of this expression from nclus = 1 to all the molecules present in the system

nclus = n results in J n X nclus=1 1 f (nclus)C(nclus) = 1 − ˆ C(n + 1) C(n + 1). (3.14)

For large n the second term on the right-hand side becomes equal to zero. Therefore, replacing the sum by an integral, as was done by Zeldovich [3], gives the following expression for the nucleation rate:

J = µZ n 1 dnclus f (nclus)C(nclus) ¶−1 . (3.15)

Due to the minimum in C(nclus) the integral can be approximated by

J = ζf (n∗)C(n∗), (3.16)

where n∗ is the size of the critical cluster, corresponding to the minimum in C(nclus) (or,

equivalently, the maximum in W (nclus))). It reads

n∗ =µ 2a0σ

3∆µ ¶3

(3.17) The well-known Zeldovich factor ζ is given by [3, 4]

ζ = · −1 2πkBT µ d2W dnclus2 ¶ nclus=n∗ ¸1/2 = 1 3 µ a0σ πkBT ¶1/2 n∗−2/3 . (3.18)

We will now discuss the second method for calculating the steady nucleation rate. It is often called the kinetic version of the classical theory. In this case it is assumed that the evaporation rate only depends on the properties of the cluster. Then, detailed balance is applied at the true equilibrium state, resulting in

e(nclus+ 1) = f (nclus) Ceq(nclus)

Ceq(nclus+ 1), (3.19)

where Ceq(nclus) is equal to equation (3.8) at equilibrium (∆µ = 0). Substituting this

expression for e(nclus+ 1) into equation (3.10), and now dividing by f (nclus)Ceq(nclus

exp(nclus∆µ/(k

BT )) results in

J

f (nclus)Ceq(nclus) exp³nclus∆µ kBT

´ =

ˆ C(nclus)

Ceq(nclus) exp³nclus∆µ kBT

´ − ˆ

C(nclus+ 1)

Ceq(nclus+ 1) exp³nclus∆µ kBT

(43)

When the vapor consist of monomers only, then ˆC(1)/Ceq(1) = exp(∆µ/(k

BT )). This is

also a good approximation when the number of dimers, trimers, etc. is negligible compared to the number of monomers. This is true for all the conditions studied, as was shown by Luijten [5]. One can then proceed in a similar way as for the constraint equilibrium to obtain J =   Z n 1 dnclus

f (nclus)Ceq(nclus) exp³nclus∆µ kBT ´   −1 ≈ ζf(n∗)Ceq(n∗) expµ n ∗∆µ kBT ¶ . (3.21)

Noticing that C(n∗) = Ceq(n) exp(n∆µ/(k

BT )), this result is exactly equal to the result

obtained from the constraint equilibrium. Historically, the term W (1) in equation (3.8) was often neglected. This causes the end result of the derivation by applying constraint equilibrium to be a factor exp(∆µ/(kBT )) larger. Therefore, it is essential to include the

term W (1) in equation (3.8). Furthermore, neglecting the term W (1) violates the Law of Mass Action for the cluster distribution, as was demonstrated by Kashchiev [1].

From equation (3.16) we can see that the steady nucleation rate equals the rate at which clusters of size n∗+ 1 are formed, times a correction factor ζ. This correction factor

depends directly on the curvature of the energy barrier at its maximum, indicating the physical meaning of the Zeldovich factor. If clusters of size n∗+ 1 have an energy difference

with clusters of size n∗ that is less than the thermal energy k

BT , they can still evaporate

by crossing the barrier again due to a thermal random ’walk’ in the region of the critical nucleus. The larger the curvature of the the top of the energy barrier is, the closer the value of ζ is to 1.

Combining equations (3.7) and (3.17) with equations (3.9), (3.18), and (3.11) and in-serting them into equation (3.16) gives

J = 1 ρclus µ p kBT ¶2µ 2σ πNAM ¶1/2 exp µ −k∆µ BT ¶ × exp µ (36π)1/3σ kBT (NAρclus)2/3 − 16πσ3 3kBT NA2ρclus 2 ∆µ2 ¶ , (3.22)

where the monomer density C1 is approximated by p/(kBT ). By introducing a

dimension-less surface energy, defined as

θ = a0σ kBT

, (3.23)

it can conveniently be rewritten as J = K exp µ −k∆µ BT ¶ exp µ θ − 274 θ 3 ∆µ2 ¶ , (3.24)

where K is a factor that depends on temperature and pressure only. This expression is known as the internally consistent classical theory (ICCT). By setting φs(1) = 0 the

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3.3 Dilute vapor in a high pressure carrier gas 33 classical nucleation theory (CNT) is obtained. Effectively, this causes the factor exp(θ) in equation (3.24) to disappear.

3.3

Dilute vapor in a high pressure carrier gas

When a vapor nucleates in the presence of a super-critical carrier gas, the carrier gas will not actively participate in the nucleation process, i.e. it does not provide any driving force for nucleation. However, it does influence the nucleation rate by changing the saturation, the surface tension, and, to a lesser extent, the density of the clusters. These effects have been discussed by Luijten [5], Luijten and van Dongen [6] and Peeters et al. [7]. We now consider the same situation as in section (3.1), only with two different components, being a carrier gas and a vapor. We can then write for the energy of formation:

W = nclusv (µclusv − µGv) + nclusg (µclusg − µGg) − Vclus(pclus− p) + φ. (3.25) Here, it has again been assumed that the interface between cluster and environment is in thermodynamic equilibrium with the cluster. Next, we will assume that for dilute vapors µclus

g −µGg = 0 for all cluster sizes. This assumption is based on the fact that the carrier gas

is much more abundant than the vapor. Therefore, the cluster-gas interactions are much more frequent than the cluster-vapor interactions, causing the chemical potential of the gas component in the cluster to be equal to the chemical potential of the gas component in the gas phase. The chemical potential of the vapor component in the cluster can be written as µclusv = µLv + Z pclus p vclusv dp0+ k BT lnµ x clus v γvclus xL vγvL ¶ , (3.26) where µL

v is the pure component chemical potential at temperature T and pressure p. The

last term on the right-hand side is due to the difference in entropy of mixing in the cluster and the bulk liquid. Combining equations (3.25) and (3.26) results in

W = −nclusv ∆µv + nclusv vclusv (pclus− p) − Vclus(pclus− p) +

nclusv kBT lnµ x clus v γvclus xL vγvL ¶ + φs. (3.27)

This result can be further simplified by assuming

Vclus = nclusv vclusv + nclusg vclusg , (3.28) giving

W = −nclus

v ∆µv + φs− nclusg vclusg (pclus− p) + nclusv kBT ln

µ xclus v γvclus xL vγvL ¶ . (3.29)

The first two terms on the right-hand side are exactly equal to the one component result, while the last two terms result from the influence of the carrier gas. If these last two terms

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