Nucleation and Condensation in Gas-Vapor Mixtures of Alkanes and Water

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

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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. Subject headings: condensation / nucleation / gas dynamics / natural gas. 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

geboren te Weert Dit proefschrift is goedgekeurd door de promotoren: prof.dr.ir. M.E.H. van Dongen en prof.dr.ir. H.W.M. Hoeijmakers A clever person solves a problem. A wise person avoids it.

Albert Einstein

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 viii Contents

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 References ...... 112 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

Curriculum Vitae 151 x Contents 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 supersaturated conditions. When the parent phase is not supersaturated, 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 maximum forms an energy barrier for the formation of the new stable phase. The maximum 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

1 2 Introduction

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 Döring [2]. The description of the nucleation rate, as formulated by Volmer and Weber, and Becker and Döring, 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 determining 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 macroscopic 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. When describing the growth of the newly formed stable clusters, one can distinguish 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 4 Introduction

w e t g a s

n o z z l e

i n l e t d r y g a s

w i n g

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 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 appropriate 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. 6 Introduction

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).

[12] F.T. Okimoto and M. Betting, in Laurence Reid Gas Conditioning Conference, 2001. 1.3 References 7

[13] M.J.E.H. Muitjens, Homogeneous condensation in a vapour/gas mixture at high pressures 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). 8 Introduction 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:

m dU = T dS pdV + µ dn , (2.1) − i i i=1 X 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) ≡ − 9 10 Phase equilibrium the change in energy of a closed system can equally well be described by the change in Gibbs energy, m dG = SdT + V dp + µ dn . (2.3) − i i i=1 X 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 = µini. (2.4) i=1 X The total diﬀerential of G is therefore also equal to:

m m

dG = µidni + nidµi. (2.5) i=1 i=1 X X Subtracting this result from equation (2.3) results in the well known Gibbs-Duhem equation

m SdT V dp + n dµ = 0, (2.6) − i i i=1 X 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 heterogeneous 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 component 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

f µ µ0 = RT ln i , (2.8) i − i f 0 µ i ¶ 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

fi φi = , (2.9) yip 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 x φLp µL = µL + RT ln i i = µL + RT ln(γ x ), (2.10) i i,pure f L i,pure i i µ i,pure ¶ 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

L p φ psat,i µL = µ0 + RT ln i,pure + vLdp. (2.11) i,pure i f 0 i Ã i ! Zpsat,i

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 y φGp µG = µ0 + RT ln i i . (2.12) i i f 0 µ i ¶ L G By applying the equilibrium condition µi = µi the following equation is obtained:

p vL ln(y φGp) = ln(φG p ) + ln(x γ ) + i dp (2.13) i i i,pure sat,i i i RT Zpsat,i L The pure component saturated liquid fugacity coeﬃcient φi,pure has been replaced by the G pure component saturated gas fugacity coeﬃcient φi,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 12 Phase equilibrium 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 mixtures 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 contribution 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 similar 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 v α Z = . (2.15) RKS v b − RT (v b) − − 1The term ”cubic” implies an eos which, if expanded, would contain volume terms raised to either the ﬁrst, second, or third power [3]. 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 T ))2, (2.16) − r where Tr is the reduced temperature defined as Tr p= 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 = xixjαij, (2.17) i j X X and b = xixjbij, (2.18) i j X X with αij = √αiαj(1 kij), (2.19) − and b + b b = i j (1 l ). (2.20) ij 2 − ij

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

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

−1

Aj Bi Aj Bi X = 1 + NAρ xiX ∆ , (2.22) Ã i B ! X Xi

Aj Bi where NA is the Avogadro constant, and ∆ is the association strength parameter given by ²Aj Bi ∆Aj Bi = g exp 1 b βAj Bi . (2.23) ij RT − ij · µ ¶ ¸ Here ²Aj Bi and βAj Bi 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

1 3ξ d d 2ξ2 d d 2 g = + 2 i j + 2 i j , (2.24) ij 1 ξ (1 ξ )2 d + d (1 ξ )3 d + d − 3 − 3 i j − 3 µ i j ¶ 14 Phase equilibrium with π ξ = N ρ x dk. (2.25) k 6 A i i i X The molecular diameter d is in this case related to the molar volume parameter via

2πN d3 b = A i . (2.26) i 3 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 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

2 1 1 n p p 2 1 n ρL ρL obj = exp,i − calc,i + exp,i − calc,i . (2.27) 2 n p n ρL  i exp,i i Ã exp,i ! X µ ¶ X   The results of the minimization are shown in table 2.1. After having obtained the pure

2 AB AB a (bar l /mol) b (l/mol) c (-) ² /kB (K) β (-) 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 interaction 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. - -

+ +

Figure 2.1: The water molecule. 16 Phase equilibrium

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 composition 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 conditions where methane is supercritical while n-nonane and water are subcritical. 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]. 18 Phase equilibrium

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]. 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. 20 Phase equilibrium

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 p yeq = fe sat,i , (2.28) i ij p 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 p p 2 p 3 ln(fe ) = B 1 + C 1 + D 1 . (2.29) ij ij p − ij p − ij p − µ sat,i ¶ µ sat,i ¶ µ sat,i ¶ 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 p y = K (p, T ) sat,w , (2.30) w w,nm p where Kw,nm is 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

Ln Ln p Ln Ln γw (p, T, xw )φw,pure(T ) vw psat,w(T ) yw = xw exp dp . (2.31) φG(p, T, y (p, T, xLn )) RT p w w w ÃZpsat,w(T ) ! 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.

Figure 2.10: Temperature-dependent parameters for the factor Kw,nm. 22 Phase equilibrium

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( D ) 1794.2 -5510.6 6032.2 -2874.9 507.40 - − nm 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 + 2 3 4 5 a1(T/Tc,m) + a2(T/Tc,m) + a3(T/Tc,m) + a4(T/Tc,m) + a5(T/Tc,m) .

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 1) the activity coeﬃcient γLn and partial molar liquid volume vLn are w ¿ w w independent of the liquid water fraction. Likewise, for small vapor fractions of water (i.e. V high (partial) methane pressure), the fugacity coeﬃcient of water φw can be assumed to be independent of the water vapor fraction. These simplifying assumptions result in

Ln Kw,nm(p, T ) psat,w yw = xw Ln,eq , (2.32) xw p Ln,eq where xw 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

Lw Lw p Lw Lw γw (p, T, xw )φw,pure(T ) vw psat,w(T ) yw = xw exp dp . (2.33) φG(p, T, y (p, T, xLw )) RT p w w w ÃZpsat,w(T ) !

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

Lw fewm psat,w yw = xw eq , (2.35) xwm p 2.3 Mixtures of methane, n-nonane, and/or water 23

eq where xwm is the equilibrium liquid water fraction for the binary methane/water system at given temperature and pressure. Similarly, for n-nonane the result is

Ln fenm psat,n yn = xn eq . (2.36) xnm p 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 calculated 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 equilibrium 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 polynomial: p p 2 xeq = a + b 1 + c 1 . (2.37) ij ij ij p − ij p − µ sat,j ¶ µ sat,j ¶ 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( c ) -119.91 105.12 -27.112 − mn ln(amw) - - - ln(bmw) -39.114 23.588 -4.8709 ln( c ) -92.900 70.780 -16.931 − mw ln(aw,nm) -31.88 25.10 -5.913 ln( b ) -87.22 72.74 -17.94 − w,nm ln(cw,nm) - - -

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

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. 24 Phase equilibrium

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. 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. 26 Phase equilibrium

[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). 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 equilibrium 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 chemical 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 vapor 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 information 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 surface 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 + V clus(p pclus) nµG. (3.1) − − 27 28 Nucleation where V clus is 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) V clus(pclus p) + φ. (3.2) − − − The chemical potential of the cluster is equal to the chemical potential of the bulk condensed 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

pclus µclus(pclus) = µL(p) + vclusdp0. (3.3) Zp Combining this equation with equation (3.2) results in

pclus W = nclus∆µ + V clusdp0 V clus(pclus p) + φ. (3.4) − − − Zp Here, the saturation ∆µ is introduced. It has a dominant role in the description of nucleation processes, and it is deﬁned as

∆µ = µG µL. (3.5) − Note that µG is the chemical potential at the supersaturated state, while µL is the chemical potential at equilibrium, at the same pressure and temperature. For incompressible condensed phases the integral in equation (3.4) can easily be calculated to give V clus(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 clus 2/3 φ = a0σ(n ) , (3.6) where σ is the size-independent surface tension. For spherical clusters with molar density clus ρ , a0 is given by 1/3 clus −2/3 a0 = (36π) (NAρ ) . (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

W (nclus) W (1) C(nclus) = C exp − , (3.8) 1 − k T µ B ¶ 3.1 Cluster distribution 29

Figure 3.1: Cluster density distribution of water at 273.15 K in case of subsaturation (∆µ/(k T ) = 2), saturation (∆µ/(k T ) = 0), and supersaturation (∆µ/(k T ) = 2. B − B B where kB is the Boltzmann constant, C1 is the density of monomers at the given thermodynamic 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: ∆µ(nclus 1) a σ((nclus)2/3 1) C(nclus) = C exp − 0 − (3.9) 1 k T − k T µ B B ¶ In figure (3.1) the cluster density distribution of water at 273.15 K is shown when the system is subsaturated (∆µ/(k T ) = 2), saturated (∆µ/(k T ) = 0), and supersaturated B − B (∆µ/(kBT ) = 2. In the case of subsaturation the cluster density drops quickly with increasing 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 appears 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 distribution? 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 30 Nucleation

c l u s c l u s J ( n - 1 ) J ( n )

c l u s c l u s f ( n - 1 ) f ( n )

c l u s c l u s c l u s n - 1 n n + 1

c l u s c l u s e ( n ) e ( n + 1 )

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 Döring [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)Cˆ(nclus) e(nclus + 1)Cˆ(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 Cˆ(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:

N 1/2 f(nclus) = a (nclus)2/3 A p, (3.11) 0 2πMk T µ B ¶ 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 C(nclus) e(nclus + 1) = f(nclus) , (3.12) C(nclus + 1) 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 Cˆ(nclus) Cˆ(nclus + 1) = . (3.13) f(nclus)C(nclus) C(nclus) − C(nclus + 1) Taking the sum of this expression from nclus = 1 to all the molecules present in the system nclus = n results in n 1 Cˆ(n + 1) J = 1 . (3.14) f(nclus)C(nclus) C(n + 1) clus − n X=1 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: − n dnclus 1 J = . (3.15) f(nclus)C(nclus) µZ1 ¶ 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 2a σ 3 n∗ = 0 (3.17) 3∆µ µ ¶ The well-known Zeldovich factor ζ is given by [3,4]

2 1/2 1/2 1 d W 1 a0σ ∗−2/3 ζ = − clus2 = n . (3.18) 2πk T dn clus ∗ 3 πk T · B µ ¶n =n ¸ µ B ¶ 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 Ceq(nclus) e(nclus + 1) = f(nclus) , (3.19) Ceq(nclus + 1) 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) clus × exp(n ∆µ/(kBT )) results in J Cˆ(nclus) = clus clus f(nclus)Ceq(nclus) exp n ∆µ Ceq(nclus) exp n ∆µ − kB T kB T ³ ´ Cˆ(nclus³+ 1) ´ (3.20) clus Ceq(nclus + 1) exp n ∆µ kB T ³ ´ 32 Nucleation

eq When the vapor consist of monomers only, then Cˆ(1)/C (1) = exp(∆µ/(kBT )). 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 −1 n dnclus J = clus  1 f(nclus)Ceq(nclus) exp n ∆µ  Z kB T  n∗∆µ ³ ´ ζf(n∗)Ceq(n∗) exp . (3.21) ≈ k T µ B ¶ ∗ eq ∗ ∗ Noticing that C(n ) = C (n ) exp(n ∆µ/(kBT )), 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 kBT , 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 inserting them into equation (3.16) gives

1 p 2 2σ 1/2 ∆µ J = exp ρclus k T πN M −k T × µ B ¶ µ A ¶ µ B ¶ (36π)1/3σ 16πσ3 exp , (3.22) k T (N ρclus)2/3 − 3k T N 2 ρclus2 ∆µ2 µ B A B A ¶ where the monomer density C1 is approximated by p/(kBT ). By introducing a dimension- less surface energy, deﬁned as a σ θ = 0 , (3.23) kBT it can conveniently be rewritten as ∆µ 4 θ3 J = K exp exp θ , (3.24) −k T − 27 ∆µ2 µ B ¶ µ ¶ 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 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 = nclus(µclus µG) + nclus(µclus µG) V clus(pclus p) + φ. (3.25) v v − v g g − g − − 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 µG = 0 for all cluster sizes. This assumption is based on the fact that the carrier gas g − g 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 clus p xclusγclus µclus = µL + vclusdp0 + k T ln v v , (3.26) v v v B xLγL Zp µ v v ¶ L where µv 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 = nclus∆µ + nclusvclus(pclus p) V clus(pclus p) + − v v v v − − − xclusγclus nclusk T ln v v + φ . (3.27) v B xLγL s µ v v ¶ This result can be further simpliﬁed by assuming

clus clus clus clus clus V = nv vv + ng vg , (3.28) giving xclusγclus W = nclus∆µ + φ nclusvclus(pclus p) + nclusk T ln v v . (3.29) − v v s − g g − v B xLγL µ v v ¶ 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 34 Nucleation are small compared to the first two terms, the energy of cluster formation is equal to the one component energy of cluster formation, when the effective surface tension and density are used. Noticing that the first two terms are of equal magnitude for the critical cluster, two sufficient conditions can be formulated for the application of a quasi-one-component expression to be valid. The first condition is obtained by comparing the third term with the second term. The pressure inside the cluster is given by the Laplace pressure pclus = p + 2σ/r, while 2 clus clus clus clus φ = 4πr σ, with the radius of the cluster r given by r = 3/(4π)(ng vg + nv vv ). The first condition then reads: clus clus 2 ng vg clus clus clus clus 1. (3.30) 3 nv vv + ng vg ¿ It states that the volume occupied by the gas in the cluster should be small compared to the total volume of the cluster. The second condition is obtained by comparing the fourth term in equation (3.29) with the first term, which results in xclusγclus k T ln v v /∆µ 1. (3.31) B xLγL v ¿ ¯µ µ v v ¶¶ ¯ ¯ ¯ It states that the energy due¯ to the difference in entrop¯ y of mixing in the cluster and ¯ ¯ the bulk liquid should be small compared to the energy due to the supersaturation of the vapor. When both criteria are fulfilled, the single-component expression of W can be used for the case of a dilute vapor in a carrier gas. In order to determine the nucleation rate, the growth rate of the clusters has to be specified again. We assume that the rate at which clusters change size is completely determined by the rate at which single vapor molecules attach and detach from the clusters. This assumption is analogous to the assumption of equal chemical potential of gas molecules in the cluster and the gas phase for all cluster sizes. The expression for the frequency of vapor monomer attachment is: N 1/2 y p f (nclus) = a (nclus + nclus)2/3 A v , (3.32) v v 0 v g 2πMk T Z µ B ¶ where non-ideal behavior of the gas phase is taken into account by the compressibility factor Z. Provided the criteria given by equations (3.30) and (3.31) are fulfilled, application of equation (3.16) gives the following result for the nucleation rate of a dilute vapor in a carrier gas: 1 y p 2 2σ 1/2 ∆µ J = v exp − v ρclus Zk T πN M k T × µ B ¶ µ A v ¶ µ B ¶ (36π)1/3σ 16πσ3 exp . (3.33) k T (N ρclus)2/3 − 3k T N 2 ρclus2 ∆µ2 µ B A B A v ¶ This is the ICCT version of the nucleation rate of a dilute vapor in a carrier gas. The CNT version is easily obtained by removing the first term in the second exponent, which 3.4 Heterogeneous nucleation 35

a B b

g C

Figure 3.3: Cross-section of a newly condensed phase B onto a condensed phase C, with contact angles α, β, and γ. corresponds to setting the surface energy of a monomer equal to zero. For actual calculations of nucleation rates in the presence of a carrier gas, one more approximation will be made, being ρclus = ρL. This assumption will be less accurate for smaller critical clusters. However, the same holds for the capillarity approximation, stating that the surface tension is independent of the cluster size. Therefore, the validity of the expressions for the ICCT (or CNT) for high supersaturation and very small clusters has been subject of debate in literature for a long time. It is noteworthy that this debate is a consequence of the introduction of a size-independent surface energy. A similar derivation underlying the classical theory could be made with a size-dependent surface energy. Unfortunately, there is no consensus yet about how this size-dependent surface energy should look like. So far, the capillarity approximation is a reasonable hypothesis.

3.4 Heterogeneous nucleation

Heterogeneous nucleation involves the formation of a cluster on a substrate, in contrast to homogeneous nucleation, which involves the formation of a cluster in the parent phase. In nature and technical applications heterogeneous nucleation is more commonly encountered than homogeneous nucleation. The process of heterogeneous nucleation can be described in a similar way as homogeneous nucleation, by determining the energy difference between two states. The first state only contains the vapor and the substrate, while in the second state, part of the vapor molecules have formed a cluster on the substrate. This state is schematically shown in figure (3.3). For an incompressible cluster the difference in energy of the two states (at isobaric and isothermal conditions) is given by

W = nclus∆µ + φ + φ ∆φ , (3.34) − s − s,0 in which φ is the total energy due to the cluster-gas interface, φs is the total energy due to the cluster-substrate interface, and ∆φs,0 is the diﬀerence in total energy due to the substrate-gas interface before and after cluster formation. By comparing equation (3.34) 36 Nucleation with equation (3.4) for incompressible clusters one can immediately derive the necessary condition for the occurrence of heterogeneous nucleation (besides, of course, the presence of a substrate), which is φ + φ ∆φ < φ . (3.35) s − s,0 hom The nucleation process will always follow the path of ’least resistance’, meaning, along the path that involves the smallest amount of energy for cluster formation. Therefore, for nucleation of incompressible clusters, the path of heterogeneous nucleation will only be preferred when the total change in interfacial energy is smaller than the change of interfacial energy in the case of homogeneous nucleation.

3.4.1 Wetting and contact angles Heterogeneous nucleation from the gas phase involves the formation of a cluster of one condensed phase onto another condensed phase. Hence, a region (line) of three-phase contact is formed. In figure (3.3) this line is reduced to a single point. The nature of the three phase-region depends on the interfacial tensions of all the two-phase contacts. In some cases, the newly condensed phase spreads over the surface of the substrate until either it is completely covered with the new phase, or the new phase has reduced to a monolayer. In that case it is said that the new phase completely wets the old phase (substrate). The wetting properties are expressed by the wetting coefficient S = σAC σAB σBC . (3.36) wet − − If the wetting coefficient is positive, phase B will completely wet phase C, since then the total interfacial energy will be the smallest. If Swet is negative, a line of three phase contact will exist, as is shown in figure (3.3). Using Neumann’s triangle [8], which can be obtained by setting the net force on the three-phase line equal to zero, the angles between the phases can be obtained. For instance, the angle α is given by 1 (σBC )2 σAC σAB cos α = . (3.37) 2 σAC σAB − σAB − σAC µ ¶ The angles β and γ are obtained using similar expressions.

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

3.5.1 Supersaturation ratio In this work we are considering the vapor to liquid nucleation when the vapor component(s) is (are) diluted in a high pressure (up to 50 bar) methane gas. In the case of a binary system the nucleation rate can then be calculated using equation (3.33), provided the saturation ∆µv is known. Therefore, the supersaturation ratio S will now be introduced. It is deﬁned as ∆µ S = exp v , (3.38) k T µ B ¶ 3.5 Mixtures of methane, n-nonane, and/or water 37

where ∆µv is again given by

∆µ = µG(p, T, y ) µL(p, T, xeq) = µG(p, T, y ) µG(p, T, yeq). (3.39) v v v − v v v v − v v For diluted vapors only gas-gas and vapor-gas interactions are important. Therefore, from equation (2.12), the supersaturation ratio can be approximated as

yi yip Sij = eq = , (3.40) yi feijpsat,i eq by setting φv = φv for diluted vapors. The actual vapor fraction yv follows from the eq experimental conditions. At given temperature and pressure, the equilibrium fraction yv can be calculated, as can all the other physical properties (see appendix A).

3.5.2 Ternary nucleation In case that both n-nonane and water are supersaturated in the methane carrier gas, different nucleation processes can occur. First, n-nonane and water could form a single nucleus together, since both components would rather be in the liquid phase. However, n- nonane and water are very poorly miscible, as was shown in section 2.3.2. Whether or not this co-nucleation will occur is subject of experimental investigation. Besides homogeneous nucleation, heterogeneous nucleation could be a possible route for the nucleation process. Two different cases can then be distinguished, nucleation of n-nonane onto water and nucleation of water onto n-nonane. First, consider the heterogeneous nucleation of n-nonane onto water. As discussed in section 3.4, heterogeneous nucleation will only take place when the total change of interfacial energy is smaller for heterogeneous nucleation than for homogeneous nucleation. The wetting coefficient of n-nonane on water is shown in figure (3.4) as a function of temperature, for two different pressures. The interfacial tension between the liquid n- nonane and water was approximated using Antonow’s rule [8]

σ = σ σ , (3.41) nw | w − n|

where σw and σn are the pure component surface tensions. As can be seen from figure (3.4), n-nonane completely wets the water substrate at temperatures above approximately 257 K. Below this temperature a three-phase line exists, and the wetting angle of the n-nonane (β in figure (3.3)) can be calculated as a function of temperature. The result is shown in figure (3.5) for two different pressures. The actual wetting angles will probably be smaller than the ones that are calculated here, because of the approximation that was made for the interfacial tension between liquid n-nonane and water. In the real system the presence of methane will most probably cause this interfacial tension to decrease. Therefore, the angles shown in figure (3.5) are maximum angles, which are true contact angles only if the methane would be completely expelled from the real n-nonane/water interface, in the methane/n-nonane/water system. In order to get an idea about the sensitivity of the wetting properties on the n-nonane/water interfacial tension, the wetting coefficient is 38 Nucleation

Figure 3.4: Wetting coeﬃcient of n-nonane on water.

Figure 3.5: Wetting angles of n-nonane on water (βn) and water on n-nonane (βw). 3.5 Mixtures of methane, n-nonane, and/or water 39

Figure 3.6: Homogeneous nucleation (HOM) of n-nonane versus heterogeneous nucleation (HEN) onto water droplets as a function of the radius of the water droplet and the radius of the critical n-nonane cluster, in mixtures of methane, n-nonane and water at 44 bar and 247 K.

shown when this interfacial tension is decreased by 10% (ﬁgure (3.4)). Then, n-nonane is completely wetting on water for temperatures above 240 K. We will therefore assume that n-nonane is completely wetting on water. If this is not the case, contact angles will be small for the cases of interest to us, so complete wetting is still a good approximation. We will now proceed to calculate the interfacial energy diﬀerence for heterogeneous versus homogeneous nucleation when n-nonane is complete wetting on water. The total interfacial energy of the water droplet and the separate critical n-nonane cluster is

2 ∗2 Φn,hom = 4π(σwmrw + σnmrn ). (3.42) When the volume of the critical n-nonane cluster is spherical, and symmetrically covering the water droplet, the total interfacial energy is

2/3 2 3 ∗3 Φn,hen = 4π σnwrw + σnm rw + rn . (3.43) µ ³ ´ ¶ The diﬀerence in total interfacial energy ∆Φn = Φn,hen Φn,hom can now be calculated. Only when it is smaller than zero, n-nonane will nucleate on−to a water droplet in a heterogeneous way. In ﬁgure (3.6) it is shown when heterogeneous nucleation can occur, depending ∗ on the radius rw of the water droplet and the critical radius rn of the n-nonane cluster, at 44 bar and 247 K. If the interfacial tension of the n-nonane/water interface is lowered by 10%, n-nonane will completely wet on water and heterogeneous nucleation is favorable for all cluster sizes. 40 Nucleation

The process of heterogeneous nucleation of water onto n-nonane is much less probable. In ﬁgure (3.5) the contact angle of water on n-nonane is also shown for the same two pressures. In all cases the contact angle is close to 180◦, indicating that water is almost completely non-wetting on n-nonane. When the interfacial tension between n-nonane and water is decreased, the angle becomes even closer to 180◦, up to the point where water becomes completely non-wetting on n-nonane.

3.6 Nucleation theorem

The nucleation theorem relates the (microscopic) composition of the critical cluster to the dependence of the nucleation rate on the macroscopic properties of the supersaturated mixture. It is a theorem that is applicable to all nucleation processes, independent of the model used. In the past years much eﬀort has been devoted to proving the theorem, both for unary as well as for multi-component nucleation. An overview of these proofs and their applications is given in the book by Kashchiev [1]. For multi-component nucleation the theorem reads ∗ ∂W ∗ = ∆ni . (3.44) ∂µi,old − We now write the nucleation rate J in the general form W ∗ J = K exp − , (3.45) k T µ B ¶ where K is a kinetic factor. Using this general expression one can write ∂ ln J ∂ ln K ∂µ ∂µ k T = ∆n∗ v + ∆n∗ g , (3.46) B ∂S − ∂S v ∂S g ∂S · ¸p,T µ ¶p,T µ ¶p,T and likewise, ∂ ln J ∂ ln K ∂µ ∂µ k T = ∆n∗ v + ∆n∗ g . (3.47) B ∂p − ∂p v ∂p g ∂p · ¸S,T µ ¶S,T µ ¶S,T This was considered by Luijten, and Luijten et al. [5,9], for the nucleation of a vapor in a carrier gas. They ﬁnally arrived at the following expressions: ∂ ln J = n∗ + 1, (3.48) ∂ ln S v µ ¶p,T and ∂ ln J = n∗xL + n∗Z∗ + 2 ln fe . (3.49) ∂ ln p − v g g g vg µ ¶S,T From this result the power of the nucleation theorem is immediately apparent. It provides a tool to determine the properties of the critical cluster directly from experimental nucleation rate data, independent of any model. The properties of the critical cluster can then 3.6 References 41 be compared to model predictions. Furthermore, the properties of the critical cluster are of key importance in the models determining the nucleation rate. Therefore, the nucleation theorem gives both information about the critical cluster, and can help to improve theoretical models. Equations (3.48) and (3.49) will be applied to our nucleation rate data, presented in chapter 6.

References

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

[2] R. Becker and W. D¨oring, Ann. Phys. 5, 719 (1935).

[3] Y.B. Zeldovich, Acta Physicochim. (URSS) 18 (1943).

[4] A.C. Zettlemoyer (ed.), Nucleation, Dekker, New York, 1969.

[5] C.C.M. Luijten, Nucleation and Droplet Growth at High Pressure, PhD thesis, Eind- hoven University of Technology, 1998.

[6] C.C.M. Luijten and M.E.H. van Dongen, J. Chem. Phys. 111, 8524 (1999).

[7] P. Peeters, J. Hruby,´ and M.E.H. van Dongen, J. Phys. Chem. B 105, 11763 (2001).

[8] J.S. Rowlinson and B. Widom, Molecular theory of capillarity, Clarendon Press, Ox- ford, 1982.

[9] C.C.M. Luijten, P. Peeters, and M.E.H. van Dongen, J. Chem. Phys. 111, 8535 (1999). 42 Nucleation Chapter 4

Droplet growth

This chapter is on the growth of droplets that are suspended in a mixture of dilute supersaturated vapors in a carrier gas. The growth of droplets (or, similarly, evaporation of droplets) is determined by the rate of mass and energy transfer from and towards the droplet. At steady state, these fluxes are constant. In most cases, the growth of the droplets can be assumed to be quasi-steady. This means that the boundary conditions which define the growth rate vary slowly compared to the time the system needs to adapt to the new conditions. In this way the whole growth process can be divided into small time intervals. In each interval the growth is assumed to be steady, at the given (averaged) conditions of the time interval. The choice of appropriate expressions describing the mass and energy flux depends on the Knudsen number Kn, which is a length-scale parameter. Here, it is defined as the ratio of the mean free path of the molecules λ to the diameter of the droplet. For very large Knudsen numbers the transfer of heat and mass is dominated by the impingement rate of molecules onto the surface of the droplet, and is accordingly described by gas kinetics. This was first done by Hertz [1] and Knudsen [2]. For very small Knudsen numbers the transfer of mass and energy is best described by continuum relations. Droplets that start to grow after the nucleation stage will grow from the region of very large Knudsen numbers to the region of very small Knudsen numbers. Therefore, a model is needed that incorporates the transition region between the Knudsen region and the continuum region. For the description of droplet growth at intermediate Knudsen numbers there exist several approaches. Gyarmathy [3] suggested an interpolating fitting function between the two limiting cases. In this way, growth rates of single component droplets can accurately be described. Fuchs and Sutugin [4] gave correction factors with which the continuum fluxes need to be multiplied in order to describe droplet growth in the transition region. In their approach the Knudsen number for mass transfer differs from the Knudsen number for heat transfer. The correction factors of heat and mass transfer are functions of their respective Knudsen numbers. Vesala and Kulmala [5], and Kulmala et al. [6] used these corrections to describe the growth of uniform multi-component droplets. A different approach to the description of transitional droplet growth is given by the flux-matching

43 44 Droplet growth

c o n t i n u u m r e g i o n K n u d s e n l a y e r

y T , i r K n

- M i

r d

d r o p l e t

T d E

y d , i

y T K n , i K n

Figure 4.1: Schematic view of a growing droplet.

method1. According to this model the droplet is surrounded by a Knudsen layer. The layer has a thickness of the order of the mean free path of the molecules, and mass and heat transfer are governed by gas kinetics within this region. Outside the Knudsen layer the transfer rates are described by continuum relations. In steady state the mass and energy fluxes in both regions are equal and constant, giving rise to a set of equations describing the growth for all values of the Knudsen number. This method was first applied for mass fluxes only by Fuchs [7], and later extended by Fukuta and Walter [8] to include the energy flux. Here, we will adopt the flux-matching model. As will be shown, it can easily be extended to include the growth of layered droplets.

4.1 Homogeneous droplet model

In figure (4.1) the growing droplet is shown schematically. The droplet is assumed to be spherical with radius rd, and to have a uniform temperature Td. The Knudsen layer surrounds the droplet, and has a radius rKn. All the fluxes are defined positive when directed away from the droplet. The velocity distribution at rKn differs from the equilibrium Maxwell velocity distribution due to the mass and energy fluxes. Young [9] assumed a Grad velocity distribution [10] at rKn in his article, describing single-component droplet growth in an ideal inert carrier gas. This model was made applicable to droplet growth in high pressure inert carrier gases by Peeters et al. [11] and Luijten [12], by incorporating real-gas

1In fact, the expressions given by Gyarmathy result from simplifications of a flux-matching model. The correction factors of Fuchs and Sutugin are obtained from solving the kinetic Boltzmann equations, in which some simplifying assumptions are made. 4.1 Homogeneous droplet model 45 effects. We will now further modify this model to include both the effect of dissolution of the carrier gas into the droplet and the condensation of more than one vapor component.

4.1.1 Continuum region In the continuum region the molar and energy fluxes can be calculated from the conservation of mass and energy. For a spherically symmetric steady flow problem the conservation of mass is given by d r2ρGu = 0, (4.1) dr where u is the mean molar (radial) velocity of the gas. In case viscous effects are neglected, the conservation of energy is given by

d M u2 r2ρGu hG + mix + r2q˙ = 0, (4.2) dr 2 µ µ ¶ ¶ G where h is the molar enthalpy of the gas, Mmix is the molar mass of the gas, and q˙ is the heat flux per unit area. The pressure is assumed to be uniform throughout the entire gas phase, which is valid for small Mach numbers (Ma = u/√γRT 1, for a perfect gas). ¿ ¯ ¯ Molar fluxes ¯ ¯ The (radius-independent) total molar flux in the continuum region is given by the integral of equation (4.1). The total molar density is given by

G G G ρ = ρg + ρj , (4.3) j X G G where ρg is the carrier gas density and ρj is the density of vapor component j. The velocity of each component is equal to the sum of the bulk velocity u and its diﬀusive velocity v. Furthermore, by deﬁnition one has

G G ρg vg + ρj vj = 0. (4.4) j X Combining equation (4.3) and (4.4) with equation (4.1) and integrating gives

˙ 2 G 2 G ˙ 2 G M = 4πr ρg (u + vg) + 4πr ρj (u + vj) = Mg + 4πr ρj (u + vj), (4.5) j j X X where M˙ is the total molar flux. When considering diluted vapors the diffusive velocities can be approximated using Fick’s law of diffusion in the form

d ρG ρGv = ρGDG j , (4.6) j j − j dr ρG 46 Droplet growth

G where Dj is the diffusion coefficient of component j in the gas phase. Substituting Fick’s G G law of diffusion into equation (4.5) and introducing the molar vapor fraction yj = ρj /ρ one obtains d M˙ 1 y = M˙ 4πr2ρG DG y . (4.7) − j g − j dr j Ã j ! j X X For small vapor fractions one has (1 y ) 1, and the equation can easily be integrated − j j ≈ from (rKn; yKn,j) to (r = ; y∞,j). The result is: ∞ P G G M˙ = M˙ 4πr ρ D (y∞ y ) M˙ + M˙ . (4.8) g − Kn j ,j − Kn,j ≡ g j j j X X

To derive an expression for the total molar ﬂux of the carrier gas molecules M˙ g we assume that the gas is constantly in equilibrium with the droplet, as was done for the description of the nucleation process in chapter 3.3. The total ﬂux of carrier gas molecules is then determined by the growth rate of the droplet times the equilibrium gas fraction in the droplet, which can be written as

d 4 M˙ = πr3x ρL . (4.9) g −dt 3 d g µ ¶ Clearly, for inert carrier gases the equilibrium liquid molar fraction equals zero, and the ﬂux of carrier gas molecules also becomes zero.

Energy ﬂux The (radius-independent) total energy ﬂux in the continuum region is given by the integral of equation (4.2), giving

M u2 E˙ = 4πr2ρGu hG + mix + 4πr2q˙ (4.10) 2 µ ¶ The heat ﬂux per unit area is given by

dT q˙ = k + h ρGv + h ρGv , (4.11) − dr g g g j j j j X where k is the heat conductivity of the gas mixture. As can be seen from equation (4.11) the heat ﬂux is a combination of heat conduction and the diﬀusion of enthalpy. The total enthalpy of the gas is given by

ρG ρG hG = g hG + j hG. (4.12) ρG g ρG j j X 4.1 Homogeneous droplet model 47

Equations (4.11) and (4.12) are substituted into equation (4.10). After neglecting the kinetic term in equation (4.10) this results in dT E˙ = M˙ hG + M˙ hG 4πr2k , (4.13) g g j j − dr j X where equation(4.5) was used for the molar ﬂuxes. When the enthalpy h is approximated by cpT , where cp is the molar heat capacity, this equation can be integrated from (rKn; TKn) to (r = ; T∞), yielding ∞

E˙ TKn(cp,gM˙ g + cp,jM˙ j) (cp,gM˙ g + cp,jM˙ j) ln − j = − j . (4.14) ˙ ˙ ˙ Ã E T∞(cp,gMg + cp,jMj) ! 4πrKnk − Pj P In the continuum region the energyPtransfer is dominated by conduction. Therefore, the term on the right hand side of equation (4.14) is small. Taking the exponent of this equation, and then the second order Taylor expansion of the remaining exponential results in 1 E˙ = (T + T∞)(c M˙ + c M˙ ) + 4πr k(T T∞). (4.15) 2 Kn p,g g p,j j Kn Kn − j X This is the ﬁnal result for the total energy ﬂux in the continuum region.

4.1.2 Knudsen layer In the Knudsen layer, surrounding the droplet, mass and energy transport are determined by gas kinetics. It is assumed that all the molecules that hit the droplet surface fully thermally accommodate. Therefore, all the droplets that leave the droplet surface are assumed to have a Maxwell velocity distribution at temperature Td:

3/2 2 2 2 M Mi(ξ + ξ + ξ ) f + = i exp ri θi φi , (4.16) i 2πRT − 2RT µ d ¶ µ d ¶ where ξri, ξθi, and ξφi are the molecular velocities of component i in the r, θ, and φ directions. The molecules coming from the Knudsen interface are not at equilibrium. The velocity distribution at the Knudsen interface should correctly describe the net molar and energy ﬂuxes. This is done by the Grad velocity distribution [10,13], which in this case is given by

3/2 2 2 2 2 M q˙ (ξ u) M Mi (ξri u) + ξ + ξ f − = i 1 i ri − i 1 − θi φi + i 2πRT − ρ R2T 2 − 5RT µ Kn ¶ Ã i Kn Ã ¡ Kn ¢! 2 2 2 v (ξ u) M 7 Mi (ξri u) + ξ + ξ i ri − i − θi φi RTKn Ã2 − ¡ 2RTKn ¢!! × 2 2 2 Mi (ξri u) + ξ + ξ exp − θi φi . (4.17) Ã− ¡ 2RTKn ¢! 48 Droplet growth

Because the Knudsen layer has a thickness of the order of the mean free path of the molecules, all the molecules that hit the droplet surface are assumed to have the Grad velocity distribution. The velocity distributions will be used in the derivation of the molar and energy ﬂuxes in the Knudsen layer.

Molar ﬂuxes The total molar ﬂux of component i in the Knudsen boundary layer can be expressed as

M˙ = α M˙ + (1 α ) M˙ − + M˙ − = α M˙ + + α M˙ −, (4.18) i ev,i i − − con,i i i ev,i i con,i i where αev and αcon are the probabilities that a molecule escapes from or sticks to the droplet surface. The total molar ﬂux of evaporating molecules of species i is given by

∞ ∞ ∞ ˙ + 2 G + 2 yd,ip 1 Mi = 4πrd ρd,iξrifi dξridξθidξφi = 4πrd . (4.19) Z √2πMiRTd Z0 −∞Z −∞Z Likewise, the total molar ﬂux of incoming molecules of species i is given by

0 ∞ ∞ ˙ − 2 G − Mi = 4πrd ρKn,iξrifi dξridξθidξφi −∞Z −∞Z −∞Z

2 yKn,ip 1 (u + vi) √πMi 4πrd 1 ≈ − Z √2πMiRTKn − √2RTkn µ 2 ¶ 2 yKn,ip 1 rd ˙ = 4πrd + 2 Mi. (4.20) − Z √2πMiRTKn 2rKn The complete derivation of this equation is given in appendix C.1. Substitution of equation (4.19) and (4.20) results in

2 rd 2 p αev,iyd,i αcon,iyKn,i M˙ i 1 αcon,i = 4πr . (4.21) − 2r2 d Z√2πM R √T − √T µ Kn ¶ i µ d Kn ¶ For very large Knudsen numbers the ratio r /r 1, and the formula of Hertz-Knudsen d Kn ¿ is retrieved [1, 2]. The total molar ﬂux of the carrier gas M˙ g can again be given by equation (4.9).

Energy flux For the calculation of the energy flux in the Knudsen boundary layer it is assumed that the translational kinetic energy of the molecules is uncorrelated to their rotational and vibrational energy. In that case the total energy flux is given by R 5R E˙ = M˙ M˙ − c T + E˙ − + M˙ − c T . (4.22) i − i p,i − 2 d i i p,i − 2 Kn i i X ½³ ´ µ ¶ ¾ X ½ µ ¶ ¾ 4.1 Homogeneous droplet model 49

This equation contains two summands on the right hand side. These are in consecutive order, the energy of the outgoing molecules, and the energy of the incoming molecules. Note ˙ − that the sum is over all components, including the carrier gas. The term Ei represents the kinetic translational energy of the incoming molecules of component i. It can be approximated by

2 2 yKn,ip 2RTKn rd 5 E˙ i = 4πr + RTKnM˙ i + Q˙ Kn,ci , (4.23) − d Z √2πM RT 2r2 2 i Kn Kn µ ¶ where Q˙ ci is the total energy transfer due to conduction of component i. The complete derivations of these equations are given in appendix C.2. When equation (4.20) and (4.23) are substituted into equation (4.22) the ﬁnal expression for the total energy ﬂux in the Knudsen layer is obtained, which reads

R 2 y p cp,i r R E˙ = 4πr2 (T T ) Kn,i − 2 + 1 d T M˙ c + d d − i Z √ − 2r2 d i p,i − 2 i 2πMiRTKn Kn i X µ ¶ X µ ¶ r2 d Q˙ + T M˙ c . (4.24) 2r2 Kn,c Kn i p,i Kn Ã i ! X

Here, the total heat ﬂux by conduction Q˙ Kn,c is given by

Q˙ = Q˙ = 4πr k (T T∞) . (4.25) Kn,c Kn,ci Kn Kn − i X Knudsen layer thickness

To deﬁne the boundary of the Knudsen layer rKn Young suggested the following relation [9]

rKn = rd + 2βKnrd, (4.26) where the Knudsen number is defined as λ Kn = . (4.27) 2rd The fluxes are weakly dependent on the choice of the free parameter β. As a best fit value Young [9] suggests β = 0.75, which we will adopt here. The value of the mean free path of the molecules λ is taken equal to the average distance a molecule has to travel before its velocity is uncorrelated to its initial velocity. This distance depends on the masses of the interacting molecules. For diluted vapors in a carrier gas the dominant molecular interactions are gas-gas (energy transfer) and vapor-gas (mass transfer) interactions. This results into j+1 different values for the mean free path, when j is the number of condensing vapor components. However, it was shown by Peeters et al. [11] that for a diluted vapor a single mean free path, calculated from the gas-gas interactions, suffices, as long as 0.43 < 50 Droplet growth

Mv/Mg < 86. It can easily be checked that this holds for the mixtures under investigation. Therefore, the mean free path is given by

1 λ = λg−g = , (4.28) G 2 ρg dgNAπ√2

where dg is the collision diameter of the carrier gas molecules.

4.1.3 Complete set of equations

In order to define a complete set of equations, the unknowns have to be defined. These are the molar fluxes M˙ j of the j condensing vapor components, the energy flux E˙ , the temperature TKn and molar fractions yKn,j of the gas at the Knudsen interface rKn, and the temperature Td of the droplet. The vapor fractions at the surface of the droplet are determined by the partial saturated vapor pressures of the droplet. These depend on the molar fractions of the vapor components in the droplet, rendering another j 1 unknowns − ( xj +xg = 1). This amounts up to 3j +2 unknowns, where j is the number of condensing vapor components. In the previous section we have derived 2j + 2 equations for the M˙ j molarP fluxes and the energy flux E˙ . Another j equations can be obtained by introducing the droplet growth rate drd/dt, increasing the number of unknowns to 3j + 3. The j equations are obtained by applying conservation of mass to the droplet, rendering

d 4 M˙ = πr3x ρL . (4.29) j −dt 3 d j µ ¶ Finally, the equation to make the set complete is obtained by coupling the molar flux of all the components to the energy flux. This can easily be done, since the energy flux originates from the latent heat of condensation of all the vapor components. When the heat release due to the dissolution of the carrier gas into the droplet is neglected one can write:

d 4 4 E˙ = πr3x ρLhL = M˙ hL + πr3x ρLh˙ L. (4.30) dt 3 d j j j j 3 d j j Ã j ! j j X X X When considering (quasi) steady growth, the droplet temperature is constant and the second summand on the right hand side equals zero. The equation above can then be rewritten to give

E˙ = M˙ hG hL + M˙ hG = M˙ L hG , (4.31) − j d,j − j j d,j − j j − d,j j j j X ¡ ¢ X X ¡ ¢ where Lj is the latent heat of condensation of vapor component j. 4.2 Layered droplet model 51

c o n t i n u u m r e g i o n K n u d s e n l a y e r

y T r , i K n

l i q u i d l a y e r

- M i r c o r e d l L l - M i

T r d d c

x E d c , i

x d l , i

y d l , i

y T K n , i K n

Figure 4.2: Schematic view of the growth of a layered droplet.

4.2 Layered droplet model

In this section the growth rate of a droplet which consists of two liquids is described. The droplet is assumed to have a layered structure. Liquid (2) covers the spherical core with a layer of uniform thickness. The core consists of liquid (1). The layered droplet is shown schematically in figure (4.2). The core has a radius rdc and the layer has an outer radius rdl. The energy flux within the droplet is neglected, and the pressure and temperature are assumed to be uniform. The molar and energy fluxes outside the droplet are described similarly as for the homogeneous droplet. The main difference with the homogeneous droplet is the occurrence of a molar flux inside the droplet through the liquid layer. We will limit the study to cases in which the main components of the liquid core are diluted in the liquid layer.

4.2.1 Molar fluxes in liquid layer The molar fluxes through the liquid layer are again controlled by diffusion, as was the case in the continuum region in the gas phase. The molar flux of vapor component k through the liquid layer is given by r r M˙ Ll = 4π dc dl ρLlDLl xLl xLl . (4.32) k r r k rdl,k − rdc,k dc − dl ¡ ¢ The vapor components k are components which are diluted in the liquid layer, and which are the main components of the liquid core. The m main (abundant) components of the liquid layer are assumed to be in constant equilibrium with the liquid core. Therefore, the 52 Droplet growth molar flux of the m components in the liquid layer is given by

d 4 M˙ Ll = πr3 xLcρLc , (4.33) m −dt 3 dc m µ ¶ similar to equation (4.9). The sum of the vapor components k and m is equal to the total number of condensing vapor components j. When the carrier gas molecules enter the droplet, their molar ﬂux is now given by

d 4 4 M˙ = πr3 xLcρLc + π r3 r3 xLlρLl , (4.34) g −dt 3 dc g 3 dl − dc g µ ¶ ¡ ¢ in which the ﬁrst term on the right hand side denotes the molar gas ﬂux through the liquid layer.

4.2.2 Complete set of equations

In the layered droplet model we have j unknown molar vapor fluxes M˙ j in the gas phase, ˙ Ll k unknown molar fluxes Mk in the liquid layer, and a total of 3j + k 2 unknown molar Ll Lc − fractions yKn,j, yrdl,j, xrdl,j, and xk . The term 2 originates from the fact that the sum of all the fractions in the liquid layer and in the liquid− core are in each case equal to (1 x ). − g Furthermore, we have the unknown temperatures TKn and Td, the unknown growth rates of the inner core drdc/dt and outer layer drdl/dt, and the unknown energy flux E˙ , rendering a total of 4j + 2k + 3 unknowns. The molar fractions in the liquid layer at the liquid core are the equilibrium fractions at the given conditions of the liquid layer. As before there are 2j+2 equations for the molar and energy fluxes in the gas phase, and now another k equations for the molar fluxes in the liquid layer. By applying conservation of mass to the droplet core,

d 4 M˙ Ll = πr3 xLcρLc , (4.35) k −dt 3 dc k µ ¶ another k equations are obtained, setting the total number of equations equal to 2j+2k+2. At the surface of the droplet there is continuity of chemical potential for each component. This is also known as local equilibrium between the gas phase and the liquid phase. It results into j equations that relate the molar fraction in the liquid layer at the surface of Ll the droplet xrdl,j to the molar fraction in the gas phase at the surface of the droplet yrdl,j. Application of conservation of mass of component j in the liquid layer, results in

d rdl M˙ Ll M˙ = xLl(r)ρLl4πr2dr. (4.36) j − j dt j Zrdc The derivation of the radius dependent fraction of component j in the liquid layer is given in appendix C.3. The total number of equations is now equal to 4j + 2k + 2. The equation that makes the set complete is again obtained from the coupling of the molar flux to the 4.3 Mixtures of methane, n-nonane and/or water 53 energy flux. The release of heat due to the dissolution of components into the inner core, and the dissolution of gas into the droplet is neglected. If it is then assumed that for all the vapor components the latent heat of condensation is released as soon as they are part of the droplet, equation (4.31) can again be used to describe the coupling between the molar flux and the energy flux. Effectively this means that the release of heat due to the dissolution of the components of the liquid core into the liquid layer is equal to their latent heat of condensation. Then, no additional heat is released when these components diffuse through the liquid layer and reach the inner core.

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

In the following section the application of the growth model to mixtures of methane, n- nonane, and/or water will be discussed. In the experiments we do not have a single growing droplet, but a cloud of equi-sized growing droplets. For all the cases studied here, these droplets are at least ﬁfty times the droplet radius apart. Therefore, the droplets in the cloud can be considered to grow independently [3, 14]. The (slowly varying) pressure is recorded during the experiment. The temperature and compressibility factor are calculated from the pressure signal, using an equation of state (see chapter 5.2). In the experiment binary mixtures of supersaturated n-nonane or water in methane, and ternary mixtures of supersaturated n-nonane and water in methane are used. In section 4.3.1 the growth calculations for the binary mixtures will be discussed, followed by the growth calculations for the ternary mixtures in section 4.3.2.

4.3.1 Binary mixtures For the experiments with the binary mixtures there is only one vapor fraction. The initial vapor fraction is obtained from the experiment. As the droplets grow, this vapor fraction is slowly depleted. Therefore, the vapor fraction is a function of time, which can be expressed as t Z∞(t0)RT∞(t0) 4 3 L , y∞ (t) = y∞ (t ) n (t ) πr (t )x (t )ρ (t ) M˙ dt , (4.37) ,v ,v 0 − d 0 p(t ) 3 d 0 v 0 0 − v 0 µ Zt0 ¶ The initial radius rd(t0) depends on the supersaturation ratio Sv of the vapor at which the nucleation of the droplet took place. Muitjens [15] showed that a cluster of molecules can be considered a stable droplet when they contain twice the number of molecules of the critical cluster. Using equation (3.17) we then have

1/3 2σ rd(t0) = 2 L . (4.38) xvρ RT ln Sv The vapor fraction at the surface of the droplet is obtained using the enhancement factor, which was derived in chapter 2.3.3. This results in 2σ y (t) = fep exp , (4.39) rd,v sat,v x ρLRT r µ v d ¶ 54 Droplet growth

where the exponential term corrects for the curvature of the droplet. At time tn the system of equations is solved. Then the droplet radius is adjusted according to dr r = r + (t t ) d , (4.40) d,tn+1 d,tn n+1 − n dt · ¸t=tn and the procedure is repeated again. All the physical properties that are directly related to the droplet (like its density) are calculated at the temperature of the droplet Td (for each time step). All the other physical properties (like the thermal conductivity) are calculated at an intermediate temperature Tm. According to Hubbard et al. [16] a one-third rule is appropriate: 1 T = (2T + T ) . (4.41) m 3 d ∞ 4.3.2 Ternary mixtures For the droplet growth in the ternary mixtures we can distinguish two cases. First we will consider the case in which the water is the dominant nucleating component (Jw > Jn). As discussed in chapter 3.5.2, the supersaturated n-nonane will then nucleate onto the water droplet, as soon as it has reached a certain minimum size. Then the droplet will grow further, having an inner core consisting mainly of water and a surrounding liquid layer, which mainly consists of n-nonane. This growth mechanism can easily be simulated by only allowing for the dissolution of n-nonane into the water droplet, until its radius is equal to the critical radius of the n-nonane at the given supersaturation ratio Sn. Then the n-nonane is allowed to grow onto the water droplet. The n-nonane vapor fraction at the surface of the droplet is now determined by

Ln fenm psat,n 2σ yr ,n = x exp , (4.42) dl rdl,n xeq p xeq ρLnRT r nm µ nm d ¶ which is an approximation based on the continuity of the chemical potential at the phase boundary, derived in chapter 2.3.3. The exponential term is again included to correct for the curvature of the droplet. The approximation for the continuity of the chemical potential of the water component is in this case given by

Ln psat,w 2σ yr ,w = x Kw,nm exp , (4.43) dl rdl,w p xeq ρLnRT r µ nm d ¶ where Kw,nm is again given in chapter 2.3.3. Prior to the heterogeneous nucleation of the n-nonane onto the water droplet, the growth can be described as binary, where the mass ﬂux of the n-nonane can be approximated by its equilibrium concentration in the liquid water droplet, rendering dr 4 M˙ = d πr3xLwρLw . (4.44) n − dt 3 d n µ ¶ The vapor depletion and the initial radius of the droplet are calculated in a similar way as was done for the growth in the binary mixtures. 4.3 References 55

The case in which n-nonane is the dominant nucleating component (Jn > Jw) is described in the same manner as the case in which water is the dominant nucleating component. Only now, the water will not heterogeneously nucleate onto the n-nonane droplet (see chapter 3.5.2). Because the water does not nucleate onto the n-nonane droplets, one can argue that there will be formed some water nuclei, since the water vapor is in a supersaturated state. However, the amount of these nuclei will be negligible compared to the amount of n-nonane droplets. Furthermore, once a water nucleus has formed, it will immediately be covered by a layer of liquid n-nonane.

References

[1] H. Hertz, Ann. Phys. 17, 177 (1882).

[2] M. Knudsen, Ann. Phys. 47, 697 (1915).

[3] G. Gyarmathy, The spherical droplet in gaseous carrier streams: review and synthesis, in Multiphase Science and Technology, volume 1, pages 99–279, Springer, Berlin, 1982.

[4] N. A. Fuchs and A. G. Sutugin, Highly dispersed aerosols, Ann Arbor Science, Ann Arbor, Michigan, 1970.

[5] T. Vesala and M. Kulmala, Physica A 192, 107 (1993).

[6] M. Kulmala, T. Vesala, and P. E. Wagner, Proc. R. Lond. A 441, 589 (1993).

[7] N.A. Fuchs, Phys. Z. Sowjet 6, 224 (1934).

[8] N. Fukuta and L.A. Walter, J. Atmos. Sci. 27, 1160 (1956).

[9] J.B. Young, Int. J. Heat Mass Transfer 36, 2941 (1993).

[10] H. Grad, Principles of the kinetic theory of gases, in Encyclopaedia of Physics, volume 12, pages 205–294, Springer, Berlin, 1958.

[11] P. Peeters, C. C. M. Luijten, and M.E.H. van Dongen, Int. J. Heat Mass Transfer 44, 181 (2001).

[12] C.C.M. Luijten, Nucleation and Droplet Growth at High Pressure, PhD thesis, Eind- hoven University of Technology, 1998.

[13] I.I. Kolodner, On the application of the Boltzmann equations to the theory of gas mixtures, Phd thesis, New York University, New York, 1950.

[14] J. M. Tishkoﬀ, Int. J. Heat Mass Transfer 22, 1407 (1979). 56 Droplet growth

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

[16] G.L. Hubbard, V.E. Denny, and A.F. Mills, Int. J. Heat Mass Transfer 18, 1003 (1975). Chapter 5

Wave tube experiments

In the previous chapters phase equilibria and processes that bring a system towards phase equilibria have been discussed. For vapor/liquid equilibria these processes are nucleation of droplets and the subsequent growth of these droplets. In this chapter we will discuss the experimental setup with which these processes have been studied. In the experiment use is made of the nucleation pulse method. The principles of the nucleation pulse method will be the subject of section 5.1. In section 5.2 it will be shown how the nucleation pulse method can be applied using a (modified) shock tube. The droplets formed in the tube are detected using an optical setup. This setup is described in section 5.3. The accuracy of nucleation rate data is, to a very large extent, determined by the accuracy to which the composition of the gas/vapor mixture is known. Therefore, much attention is focussed on preparing the gas/vapor mixture for an experiment. In collaboration with Hruby´1, the existing method of mixture preparation was modified. The purpose of this modification is twofold. Besides obtaining a higher accuracy in the desired composition of the mixture, now, a gas/vapor mixture containing two different vapor components can be prepared (instead of just one vapor component). The setup and method of mixture preparation will be discussed in section 5.4. In the final section of this chapter the experimental procedure will be outlined.

5.1 Nucleation pulse method

The main purpose of the nucleation pulse method is separating the nucleation process and the process of droplet growth. In this way, a cloud of droplets can be created in which all the droplets have the same size. This facilitates the characterization of the cloud of droplets, i.e. the determination of the number density and the size of the droplets. Furthermore, steady nucleation rates can be determined. The nucleation pulse method was ﬁrst described by Allard and Kassner [1]. The basic idea of the method is to create a small pulse in the nucleation proﬁle, as

1Dr. J. Hruby´ from the Institute of Thermomechanics in Prague resided as an invited guest at the Gas Dynamics Department of the University of Technology in Eindhoven from January 1999 to February 2000.

57 58 Wave tube experiments is shown in figure (5.1). During the nucleation pulse the nucleation rate is orders of magnitude larger than after the pulse. The new droplets formed in the nucleation pulse need to grow to an optically detectable size after the pulse. Therefore, the system still has to be in a supersaturated state after the nucleation pulse. Hence, a small amount of new droplets will be formed after the pulse. As long as the nucleation rate after the pulse is orders of magnitude smaller than during the pulse, the number of droplets formed after the pulse is negligible compared to the number formed during the pulse. The typical time needed for the nucleation process to reach a steady state is of the order of 10−8 s [2]. In order to achieve a steady nucleation rate, the duration of the nucleation pulse has to be several times longer than this relaxation time. However, the pulse cannot be too long, since the nucleation process will then deplete the vapor and the supersaturation can no longer be considered constant. Therefore, the duration of the nucleation pulse needs to be chosen such that there is sufficient time to establish steady nucleation, but not enough time to alter the supersaturation by depleting the vapor. The nucleation pulse profile can be obtained by making a similar saturation profile, which is also shown in figure (5.1). During the nucleation pulse the system is in a state of high supersaturation. To quench the nucleation process the supersaturation is lowered. As can be seen in figure (1.1), the nucleation rate is an extremely steep function of the supersaturation. Therefore, by decreasing the supersaturation by, for example, a factor two, the nucleation rate will decrease by several orders of magnitude, effectively terminating the formation of new droplets. As shown in chapter 3.5, the supersaturation is given by y S = , (5.1) yeq which is the ratio of the actual vapor fraction to the equilibrium vapor fraction at the given thermodynamic conditions. Therefore, the pulse in the supersaturation can either be generated by making a pulse in the actual vapor fraction y, or a ’dip’ (negative pulse) in the equilibrium vapor fraction yeq. The equilibrium vapor fraction yeq is exponentially dependent on the temperature, making it most convenient to create a dip in the temperature of the system. This dip in the temperature can be obtained by exposing the system to an adiabatic pressure change, as shown in figure (5.2). The experimental task is therefore to create a pressure history as shown in figure (5.2). Until now, this has been achieved in two different ways. One is using so-called pulse-expansion cloud chambers, in which the desired pressure history is obtained by either fast moving pistons or by opening valves to (de-)pressurized neighboring chambers [3–5]. A combination of the two is also possible. Another way to obtain the desired pressure history is by the use of a (modified) shock tube, as will be shown in the next section.

5.2 Pulse-expansion wave tube

The idea of using a shock tube to obtain the nucleation pulse was ﬁrst introduced by Peters [6]. Later, the method was slightly modiﬁed by Looijmans et al. [7, 8], resulting in 5.2 Pulse-expansion wave tube 59

l o g J S

t t

Figure 5.1: Schematic nucleation pulse proﬁle and saturation proﬁle.

P T

t t

Figure 5.2: Schematic pressure proﬁle and resulting adiabatic temperature proﬁle. 60 Wave tube experiments

4 1

H P S L P S

Figure 5.3: Schematic x-t plot of the wave patterns in a shock tube.

the pulse-expansion wave tube that was also used in this study. Alternative ways to obtain a nucleation pulse using a shock tube are given by Hruby´ [9].

5.2.1 Pressure profile A standard shock tube consists of a high pressure section (HPS) and a low pressure section (LPS), which are separated by a diaphragm. When the diaphragm is burst, the gas in the LPS will be compressed and the gas in the HPS will be expanded. In figure (5.3) the wave propagations in a standard shock tube are schematically shown in an x-t plot. Four regions of constant thermodynamic states can be distinguished. The uniform thermodynamic state in the HPS (region 4) is disturbed by the expansion fan, resulting in the uniform thermodynamic state in region 3. Region 3 and 4 can be related to each other by using the so-called Riemann invariants u Γ, where u is the velocity of the gas and § 1 Γ = dp. (5.2) ρc Z Here, c is the velocity of sound in the gas. The Riemann invariants are obtained from the solution of the one-dimensional Euler equations for inviscid isentropic flow. The following characteristic equations can then be obtained: ∂ ∂ + (u c) (u Γ) = 0. (5.3) ∂t § ∂x § µ ¶ These equations state that u + Γ and u Γ are constant along the trajectories in the x-t plane with slope dx/dt = u + c and dx/d−t = u c, respectively. − 5.2 Pulse-expansion wave tube 61

t t 6

5 7 e 2 1

d x c

a 3

4 2 1

p x

H P S L P S

Figure 5.4: Right: Schematic x-t plot of the wave patterns in the pulse-expansion wave tube. Left: resulting pressure history at the end wall of the HPS.

The initial uniform thermodynamic state in the LPS (region 1) is changed to the different uniform state in region 2 after the passage of the shock wave. The states in these two regions are connected by the Rankine-Hugoniot equations, which result from solving the integral conservation laws for mass, energy, and momentum across the shock front. The (hot) compressed gas in region 2 is separated by the (cold) expanded gas in region 3 by a contact discontinuity. Across this discontinuity the pressure and velocity of the gas are continuous. A full treatment of this shock tube problem is given in many standard textbooks on gas-dynamics, see for instance [10].

The relevant wave patterns in our modified shock tube, or pulse-expansion wave tube, are schematically shown in figure (5.4). Not shown are the wave-wave interactions, nor the reflections at the contact discontinuities, nor the reflection of the weak shock at the widening. The modification of the tube consists of the local widening in the LPS of the tube, just to the right of the diaphragm. Furthermore, the total length of the LPS is large compared to the total length of the HPS, for reasons which will become apparent below. The region in the xt-diagram where the shock wave encounters the increase in the diameter of the LPS has been enlarged in figure (5.4). Five different regions of uniform thermodynamic conditions can then be distinguished. The conditions in region 1 are the initial conditions of the gas in the LPS, while the conditions in region 2 are obtained from the standard shock tube problem, as discussed above. Likewise, the relations between regions 2 and 5, 1 and 7, and 6 and 7, are similar to the relations discussed for the standard shock tube problem. What remains to be specified are the relations between the 62 Wave tube experiments conditions in region 5 and 6. These relations are given by the continuity of mass flux

ρ5u5A5 = ρ6u6A6, (5.4) and Bernoulli’s equation 1 1 u2 + h = u2 + h , (5.5) 2 5 5 2 6 6 together with the assumption of conservation of entropy. In these equations A denotes the area of the cross-section of the tube, and h is the (specific) enthalpy. At the increase of the diameter of the tube the shock reflects as a small expansion fan in the direction of the HPS, while the shock is transmitted in the direction of the LPS. When the diameter of the tube decreases again a small compression wave is reflected into the HPS, while the shock continues in the LPS. The dependency of the depth of the pressure pulse on the ratio of the tube cross-section is described by Looijmans et al. [8]. In the plot on the left-hand side of figure (5.4) the resulting pressure history at the end-wall of the HPS is shown. Before the start of an experiment, the pressure in the high pressure section (HPS) is typically twice as large as in the LPS. The experiment starts by bursting the diaphragm. As a result a strong expansion fan will travel into the HPS and a shock wave will travel into the LPS. The shock wave travelling into the LPS first encounters the local widening of the tube, resulting in the weak expansion fan travelling into the HPS. Next, the shock wave encounters the narrowing in the tube, resulting in the weak compression wave propagating into the HPS. The waves travelling into the HPS will reflect at the end wall. When the strong expansion fan reaches the end wall (a), the pressure at the end wall will start to drop, as can be seen in the left plot of figure (5.4). The pressure will keep decreasing until the tail of the strong expansion fan has reflected at the end wall (b). The pressure at the end wall will then remain constant until the weak expansion fan reflects at the end wall, causing an extra decrease in the pressure (c-d). The pressure is increased again when the weak compression wave reflects at the end wall (e). The pressure will then remain constant for a fairly long time until the strong shock wave reflected from the end wall of the LPS (not shown in figure (5.4)) hits the end wall of the HPS, increasing the pressure and temperature again, and terminating droplet growth. Hence, the desired dip in the pressure profile c-d-e is obtained by the local widening in the LPS, and the duration of the experiment is determined by the length of the tube. In our case the HPS has a length of 125 cm. The length of the LPS was recently extended from 642 cm to 923 cm. The tube has an inner diameter of 36 mm. The local widening is positioned 14 cm behind the diaphragm and has a length of 15 cm. The inner diameter of the widening is 41 mm. Note that the depth of the pressure dip can be varied by varying the diameter of the widening. The duration of the dip can be varied by varying the length of the widening.

5.2.2 Bursting of the diaphragm The diaphragms separating the HPS from the LPS are made of polyester (polyethy- lenterephtalate), having different thicknesses for different pressure differences. The di- 5.2 Pulse-expansion wave tube 63 aphragms are mounted between the HPS and LPS in a contraption specially designed for the bursting of the diaphragm [11]. In the contraption the diaphragm presses against a metal wire. The wire forms a nearly complete circle of a radius of 40 mm. The wire can be heated by sending an electric current through it. The heating of the wire weakens the diaphragm, causing it to burst. The remains of the diaphragm disappear into the LPS, and hence do not disturb the experimental observations in the HPS. Because the diaphragm ruptures at the position of the metal wire, there are no obstructions left in the diaphragm section after the diaphragm has burst. The current through the metal wire is obtained by means of a transformer. By reversing the current in the first coil of the transformer, a current is induced in the second coil, which is connected to the metal wire. In this way, a strong current pulse is produced, which causes the diaphragm to burst almost instantly. The opening time of the diaphragm is approximately 100 µs [11].

5.2.3 Thermodynamic state The pressure at the end wall of the HPS is recorded using a calibrated Kistler 603B dynamic pressure transducer, in combination with a Kistler 5001 charge amplifier. The transducer is mounted in the tube at the end wall of the HPS. The pressure signal is recorded with a Le Croy 6810 waveform recorder. The temperature is calculated from the pressure signal assuming an isentropic adiabatic expansion: T (t ) p(t ) α i+1 = i+1 (5.6) T (t ) p(t ) i µ i ¶ For a calorically ideal gas the coefficient is given by α = (γ 1)/γ, where γ is the ratio of − the heat capacities, γ = cp/cv. For non-ideal gasses the coefficient α reads

RZ T ∂Z α = 1 + . (5.7) c Z ∂T p Ã µ ¶p!

For a real gas the compressibility factor Z and the molar heat capacity cp are functions of temperature and pressure. They can be calculated using an appropriate equation of state. In our case, the vapor fractions are always very small. Therefore, the equation of state proposed by Sychev [12] for pure methane is used for the adiabatic calculations. The equation is given in appendix A. After the nucleation pulse, the formed droplets will start to grow. As a consequence, the released latent heat of condensation will increase the temperature of the mixture, and the calculated isentropic temperature has to be corrected. Using the growth calculations described in chapter 4, the temperature is corrected by adding n (t )Z(t )RT (t ) E˙ dT = d 0 0 0 dt. (5.8) p(t0) cp The total temperature correction is always small for our experiments (less than 1 K), due to the high pressure of the carrier gas. 64 Wave tube experiments

Figure 5.5: Time-dependent proﬁles of the pressure, supersaturation, and nucleation rate of experiment 108.

The initial composition of the gas/vapor mixture is determined prior to the experiment. In ﬁgure (5.5) the time-dependent proﬁles of the pressure, supersaturation and nucleation rate are shown, of an actual experiment with a water/methane mixture. The supersaturation is calculated using the equilibrium properties given in chapter 2.3.3. The nucleation rate is calculated using the CNT expression, derived in chapter 3.3. As can be seen, the supersaturation in the pressure dip is about twice as large as the supersaturation after the pressure dip. As a result, the nucleation rate in the pressure dip is about 10 orders of magnitude larger than the rate after the pressure dip. The duration of the nucleation pulse is 0.368 ms, while the duration of the following low pressure plateau is 23 ms. Hence, the number of droplets per m3 formed during the (low) pressure plateau, 5 Jplateautplateau = 2.3 10 , is indeed negligible compared to the number of droplets formed per m3 in the nucleation× pulse, which is J t = 3.5 1014. pulse pulse ×

5.3 Droplet detection

The droplets at the end wall of the HPS are detected using an optical setup. A schematic view of the setup is shown in figure (5.6). The laser used in the setup is an argon-ion laser. It produces two light beams of different wavelengths, being 488.0 nm and 514.2 nm. In the experiments, the 514.2 nm beam is used, and the 488.0 nm beam is blocked by the filter. As the light leaves the laser, it first passes the filter F, which blocks the 488.0 nm wavelength light beam. Then, the light passes the polarizer P1, polarizing the light in the vertical direction. Before the light enters the tube, part of it is deflected by a semi- 5.3 Droplet detection 65

Figure 5.6: Schematic view of the optical setup at the end wall of the HPS. permeable mirror. The intensity of the deflected part is measured by a Telefunken BPW 34 photodiode (D1). The light enters the tube through the side wall, 5 mm from the end wall of the HPS. The small windows are positioned slightly inclined in the side wall of the tube, in order to separate multiple reflections in the windows. The transmitted light leaving the tube first passes the pinhole d3, removing the forward scattered light. Then the light passes through a lens before it enters a glass plate. The glass plate is used to decrease the intensity of the transmitted light. This is achieved by selecting the second internal reflection from the glass plate, and focussing this light beam on the photodiode D2 (Telefunken BPW 34). The focussing of the light on the photodiode is done by both lens L3 and L4. The light scattered by the droplets leaves the tube through the end wall, ◦ which is made of glass. Lens L1 focusses the light that is scattered at an angle of 90 onto the Hamamatsu 1P28A, red extended photomultiplier. In total, three optical signals are measured during the experiment. These are the reference signal, the transmission and the light scattered at an angle of 90◦. The reference signal is used to filter out any fluctuations in the transmitted and scattered light that are caused by the laser. From the scattered signal the radius of the droplets is determined. Once the radius of the droplets is known, the measured transmission is used to determine the number density of the droplets. All three optical signals are recorded with the same Le Croy 6810 waveform recorder, that also records the pressure signal at the end wall of the HPS.

5.3.1 Light scattering by dielectric particles The theoretical description of light scattered by small dielectric particles is also called Mie theory [13–15]. Here, we will highlight some aspects of this theory. In ﬁgure (5.7) the 66 Wave tube experiments

Figure 5.7: The scattering of a polarized monochromatic plane wave. scattering of a polarized monochromatic plane wave is shown. The plane wave is incident from the positive x-axis and is scattered in the xy-plane, at an angle θ with the negative x-axis (direction of propagation of the incident light). The incident wave Ei is linearly polarized at an angle φ with the xy-plane (scattering plane). It can be decomposed in a component Ei,l which is in the scattering plane and a component Ei,r which is perpendicular to the scattering plane. Similarly, the scattered wave can also be decomposed in a parallel and perpendicular component. The scattered wave is a linear transformation of the incident wave, given as

E S S E S E + S E s,l = 2 3 i,l = 2 i,l 3 i,r . E S S E S E + S E µ s,r ¶ µ 4 1 ¶ µ i,r ¶ µ 4 i,l 1 i,r ¶ The elements of the scattering matrix depend on the properties of the dielectric scattering particle, the wavelength of the incident wave, and the angles θ and φ. They result from solving the Maxwell equations with appropriate boundary conditions at the surface of the scattering particle. For spherical symmetry, the scattering coefficients S3 and S4 are equal to zero [13–15]. In that case, the scattered parallel (perpendicular) component is completely determined by the incident parallel (perpendicular) component. The scattering coefficients S1 and S2 can then be written as functions of the angles θ and φ, the relative index of refraction m between the scatterer and the environment, and a size parameter α, which is defined as 2πr α = d , (5.9) λ where rd is the radius of the scatterer, and λ is the wavelength of the incident light. Often, it is not the electric field of the light that is measured, but rather its intensity. In case of a 5.3 Droplet detection 67

spherical symmetric scatterer, the (far ﬁeld) intensity Is of the scattered light in the solid angle (dθ, dφ) is given by I λ2 I I = 0 cos2 φ S (θ, α) 2 + sin2 φ S (θ, α) 2 sin θdθdφ = 0 C , (5.10) s 2πr2 | 2 | | 1 | r2 scat Z ¡ ¢ where r is the distance from the scatterer. Here, the scattering cross-section Cscat is introduced. It has the dimension of an area. The physical signiﬁcance of the scattering cross section is as follows. The amount of energy scattered in the solid angle (dθ, dφ) is equal to the amount of energy of the incident wave falling on an area Cscat. Likewise, the energy absorbed by a particle may be described by a cross-section Cabs. Conservation of energy then gives the scattering cross-section for extinction Cext as

Cext = Cscat + Cabs, (5.11) where Cscat is now obtained by integrating over all angles. For the substances studied in this work, the imaginary part of the refractive index is approximately zero. Therefore, the absorption by the particles can be neglected, and Cext = Cscat. Often, scattering efficiencies are given instead of scattering cross-sections. The scattering efficiencies are defined as, e.g. C Q = ext , (5.12) ext G 2 where G is the geometric area of the particle. For a spherical particle it is equal to πrd.

5.3.2 Scattering intensity In the experiment the incident light is vertically polarized and scattered through the end wall of the HPS, which is made of glass. The scattered light is focussed on the photomultiplier, and reaches it through a vertical slit. The slit is 2 mm wide and has a height of 11 mm (d1 in figure (5.6). Therefore, the measured intensity is the intensity of the light that is scattered in the solid angle 88.85o < θ < 91.15o and 84o < φ < 96o. In the experiment, the measured intensity is not the light intensity scattered by one particle, but by many particles. In a review article by Mishchenko et al. [16] it is shown that all particles can be considered as independent incoherent scatterers if they are more than 4 times their diameter apart. This is certainly the case for all the experiments performed in this work. Therefore, the total scattered intensity is proportional to the number of droplets. In figure (5.8) the calculated scattered intensity of n-nonane is shown as a function of the size parameter α2, using the geometries of our setup. The top axis shows the corresponding squared radius of the scattering droplet. It is immediately apparent that the scattering efficiency (and, hence, scattering intensity) has a very distinct form as a function of the size of the droplets. During an experiment, the scattered intensity is measured as a function of time. In figure (5.9) an example of such a signal is shown for n-nonane droplets. It has the same specific form as the calculated size-dependent scattering intensity. The first extremum in the time-dependent plot corresponds to the first extremum in the size-dependent plot. Hence, by relating each extremum in the time-dependent plot to its extremum in the size-dependent plot the growth curve of the droplets is obtained. 68 Wave tube experiments

Figure 5.8: The scattering intensity of n-nonane for the geometry of our setup.

Figure 5.9: The scattering intensity of n-nonane droplets in methane. 5.4 Mixture preparation 69

5.3.3 Light extinction Besides the scattered-light intensity, the intensity of the transmitted light is measured. Ac- cording to the law of Lambert-Beer, for single scattering spherical non-absorbing droplets this intensity is given by 2 Itrans = I0 exp πrdQextndl , (5.13) where l is the length of the optical path through¡ the cloud¢ of droplets. The extinction efficiency Qext depends on the relative index of refraction of the droplet and the size parameter. At the instant of time corresponding to an extremum in the plot of the scattered intensity, the radius of the droplets is known. The value of the extinction efficiency can then be calculated from Mie theory. The single remaining unknown in equation (5.13) then is the droplet number density nd. Therefore, the number density of droplets can be accurately obtained at each time corresponding to an extremum in the plot of the scattered intensity. The value of the number density should be approximately equal at each extremum. Ideally, it should only vary proportional to the small variations in the density of the gas. Effects of multiple scattering are discussed by Lamanna et al. [17]. For our conditions these effects can be neglected.

5.3.4 Layered droplets For the binary experiments of n-nonane in methane or water in methane, the formed droplets will be homogeneous and consist (mainly) out of n-nonane or water, respectively. However, as was shown in chapter 4.3.2, for experiments with ternary mixtures of n-nonane and water in methane, we also expect the formation of layered droplets. These droplets will have a core that is rich in water and an outer layer that is rich in n-nonane. The scattering properties of these (spherical symmetric) particles will be different for different ratios rn/rw with rw the radius of the inner water core and rn the radius of the outer n-nonane layer. Therefore, the scattering and extinction efficiencies were calculated for different ratios of the inner and outer radius. The results are shown in figures (5.10) and (5.11). For the calculations use was made of the program DMiLay by Wiscombe [18]. From figures (5.10) and (5.11) it can be seen that the scattering and the extinction efficiencies of the layered droplets are almost indistinguishable of those of pure n-nonane, when the outer radius of the n-nonane layer is twice as large as the radius of the water core. We expect that in most cases the n-nonane layer will largely block the growth of the water core, causing the n-nonane layer to grow much thicker than the radius of the water core. Therefore, we will use the scattering efficiencies of pure n-nonane in all experiments where we expect to observe layered droplets.

5.4 Mixture preparation

As was already shown, the nucleation rate is an extremely steep function of the supersaturation. Small inaccuracies in the initial composition of the gas/vapor mixture will result in 70 Wave tube experiments

Figure 5.10: The 90◦ scattering intensity of vertical polarized light for layered droplets with a water core and a n-nonane layer, as a function of the droplet radius.

Figure 5.11: The extinction eﬃciency of vertical polarized light for layered droplets with a water core and a n-nonane layer, as a function of the droplet radius. 5.4 Mixture preparation 71

MFC 0 heated box MFC 1 Psat d SAT 1 l o f i MFC 2 n SAT 2 a m gas control MFC 3 supply unit

MFC : mass flow controller GC SAT : saturator P : pressure UPC : upstream pressure controller waste RH : humidity sensor RH GC : gas chromatograph HPS : high pressure section P UPC HPS LPS : low pressure section ----- : electronics

HPS LPS

Figure 5.12: Schematic view of the setup for the preparation of the gas/vapor mixture.

large errors in the nucleation rates. Therefore, when performing nucleation experiments, the controlled preparation of the gas/vapor mixture is of crucial importance for obtaining good experimental results. The preparation of the gas/vapor mixture is therefore a very challenging part of the experimental procedure, especially when preparing mixtures containing more than one vapor component. In figure (5.12) the setup for the preparation of the gas/vapor mixture is schematically shown. The flow of dry methane gas is split into three different branches before it is combined again. The flow through each branch is controlled by a mass flow controller (MFC). In two branches the dry methane gas is saturated with either n-nonane or water, at a constant pressure and a constant temperature, which is lower than the ambient temperature. The third branch contains the dry methane gas. It can be used to dilute the gas/vapor mixture, when the three branches are brought together. The mixture will then pass a needle valve and flow through the HPS of the pulse-expansion wave tube, which is at a lower pressure than the saturation section. By keeping the pressure, temperature, and ratios of the flows in the three branches fixed, the composition of the gas/vapor mixture is also fixed. A detailed description of the setup is given in a report by Hruby´ [19].

5.4.1 Saturation section The pressure in the saturation section, which begins at the exit of the MFC’s, and ends at the needle valve in the heated box (see figure (5.12)), must be held at a constant value in order not to change the composition of the gas/vapor mixture. This is achieved by means of the Brooks 0154 control unit, in combination with a PID-controller. Before the start of the filling procedure, the needle valve and the MFC’s are closed. The pressure in the 72 Wave tube experiments saturation section is at a set point value, which is higher than the desired initial pressure in the HPS. After evacuating the HPS the needle valve is slightly opened. As a result gas will start to flow into the HPS and the pressure in the saturation section, measured with a Druck PMP1400 pressure transducer (range 100 bar), will drop below the set point value. This is recorded by the control unit, which will respond by increasing the flow of one preselected MFC. The other MFC’s are connected to this preselected MFC (via the control unit), and will also increase their flow, keeping the ratio of the flows constant. In this way, the pressure in the saturation section will increase again to the set point value. With this control loop, the pressure in the saturation section and the ratio of the flows are fixed. The needle valve is gradually opened further by the experimentator, till the MFC’s operate in their upper range, were they have the highest accuracy.

Flow rates The Brooks mass flow controllers 1 to 3 have a maximum operating range of 3, 1.5, and 0.3 normal liters of nitrogen per minute, respectively. Each MFC can be connected to either the n-nonane or water saturator, by means of the manifold. This ensures a high accuracy over a large flow range through the saturators. The fourth MFC controls the dilution of the gas/vapor mixture, and has a maximum range of 3 normal liters of nitrogen per minute. The maximum inlet pressure of the MFC’s is 100 bar. It is measured by a Druck PMP1400 pressure transducer with a range of 160 bar. The outlet pressure should be at least 5% smaller than the inlet pressure in order to ensure proper operation of the MFC’s. This sets the maximum operating pressure in the saturation section to 95 bar. In order to obtain the flow rate in normal liters of methane instead of normal liters of nitrogen, the rate can be multiplied by a standard conversion factor. However, this will cause a decrease in the accuracy of the indicated flow rates of the MFC’s. Therefore, the MFC’s were calibrated with a Brooks 1067 Vol-U-Meter Gascalibrator using methane gas before each series of experiments. The flow through the MFC’s is slightly dependent on the inlet pressure and the temperature of the gas. Therefore, during experiments the inlet pressure of the MFC’s must be the same as the inlet pressure at which the calibration was performed. The tubing upstream of the MFC’s are located inside an outer tube, through which water of constant temperature (20◦ C) flows, keeping the inlet temperature constant.

Saturators In figure (5.13) the saturators, and the way they are connected in the saturation section, are shown. As can be seen in the figure, the setup is symmetric. The left side of the setup is used for the saturation with n-nonane, and the right side is used for the saturation with water. The dilution flow passes through the middle. The saturators consist of two vessels which are completely filled with three layers of glass beads, which have a diameter of 2, 4, and 6 mm. Then, the vessels are approximately half filled with either n-nonane or water. The vessels are submerged in a temperature-controlled bath. The temperature within each vessel is measured using a platina-resistor-thermometer of Tempcontrol, giving 5.4 Mixture preparation 73

f r o m f r o m f r o m

M F C M F C M F C

B n A n A w B w

C n E n E w C w

t o

h e a t e d

b o x

D n D w

T s a t n - n o n a n e w a t e r

Figure 5.13: The saturators in the saturation section.

temperatures within an accuracy of 0.02 K. Saturation of the methane gas is accomplished by bubbling the gas through the liquid in the vessels. During the filling and flushing of the HPS valves A, C, and D are closed and valves B and E are open. By opening valve D, the gas will not flow through the second vessel, and saturation will only occur in the first vessel. This provides a means to check whether the saturation of the methane gas is complete. Before the start of the filling procedure the vessels have to be carefully pressurized, to prevent that liquid will flow out of the vessels. Before the filling procedure, valves A, B, and E are closed and valves C and D are open. In this way, the vessels are isolated from the rest of the setup, and are at uniform pressure. Then, the pressure in the saturation section is made slightly higher than the pressure in the vessels. Valve B can then safely be opened slightly. The pressure in the saturation section will now decrease until the pressure in the vessels equals the pressure in the saturation section. This procedure has to be performed separately for the n-nonane side and the water side, obtaining the same pressure everywhere in the saturation section. Then valves A, C, and D are closed and valves B and E are opened, and the saturation section is ready for the start of the filling procedure.

Heated box As the gas/vapor mixture leaves the saturation section through the needle valve, it can undergo a large pressure drop (maximum of 95 bar). As a result, the temperature of the gas/vapor mixture will also drop. In order to prevent premature condensation, the needle valve is placed in a heated box (see ﬁgure 5.12). The box can be heated up to 200◦ C. For the experiments presented in this work, the temperature of the box was set to 90◦ C. 74 Wave tube experiments

LPS HPS C B from heated box A G vacuum P to GC D F

RV UPC waste E RH P control unit H

waste

Figure 5.14: Schematic view of the setup for the ﬂushing of the HPS.

Downstream of the needle valve, the gas/vapor mixture is gathered in a ’mixing’ vessel, to ensure a homogeneous mixture. The hot gas mixture leaving the heated box is cooled before it enters the HPS. This is achieved by locating the tubing coming out of the heated box inside an outer tube, through which water ﬂows.

5.4.2 Flushing through the HPS Before the start of each experiment the HPS is evacuated. Then, it is filled with the gas/vapor mixture coming out of the saturation section. This mixture has a constant composition. However, when the HPS is filled up to the initial pressure of the experiment, the composition of the gas/vapor mixture will not have the same composition as the gas/vapor mixture leaving the saturation section. This is due to the adsorption of vapor by the walls of the HPS, and the tubing connected to the HPS. Therefore, the HPS is flushed with the gas/vapor mixture coming from the saturation section, at the desired initial pressure in the HPS. A schematic drawing of the setup used for the flushing is shown in figure (5.14). When a new diaphragm has been placed in the tube, it is first evacuated. The reducing valve RV, and the valves F and H (and the needle valve in the heated box) are then closed, while the rest of the valves are open. When proper vacuum conditions have been estab- lished (p 10−4 bar), valves D, E, and G are closed. Then filling of the HPS is started by opening≈the needle valve in the heated box. The gas/vapor mixture will then flow past the end wall of the HPS and will leave the HPS again near the diaphragm. Then it will flow past the humidity sensor (RH) and the branch going to the gas chromatograph (GC). The pressure is reduced by the reducing valve to a value below 7 bar, before it ends at the Brooks 5866 upstream pressure controller (UPC). The maximum inlet pressure of this 5.5 Experimental procedure 75 pressure controller is 7 bar. The pressure in the HPS is measured with a Druck PMP4070 pressure transducer, with a maximum range of 3.5, 10, 35, 70, or 135 bar, depending on the desired initial pressure in the HPS. Both the Druck pressure transducer and the Brooks pressure controller are connected to the Brooks 0154 control unit, which records the pressure and compares it to a set point (i.e. the desired initial pressure in the HPS). As soon as the desired initial pressure in the HPS has been reached, the pressure controller will slightly open, and the flushing of the HPS with the gas/vapor mixture starts. Dur- ing filling and flushing, the water content of the gas/vapor mixture is constantly recorded by means of a Vaisala humidity sensor (RH). The fraction of n-nonane can be monitored by opening valve F and taking samples with the gas chromatograph (GC), which uses a flame ionization detector (also see reference [20]). The humidity sensor and the gas chromatograph are used as relative indicators only. The actual vapor fractions are determined from the conditions in the saturator section. When the values indicated by the humidity sensor and the gas chromatograph take on a constant value, the walls of the setup are saturated with the gas/vapor mixture. Then, the composition of the mixture in the HPS is equal to the composition of the mixture coming out of the saturation section. As an extra check, the composition of the mixture coming out of the HPS near the diaphragm can be compared to the composition of the mixture coming out of the saturation section. This is simply achieved by opening valve E and closing valve A. Before the start of the experiment, after flushing is completed, valve E is opened and valves B and C are closed. Typically, the volume of the HPS needs to be replaced two times at the initial pressure before the composition of the mixture in the HPS is constant. For an initial pressure of 90 bar in the HPS the whole procedure of mixture preparation then takes up to 4 hours.

5.5 Experimental procedure

Before the experiment the saturation section has to be made operational, as was described in section 5.4.1. The ratios of the mass flows have to be set. The choice of the ratios depends on the desired initial vapor fractions in the HPS. The molar vapor flow coming from a saturator can be obtained by applying mass balance, stated as N˙ = (1 yeq)(N˙ + N˙ ), (5.14) g − v g v eq where yv is the equilibrium molar vapor fraction at the conditions in the saturator, and N˙ g and N˙ v are the molar fluxes of the gas and the vapor. The molar flux of the gas N˙ g is controlled by the MFC. In the saturation section three flows are combined, a dilution flow (subscript 0), a methane/n-nonane flow (subscript n), and a methane/water flow (subscript w). The vapor fraction yi of component i in the mixture leaving the saturation section is therefore given by N˙ v,i yi = . (5.15) N˙ g,0 + (N˙ g,n + N˙ v,n) + (N˙ g,w + N˙ v,w) Next, the desired initial pressure in the HPS is set with the control unit 0154. The diaphragm is positioned in the tube, and both the LPS and HPS (including the tubing up to 76 Wave tube experiments the needle valve in the heated box) are evacuated. Then, the filling procedure of the HPS is started. The LPS is filled simultaneously with the HPS, up to the desired initial pressure of the LPS. For the filling of the HPS high purity methane gas (purity 4.5) is used. For the filling of the LPS dry methane gas of 2.5 purity is used. When flushing is completed, the pulse-expansion wave tube is isolated from the rest of the setup and the initial temperature is measured at the end wall of the HPS using a Keithley 871A thermocouple. The laser is switched to full power and the Kistler dynamic pressure transducer is reset. Then, the diaphragm is burst and the pressure and optical signals are recorded with the LeCroy 6810 waveform recorder, after which they are saved to a disc for further processing. Then the needle valve in the heated box is closed, stopping the flow from the saturation section. The pressure in the rest of the setup released, and the tube is opened to remove the remains of the diaphragm. A new diaphragm is placed in the tube and the whole procedure can be repeated for a new experiment. The nucleation rate is obtained from further processing the measured data. From the analysis of the optical signals the droplet number density nd can be obtained in two different ways. First, it can be obtained from the extinction signal, as described in section 5.3.3. Second, it can be obtained by comparing the height of the extrema in the scattering signal, for two different experiments. As discussed in section 5.3.2, the height of the extrema is proportional to the number density of droplets. The duration ∆t of the nucleation pulse is determined from the pressure signal. The nucleation rate J is easily obtained as n J = d . (5.16) ∆t The temperature is calculated from the pressure signal, and the supersaturation is calculated from the settings of the saturation section, and the equilibrium conditions during the nucleation pulse.

References

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[5] J.L. Schmitt, Rev. Sci. Instrum. 52, 1749 (1981).

[6] F. Peters, Exp. Fluids 1, 143 (1983).

[7] K.N.H. Looijmans, P.C. Kriesels, and M.E.H. van Dongen, Exp. Fluids 15, 61 (1993).

[8] K.N.H. Looijmans and M.E.H. van Dongen, Exp. Fluids 23, 54 (1997). 5.5 References 77

[9] J. Hruby,´ in Nucleation and Atmospheric Aerosols 2000, edited by B.N. Hale and M. Kulmala, pages 237–240, American Institute of Physics, 2000.

[10] P.A. Thompson, Compressible–Fluid Dynamics, The Maple Press Company, Rensse- laer, 1984.

[11] 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.

[12] V.V. Sychev et al., Thermodynamic Properties of Methane, Hemisphere publishing corporation, Washington, 1987.

[13] H.C. Van de Hulst, Light scattering by small particles, Dover, New York, 1981.

[14] C.F. Bohren and D.R. Huﬀman, Absorption and scattering of light by small particles, Wiley, New York, 1983.

[15] M.I. Mishchenko, J.W. Hovenier, and L.D. Travis, Light Scattering by Nonspherical Particles, Academic Press, San Diego, 2000.

[16] M.I. Mishchenko, L.D. Travis, and D.W. Mackowski, J. Quant. Spectrosc. Radiat. Transfer 55, 535 (1996).

[17] G. Lamanna, J. van Poppel, and M.E.H. van Dongen, Exp. Fluids 32, 381 (2002).

[18] W.J. Wiscombe, Fortran subroutine for the calculation of light scattering by coated spheres, ftp://climate.gsfc.nasa.gov/pub/wiscombe/Single Scatt/Coated Sphere/, 1993.

[19] J. Hruby,´ New mixture-preparation device for investigation of nucleation and droplet growth in natural gas-like systems, Internal report, R-1489-D, Eindhoven University of Technology, 1999.

[20] C.C.M. Luijten, Nucleation and Droplet Growth at High Pressure, PhD thesis, Eind- hoven University of Technology, 1998. 78 Wave tube experiments Chapter 6

Experimental results and discussion

In this chapter the results of the experiments conducted with the expansion wave tube, using mixtures of methane, n-nonane, and/or water, will be presented. As discussed in chapter 5, both nucleation rate and droplet growth rate data are obtained with this setup. Therefore, this chapter will be divided into two parts. In the ﬁrst part the nucleation rates will be analyzed, while the second part will deal with the droplet growth rates.

6.1 Nucleation

In this section the nucleation rate data in supersaturated mixtures of n-nonane and/or water in methane will be given. The results have been published before [1], although some minor modifications have been made1. First, the binary systems n-nonane in methane and water in methane will be treated. The experimental results will be compared to the predictions of the CNT and ICCT models, given in chapter 3. The applications of the nucleation theorem, as discussed in chapter (3.6), will be applied to our experimental data. This will give the actual composition of the critical clusters. Once the composition of the critical clusters is known, an effort will be made to assess whether or not the use of a quasi-one-component theory is still justified, using the conditions derived in chapter 3.3. Then, the nucleation rates in the ternary mixtures methane, n-nonane, and water will be analyzed, using the results of the binary nucleation.

6.1.1 Binary mixtures n-nonane in methane Nucleation experiments of n-nonane in methane have been performed at two different conditions, being 10 bar and 235 K , and 40 bar and 240 K. In figure (6.1) the results for 10 bar and 235 K are shown in a JS-plot. Also shown in this figure are the data by

1In this work improved enhancement factors are used, giving diﬀerent equilibrium properties. Further- more, equilibrium bulk liquid properties are now calculated for both n-nonane/methane and water/methane systems.

79 80 Experimental results and discussion

Figure 6.1: Nucleation rates of n-nonane in methane as a function of the n-nonane supersaturation. Squares: new data at 10 bar and 235 K. Triangles: data by Luijten et al. [2] at 10 bar and 240 K. : new data at 10 bar and 240 K. ×

Figure 6.2: Nucleation rates of n-nonane in methane as a function of the n-nonane supersaturation, at 40 bar and 240 K. Squares: new data. Triangles: Luijten et al. [2]. 6.1 Nucleation 81

Luijten et al. [2], which are obtained at 10 bar and 240 K. In order to make a comparison we also performed one experiment at this condition, which is indicated by the . As can × be seen in the figure, our data compare well to the data by Luijten et al.. The results from the models are in qualitative agreement with the data. However, the ICCT model gives too large values for the nucleation rates while the CNT model gives too small rates. In figure (6.2) the results at 40 bar and 240 K are shown, again together with the data by Luijten et al. [2], at the same conditions. At these high pressures the data by Luijten et al. show a large scatter, and no real trend can be seen. This scatter at high pressures was also observed by Looijmans and Van Dongen [3]. The new data show that this scatter was most likely caused by inaccuracies in the composition of the supersaturated mixture, due to the very small vapor fractions at these high pressures. With the new mixture preparation device, discussed in chapter 5.4, the vapor fractions are more accurately known, and a clear trend in the nucleation rate data is visible. Also shown in this figure are the predictions by the CNT and ICCT models. Both models clearly fail to predict the nucleation rates at these conditions. The slope of the experimental results is far less steep than that of the predicted results. This also introduces an experimental uncertainty. As discussed in chapter 5.1, the experiment is based on the nucleation pulse method, meaning that the nucleation rate after the pulse is negligible compared to the nucleation rate during the pulse. When the slope of J versus S decreases, as indicated in the results of the 40 bar methane/n-nonane experiments, this assumption is no longer valid. Effectively, this means that the cloud of growing droplets is no longer a mono-disperse mixture, as is assumed here. Therefore, an extra uncertainty is introduced into these quantitative results. However, the decreased slope of J versus S is real, otherwise we would not have the problem just discussed.

water in methane The nucleation rates of water in methane are obtained at three diﬀerent conditions, being 10 bar and 235 K, 25 bar and 235 K, and 40 bar and 240 K. The results are shown in ﬁgure (6.3), together with the results from the CNT and ICCT models. For the methane/water system the CNT model clearly gives the best results, while the ICCT model gives again too high nucleation rates. Note that for the methane/water system the high-pressure nucleation rates are in qualitative agreement with the models, which was clearly not the case for the methane/n-nonane system.

cluster composition Using equations (3.48) and (3.49) the number of vapor and gas molecules in the critical cluster (bulk + surface) can be determined. The equations have been applied to the experimental results, and the results are shown in table 6.1. In order to obtain the number of gas molecules in the critical cluster the equilibrium fraction of methane in the bulk liquid was calculated from the CPA eos, given in chapter 2. The same holds for the enhancement factor. In the case of water in methane, the number of gas molecules were determined at a constant supersaturation of S = 10.4. Overall, the obtained number of gas molecules in 82 Experimental results and discussion

Figure 6.3: Nucleation rates of water in methane, as a function of the supersaturation of water. The rates are obtained at 10 bar and 235 K, 25 bar and 235 K, and 40 bar and 240 K. the critical cluster are less accurate than the number of vapor molecules, due to the large steps in pressure in the experimental data. Most striking is the composition of the critical cluster of supersaturated n-nonane in methane at 40 bar and 240 K, which consists of only 4 vapor molecules, together with 22 gas molecules. This might be an important clue for the explanation of the qualitative deviation of J versus S at these conditions. A cluster of this composition will most likely have a very diﬀerent surface energy than calculated here. The use of density functional theory in conjunction with an appropriate eos might be able to give more information about the surface energy of these clusters.

n C9H20 CH4 H2O CH4 −∗ − ∗ ∗ − ∗ nv ng nv ng 10 bar, 235 K 20 1 15+ 23 1 4 2 § § § 25 bar, 235 K - - 16 1 4 2 § § 40 bar, 240 K 4 22++ 15 1 - § + Assumed to be equal to the result by Luijten et al. [2] at 240 K and 10 bar. ++ taken from Luijten et al. [2].

Table 6.1: Composition of the critical cluster. 6.1 Nucleation 83 quasi-one-component theory In the previous parts the experimental nucleation rates of n-nonane in methane and water in methane were compared with the predictions of the quasi-one-component versions of the CNT and ICCT models. This is justified if the sufficient conditions, derived in chapter 3.3, are satisfied. To calculate the condition expressed by equation (3.30) the molecular volumes were approximated by the fit parameter b of the CPA eos, calculated in chapter 2.3.1. The number of gas and vapor molecules in the critical cluster were taken from table 6.1. To clus L evaluate the condition expressed by equation (3.31) the ratio γij /γij was set equal to one. The fraction of vapor molecules in the critical cluster was calculated from the values given in table 6.1. The range of ln(Sij) is small, therefore a mean value was taken. The evaluation of the conditions for each set of nucleation rate data is shown in table 6.2. For

n C9H20 CH4 H2O CH4 − −clus − clus 2 ngbg xv 2 ngbg xv ln L / ln Sv ln L / ln Sv 3 ngbg+nvbv xv 3 ngbg+nvbv xv 10 bar, 235 K 0.080 ¯ ³ 0.12´ ¯ 0.16 ¯ ³ 0.065´ ¯ ¯ ¯ ¯ ¯ 25 bar, 235 K - ¯ - ¯ 0.21 ¯ 0.10 ¯ 40 bar, 240 K 0.33 0.88 - -

Table 6.2: Veriﬁcation of equations (3.30) and (3.31).

the case of n-nonane in methane at 40 bar and 240 K, both values of the ratios are close to one, and the conditions given by equations (3.30) and (3.31) are not fulfilled. The quasi- one-component theory can no longer be applied to this mixture at these conditions, which was already obvious from figure (6.2). The values of the conditions for the mixture of methane and water are also quite large (close to one). However, looking at figure (6.3) the quasi-one-component theories are performing well. This can be explained by considering the actual value of both terms. They are of opposite sign for all cases considered. The combined effect of the volume change of the cluster and the entropy of mixing, due to the incorporation of the methane, is still quite large in the case of n-nonane in methane at 40 bar and 240 K. In all the other cases the effects largely cancel. This can explain why the quasi-one-component theories give (qualitative) good results for all cases, except for the high pressure n-nonane/methane case.

6.1.2 Ternary mixtures The nucleation rates of supersaturated mixtures of n-nonane and water in methane have been obtained at two different conditions, being 10 bar and 235 K, and 40 bar and 240 K. The small difference in temperature was necessary in order to study the full range of compositions at both pressures. The interesting question to be answered is wether or not the supersaturated n-nonane influences the nucleation rate of the supersaturated water, and vice versa. Therefore, for each condition, the experiments are divided into two sets. In one 84 Experimental results and discussion set, the supersaturation of water is held constant, while the supersaturation of n-nonane is increased stepwise. In the other set the supersaturation of n-nonane is held constant, while the supersaturation of water is increased stepwise. In order to measure nucleation rates, the value of the supersaturation ratio of the vapor component with constant supersaturation has to coincide with the experimental window of measurable nucleation rates. The obtained sets of nucleation rates are then analyzed by comparing the nucleation rates in the ternary system (methane + n-nonane + water) to the sum of the nucleation rates of the binary systems (methane + n-nonane and methane + water), at the same conditions. In order to scale the ternary nucleation rates with the sum of the binary rates, the nucleation rates of the binary systems were fitted to the CNT expression, given by equation (3.24) (without the factor exp(θ)). The parameters K and θ are functions of temperature and pressure only. At isothermal and isobaric conditions they can therefore conveniently be used as fitting parameters. In cases for which application of a quasi-one- component theory is not justified, equation (3.24) has no physical significance, other than being a fit function. The results of the fitting procedure are listed in table 6.3. Also shown in this table are the values of the theoretical values of the parameters K and θ, when the relevant physical properties are used. To determine the accuracy of the fit functions, the

n-nonane in methane water in methane θ log K θ log K fit theory fit theory fit theory fit theory 10 bar, 235 K 15.85 15.60 30.70 26.26 10.38 10.81 26.78 27.40 25 bar, 235 K 8.13 9.67 22.22 27.56 40 bar, 240 K 3.93 10.25 17.07 28.19 7.12 9.30 22.03 28.14

Table 6.3: Fitted and theoretical parameters of CNT.

experimentally obtained nucleation rates are scaled with their fit functions. The results are shown in figure (6.4) for the relevant systems. As can be seen from figure (6.4), the fit functions reproduce the experimental data within a factor 2. Assuming that the scatter is entirely due to the uncertainty in the vapor fraction, the factor 2 corresponds to an inaccuracy of about 4% in the vapor fraction. Remembering that the vapor fractions are of the order 10−4 to 10−5, the maximum absolute inaccuracy in the vapor fraction is a few ppm. We can now proceed to analyze the nucleation rates in the ternary system. The experimentally obtained nucleation rates of the ternary system Jnwm,exp are scaled with the sum of the binary fitted nucleation rates Jnm,fit + Jwm,fit, at the same conditions. The scaled nucleation rates are plotted as a function of the supersaturation of the component of which the supersaturation was varied. The results are shown in figure (6.5). For all the cases studied here, the scaled ternary nucleation rates are scattered around the value one, without any obvious trend. This indicates that n-nonane and water nucleate independently in supersaturated mixtures of n-nonane and water in methane. The scatter in the data is 6.1 Nucleation 85

Figure 6.4: Experimental binary nucleation rates scaled with their ﬁt function, for n- nonane in methane and for water in methane.

larger than the scatter in the scaled binary nucleation rates. This can be explained by the fact that the gas mixture is now composed of three different gas flows, instead of two (see chapter 5.4.1), giving rise to a larger uncertainty in the vapor fraction. Independent nucleation of supersaturated n-nonane and water was also observed by Wagner and Strey [4,5] and Viisanen and Strey [6]. They studied the nucleation rates of supersaturated mixtures of n-nonane and water in argon, at atmospheric pressures and two different temperatures, being 230 K and 240 K. Furthermore, Ten Wolde and Frenkel [7] numerically studied the nucleation of two supersaturated vapors of highly immiscible liquids, using a Monte Carlo technique. They found that the critical clusters in these mixtures were always rich in one of the two components. This again indicates independent nucleation of the supersaturated vapors. 86 Experimental results and discussion

Figure 6.5: Experimental ternary nucleation rates scaled with the sum of the binary rates.

6.2 Droplet growth

In this section the droplet growth rates will be analyzed. From the analysis of the binary growth rates diffusion coefficients can be obtained. Furthermore, the binary equilibrium vapor fractions, calculated with the CPA eos (chapter 2), can be evaluated. From the analysis of the growth rates in the ternary mixture, one can conclude whether or not heterogeneous nucleation occurs. The diffusion coefficients obtained from the first analysis will be used in the droplet growth model discussed in chapter 4. The experimental results will be compared with the predictions of this droplet growth model. 6.2 Droplet growth 87

Figure 6.6: The droplet radius squared as a function of time, for a typical experiment of supersaturated n-nonane in methane, at 11 bar and 242 K.

6.2.1 Droplet growth rates binary mixtures As discussed in chapter 5.3.2, droplet growth curves have been obtained from the Mie scattering signals, for each individual experiment. In figure (6.6) a typical growth curve obtained from the Mie scattering signal is shown. The nucleation pulse is positioned at about 7 ms, after which the droplets start to grow. The smallest detectable droplet radius corresponds to the maximum of the first Mie peak. The droplet radius at this first Mie peak is about 0.16 µm, for both n-nonane and water droplets. Because of the high carrier gas pressure (> 10 bar), the Knudsen number is small at this condition (Kn < 0.015), and the growth of the droplets is diffusion controlled. For diffusion controlled growth the increase of the droplet surface is proportional to time, as can be seen in figure (6.6). The factor of proportionality can be obtained by equating the molar flux M˙ j expressed by equation (4.8) to the molar flux expressed by equation (4.29), resulting in

2 G G drd 2ρ Dj = L (y∞,j yrd,j), (6.1) dt xjρ − where rKn is approximated by rd for diffusion-controlled growth. The proportionality holds at isothermal and isobaric conditions, as long as there is no significant depletion. The con- G G L stant of proportionality is given by 2ρ Dj /(xjρ ). In figures (6.7) to (6.10) the surface growth rates of n-nonane in methane and water in methane are shown as a function of the vapor fraction, at the two different conditions studied. The surface growth rate 88 Experimental results and discussion

Figure 6.7: Surface growth rate of water in methane at 11 bar and 242 K as a function of the molar water fraction.

Figure 6.8: Surface growth rate of water in methane at 44 bar and 247 K as a function of the molar water fraction. 6.2 Droplet growth 89

Figure 6.9: Surface growth rate of n-nonane in methane at 11 bar and 242 K as a function of the molar n-nonane fraction.

Figure 6.10: Surface growth rate of n-nonane in methane at 44 bar and 247 K as a function of the molar n-nonane fraction. 90 Experimental results and discussion equals zero when the vapor fraction equals the equilibrium vapor fraction, at the given temperature and pressure. The temperature and pressure are obtained by averaging over the growth stage of all the relevant experiments. From this averaged temperature and pressure an average equilibrium fraction is calculated using the CPA eos, for each condition considered. Then, a line is drawn from this equilibrium fraction through the experimental data. Within the accuracy of the experiments, this is possible for all experimental conditions investigated. This provides an independent check of the calculated vapor fractions, and therefore of the accuracy of the CPA eos, as it is used in this work. Especially at 44 bar and 247 K for n-nonane in methane the agreement is very good. At the other conditions the supersaturation is higher, making the determination of the equilibrium vapor fraction from the experimental data less accurate, due to the necessary extended range of extrapolation. The diffusion coefficient can now be determined experimentally, since it is directly related to the slopes of the lines drawn in figures (6.7) to (6.10). The experimentally obtained diffusion coefficients are listed in table 6.4 for each binary mixture and each condition. Also shown in this table are values for the diffusion coefficient when using the semi-empirical Fuller correlation [8]. The values obtained from the Fuller correlation differ less than 15% from the experimentally obtained values, which is in agreement with the given accuracy of the Fuller correlation.

n C9H20 CH4 H2O CH4 −2 −1 − 2 −1 2 −1 − 2 −1 Dexp (m s ) DF ul (m s ) Dexp (m s ) DF ul (m s ) 11 bar, 242 K 38.32E-8 45.03E-8 13.88E-7 15.76E-7 44 bar, 247 K 10.17E-8 10.46E-8 31.52E-8 36.42E-8

Table 6.4: Experimentally determined diﬀusion coeﬃcients of n-nonane in methane and water in methane.

ternary mixtures For the experiments with the ternary mixtures of supersaturated n-nonane and water in methane similar growth curves are obtained as for the binary experiments. As was already discussed during the analysis of the nucleation results, for the ternary experiments the molar fraction of one supersaturated vapor component was kept approximately constant, while the molar fraction of the other component was increased stepwise. We will first consider the case in which the supersaturation (i.e. molar vapor fraction) of n-nonane is kept constant, while the molar fraction of water is increased. In figures (6.11) and (6.12) the surface growth rates of the droplets in these mixtures is shown as a function of the molar fraction of water, at 11 bar and 242 K and 44 bar and 247 K, respectively. As we have seen, n-nonane and water nucleate independently. Therefore, when n-nonane is the dominating nucleating vapor (i.e. it has the highest nucleation rate at the given conditions), there are orders of magnitude more n-nonane droplets than water droplets at the beginning of the growth stage. From figures (6.11) and (6.12) one can then conclude 6.2 Droplet growth 91

Figure 6.11: Surface growth rate of droplets in mixtures of supersaturated n-nonane and water in methane as a function of the molar water fraction, at 11 bar and 242 K. Also shown is the surface growth rate of the binary water/methane mixture.

Figure 6.12: Surface growth rate of droplets in mixtures of supersaturated n-nonane and water in methane as a function of the molar n-nonane fraction, at 44 bar and 247 K. Also shown is the surface growth rate of the binary water/methane mixture. 92 Experimental results and discussion that the supersaturated water vapor does not grow onto the n-nonane droplets, since the droplet growth rate is independent of the molar fraction of water. This is in agreement with the theoretical analysis made in chapter 3.5.2, stating that water will not heterogeneously nucleate onto n-nonane droplets. Next, we will consider the growth rates in supersaturated mixtures of n-nonane and water when the supersaturation of water is kept fixed. Hence, water is the dominant nucleating vapor, and the number of water droplets at the beginning of the growth stage is orders of magnitude larger than the number of n-nonane droplets. In figures (6.13) and (6.14) the surface growth rates of the droplets are now shown as a function of the molar n-nonane fraction, at 11 bar and 242 K and 44 bar and 247 K, respectively. In these figures something very interesting can be seen. The droplet growth rates in the mixtures of supersaturated n-nonane and water in methane increase when the molar fraction of n- nonane is increased beyond a minimum value. As the molar vapor fraction of n-nonane increases, the droplet growth rate in the ternary mixture approaches the droplet growth rate of the supersaturated binary n-nonane/methane mixture. Apparently, the supersaturated n-nonane does grow onto water droplets, again in agreement with the analysis made in chapter 3.5.2. The n-nonane slowly covers the water droplet. At first, the water can still diffuse through this n-nonane shell, and reach the water core. As the n-nonane shell grows thicker, the water vapor is completely shielded from the water core. The water no longer contributes to the droplet growth, since water cannot grow onto n-nonane. This explains why the droplet growth rates in the ternary mixture approach the growth rate of the binary n-nonane/methane mixture. As the supersaturation of the n-nonane in the ternary mixture increases, the n-nonane shell will grow thicker faster. So, higher n-nonane supersaturation will result in less influence of water on the droplet growth rate.

6.2.2 Droplet growth model In this section we will compare the experimentally obtained growth results to the predictions of the model presented in chapter 4. The physical properties used for the growth calculations are given in appendix A. In order to obtain values for the diffusion coefficient, the Fuller correlation is multiplied by the ratio of the experimental to the calculated value of the diffusion coefficient, which are listed in table 6.4. In this way, small temperature and pressure variation are still accounted for, while the mean value is equal to the experimentally obtained value.

Binary mixtures In figure (6.15) four plots are shown. Each plot contains an example of a typical growth curve of either n-nonane in methane or water in methane, at 11 bar and 242 K or 44 bar and 247 K. The agreement is as expected, since the diffusion coefficients were fitted to the experimental results. The growth curves intersect the time axis at about 6.6 ms, which is the time at which the nucleation pulse occurs. After about 30 ms the shock wave, which has reflected from the end wall of the low pressure section, reaches the end wall 6.2 Droplet growth 93

Figure 6.13: Surface growth rate of droplets in mixtures of supersaturated n-nonane and water in methane as a function of the molar n-nonane fraction, at 11 bar and 242 K. Also shown is the surface growth rate of the binary n-nonane/methane mixture.

Figure 6.14: Surface growth rate of droplets in mixtures of supersaturated n-nonane and water in methane as a function of the molar n-nonane fraction, at 44 bar and 247 K. Also shown is the surface growth rate of the binary n-nonane/methane mixture. 94 Experimental results and discussion

Figure 6.15: Experimentally obtained growth curves together with model predictions. of the high pressure section. Then, temperature and pressure increase again, causing the droplets to evaporate. The decrease in the slope of the growth curve of water in methane at 11 bar and 242 K is caused by the depletion of the vapor. The growth curves of n- nonane in methane show a small jump at the initial state. This is due to the fact that the experimental time-step is slightly too large for the initial stage of the droplet growth. The dissolution of methane into the water droplets is in all cases quite small. However, the solubility of methane into liquid n-nonane at 44 bar and 247 K is about 25 mol percent. In order to show the influence of the methane solubility on the growth rate of n-nonane droplets, the theoretical growth curve when ignoring the presence of methane is also shown in figure (6.15), for the condition of 44 bar and 247 K. The growth curves with and without methane solubility differ little. This can be explained by the fact that the dissolution of methane into liquid n-nonane has little influence on the liquid volume. Or, equivalently, the partial liquid n-nonane density in methane/n-nonane mixtures is approximately equal to the pure liquid n-nonane density. 6.2 Droplet growth 95

Figure 6.16: Example of droplet growth curves of n-nonane and water in methane, in the case n-nonane has a nucleation rate larger than water.

In appendix E an overview is given of all the binary growth curves.

Ternary mixtures For the description of droplet growth in supersaturated mixtures of n-nonane and water in methane two cases can be distinguished, depending on which vapor is predominantly present. As before, we will ﬁrst consider the case in which the nucleation rate of n-nonane is much larger than the nucleation rate of water. To a very good approximation, the initial cloud of droplets can then be considered to exist of n-nonane droplets only. As discussed in chapter 4.3.2, it is assumed that the water will not grow onto the n-nonane droplets. This is also in agreement with the observations made in section 6.2.1. The solubility of water into the liquid n-nonane is incorporated via equation (2.32), where yw is taken equal to the supersaturated water vapor fraction. The solubility of methane into the liquid n- nonane/water mixture is taken equal to the solubility of methane into pure liquid n-nonane, which is a good approximation due to the very small water fractions in the ternary liquid. For the same reason, the density of the ternary liquid is approximated by the density of liquid n-nonane/methane mixtures, at the same conditions. Two examples of the growth calculations are shown in ﬁgure (6.16). The agreement between the experimental results and the model calculations is again very good. This indicates that water indeed does not contribute to the growth of the n-nonane droplets (the solubility of water into liquid n-nonane is negligible). Now we will consider the case in which water is the dominant supersaturated vapor fraction. The initial cloud of droplets then almost completely consists of water droplets. As discussed in chapter 4.3.2 and shown in section 6.2.1, the supersaturated n-nonane can grow onto the water droplets. In this case we will therefore use the layered droplet 96 Experimental results and discussion

Figure 6.17: Curves of droplet growth of n-nonane and water in methane, where water has the largest nucleation rate, at 11 bar and 242 K. Line: radius of outer (n-nonane) layer. Dotted line: radius of inner (water) core. model for the growth calculation. It is then assumed that the n-nonane completely wets the water droplet as soon as it has grown larger than the critical size of homogeneously nucleating n-nonane droplets. The solubility of n-nonane into liquid water is very small and is therefore neglected. The solubility of methane into the ternary liquids, which are rich in either n-nonane or water, is approximated by the solubility of methane into the pure liquid, being either n-nonane or water. Likewise, the density of the ternary liquid which is rich in n-nonane is approximated by the density of liquid n-nonane/methane and the density of the ternary liquid which is rich in water is approximated by the density of liquid water/methane. In figure (6.17) four examples of growth curves are shown for the condition of 11 bar and 242 K. In the first plot of figure (6.17) the n-nonane is subsaturated during the growth of the droplet. Hence, it does not form a layer around the water core. For conditions for which the n-nonane is supersaturated the model predicts a too small 6.2 Droplet growth 97 growth rate of the droplets. The difference with the experiments becomes smaller as the vapor fraction of n-nonane becomes larger. This holds for all the experiments at 11 bar and 242 K. At 44 bar and 247 K the agreement between the layered droplet calculations and the experiments is somewhat better, as can be seen in appendix E. The origin of the deviation for the smaller n-nonane vapor fractions becomes more clear when we look again at the surface growth rates as a function of the n-nonane fraction. The results of the surface growth calculations are shown in figures (6.18) and (6.19), for the two different conditions studied. The calculations are performed at the fixed averaged conditions (i.e. pressure, temperature and water vapor fraction) of the relevant experi- 12 −3 mental series. The droplet number density is fixed at nd = 10 m . The surface growth rate is evaluated 5 ms after the start of the growth, when the depletion of water is still negligible. This roughly corresponds to the appearance of the first Mie peak in the experiments. At this instant the layered model predicts constant surface growth rates for all the 2 experimental conditions considered, as can be seen in the growth plots (rdl versus t). The experimental surface growth rates which are also plotted in figures (6.18) and (6.19), are time averaged surface growth rates. From figure (6.18) it becomes clear that according to the layered droplet model the growth rate of the droplets can decrease when the supersaturation of n-nonane is small. This can be explained as follows. The volumetric growth rate of the water core is mainly determined by the diffusive flux of water molecules through the n-nonane layer. This flux is much smaller than the flux of water molecules through the gas phase towards an uncovered water droplet. Therefore, the volumetric growth rate of the water core is very small compared to the volumetric growth rate of an uncovered water droplet. And, when the supersaturation of the n-nonane is small, the volumetric growth rate of the n-nonane layer can also be smaller than the volumetric growth rate of the uncovered water droplet, resulting in a smaller overal growth rate for the layered droplet. When the (carrier gas) pressure is increased at constant supersaturation, the ratio of the molar flux of water molecules through the n-nonane layer to the molar flux of water molecules through the gas phase becomes closer to one, and the effect of the decrease in the growth rate becomes less pronounced. This can be seen in figure (6.19). The predicted decrease in the droplet growth rates in figure (6.18) is not observed in the experiments. Apparently, in the experiments the water core can grow faster. This cannot be explained by a difference between the actual and calculated value of the diffusion coefficient of water through liquid n-nonane. Even increasing this diffusion coefficient by a factor 10 can not explain the deviation. Our hypothesis is that the observed difference is caused by n-nonane not completely wetting the surface of the water droplet, as is assumed in the layered model. In that case, the n-nonane forms one or several caps on the water droplet. The angles between the phases are fixed by the interfacial tensions at the given conditions. Due to cap formation, parts of the water droplet remain uncovered, and the growth of the water droplet is only slightly decreased. As the n-nonane caps grow, they slowly cover increasingly larger parts of the water surface, until eventually the water droplet is completely covered. Then, the growth can accurately be described by the layered model. The agreement between the experimentally obtained growth rates in the ternary mix- 98 Experimental results and discussion

Figure 6.18: Surface growth rate of droplets in mixtures of supersaturated n-nonane and water in methane as a function of the molar n-nonane fraction, at 11 bar and 242 K. Also shown is the surface growth rate of the binary n-nonane/methane mixture.

Figure 6.19: Surface growth rate of droplets in mixtures of supersaturated n-nonane and water in methane as a function of the molar n-nonane fraction, at 44 bar and 247 K. Also shown is the (experimentally obtained) surface growth rate of the binary n-nonane/methane mixture. 6.2 References 99 ture at 44 bar and 247 K and the layered growth calculations is better, as can be seen from figure (6.19). Possibly, the complete wetting of n-nonane on water is a better approximation at this higher methane pressure. However, the decrease of the growth rate was also expected to become less due to the higher carrier gas pressure. An overview of all the growth curves in the ternary mixture is given in appendix E. At this point it is noteworthy that it can be confirmed experimentally whether or not n- nonane forms a cap on water droplets. This could be achieved by measuring the polarization of the scattered light. For spherical scattering particles the scattering coefficients S3 and S4 in equation (5.3.1) are equal to zero. Therefore, when the incident beam is vertically polarized, the scattered light is also vertically polarized. For non-spherical particles the scattering coefficients S3 and S4 are generally not equal to zero. When the vertically polarized incident beam is then scattered by non-spherical particles, the scattered light will no longer be completely vertically polarized. Therefore, by measuring the intensity of the horizontally polarized light of the scattered light, cap formation can be confirmed. Ideally, this intensity would be non-zero when n-nonane has heterogeneously nucleated onto the water droplet, and formed caps, and become zero again when the caps completely cover the water droplet. Measuring the polarization of the scattered light will be subject of future experiments.

References

[1] P. Peeters, J. Hruby,´ and M.E.H. van Dongen, J. Phys. Chem. B 105, 11763 (2001).

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

[3] K.N.H. Looijmans and M.E.H. van Dongen, Exp. Fluids 23, 54 (1997).

[4] P.E. Wagner and R. Strey, in Proceedings of the 3rd International Aerosol conference, Kyoto, edited by S. Masuda and K. Takahashi, page 201, Oxford, U.K., 1990, Springer.

[5] P.E. Wagner and R.Strey, J. Phys. Chem. B 105, 11656 (2001).

[6] Y. Viisanen and R. Strey, J. Chem. Phys. 105, 8293 (1996).

[7] P.R. ten Wolde and D.J. Frenkel, J. Chem. Phys. 109, 9919 (1998).

[8] R.C. Reid, J.M. Prausnitz, and B.E. Poling, The Properties of Gases and Liquids, McGraw–Hill Book Company, New York, 1987. 100 Experimental results and discussion Chapter 7

Nucleation of ice

This chapter is written in the form of an article, that has been accepted for publication in The Journal of Chemical Physics.

7.1 Introduction

The experimental results presented in this paper are part of an ongoing research program, focussed on the nucleation behavior of natural gas [1]. Apart from the main component methane, natural gas contains many other (vapor) components. These other vapor components can roughly be separated into two different categories: water and hydrocarbons. One of the aims in the program is trying to establish whether or not nucleation of ice and/or the nucleation of methane hydrates occurs. As a first step towards this goal, the nucleation behavior of water in helium was studied. In this way, hydrate formation is excluded. What remains are three possible nucleation processes. These are: vapor/liquid, liquid/solid and vapor/solid nucleation. The nucleation behavior of water vapor (in an inert carrier gas) has been studied before, and a recent and extensive study was presented in a paper by Wölk and Strey [2]. They measured nucleation rates in the temperature range of 219 to 260 K. In their paper only vapor/liquid nucleation is considered, and no indications for other nucleation processes are given. Another extensive study has been performed by Pe- ters and Paikert [3]. They measured nucleation rates of water in argon in the temperature range of 200 to 250 K. The results presented here are in the temperature range from 200 to 235 K. Attention will be focussed on the detection of vapor/solid nucleation.

7.2 Nucleation

Through the years several expressions for the nucleation rate J have been developed [1, 4]. For analyzing our experimental results we will use an expression inspired by classical nucleation theory: W J = K exp , (7.1) −k T µ B ¶ 101 102 Nucleation of ice in which K is a kinetic factor, T is the temperature, and W is the energy needed to form a critical cluster, at isothermal and isobaric conditions. When a vapor is supersaturated, it can lower its Gibbs free energy by forming a liquid phase. However, in order to form a liquid phase, an interface between the vapor and the liquid has to be formed, which increases the energy of the system. The lowering of the free energy due to the bulk liquid formation is proportional to the volume (r3) of the liquid, while the increase of the free energy due to the formation of the interface is proportional to the area of this interface (r2). Therefore, the energy of formation of a cluster of molecules has a maximum as a function of the cluster size. The critical cluster is the cluster at this maximum and W is its corresponding work of formation. For nucleation from the vapor phase classical nucleation theory gives [5]

16π M 2 W = σ3, (7.2) 3 N ρk T ln(S) µ A B ¶ and p 2 2σM 1/2 S K = sat . (7.3) k T πN ρ µ B ¶ µ A ¶ The work of formation of a critical nucleus W and the rate factor K depend on the molar mass M, the speciﬁc density ρ of the condensed phase, the surface tension σ and the saturated vapor pressure psat. At near-atmospheric pressures the supersaturation S of the vapor in a carrier gas can be expressed as [5] p yp S = v = , (7.4) psat psat where y is the vapor fraction. Expressions for the physical properties are given in the appendix. In the original expression of the CNT an extra factor S appeared in the rate factor K. However, this is now generally believed to be incorrect. More details are given in the book by Kashchiev [6]1.

7.3 Surface energy of ice

When will vapor/ice nucleation become more probable than vapor/liquid nucleation? In terms of the classical nucleation theory this means that the barrier height in the energy of formation (W ) of a critical ice nucleus becomes smaller than the barrier for the formation of a critical water nucleus. Using Eq. (7.2), these energies of formation can be compared when the physical properties of ice are known. The density and saturated vapor pressure of ice are readily available. Obtaining an expression for the surface tension (or rather surface energy) of ice is less straightforward. Ice at temperatures above about 235 K is covered with a quasi liquid layer (QLL) of microscopic size [7–9]. In this case the ice/vapor interface actually consists of two interfaces, an ice/water interface and a water/vapor interface.

1Also see chapter 3.2 of this thesis 7.4 Experiment 103

Therefore, for temperatures above about 235 K the equilibrium ice/vapor surface energy can be obtained using Antonow’s rule [10,11]

σsolid/vapor = σsolid/liquid + σliquid/vapor. (7.5)

For temperatures well below 235 K the QLL has disappeared and the ice/vapor surface energy can be approximated using the latent heat of evaporation [12]. When it is assumed that a water molecule that moves from the bulk ice to the surface loses half of its bonding energy, the surface energy of the ice/vapor interface corresponds approximately to half of the energy of evaporation. This energy of evaporation is calculated from the entropy diﬀerence between the bulk ice and the vapor

Lsolid/vapor = T ∆Ssolid/vapor. (7.6)

We then have the surface energy per mole of ice. To obtain the surface energy per unit of area, the number of moles (of ice) per unit area is estimated from the density of ice. The molecules are assumed to occupy a spherical volume of radius

3M 1/3 r = (7.7) sphere 4πN ρ µ A solid ¶ The square of the diameter of the sphere gives the area per molecule on the ice surface. This corresponds to −1 3M 2/3 a = 4N (7.8) A 4πN ρ Ã µ A solid ¶ ! moles of ice per unit area. Finally, we have for the surface energy of ice below approximately 235 K the following expression 1 σ = CL a, (7.9) solid/vapor 2 solid/vapor in which C is a correction factor of order one.

7.4 Experiment

The experimental setup has been described in detail elsewhere [13, 14]. The improved procedure of mixture preparation has been described in the paper by Peeters et al. [1]. The nucleation rates are measured using the nucleation pulse method. The method is based on the fact that the nucleation rate is a very steep function of the supersaturation. First, the gas/vapor mixture is rapidly brought into a state of high supersaturation, in which signiﬁcant nucleation takes place. This is generally achieved by adiabatically expanding the mixture. Shortly after the expansion the mixture is slightly compressed, decreasing the supersaturation somewhat. In this new less supersaturated state no signiﬁcant nucleation 104 Nucleation of ice

w a s t e g a s s u p p l y L

A P C

S M P D E R H R

P P M k H P S L P S

P D

H P S : H i g h p r e s s u r e s e c t i o n P k : D y n a m i c p r e s s u r e t r a n s d u c e r

L P S : L o w p r e s s u r e s e c t i o n M P D : M i x t u r e p r e p a r a t i o n d e v i c e

P M : P h o t o m u l t i p l i e r R H : H u m i d i t y s e n s o r

P D : P h o t o d i o d e P C : P r e s s u r e c o n t r o l l e r

Figure 7.1: Schematic view of the setup.

occurs, while the clusters formed during the state of high supersaturation (the nucleation pulse) can grow to optically detectable sizes. In this way, the formation and the growth of the droplets is effectively decoupled. The nucleation pulse can be obtained using a pulse expansion wave tube. This is basically a shock tube, the high pressure section of which is used as the test section. A schematic view of the setup is shown in Fig. (7.1). Just behind the right-hand side of the diaphragm, separating the high pressure section from the low pressure section, there is a local increase in the diameter of the tube. When the diaphragm is burst, an expansion fan will travel into the high pressure section and reflect at the end wall, causing a rapid decrease of the pressure. The shock wave travelling into the low pressure section will partially reflect at the local widening. This will result in the formation of a small pressure dip at the end wall of the high pressure section, i.e. the nucleation pulse. The droplets formed in the pulse will keep growing until the shock wave, reflected from the end wall of the low pressure section, reaches the end wall of the high pressure section. The low pressure section has recently been extended from 6.42 m to 9.23 m, effectively increasing the measuring time by about 40%. The pressure at the end wall of the HPS is measured using a Kistler 603B dynamic pressure transducer. The temperature is calculated from the pressure assuming an adiabatic isentropic expansion. The pressure transducer was calibrated in the shock tube before as well as after the experiments. Both calibrations gave the same result, within the experimental accuracy (less than 0.2% relative difference). The number density and the size of the droplets are determined using a combination of light extinction and 90 degree light scattering (CAMS). The nucleation rate is obtained by taking the ratio of the number density and the time duration of the nucleation pulse. With this setup nucleation rates between 1014 and 1017 m−3s−1 can be measured. The test gas mixture originates from a combination of two different gas streams. One is 7.5 Results and discussion 105 a ”wet” helium gas stream. This gas stream is saturated with water by bubbling it through two containers half filled with water, at a constant pressure (5.17 bar) and temperature (291.2 K). In this way the wet gas stream has a constant vapor fraction. It can be diluted by the second gas stream, which consists of dry helium only. The gas streams are controlled by mass flow controllers. The composition of the gas/vapor mixture is altered by setting different ratios of the mass flow controllers. Before the start of an experiment, the high pressure section is flushed (at the initial pressure of the experiment) with the gas/vapor mixture. When the humidity sensor takes on a constant value, equilibrium has been reached between the walls of the setup and the gas/vapor mixture, and the experiment can begin.

7.5 Results and discussion

In Fig. (7.2) the results of the nucleation experiments are shown as a function of the temperature. The experimental determined rates are scaled with the rates from classical nucleation theory. It is apparent that around 207 K there is a jump in the scaled rates of about 4 orders of magnitude. This jump is not caused by a peculiar behavior of the analytical expression for the nucleation rate. Rather, it is caused by a change in the nucleation process. This also becomes apparent when we look at the supersaturation of water as a function of the temperature, shown in Fig. (7.3). In order to measure nucleation rates within our experimental nucleation rate ’window’ (O(14) O(18)), the supersaturation of water has to be decreased again at 207 K. The jump−in the scaled nucleation rates can be explained by the onset of vapor/ice nucleation. Onset of liquid/ice nucleation cannot explain the jump in the scaled nucleation rates. It would mean that the formed liquid droplets would freeze, rendering the same amount of measured particles (droplets) per unit volume. Effectively, this gives a small error in the measured number density, since the scattering properties of (spherical) ice particles are slightly different from those of water droplets. This gives an error in the measured number density (and nucleation rates) of about 15%, which is well within the experimental error of a factor 3. Because of the strong dependency of the nucleation rate on the supersaturation, the experimental vapor fractions form a narrow band in the yp T diagram, as shown in Fig. (7.4). To compare the energy of formation of a critical water− nucleus to that of a critical ice nucleus for our experiments, the narrow band of experimental partial vapor pressures was approximated by the curve also shown in Fig. (7.4). The partial vapor pressures and the physical properties can now be put into the expression for the energy of formation for a critical nucleus (Eq. (7.2)). This results in different temperature dependent curves for the energy of formation (for our experiments), which are shown in Fig. (7.5). When the value C = 0.6 is taken for the correction factor in the expression for the ’dry’-ice/vapor surface energy (Eq. (7.9)), the vapor/’dry’-ice curve intersects the vapor/liquid curve at 207 K, making vapor/’dry’-ice nucleation more probable below this temperature. A transition to vapor/’wet’-ice nucleation will not occur, as the energy of formation of such a cluster is always larger than at least one of the other two. In Fig. (7.6) the ’dry’-ice and ’wet’-ice approximations for the surface energy are shown in one figure. For C = 0.6 the ’dry’-ice 106 Nucleation of ice

Figure 7.2: Ratio of experimental nucleation rates to theoretical classical nucleation rates of liquid water, as a function of the temperature.

Figure 7.3: Experimental supersaturation of water vapor with respect to liquid water, as a function of temperature. 7.5 Results and discussion 107

Figure 7.4: Experimental partial vapor pressure versus nucleation temperatures. The line represents a ﬁt through the data.

Figure 7.5: The energy of formation of a critical nucleus as a function of temperature. Line: vapor/liquid nucleation. Dashed line: vapor/’dry’-ice nucleation (using equation (7.9) with C = 0.6). Dotted line: vapor/’wet’-ice nucleation (using equation (7.5)). 108 Nucleation of ice

Figure 7.6: Surface energy of ice as a function of temperature. Line: estimated ’dry’- ice surface energy, with C = 0.6. Dotted line: ’wet’-ice surface energy, obtained using Antonow’s rule. surface energy intersects the ’wet’-ice surface energy at 220 K. This temperature is within the range of experimentally observed temperatures at which the ice surface becomes completely ’dry’ [7–9]. Above this temperature the equilibrium surface consist of an ice/liquid interface, plus a very thin liquid layer (which is not stable in bulk), and a liquid/gas interface. Below the transition temperature the surface consists of an ice/gas interface only. The derivative with respect to temperature of the surface energy is discontinuous at the transition temperature. The transition is therefore a first-order phase transition of the surface. This is also known as a Cahn transition [11,15]. Nucleation rates of water in argon have recently been measured by Wölk and Strey [2]. They used a pulse expansion cloud chamber, with which they can measure nucleation rates between 1011 and 1015 m−3s−1. So their upper range of nucleation rates corresponds to our lower range. Their cylindrical measuring chamber has a diameter of 3 cm and a total volume of about 25 cm3. The measurements were performed in the temperature range of 219 K to 260 K. Peters and Paikert [3] also measured nucleation rates of water in argon, in the temperature range of 200 K to 245 K, using an expansion wave tube, similar to ours. The scatter in their data amounts to several orders of magnitude, especially in the lower temperature range. In Fig. (7.7) our data are shown together with the data by Wölk and Strey [2]. At each temperature, the data by Wölk and Strey show a range of scaled nucleation rates. The lower values of the scaled (isothermal) nucleation rates correspond to a higher supersaturation. The temperature dependency of the nucleation rates measured by Wölk and Strey differs from our measured dependency. 7.5 Results and discussion 109

Figure 7.7: Experimental nucleation rates as a function of temperature. The experimental rates are scaled with the values obtained from classical nucleation theory (for vapor/liquid nucleation).

In order to investigate this difference we considered the possibility of the influence by thermal boundary layers in our setup. Our measuring volume is located 5 mm from the end wall of the HPS. To calculate the influence of time-dependent conductive heat transport from the ’hot’ end wall of the HPS to the cold gas the following (one-dimensional) coupled differential equations have to be solved:

∂ρ ∂ρu + = 0, (7.10) ∂t ∂x and ∂T ∂T dp ∂ ∂T ρc + u = + k , (7.11) p ∂t ∂x dt ∂x ∂x µ ¶ which represent the conservation of mass and energy, respectively. Here, u is the velocity, k is the thermal conductivity, t is the time, and x is the position coordinate. Before the expansion (t = 0) the temperature is uniform (T0). The temperature at the end wall (x = 0) remains equal to T . The temperature far from the end wall (x ) is equal 0 → ∞ to the time-dependent isentropic temperature T∞(t), which is calculated from the known uniform time-dependent pressure p(t). It was shown by Keck [16] and Van Dongen [17] that for an ideal gas the coupled diﬀerential equations can be rewritten in the form

∂ T ∂2 T = , (7.12) ∂s T ∂η2 T µ ∞ ¶ µ ∞ ¶ 110 Nucleation of ice where the coordinate transformations x ρ(x0, t) η(x, t) = dx0, (7.13) ρ Z0 0 and t p(t0) s(t) = α dt0, (7.14) 0 p Z0 0 are used. Furthermore, it is assumed that the thermal conductivity k is proportional to the temperature, resulting in T T k = k0 = ρ0cpα0 , (7.15) T0 T0 where α0 is the initial thermal diﬀusivity. For the initial condition T (η, 0) = T∞(0), and boundary conditions T (0, s) = T and T ( , s) = T∞(s) the solution of Eq. (7.12) is [16]: 0 ∞ t t 00 −1/2 T (η, t) η p(t ) 00 d T0 0 = 1 + erfc α0 dt 0 0 dt . (7.16) T (t) 2 0 p dt T (t ) ∞ Z0 Ã µ Zt 0 ¶ ! µ ∞ ¶ This integral can be solved numerically using the pressure p(t) obtained from the experiment. One then obtains the temperature as a function of the transformed position coordinate η and time t. The actual position coordinate x can easily be obtained, using

p η T (η0, t) x(η, t) = 0 dη0, (7.17) p(t) T Z0 0 which is the inverse of Eq. (7.13). The boundary-layer calculations have been performed for experiment numbers 482 and 500. Experiment 482 is the one with the lowest nucleation temperature (201.66 K), while experiment 500 has a nucleation temperature of 231.06 K. In Fig. (7.8) the temperature profile resulting from both experimental pressure signals is shown. The time corresponds to 3 ms after the start of the expansion. The nucleation pulse has already occurred within these 3 ms. As can be seen from Fig. (7.8), there is no influence of the thermal boundary layers at the measuring position (5 mm) before the end of the nucleation pulse. From this we can conclude that the thermal boundary layers do not influence the nucleation process. In Fig. (7.9) the temperature profile of these experiments at the measuring position (5 mm) is shown as a function of time, scaled with the isentropic temperatures. The thermal boundary layers reach the measuring position after about 5 ms, measured from the end of the nucleation pulse (which end at t 5 ms). Therefore, the thermal boundary layers influence the droplet growth process. Ho≈wever, this will only give an error in the measured number density of less than one percent, and can certainly not explain a difference of several orders of magnitude in the measured nucleation rates. The difference in our measurements and those of Wölk and Strey [2] could also originate from thermal influences due to the separation of viscous and thermal boundary layers formed along the side walls of the tube, caused by the space- and time-dependent flows 7.5 Results and discussion 111

Figure 7.8: Development of the thermal boundary layers at the end wall of the HPS, assuming one-dimensional unsteady heat conduction.

Figure 7.9: Development of the thermal boundary layers as a function of time, at the measuring position (5 mm from the end wall of the HPS). The temperatures are scaled with the isentropic temperatures. 112 Nucleation of ice induced by the expansion. In order to investigate this, the full three-dimensional Navier- Stokes equations have to be solved. This is a subject of current investigation. Possibly, this can explain the difference in temperature dependency of the nucleation rates, measured with the different devices. However, it is highly unlikely that this can explain the jump in the nucleation rates we measured at the nucleation temperature of 207 K. Therefore, the proposition of the onset of a different nucleation process at this temperature, being vapor/ice nucleation, remains valid.

7.6 Conclusions

We have presented new experimental results of nucleation rates of water vapor in helium in the temperature range of 200 to 235 K. In this temperature range a transition in the nucleation process is observed at 207 K, which we suggest is due to the change of vapor/liquid to vapor/ice nucleation. A qualitative theoretical explanation of this transition has been given, based on classical nucleation theory. In this theory, the surface energy of ice has a dominant role. Therefore, an expression for the surface energy of ice at these low temperatures was derived, which includes one free parameter. This parameter has been fitted to the transition in the nucleation rates. The resulting temperature-dependent surface energy has a first-order surface phase transition (Cahn transition) at 220 K, which is within the range of experimentally observed temperatures of this surface phase transition. The overall difference in temperature dependency of the nucleation rates of our measurements compared to the measurements of Wölk and Strey [2] cannot be explained by the influence of one-dimensional heat conduction from the end wall in our setup. The differences in the measurements might be explained by the separation of boundary layers, induced by the flows resulting from the expansion. This is subject of current investigation.

References

[1] P. Peeters, J. Hruby,´ and M.E.H. van Dongen, J. Phys. Chem. B 105, 11763 (2001).

[2] J. W¨olk and R. Strey, J. Phys. Chem. B 105, 11683 (2001).

[3] F. Peters and B. Paikert, Exp. Fluids 7, 521 (1989).

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

[5] C.C.M. Luijten and M.E.H. van Dongen, J. Chem. Phys. 111, 8524 (1999).

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

[7] J.G. Dash, Fu Haiying, and J.S. Wettlaufer, Rep. Prog. Phys. 58, 115 (1995). 7.6 References 113

[8] A. Do¨ppenschmidt and H.-J. Butt, Langmuir 16, 6709 (2000).

[9] X. Wei, P.B. Miranda, and Y.R. Shen, Phys. Rev. Lett. 86, 1554 (2001).

[10] G.N.J. Antonow, J. Chim. Phys. 5, 372 (1907).

[11] J.S. Rowlinson and B. Widom, Molecular theory of capillarity, Clarendon Press, Oxford, 1982.

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

[13] K.N.H. Looijmans and M.E.H. van Dongen, Exp. Fluids 23, 54 (1997).

[14] K.N.H. Looijmans, P.C. Kriesels, and M.E.H. van Dongen, Exp. Fluids 15, 61 (1993).

[15] J.W. Cahn, J. Chem. Phys. 66, 3667 (1977).

[16] J.C. Keck, Letters in Heat and Mass Transfer 8, 313 (1981).

[17] M.E.H. van Dongen, Thermal Diﬀusion Eﬀects in Shock Tube Boundary Layers, Phd thesis, Eindhoven University of Technology, Eindhoven, 1978. 114 Nucleation of ice Chapter 8

Conclusions and recommendations

This project has been motivated by the need to get a better understanding of the nucleation and growth processes as they occur in (supersaturated) natural gas, at elevated pressures. To that end, an existing pulse-expansion wave tube has been used, suited for studying nucleation and droplet growth at high pressures. The pulse expansion wave tube is designed to work according to the nucleation pulse principle. This means that in the setup, during a very short period of time droplets are formed (the nucleation pulse). After this period the nucleation process is terminated, while the formed droplets simultaneously start to grow to optically detectable sizes. With the existing setup, nucleation rates in the range of 1013 to 1017 m−3s−1 can be measured at nucleation pressures of 1 to 50 bar. The droplet number density and the size of the droplets are determined using a combination of 90◦ Mie scattering and light extinction techniques. One characteristic aspect of nucleation and droplet growth in natural gas is that they occur at elevated pressures. Another characteristic aspect is that natural gas consists of many diﬀerent components. These components can be divided into 3 categories. These are: methane gas (the main component), water vapor, and, a large collection of hydrocarbon vapors. Therefore, in this thesis mixtures of methane, water and n-nonane have been studied, where n-nonane is chosen as a typical hydrocarbon vapor. Often, carbon-dioxide is also present, but this is not included in the present study. In order to measure nucleation rates in supersaturated mixtures of n-nonane and/or water in methane a new mixture preparation device has been build. With this newly designed setup binary and ternary gas/vapor mixtures of accurately known composition can be made in a controllable and reproducible way. Nucleation and droplet growth are processes that bring a system to a new equilibrium. Therefore, in order to describe nucleation and droplet growth, it is important that the new equilibrium condition is known. Equilibrium conditions can be calculated using equations of state. In this work the CPA (cubic plus association) equation of state (eos) is used to calculate the equilibria of systems containing methane, n-nonane, and/or water. The CPA eos combines the cubic RKS (Redlich-Kwong-Soave) eos with the association term from of the statistical associating ﬂuid theory. This term is needed to describe the equilibria in mixtures that contain molecules (like water) that can have strong bonding interactions with each other. The CPA eos is chosen because it is still relatively simple, and has proven

115 116 Conclusions and recommendations to describe vapor/liquid equilibria in methane/n-nonane/water mixtures equally well as more complicated equations of state. The CPA eos contains a number of pure-component parameters which have been estimated on the basis of vapor pressure data and liquid density data, in the temperature range of interest. To describe equilibria in mixtures, one temperature-dependent parameter is used. This parameter has been determined by comparison with experimentally obtained equilibrium compositions of the liquid phase. It is shown that ”fitting” the eos to liquid compositions is much more effective than ”fitting” the eos to the equilibrium composition of the gas phase. Furthermore, allowing the mixture parameter to vary with temperature gives more accurate results than assuming a constant mixture parameter, as is often done. Using the temperature-dependent mixture parameter the qualitativily correct temperature-dependent liquid/liquid equilibria of n-nonane and water can be obtained. This is not possible when assuming a constant mixture parameter. For further use in nucleation and growth calculations, the equilibrium compositions and liquid densities have been expressed as functions of pressure and temperature. The calculated equilibrium vapor fractions of n-nonane or water in methane could be checked using the experimentally obtained droplet growth rates. In all cases the calculated values agree well with the experimental results. For the calculation of the homogeneous nucleation rates use is made of two different analytical expressions, based on the classical nucleation theory. These are the ICCT version and the CNT version, respectively. The homogeneous nucleation rates of a vapor in a high pressure carrier gas are calculated with the unary theory, using the equilibrium properties (density, surface energy, etc.) of the mixture. Two sufficient criteria are derived for this approximation to be valid. One of the criteria depends on the partial volume of the gas in the cluster, the other criteria depends on the entropy of mixing, due to the dissolution of the gas into the cluster. Homogeneous nucleation rates of water in methane have been experimentally determined as a function of supersaturation, at three different conditions, being 10 bar and 235 K, 25 bar and 235 K, and 40 bar and 240 K. In all cases the results of the CNT version of the classical theory agrees better with the experimental results than the results of the ICCT version. Nucleation rates of n-nonane in methane have been determined at two different conditions, being 10 bar and 235 K, and 40 bar and 240 K. The experimental nucleation rates obtained at 10 bar are in qualitative agreement with the predictions of the classical theory. However, the ICCT model gives too high nucleation rates, while the CNT model gives too low nucleation rates. The experimental nucleation rates obtained at 40 bar show a clear trend as function of the supersaturation, which strongly deviates from the measurements at 10 bar. The measured trend at 40 bar is not predicted by the classical theory. The experimentally obtained slope of the curve of the nucleation rate versus the supersaturation is much smaller than predicted by the theory. When applying the two criteria for a quasi-one-component theory to the 40 bar n- nonane/methane measurements, it is clear that both criteria are not satisfied. When the criteria are applied to the other nucleation rate data, they are also not satisfied. Yet, the quasi-one-component classical theory performs reasonably well, when comparing its results to the experimental results. This is explained as follows. The two criteria take into account a volume effect due to the dissolution of carrier gas into the cluster, and the 117 entropy of mixing due to the dissolution of carrier gas into the cluster. Both effects are counteracting and largely cancel each other in all cases considered, except for the 40 bar n-nonane/methane measurements. For future research it might be interesting to keep the full expression for the energy of cluster formation, i.e. without neglecting the volume and entropy effect of the dissolution of the carrier gas into the cluster. With the new setup, nucleation rates in ternary mixtures have also been studied. The mixtures consist of supersaturated n-nonane and supersaturated water in methane. Nucle- ation rates have been measured as a function of supersaturation at two different conditions, being 10 bar and 235 K, and 40 bar and 240 K. At each condition the nucleation rates in the ternary mixtures are compared to the sum of the (experimentally obtained) binary rates of n-nonane in methane and water in methane. From the comparison it is concluded that n-nonane and water nucleate independently, for all conditions considered. Besides nucleation rate data droplet growth curves have been obtained from the experiments. From the analysis of the surfaces growth rates as a function of the vapor fraction within each experimental series of the binary mixtures, the calculated equilibrium fractions could be validated. Furthermore, new experimental diffusion coefficients of n-nonane in methane and (supercooled) water in methane have been obtained. In future experiments the range of measured droplet growth rates can be extended by applying a deeper nucleation pulse. This can easily be achieved by putting a widening with a larger diameter in the low pressure section of the pulse-expansion wave tube. In this way, by selecting the same nucleation conditions, the growth of the droplets will occur at a smaller (supersaturated) vapor fraction. Low temperature and high pressure equilibrium vapor fractions can then be determined experimentally from analysis of the growth rates. From the analysis of the growth rates in the ternary mixture it becomes clear that supersaturated n-nonane can grow onto water droplets, but supersaturated water cannot grow onto n-nonane droplets. This can be explained by another effect that can occur (besides co-nucleation) in a system containing two different supersaturated vapors. One vapor can heterogeneously nucleate onto the droplet formed by the other supersaturated vapor. Whether or not this will happen can be deduced from theoretical considerations. Heterogeneous nucleation is more probable than homogeneous nucleation when the energy of formation of a critical cluster is smaller for the heterogeneous process than for the homogeneous process. This depends on the change of the total interfacial energy for both processes. The process for which the critical cluster ’produces’ the smallest change in the interfacial energy of the system is the most probable process. Using this criterion, it is shown that water will most probably not heterogeneously nucleate onto n-nonane, while heterogeneous nucleation of n-nonane onto water droplets is very likely to occur. This has indeed been confirmed by the analysis of the droplet growth rates. In order to describe the droplet growth rates a model has been developed based on the flux matching method. This model is valid for droplet growth in the Knudsen regime, continuum regime, as well as the transition regime. It can be applied to multi-component mixtures, in which the droplet is a uniform mixture of the condensing vapor components. It also includes the dissolution of the carrier gas into the droplet. The depletion of condensing vapor components is taken into account, as well as temperature effects due to the release 118 Conclusions and recommendations of heat of condensation. The model has been extended further to describe the growth of heterogeneous droplets, consisting of an inner liquid core and an outer (different) liquid layer. To describe the growth of water droplets on which n-nonane heterogeneously nucleates, it has been assumed that n-nonane completely wets the water droplet as soon as its size equals the size of a critical n-nonane cluster. Then, as soon as the n-nonane participates in the growth process, the outer n-nonane layer will form an increasingly large obstruction for the growth of the water core as the layer grows thicker. This can cause a decrease in the growth rate of the droplet when the supersaturation of the n-nonane is small. The effect of decreasing growth rate becomes smaller when the carrier gas pressure is increased at constant supersaturation. When the experimental growth results obtained at 11 bar and 242 K are compared with the growth rate predictions of the layered droplet model it is apparent that the predicted decrease in the growth rate at small supersaturation of n-nonane is not observed in the experimental results. We think that this is caused by n-nonane not completely wetting the water surface at these conditions. Rather, it heterogeneously nucleates onto the water droplet forming one or more caps. In this way, part of the water droplet remains uncovered and its growth is only slightly inhibited by the n-nonane cap(s). At a certain point the cap(s) will completely cover the water core and the layered model provides an accurate description of the droplet growth. The experimental growth results obtained at 44 bar and 247 K are in better agreement with the predictions of the layered droplet model. At this higher pressure the decrease of the growth rate was expected to be less pronounced. Furthermore, at these conditions the assumption that n-nonane completely wets the water surface is probably better too. Whether or not the n-nonane forms a cap (or caps) on water droplets can be determined experimentally in future experiments by measuring the polarization of the scattered light. The vertical polarization of the incident light is not changed when the light is scattered by spherical particles. This does not hold for non-spherical particles. In the measurements of the nucleation rates of supersaturated water vapor in helium as a function of temperature, a sharp transition has been observed at 207 K. We suggest that this transition is due to the change from vapor/liquid to vapor/solid nucleation. A qualitative theoretical explanation has been given based on the energy of formation of a critical cluster. To conclude, it is noteworthy that the setup can also be used to obtain vapor/liquid equilibria at higher temperatures (0◦ to 18◦ C) and pressures up to 95 bar. This is done as follows. First, a low pressure (1 bar) nucleation experiment is performed, for which the mixture is also prepared at low pressure. The composition of the mixture is then easily obtained from the saturated vapor pressure. Next, a series of experiments is performed for which the gas/vapor mixture is prepared at a constant temperature and constant high pressure. Then, from the nucleation rate data, the composition of the mixture can be deduced. Appendix A

Physical properties

This appendix summarizes the physical properties of the substances and mixtures used in all the equilibrium, nucleation, and growth calculations.

Helium

M = 4.003 kg kmol−1 [1] pc = 2.27 bar [1] Tc = 5.19 K [1] 3 −1 Vc = 57.4 cm mol [1] d = 0.2551 nm [1] −1 −1 cp = 5R/2 J mol K [1] k = 2.449 10−2 + 1.124 10−3T 2.929 10−6T 2 −+4.493 × 10−9T 3 2.518× 10−12−T 4 × W m−1 K−1 [2] × − × Methane

M = 16.043 kg kmol−1 [1] pc = 46.0 bar [1] Tc = 190.4 K [1] 3 −1 Vc = 99.2 cm mol [1] d = 0.3758 nm [1] −2 −5 2 cp = 19.25 + 5.213 10 T + 1.197 10 T 1.132 10−8×T 3 × J mol−1 K−1 [1] k◦ = 1−.8436 ×10−3 + 8.2265 10−5T + 8.6143 10−8T 2 W m−1 K−1 [3] × × × k = k◦(T ) + 0.03918 exp 0.535 pVc 1 W m−1 K−1 [1] ZRT − £ ¤

119 120 Physical properties n-Nonane

M = 128.259 kg kmol−1 [1] pc = 22.9 bar [1] Tc = 594.6 K [1] 3 −1 Vc = 548 cm mol [1] d = 0.683 nm [4] m = 1.405 (-) [5] c = 8.374 + 0.8729T 4.823 10−4T 2 + 1.031 10−7T 3 J mol−1 K−1 [1] p − − ∗ ∗ psat = 133.322 exp ( 17.56832 ln T + 0.0152556T 9467.4/T +−128.77889) Pa [2] − ρL = 733.5 0.788(T 273.15) 9.689− 10−5(T− 273.15)2 kg m−3 [2] L = 46− .43d3× ((1 T−/594.6)/(1 424/594.6))0.375 J mol−1 [1] σ = 0.02472 × 9.347− 10−5(T 273− .15) N m−1 [6] 0 − × − n-Nonane in methane

D = 0.356125 10−4T 1.75Z/p m−2 s−1 [1] × −1 σ = σ0 nakBT ln((p + pL)/pL) N m [7] − 18 na = 6.0 10 m 2 [7] × 7 5 2 2 − pL = 2.585 10 2.731 10 T + 7.781 10 T Pa [7] L × − × 2 × −3 ρ = 1825 (c0 + c1p/pc,n + c2(p/pc,n) ) mol m × 2 c0 = 22.482 + 224.482T/Tc,n 731.160(T/Tc,n) − 3 − 4 +1054.87(T/Tc,n) 571.918(T/Tc,n) (-) − 2 c1 = 78.2664 657.378T/Tc,n + 2094.70(T/Tc,n) − 3 4 2983.18(T/Tc,n) + 1598.36(T/Tc,n) (-) − 2 c2 = 17.7811 + 154.590T/Tc,n 504.155(T d/Tc,n) − 3 − 4 +730.735(T/Tc,n) 397.030(T/Tc,n) (-) The liquid density of the mixture is− obtained from the CPA eos given in chapter 2.

Water

M = 18.015 kg kmol−1 [1] pc = 221.2 bar [1] Tc = 647.3 K [1] 3 −1 Vc = 57.1 cm mol [1] d = 0.2641 nm [1] m = 1.334 (-) [5] cG = 32.24 + 1.924 10−3T + 1.055 10−5T 2 p × × 121

3.596 10−9T 3 J mol−1 K−1 [1] cL = 717− .88 ×7.626T + 0.03396T 2 6.730 10−5T 3 p − − × +5.014 10−8T 4 J mol−1 K−1 [8] Ice × −1 −1 cp = 1.884 + 0.1320T J mol K [9] L psat = 610.8 exp [ 5.1421 ln (T/273.15) 6828.77 (−1/T 1/273.15)] Pa [10] Ice − − psat = exp (( 2663.5/T + 12.537) ln(10)) Pa [11] ρL = 999.84−+ 0.086(T 273.15) 0.0108(T 273.15)2 kg m−3 [9] ρIce = 916.7 0.175(T −273.15) −5.0 10−4(−T 273.15)2 kg m−3 [9] L = 40.65d−3 ((1 T−/647.3)/(1− 373× .15/647.−3))0.375 J mol−1 [1] × − −4 − −1 σ0 = 0.127245 1.89845 10 T (for T < 268 K) N m [12] − −×4 −1 σwetice = 0.02850 + 2.5 10 (T 273.15) N m [9] Lmelt = 6.010 103 × (at T =− 273.15 K) J mol−1 [9] Levap = 4.065 × 104 (at T = 373.15 K) J mol−1 [8] × ∆Sice/vapor = Svapor Sice − Lmelting 373.15 Cp,liquid = Sice(273.15K) + 273.15 + 273.15 T dT + Levaporation + T Cp,vapor dT S (273.15K) 373.15 373.15 T R − ice T Cp,solid dT 273.15 T R = 8−.8649 0.13004T + 0.52750 10−5T 2 R 0.11987− 10−8T 3 + 30.356 ln×(T ) (J mol−1) − × Water in methane

D = 1.241888 10−4T 1.75Z/p m−2 s−1 [1] × −1 σ = σ0 nakBT ln((p + pL)/pL) N m [13] − 18 −2 na = 5.4 10 m [13] × 7 5 pL = 4.8195 10 + 2.1211 10 T Pa [13] ρL = −17.5 10×3(c + c p/p ×+ c (p/p )2) mol m−3 × 0 1 c,w 2 c,w c0 = 3.799 1.574T/Tc,w (-) − 2 c1 = 55.0431 + 512.856T/Tc,w 1797.13(T/Tc,w) − 3 − 4 +2804.23(T/Tc,w 1642.56(T/Tc,w) (-) − 2 c2 = 78.5506 738.809T/Tc,w + 2609.44(T/Tc,w) − 3 4 4098.90(T/Tc,w) + 2414.83(T/Tc,w) (-) The liquid densit−y of the mixture is obtained from the CPA eos given in chapter 2.

Water in n-nonane and methane

−12 2 1/6 0.6 Ln 2 −1 D = 8.93 10 (Vw/Vn ) (Pn/Pw) T/η m s [1] × 3 −1 Vw = 37.4 cm mol [1] 3 −1 Vn = 209.4 cm mol [1] 122 Physical properties

1/4 L 2 −1 Pw = (2Mwσ0,w)/ρw dyn cm mol [1] 1/4 Ln 2 −1 Pn = ((xmnMm + (1 xmn)Mn)σn )/ρ dyn cm mol [1] Ln − η = exp((1 xmn) ln(ηn) + xmn ln(ηm) +x (1− x )G ) cP [1] mn − mn nm Gnm = 1.24736(1.343 0.685(1 xmn)) cP [1] − − 3 − ηn = exp( 4.447 + 1.21 10 /T ) cP [1] − × −2 ηm = (0.3327654516 0.2479295082 10 T +0.4621290334− 10−5T 2) × cP [1] ×

Isothermal compressibility In the isentropic expansion temperature calculations, the isothermal compressibility factor Z and the isobaric heat capacity cp of the carrier gas are needed. These are calculated according to the equation of state proposed by Sychev et al. [14]:

10 7 ωi Z = 1 + A = 1 + b (A.1) 0 ij τ j i=1 j=0 X X 2 cp cp,0 (1 + A2) = 1 + A5 + (A.2) R R − 1 + A1 c 10 T j 6 T −j p,0 = α + β (A.3) R j 100 K j 100 K j=0 j=1 X µ ¶ X µ ¶ 10 7 ωi A = (i + 1)b (A.4) 1 ij τ j i=1 j=1 X X 10 7 ωi A = (j 1)b (A.5) 2 − − ij τ j i=1 j=0 X X 10 7 j(j 1) ωi A = − b (A.6) 5 − i ij τ j i=1 j=0 X X The above relations are given in terms of the reduced density ω = ρ/ρc and the reduced temperature τ = T/Tc. Therefore, the density is calculated iteratively from (A.1) and the deﬁnition Z = p/(ρRT ), before cp can be evaluated. For the coeﬃcients bij, αj and βj (many of which are zero, due to the general notation given above) the reader is referred to [14].

References

[1] R.C. Reid, J.M. Prausnitz, and B.E. Poling, The Properties of Gases and Liquids, McGraw–Hill Book Company, New York, 1987. A.0 References 123

[2] C.-H. Hung, M.J. Krasnopoler, and J.L. Katz, J. Chem. Phys. 90, 1856 (1989).

[3] Y.S. Touloukian, P.E. Liley, and S.C. Saxena, Thermophysical Properties of Matter, volume 3, IFI/Plenum, New York, 1970.

[4] D.V. Matyushov and R. Schmid, J. Chem. Phys. 104, 8627 (1996).

[5] Landolt–B¨ornstein, Zahlenwerte und Funktionen, volume II Band, 8. Teil, Springer Verlag, Berlin, 1962.

[6] J.J. Jasper and E.V. Kring, J. Phys. Chem. 59, 1019 (1955).

[7] C.C.M. Luijten and M.E.H. van Dongen, J. Chem. Phys. 111, 8524 (1999).

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

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

[10] N.B. Vargaftik, Tables on the thermophysical properties of liquids and gases 2nd edition, Wiley, New York, 1975.

[11] J. Marti and K. Mauersberger, Geophysical Research Letters 20, 363 (1993).

[12] P.T. Hacker, Experimental values of the surface tension of supercooled water, National Advisory Committee for Aeronautics, 1951, technical note 2510.

[13] P. Peeters, J. Hruby,´ and M.E.H. van Dongen, J. Phys. Chem. B 105, 11763 (2001).

[14] V.V. Sychev et al., Thermodynamic Properties of Methane, Hemisphere publishing corporation, Washington, 1987. 124 Physical properties Appendix B

Energy of cluster formation

By combining the ﬁrst and second law of thermodynamics one can obtain

dU + pdV T dS 0 (B.1) − ≤ For two states of the same system one can therefore write

W = ∆(U + pV T S) 0. (B.2) − ≤ Consider two states of the same system which contains n molecules. In state I the system is at pressure p and temperature T . In state II the system is at the same pressure and temperature, only now it contains a single cluster. The cluster consists of nL bulk molecules and nS surface molecules, leaving nG = n nL nS molecules in the gas phase. The energy W is now given by − −

W = U II U I + p(V II V I ) T (SII SI ) (B.3) − − − − For each subsystem (gas, liquid or surface) the internal energy is given by U = µn pV +T S. Therefore, the internal energy in state I is −

U I = nµG pV I + T SI (B.4) − and the internal energy of state II is

U II = U G + U L + U S = nGµG pV G + T SG − +nLµL pclusV clus + T SL − +nSµS + φ + T SS, (B.5) where the surface term φ can be expressed as σA, which is the surface energy times the area. Now we have for the energy W :

W = nGµG + nLµL + nSµS nµG + φ + V clus(p pclus). (B.6) − − 125 126 Energy of cluster formation

The entropy terms cancel due to the fact that SII = SG + SL + SS. Furthermore, V clus = V II V G was used. In literature different terminologies are used for the energy difference W − in the case of cluster formation. Inspired by the fact that the pressure and temperature are constant it is often called the change in Gibbs free energy. Strictly spoken, this is incorrect since the Gibbs free energy of state II cannot be defined, since the pressure is not uniform. It is also often called the work of formation, or the work of cluster formation. However, the energy W incorporates more than just terms describing work in a thermodynamic sense. Therefore, we prefer to call W the energy of (cluster) formation, which is also encountered in literature. Appendix C

Droplet growth; derivation of equations

C.1 Incoming mass ﬂux

The exact solution of the integral given by equation (4.20) can be obtained using standard integrals (or, more conveniently, a standard software package like Maple), resulting in

0 ∞ ∞ ˙ − 2 − Mj = 4πrd ρKn,jξrjfj dξrjdξθjdξφj −∞Z −∞Z −∞Z 4πr2 d √ 2 2 √ 2 2 = 2 2 10 πuρKn,jR TKn 10 πvjρKn,jR TKn + −20√πR TKn − − © 2RTKn 2 2 5 2MjRTKnRTKnuvjρKn,j 2 2MjRTKnuq˙Kn,j + 10 R TKnρKn,j Ã − s Mj ! × p p u2M exp j + −2RT · Kn ¸ 2 2 2 2 u Mj 10√πuρKn,jR TKn + 10√πvjρKnjR TKn erf . (C.1) "√2pRTKn #) ¡ ¢ This exact solution can be simpliﬁed, using the fact that the Mach number is much smaller than unity. Then, x = u√M/√2RT < u√M/√γRT = Ma 1, and exp x 1+x and ¿ ≈ erfx 2x/√π. Using this¯ approximation¯ ¯and introducing¯ the eos y p = Zρ RT results in ≈ ¯ ¯ ¯ ¯ j j ¯ ¯ ¯ ¯ (u + v ) πM ˙ − 2 yKn,jp 1 j j Mj = 4πrd 1 . (C.2) − Z 2πMjRTKn Ã − √2RTpKn ! Using equation (4.5) p ˙ 2 Mj = 4πrKnρKn,j(u + vj), (C.3)

127 128 Droplet growth; derivation of equations this can be rewritten as

2 ˙ − 2 yKn,jp 1 rd ˙ Mj = 4πrd + 2 Mj. (C.4) − Z 2πMjRTKn 2rKn p C.2 Energy ﬂux

For the determination of the energy flux the gas is first treated as an ideal gas. The properties of the actual gas are then inserted into the final result. The total molar internal energy of component i in the gas phase is given by

e + e + e = c T RT. (C.5) rot,i vib,i tran,i p,i − When the system is at rest the translational energy is given by 3 e = RT, (C.6) tran,i 2 rendering the following result for the rotational and vibrational energy:

5 e + e = c R T. (C.7) rot,i vib,i p,i − 2 µ ¶ If the translational energy is uncorrelated to the rotational and vibrational energy the last result also holds when the system is not at rest. The total outgoing energy ﬂux of component i is then given by

∞ ∞ ∞ 1 E˙ + = 4πr2 ρ ξ ξ2 + ξ2 + ξ2 + e + e f +dξ dξ dξ i d d,i ri 2 ri θi φi rot,i vib,i i ri θi φi Z Z Z · ¸ 0 −∞ −∞ ¡ ¢ 2 ρd,i 2 2 ρd,i = 4πr (erot,i + evib,i) RTd + 2R T . (C.8) d √2πM RT d √2πM RT ½ i d i d ¾ Substitution of ˙ + 2 ρd,iRTd Mi = 4πrd , (C.9) √2πMiRTd and equation (C.7) results in

R E˙ + = M˙ + c T . (C.10) i i p,i − 2 d µ ¶ Of course, M˙ + = M˙ M˙ −, which then gives the ﬁrst summand of equation (4.22). i i − i If it is again assumed that the translational energy of the molecules is uncorrelated to the rotational and vibrational energy, the total incoming molar translational energy of C.3 Liquid layer 129 component i is given by

0 ∞ ∞ 1 E˙ − = 4πr2 ρ ξ ξ2 + ξ2 + ξ2 f −dξ dξ dξ i d 2 Kn,i ri ri θi φi i ri θi φi Z Z Z −∞ −∞ −∞ ¡ ¢ 2 rd 2 3 2 2 = 15πMiviu ρKn,iRTKn + 5πMiu ρKn,iRTKn + 25πuρKn,iR TKn+ 5RTKn 10πq˙ ©RT + 15πM v u2ρ RT 5πM u3ρ RT Kn,i Kn − i i Kn,i Kn − i Kn,i Kn− √ 2 2 £ u Mi 25πuρKn,iR T 10πq˙Kn,iRTKn erf + Kn − √2πRT µ Kn ¶ 2πRT ¤ 20 Kn ρ R2T 2 + 2 2πM RT q˙ u M Kn,i Kn i Kn Kn,i "− r i − p 20 2πM RT v uρ RT 5 2πM RT u2ρ RT i Kn i Kn,i Kn − i Kn Kn,i Kn × p u2M p i exp i . (C.11) −2RT µ Kn ¶¾ This exact solution can again be simplified, since u√M /√2RT 1. First order Taylor i ¿ expansion of the exponential and error function then¯ results in ¯ ¯ ¯ 2ρ R2T 2 ρ ¯ u 5RT ¯ q˙ E˙ − = 4πr2 Kn,i Kn Kn,i Kn Kn,i . (C.12) i − d √2πM RT − 2 2 − 2 · i Kn µ ¶ ¸ The heat flux per unit area is due to conduction and the diffusion of enthalpy, 5RT q˙ = q˙ + ρ v Kn . (C.13) Kn,i Kn,ci Kn,i i 2 µ ¶ Substituting this into equation (C.12), together with ˙ 2 Mi = 4πrKnρKn,i (u + vi) , (C.14) then results in 2 2 2 − 2 2ρKn,iR TKn rd 5RTKn E˙ = 4πr + M˙ i + Q˙ ci , (C.15) i − d √2πM RT 2r2 2 i Kn Kn ·µ ¶ ¸ ˙ 2 where Qci = 4πrKnq˙Kn,ci was used.

C.3 Liquid layer

The conservation of mass of component j in the liquid layer is given by d rdl M˙ Ll M˙ = xLl(r)ρLl4πr2dr, (C.16) j − j dt j Zrdc 130 Droplet growth; derivation of equations

We assume a that the molar density of the liquid layer does not depend on the radius. The molar ﬂux through the liquid layer is given by

dxLl M˙ Ll = 4πr2ρLlDLl j . (C.17) j − j dr At stationary conditions the mass ﬂux is constant. Therefore, integration of

Ll x ˙ Ll rdl rdl,j M 1 Ll − j , dxj = Ll Ll , 2 dr , (C.18) xLl(r) 4πρ D r (r ) Z j j Z results in the following radius dependent fraction in the liquid layer:

M˙ Ll 1 M˙ Ll xLl(r) = j + − j + xLl . (C.19) j Ll Ll Ll Ll rrl,j 4πρ Dj r Ã4πρ Dj rdl !

Inserting this result into the integral of equation (C.16) yields

rdl M˙ Ll 1 1 1 r3 4 xLl(r)ρLl4πr2dr = j r2 r2 + dc + πxLl ρLl r3 r3 . (C.20) j DLl 6 dl − 2 dc 3 r 3 rdl,j dl − dc Zrdc j µ dl ¶ ¡ ¢ Appendix D

Tables of experimental data

This appendix summarizes the results of nucleation and growth experiments.

2 exp p0 T0 yn p T J ∆t dr /dt 105 10−3 1012 (bar) (K) × (bar) (K) (m−3s−1) ×(s) (m×2 s−1) 302 24.71 293.75 23.79 10.02 234.57 6.0E15 0.400 17.8 303 24.71 294.65 24.96 9.98 235.13 5.5E15 0.416 18.4 304 24.73 294.15 25.61 10.02 234.87 2.0E16 0.400 19.0 305 24.74 293.85 26.65 10.15 235.32 2.9e16 0.368 - 306 24.73 294.85 27.74 10.02 235.48 4.9E16 0.368 - 307 24.70 294.15 22.97 9.96 234.57 2.5E15 00.400 17.1 308 24.71 293.95 22.19 10.03 234.81 1.0E15 0.368 16.7 309 24.70 294.85 21.49 9.99 235.36 2.3E14 0.400 16.2 310 24.71 294.35 20.81 9.95 234.70 3.0E14 0.400 15.6 311 24.73 294.15 20.17 10.06 235.10 8.8E13 0.400 15.2

340 88.13 296.05 10.32 40.05 240.38 2.8E17 0.320 9.40 341 88.43 296.35 7.06 39.95 240.28 4.0E16 0.336 4.90 342 88.43 296.45 5.76 39.94 240.34 1.3E16 0.336 3.28 343 88.43 296.15 4.48 40.04 240.24 5.3E15 0.352 1.85 344 88.43 296.65 8.43 39.95 240.53 7.0E16 0.384 6.17

028 23.05 294.85 36.30 10.17 240.69 1.0E17 0.416 -

Table D.1: Data of n-nonane in methane.

131 132 Tables of experimental data

2 exp p0 T0 yw p T J ∆t dr /dt 105 10−3 1012 × × × (bar) (K) (bar) (K) (m−3s−1) (s) (m2 s−1) 103 24.57 293.85 23.87 9.95 234.57 1.6E14 0.368 5.56 104 24.60 294.25 28.65 9.95 234.86 4.6e15 0.352 6.84 105 24.57 294.55 32.78 10.01 235.54 4.5E16 0.352 - 106 24.58 294.45 32.72 9.98 235.28 5.0E16 0.368 - 107 24.50 294.75 30.79 10.02 235.98 2.2E15 0.368 7.65 108 24.55 294.55 26.35 9.97 235.32 1.7E14 0.368 6.39 170 24.92 295.45 32.57 10.06 235.76 2.4E16 0.368 - 195 24.50 293.95 32.13 9.85 234.28 1.7e17 0.272 - 201 24.65 295.25 29.90 10.01 235.92 1.1E15 0.368 7.00

114 60.02 294.75 11.84 24.89 234.49 1.5E17 0.384 - 115 60.07 295.15 11.82 25.22 235.59 3.3E16 0.400 - 116 60.09 294.85 11.30 25.09 235.00 2.0E16 0.384 - 117 60.12 295.15 10.10 25.07 235.15 4.6E15 0.368 - 118 60.06 294.75 8.87 25.33 235.52 2.1E14 0.400 - 119 60.05 294.85 9.49 25.03 234.88 2.0E15 0.384 - 120 60.10 295.05 10.68 25.06 235.07 8.6E15 0.384 -

173 87.39 295.45 9.24 40.13 240.55 1.3E16 0.400 2.09 174 87.29 294.75 10.03 40.00 239.81 1.3E17 0.416 2.06 185 87.40 295.15 8.93 39.94 239.99 1.3E16 0.416 1.90 186 87.72 295.55 9.44 40.17 240.45 2.5E16 0.272 2.05 187 87.80 295.95 8.82 39.93 240.35 7.7E15 0.352 1.87 188 87.89 295.85 9.69 39.75 239.91 4.2E16 0.400 2.05 189 87.91 296.15 10.07 40.06 240.66 2.5E16 0.368 2.17 190 86.97 294.55 10.06 40.07 239.99 1.4E17 0.352 2.23 191 87.53 295.35 10.36 40.21 240.48 1.1E17 0.304 2.27 192 87.99 295.55 9.98 40.10 240.13 8.0E16 0.352 2.16

Table D.2: Data of water in methane. 133

2 exp p0 T0 yw yn p T J ∆t dr /dt 105 105 10−3 1012 (bar) (K) × × (bar) (K) (m−3s−1) ×(s) (m×2 s−1) 253 24.76 294.45 31.23 0.96 10.02 235.03 4.0E16 0.224 7.45 252 24.56 294.45 31.09 1.91 9.95 235.13 2.2E16 0.352 7.37 251 24.50 294.25 31.10 2.87 9.86 234.60 4.3E16 0.352 7.10 203 24.67 295.25 30.60 4.92 9.97 235.63 7.5E15 0.368 7.56 204 24.70 295.15 30.74 7.45 9.94 235.34 2.0E16 0.384 7.60 205 24.71 295.25 30.86 9.98 10.00 235.74 1.0E16 0.432 8.80 206 24.69 295.35 30.23 12.78 10.00 235.86 5.3E15 0.432 10.3 208 24.72 294.55 30.18 17.93 10.15 235.98 7.0E15 0.384 13.9 209 24.72 294.05 30.06 20.48 9.99 234.60 6.1E16 0.384 14.7

225 24.73 294.55 29.33 22.62 10.00 235.08 6.6E15 0.400 17.1 224 24.73 294.75 23.83 23.01 9.99 235.17 1.7E15 0.400 17.5 223 24.74 295.25 19.10 23.02 10.00 235.69 6.7E14 0.416 17.5 222 24.74 295.35 15.30 22.69 10.01 235.82 4.4E14 0.416 17.2 221 24.76 295.35 9.44 23.01 9.95 235.41 1.0E15 0.400 17.2 220 24.76 295.15 4.73 22.95 10.10 236.10 4.3E14 0.368 17.4

230 88.70 296.65 1.83 7.51 40.03 240.45 6.3E16 0.304 5.51 231 88.80 296.65 3.72 7.30 39.95 240.25 5.8E16 0.352 5.54 232 87.59 295.25 5.76 7.18 39.85 239.78 2.5E16 0.368 5.18 234 87.80 295.35 9.65 7.25 39.97 239.89 6.6E16 0.352 5.49 240 88.10 295.65 9.63 5.66 39.89 239.81 7.3E16 0.368 3.90 241 87.99 295.55 9.44 4.28 40.14 240.20 3.8E16 0.272 2.48 242 88.02 295.95 9.44 2.89 40.24 240.69 1.2E16 0.304 1.89

Table D.3: Data of water and n-nonane in methane. 134 Tables of experimental data

run p (bar) T (K) y 104 p (bar) T (K) J (m−3s−1) 0 0 × 425 1.678 295.75 40.20 0.9692 237.49 3.9E15 429 1.719 296.65 36.58 0.9721 236.23 3.1e14 430 1.738 296.65 36.66 0.9644 234.414 7.6E15 432 1.758 296.55 32.10 0.9484 231.70 3.2E15 433 1.778 296.45 29.65 0.9556 231.30 4.02E14 434 1.799 296.55 27.58 0.9554 230.27 1.97E14 435 1.819 296.45 27.66 0.9507 228.75 4.5E15 436 1.838 296.55 26.72 0.9371 226.54 7.4e16 437 1.859 296.55 24.96 0.9413 225.95 2.4E16 438 1.878 296.55 22.72 0.9482 225.66 3.8E15 439 1.938 296.45 20.35 0.9358 221.59 8.0E16 440 1.978 296.35 18.90 0.9326 219.42 4.0E17 442 1.978 296.65 16.49 0.9204 218.49 9.1E16 444 1.999 296.65 14.87 0.9359 219.05 4.3E15 448 2.099 296.45 11.24 0.9316 214.24 3.2E16 450 2.099 296.15 11.58 0.9741 217.90 8.2E14 451 2.099 296.15 11.70 0.9386 214.70 4.1E16 452 2.099 296.15 11.54 0.9553 216.22 3.9E15 455 2.113 295.95 10.23 0.9213 212.37 1.1E17 469 2.303 295.25 6.425 0.9725 209.19 2.0E15 470 2.303 295.35 6.649 0.9605 208.22 1.7E16 471 2.303 295.45 6.455 0.9583 208.10 2.2E16 477 2.503 294.85 2.872 1.005 204.74 9.1E15 480 2.503 295.15 2.612 0.9855 203.34 1.6E16 482 2.518 295.45 2.391 0.9685 201.66 3.0E16 484 2.343 295.65 4.584 0.9554 206.85 9.1E16 486 2.343 295.45 4.093 0.9580 206.65 4.4E16 487 2.383 295.55 3.434 0.9629 205.75 1.4E16 491 2.108 295.65 14.33 0.9851 218.14 1.2E16 493 2.048 295.75 17.19 0.9817 220.42 1.6E16 494 1.999 295.85 19.86 0.9809 222.60 1.2E16 496 2.192 294.45 9.723 0.9721 212.73 5.0E16 498 1.737 295.45 40.19 1.008 237.67 2.2E15 499 1.798 294.85 30.81 0.9994 233.15 1.7E14 500 1.838 295.05 30.77 0.9968 231.06 3.3E15 505 5.054 295.25 1.974 2.1135 208.37 1.5E16

Table D.4: Data of water in helium. Appendix E

Droplet growth curves

135 136 Droplet growth curves

Figure E.1: Experimental growth curves together with model calculations, for n-nonane in methane at 11 bar and 242 K. 137

Figure E.2: Experimental growth curves together with model calculations, for water in methane at 11 bar and 242 K. 138 Droplet growth curves

Figure E.3: Experimental growth curves together with model calculations, for n-nonane in methane at 44 bar and 247 K. 139

Figure E.4: Experimental growth curves together with model calculations, for water in methane at 44 bar and 247 K. 140 Droplet growth curves

Figure E.5: Experimental growth curves together with model calculations, for water in methane/n-nonane at 11 bar and 242 K. 141

Figure E.6: Experimental growth curves together with model calculations, for water in methane/n-nonane at 44 bar and 247 K. 142 Droplet growth curves

Figure E.7: Experimental growth curves together with model calculations, for n-nonane in methane/water at 11 bar and 242 K. The dotted line represents the calculated squared radius of the inner water core. 143

Figure E.8: Experimental growth curves together with model calculations, for n-nonane in methane/water at 44 bar and 247 K. The dotted line represents the calculated squared radius of the inner water core. 144 Droplet growth curves Summary

The vapor-to-liquid nucleation of supersaturated n-nonane and/or water in methane, and subsequent droplet growth, have been studied experimentally using a pulse-expansion wave tube. The pulse-expansion wave tube is basically a modified shock tube, and it operates according to the nucleation-pulse principle. With the nucleation-pulse method the nucleation stage and growth stage of the droplets are effectively separated in time. During a very short period of time droplets are formed, which then start to grow simultaneously, leading to a monodisperse droplet cloud. The growing droplets are detected by measuring both the intensity of the transmitted laser beam and the intensity of light scattered at an angle of 90◦. The gas/vapor mixtures are prepared by a newly designed mixture preparation device. With this new setup, mixtures of accurately known composition containing a carrier gas and two vapors can be prepared. For the analysis of the experimental results vapor/liquid equilibria need to be known. These are calculated using the cubic plus association (CPA) equation of state (eos). The pure component parameters of the CPA eos for methane, n-nonane, and water, have been derived from saturated vapor pressure data and liquid density data. A single temperature- dependent mixture parameter is used, obtained from comparison of computed and experimental equilibrium liquid phase compositions. The experimental nucleation rate data of the binary systems have been compared with the CNT (classical nucleation theory) and ICCT (internally consistent classical theory) version of the classical nucleation theory. The experimental nucleation rate data of supersaturated water in methane have been obtained at 10 bar and 235 K, 25 bar and 235 K, and 40 bar and 240 K. In all cases results from CNT agree reasonably with the experimental results, while results from ICCT predict too large nucleation rates as a function of supersaturation. Nucleation rates of supersaturated n-nonane in methane have been obtained at 10 bar and 235 K, and 40 bar and 240 K. Results from CNT and ICCT are in qualitative agreement with the 10 bar measurements. The 40 bar measurements show a much smaller supersaturation dependence, than predicted by CNT and ICCT. At these conditions the quasi-one-component theory should be adapted to include the volumetric effect and the effect of entropy of mixing due to the dissolution of the carrier gas into the cluster. The experimental nucleation rates of supersaturated n-nonane and water in methane have also been obtained at 10 bar and 235 K, and 40 bar and 240 K. From comparison with the experimentally obtained nucleation rates in the binary systems it can be concluded that supersaturated n-nonane and water nucleate independently.

145 146 Summary

From the analysis of the droplet growth rates in the binary systems, the calculated equilibrium vapor fractions are validated. Furthermore, diffusion coefficients of (supercooled) water in methane and n-nonane in methane have been determined experimentally. From the analysis of the droplet growth rates in the systems in which both n-nonane and water are supersaturated in methane, it is concluded that n-nonane does grow onto water droplets, while water does not grow onto n-nonane droplets. A droplet growth model for the growth of multi-component droplets in real (i.e. not inert) carrier gasses has been described based on the flux-matching method. The model has been extended to describe the growth of heterogeneous droplets, consisting of a liquid core surrounded by a (different) liquid layer. This model has been used to describe the growth of n-nonane onto water droplets in systems consisting of supersaturated n-nonane and water in methane. It was assumed that n-nonane completely wets the water surface. The model predicts too small growth rates for small values of the n-nonane supersaturation. This can be due to the incomplete wetting of n-nonane on water. The n-nonane then first heterogeneously nucleates onto the water droplet forming one or more caps. These caps then grow larger until they eventually merge and cover the whole surface of the water droplet. For larger carrier gas pressures (and smaller vapor fractions) the agreement between the predictions of the layered droplet model and the experimental results becomes better. The temperature-dependent nucleation behavior of supersaturated water vapor in helium has been investigated experimentally. With decreasing nucleation temperatures a sharp transition in the nucleation rates is observed at 207 K. It is hypothesized that this transition is caused by a change in the nucleation process. For temperatures above 207 K vapor/liquid nucleation occurs while for temperatures lower than 207 K vapor/solid nucleation occurs. This hypothesis is supported by a qualitative analysis of the energy of formation of a critical ice cluster and a critical (liquid) water cluster. Samenvatting

De vloeistofnucleatie in systemen bestaande uit oververzadigd n-nonaan en/of oververzadigd water in methaangas is experimenteel bestudeerd. Hierbij is gebruik gemaakt van een puls- expansiegolfbuis. Dit is een aangepaste schokbuis, waardoor de experimenten volgens het nucleatiepulsprincipe uitgevoerd kunnen worden. Volgens het nucleatiepulsprincipe worden het nucleatieproces en het druppelgroeiproces gescheiden in de tijd. Eerst worden er gedurende een korte tijd stabiele vloeistofclusters gevormd, waarna deze clusters gelijk- tijdig beginnen te groeien tot grotere druppels. Hierdoor ontstaat een wolk van druppels die allemaal vrijwel even groot zijn. De wolk met groeiende druppels wordt belicht met een laserbundel. De druppels worden gedetecteerd door de lichtintensiteit van het doorgelaten licht te meten, samen met de intensiteit van het licht dat onder een hoek van 90◦ wordt ver- strooid. De gas-/dampmengsels worden geprepareerd met een nieuw ontworpen opstelling. Met deze nieuwe opstelling kunnen mengsels van nauwkeurig bekende samenstelling worden gemaakt, bestaande uit een gas plus twee dampcomponenten. Voor de analyse van de experimentele resultaten moeten de vloeistof-gas-evenwichten bekend zijn. Deze evenwichten zijn berekend met behulp van de CPA (cubic plus association) toestandsvergelijking. De parameters van de CPA toestandsvergelijking voor de pure componenten methaan, n-nonaan en water zijn bepaald aan de hand van data voor de verzadigde dampdrukken en vloeistofdichtheden. Voor de beschrijving van mengsels is gebruik gemaakt van een temperatuurafhankelijke mengselparameter, die bepaald is aan de hand van experimenteel bepaalde vloeistofcomposities. De experimenteel verkregen nucleatiesnelheden in de binaire systemen zijn vergeleken met de CNT (classical nucleation theory) en ICCT (internally consistent classical theory) versie van de klassieke nucleatietheorie. De nucleatiesnelheden van oververzadigd water in methaan zijn bepaald bij 10 bar en 235 K, 25 bar en 235 K, en 40 bar en 240 K. In al deze gevallen geeft CNT een redelijke beschrijving van de experimentele resultaten, terwijl ICCT te grote nucleatiesnelheden geeft. De nucleatiesnelheden van oververzadigd n-nonaan in methaan zijn bepaald bij 10 bar en 235 K, en 40 bar en 240 K. De beschrijvingen van CNT en ICCT zijn in kwalitatieve overeenkomst met de experimentele resultaten verkregen bij 10 bar. De experimentele 40 bar resultaten vertonen een veel kleinere afhankelijkheid van de oververzadiging dan voorspeld door zowel CNT als ICCT. Bij deze conditie moet de quasi- één-component-theorie aangepast worden, zodat het volume- en mengtropie-effect van het methaangas in het vloeistofcluster verdisconteerd wordt. De experimenteel bepaalde nucleatiesnelheden van oververzadigd n-nonaan en oververzadigd water in methaangas zijn

147 148 Samenvatting ook bepaald bij 10 bar en 235 K, en 40 bar en 240 K. Uit de vergelijking met de experimenteel bepaalde nucleatiesnelheden in de binaire systemen kan geconcludeerd worden dat oververzadigd n-nonaan en oververzadigd water onafhankelijk van elkaar nucleëren. De berekende evenwichtsdampfracties zijn gevalideerd aan de hand van de experimenteel verkregen druppelgroeisnelheden. Verder zijn ook de diffusiecoëfficiënten van (on- derkoeld) water in methaangas en n-nonaan in methaangas experimenteel bepaald. Uit de analyse van de druppelgroeisnelheden in systemen waarin zowel n-nonaan als water oververzadigd aanwezig zijn in methaangas, kan geconcludeerd worden dat n-nonaan wel op waterdruppels groeit, terwijl water niet op n-nonaandruppels groeit. Er is een druppelgroeimodel geformuleerd dat de groei van multi-component-druppels in een reëel (d.w.z. niet inert) gas beschrijft. Het model is gebaseerd op de methode van ”flux-matching”.Het model is uitgebreid om de groei van heterogene druppels te beschrijven, bestaande uit een kern van vloeistof I, omringd door een mantel van vloeistof II. Dit model is toegepast op systemen bestaande uit oververzadigd n-nonaan en oververzadigd water in methaangas, waarbij n-nonaan op waterdruppels groeit. Hierbij is veronder- steld dat n-nonaan ”complete wetting” is op het wateroppervlak. Dit model geeft te lage druppelgroeisnelheden voor kleine n-nonaanfracties. Dit komt waarschijnlijk doordat de aanname van ”complete wetting” niet geldig is. Dit betekent dat in eerste instantie het n-nonaan heterogeen nucleëert op de waterdruppel waarbij het één of meerdere ”lenzen” op de druppels vormt. Deze ”lenzen” groeien dan vervolgens totdat ze uiteindelijk de hele waterdruppel omringen. Bij hogere drukken van het methaangas (en kleinere dampfrac- ties) wordt de overeenstemming tussen de met het model berekende druppelgroeisnelheden en de experimenteel bepaalde druppelgroeisnelheden beter. Het temperatuurafhankelijke nucleatiegedrag van oververzadigd waterdamp in helium- gas is experimenteel onderzocht. Bij 207 K is een scherpe overgang in het nucleatiegedrag waargenomen. De verklaring is dat deze overgang veroorzaakt wordt door een verandering van het nucleatieproces. Boven 207 K vind nucleatie van damp naar vloeistof plaats, terwijl voor temperaturen lager dan 207 K nucleatie van damp naar vaste stof plaatsvindt. Deze verklaring wordt ondersteund door een kwalitatieve analyse van de vormingsenergie van een kritisch ijscluster en een kritisch (vloeibaar) watercluster. Nawoord

Dit onderzoek is mede tot stand gekomen dankzij de financiële steun van Twister B.V., producent van supersone gasscheiders voor de aardgasindustrie. Met name de laatste jaren fungeerde Marco Betting als contactpersoon tussen Twister B.V. en de Technische Universiteit Eindhoven. Tijdens de gesprekken over de projectvoortgang ontstonden vaak interessante discussies, waarbij altijd voldoende ruimte was voor het uitwisselen van nieuwe ideeën. Dit project is denk ik een mooi voorbeeld waarbij industriële en wetenschappelijke ontwikkeling hand in hand gaan. Algemeen kan gesteld worden dat het doel van een promotie-onderzoek is het vergaren van meer kennis op een bepaald gebied. Het is de taak van de promovendus om dit te bewerkstelligen. Meestal doet hij dit echter niet alleen. In eerste instantie rapporteert de promovendus aan zijn promotor. In mijn geval was dit Rini van Dongen, iets waar ik mezelf zeer gelukkig mee prijs. Zijn deur stond altijd open, niet alleen als ik iets te rapporteren had, maar ook als ik vragen had. Hij heeft me vrij gelaten om de richting van het onderzoek binnen het gestelde doel voor een groot deel zelf te bepalen. Tegelijkertijd heeft hij ertoe bijgedragen om de gekozen richtingen tot een goed einde te brengen. Another person whom I want to thank for interesting discussions about nucleation is Jan (or, Honza) Hruby from the Institute of Thermomechanics in Prague, who resided in our group for over one year, and with whom I shared the office during that time. I am especially thankull to him for his contribution to the improvement of the experimental setup. He is largely responsible for the design of the new and improved setup for the preparation of the gas/vapor mixture. Dit werk leunt voor een groot deel op de verkregen experimentele resultaten. Be- trouwbare experimentele resultaten kunnen alleen verkregen worden met een betrouw- bare opstelling. Er is een hele groep van bekwame technici die hiervoor gezorgd hebben. Gedurende het eerste jaar van mijn promotie zijn we verhuisd naar een nieuw laborato- rium. Harm Jager heeft voor een groot deel de verhuizing en de opbouw van mijn opstelling voor zijn rekening genomen, en ik ben hem daar erg dankbaar voor. Mijn dank gaat ook uit naar Herman Koolmees, die het ontwerp van de opstelling voor de mengselpreparatie heeft verwezenlijkt. Ad Holten bood altijd een helpende hand bij problemen met de optiek en de elektronica. Met dat laatste heeft Freek van Uitert ook nog zijn steentje bijgedragen. Andere technici die ook bijgedragen hebben aan het soepele verloop van de experimenten zijn Jan Willems, Gerard Trines, Gerald Oerlemans, Joachim Tempelaar en Eep van Voorthuizen.

149 150 Nawoord

Zonder studenten is er geen universiteit. Mijn eigen afstudeerwerk heb ik gedaan bij Carlo Luijten. Mede door zijn enthousiasme voor het wetenschappelijk onderzoek heb ik besloten zelf ook te gaan promoveren. En ook bij dit onderzoek zijn een aantal studenten betrokken geweest. Allereerst was er Corine Fabrie, die zich tijdens haar eerste stage heeft ingezet voor het meten van evenwichtsdampfracties. Daarna kwam afstudeerder Gerben Pieterse, die me geholpen heeft met het zoeken naar nieuwe evenwichten, zodat we vervolgens de groei van de druppels konden gaan beschrijven. Tenslotte is Joost Gielis samen met mij op zoek gegaan naar ijs. En uiteindelijk hebben we dat ook gevonden. Naast hun inhoudelijke bijdrage hebben deze mensen ook gezorgd voor een gezellige werksfeer. Dit geldt ook voor alle collega promovendi, technici en de wetenschappelijke staf. De goede herinneringen aan de (soms bizarre) gesprekken gedurende de koffie- en lunchpauzes zal ik me nog lang herinneren. Allemaal bedankt voor het feit dat ik naar vier plezierige jaren kan terugkijken. Er bestaat natuurlijk ook nog leven na de universiteit. Familie en vrienden helpen door hun steun en luisterend oor. Een aantal mensen wil ik in het bijzonder bedanken. Allereerst zijn dat mijn ouders. Voor hun onvoorwaardelijke steun en hulp om het beste uit mezelf te halen blijf ik hen altijd dankbaar. Als laatste, maar zeker niet op de laatste plaats, wil ik jou bedanken Lucille. Door jou weet ik zeker dat er altijd één persoon zal zijn die in mij gelooft. Op die manier kan ik dus eigenlijk nooit falen. Dank je voor al jouw steun en vertrouwen. Curriculum Vitae

29 oktober 1973 : geboren te Weert

september 1986 - juni 1992 : Voorbereidend Wetenschappelijk Onderwijs, Philips van Horne Scholengemeenschap, Weert

september 1992 - augustus 1998 : Technische Natuurkunde, Technische Universiteit Eindhoven

september 1998 - augustus 2002 : Assistent in Opleiding in dienst van TUE, faculteit Technische Natuurkunde vakgroep Gasdynamica

151 Stellingen behorende bij het proefschrift “Nucleation and Condensation in Gas-Vapor Mixtures of Alkanes and Water”

door P.Peeters

31 oktober 2002

1. Oververzadigde waterdamp en oververzadigde n-nonaan-damp nucle¨eren onafhankelijk van elkaar. – P. Peeters, J. Hruby,´ and M. E. H. van Dongen, J. Phys. Chem. B 105(47), 11763 (2001)

2. Oververzadigde waterdamp condenseert niet op n-nonaan-druppels, maar oververzadigde n-nonaan-damp condenseert wel op waterdruppels. – Dit proefschrift, hoofdstuk 6

3. Rechtstreekse nucleatie van waterdamp naar ijs treedt op onder een kritische temperatuur. De waarde van de kritische temperatuur hangt af van de oververzadiging. – P. Peeters, J. J. H. Gielis, and M. E. H. van Dongen, J. Chem. Phys., to be published

4. Depolarisatie van licht biedt een uitstekende mogelijkheid om de heterogene nucleatie en lensvorming van alkanen op microscopische waterdruppels aan te tonen. – B. Krämer, O. Hubner,¨ H. Vortisch, L. Wöste, T. Leisner, M. Schwell, E. Ruhl,¨ and H. Baumgärtel, J. Chem. Phys. 111(14), 6521 (1999) 5. De door Schmidt (1962) voor het eerst waargenomen pulsaties in een supersone ex- pansiestroming in vochtige lucht kunnen worden begrepen met een eenvoudig model gebaseerd op het plotseling vrijkomen van latente warmte in een compressibele stro- ming. – B. Schmidt, Beobachtungen ub¨ er das Verhalten der durch Wasserdampf-Kondensation ausgelösten Störungen in einer Ub¨ erschall-Windkanalduse¨ , proefschrift, Fakultät fur¨ Ma- schinenbau, Universität Karlsruhe (TH), 1962 6. Het feit dat het besturingsprogramma Windows melding maakt van “unexpected errors” bewijst dat de ontwerpers weten dat het programma fouten bevat. Dit zijn dan de “expected errors”, waarvan geen melding wordt gemaakt.

7. Het echte “debuggen” van een programma begint pas na het succesvol compileren.

8. Perfectie is een asymptoot.

9. Het verbeteren van iets (bijv. een product of idee) is statistisch gezien altijd mogelijk.

10. De stelling “de som van de delen is meer dan alle delen apart” is aanvechtbaar en dus feitelijk onjuist. 11. Het direct meten van de maximale Doppler verschuiving van een naderende trein is een gevaarlijke bezigheid.

12. De beste manier om een droom uit te laten komen is door wakker te worden. – Film ’Excess Bagage’, 1997