NUCLEATION THEORY USING EQUATIONS OF STATE
by
ABDALLA A. OBEIDAT
A THESIS
Presented to the Faculty of the Graduate School of the
UNIVERSITY OF MISSOURI-ROLLA
in Partial Fulfillment of the Requirements for the Degree
DOCTOR OF PHILOSOPHY
in
PHYSICS
2003
Gerald Wilemski, Advisor
Barbara N. Hale
Jerry L. Peacher
Paul E. Parris
Daniel Forciniti
ii
ABSTRACT
Various equations of state (EOS) have been used with the most general Gibbsian
form (P − form) of classical nucleation theory (CNT ) to see if any improvement
could be realized in predicted rates for vapor-to-liquid nucleation. The standard or
S −form of CNT relies on the assumption of an incompressible liquid droplet. With
the use of realistic EOSs, this assumption is no longer needed. The P −form results
for water and heavy water were made using the highly accurate IAP W S − 95 EOS
and the CREOS. The P − form successfully predicted the temperature (T ) and
supersaturation (S) dependence of the nucleation rate, although the absolute value
was in error by roughly a factor of 100. The results for methanol and ethanol using
a less accurate CP HB EOS showed little improvement over the S − f orm results.
Gradient theory (GT ), a form of density functional theory (DF T ), was applied to
water and alcohols using the CP HB EOS. The water results showed an improved
T dependence, but the S dependence was slightly poorer compared to the S − form
of CNT .
The methanol and ethanol results were improved by several orders of
magnitude in the predicted rates. GT and P − f orm CNT were also found to be in
good agreement with a single high T molecular dynamics rate for T IP 4P water.
The P −f orm of binary nucleation theory was studied for a fictitious water-ethanol
system whose properties were generated from DF T and a mean-field EOS for a hard
sphere Yukawa fluid. The P − form was not successful in removing the unphysical
behavior predicted by binary CN T in its simplest form.
greatly superior to all forms of classical theory.
The DF T results were
iii
ABSTRACT
Various equations of state (EOS) have been used with the most general Gibbsian
form (P − form) of classical nucleation theory (CNT ) to see if any improvement
could be realized in predicted rates for vapor-to-liquid nucleation. The standard or
S −form of CNT relies on the assumption of an incompressible liquid droplet. With
the use of realistic EOSs, this assumption is no longer needed. The P −form results
for water and heavy water were made using the highly accurate IAP W S − 95 EOS
and the CREOS. The P − form successfully predicted the temperature (T ) and
supersaturation (S) dependence of the nucleation rate, although the absolute value
was in error by roughly a factor of 100. The results for methanol and ethanol using
a less accurate CP HB EOS showed little improvement over the S − f orm results.
Gradient theory (GT ), a form of density functional theory (DF T ), was applied to
water and alcohols using the CP HB EOS. The water results showed an improved
T dependence, but the S dependence was slightly poorer compared to the S − form
of CNT .
The methanol and ethanol results were improved by several orders of
magnitude in the predicted rates. GT and P − f orm CNT were also found to be in
good agreement with a single high T molecular dynamics rate for T IP 4P water.
The P −f orm of binary nucleation theory was studied for a fictitious water-ethanol
system whose properties were generated from DF T and a mean-field EOS for a hard
sphere Yukawa fluid. The P − form was not successful in removing the unphysical
behavior predicted by binary CN T in its simplest form.
greatly superior to all forms of classical theory.
The DF T results were
iv
ACKNOWLEDGEMENTS
I would like to express my gratitude and appreciation to my research advisor, Dr.
Gerald Wilemski, for his constant support and patience through the time I spent at
UMR. Without his guidance and motivation this work would have never been done.
I would like also to thank the members of my Ph.D. committee Dr. B. Hale, D. P.
Parris, Dr. J. Peacher, and Dr. D. Forciniti for their help and support. I also would
like to thank Dr. J-S. Li for his support and suggestions during my thesis work.
I am very thankful to my parents, without their endless love and support, I would
not have been neither in UMR nor in this life.
I would like also to thank my roommates and friends, Eyad, Malik, Ahmad, Abdul,
Vikas, and Sabrina, who made my life much easier while staying in the US.
This dissertation is dedicated to the wonderful lady Enas.
This work was supported by the Engineering Physics Program of the Division
of Materials Sciences and Engineering, Basic Energy Sciences, U. S. Department of
Energy.
v
TABLE OF CONTENTS
Page
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
LIST OF ILLUSTRATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
SECTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1. NUCLEATION PHENOMENOLOGY AND BASIC THEORY 1
2.
3.
1.2. BRIEF OVERVIEW OF BINARY NUCLEATION . . . . .
8
1.3. MOTIVATION . . . . . . . . . . . . . . . . . . . . . . . . .
10
EQUATION OF STATE APPROACH FOR CLASSICAL
NUCLEATION THEORY . . . . . . . . . . . . . . . . . . . . .
12
2.1. THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.1.1. Work of Formation . . . . . . . . . . . . . . . . . . . .
12
2.1.2. Gibbs’s Reference State . . . . . . . . . . . . . . . . .
15
2.3.1. Number of Molecules in Critical Nucleus . . . . . . . .
17
EQUATIONS OF STATE FOR UNARY SYSTEMS . . . . . . .
19
3.1. WATER . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.1.1. IAPWS-95 . . . . . . . . . . . . . . . . . . . . . . . .
19
3.1.2. Cross Over Equation of State (CREOS-01) . . . . . .
20
3.1.3. Jeffery and Austin EOS (JA—EOS) . . . . . . . . . .
21
3.1.4. Cubic Perturbed Hard Body (CPHB) . . . . . . . . .
22
vi
4.
5.
6.
7.
3.1.5. Peng-Robinson (PR) . . . . . . . . . . . . . . . . . .
23
3.2. HEAVY WATER: CREOS-02 . . . . . . . . . . . . . . . . .
24
3.3. METHANOL AND ETHANOL: CPHB . . . . . . . . . . .
24
RESULTS OF EOS APPROACH FOR UNARY SYSTEMS . .
25
4.1. WATER . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4.2. HEAVY WATER . . . . . . . . . . . . . . . . . . . . . . . .
29
4.3. DISCUSSION OF WATER RESULTS . . . . . . . . . . . .
32
GRADIENT THEORY OF UNARY NUCLEATION . . . . . .
36
5.1. BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . .
36
5.2. THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
RESULTS OF GRADIENT THEORY FOR UNARY
NUCLEATION . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
6.1. WATER AND TIP4P WATER . . . . . . . . . . . . . . . .
40
6.1.1. Planar and Droplet Density Profiles from GT . . . . .
40
6.1.2. Water Nucleation Rates . . . . . . . . . . . . . . . . .
44
6.1.3. TIP4P Water Nucleation . . . . . . . . . . . . . . . .
47
6.2. COMPARISON OF THE WATER EOS . . . . . . . . . . .
48
6.3. RESULTS FOR METHANOL AND ETHANOL . . . . . .
52
BINARY NUCLEATION THEORY . . . . . . . . . . . . . . . .
55
7.1 CLASSICAL NUCLEATION THEORY . . . . . . . . . . .
55
7.2 DENSITY FUNCTIONAL THEORY (DFT) . . . . . . . .
58
7.3 SURFACE TENSION AND REVERSIBLE WORK . . . . .
59
vii
7.4 DFT FOR HARD SPHERE-YUKAWA FLUID . . . . . . .
60
PROPERTIES OF THE MODEL BINARY HARD-SPHERE
YUKAWA (HSY) FLUID . . . . . . . . . . . . . . . . . . . . . .
61
8.1 EQUATION OF STATE . . . . . . . . . . . . . . . . . . . .
61
8.2 FITTED PROPERTY VALUES . . . . . . . . . . . . . . .
66
RESULTS OF THE HSY BINARY FLUID . . . . . . . . . . . .
67
9.1. CRITICAL ACTIVITIES AT CONSTANT W* . . . . . . .
67
9.2. NUMBER OF MOLECULES IN CRITICAL DROPLET . .
69
10. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . .
71
8.
9.
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. IMPORTANT THERMODYNAMIC RELATIONS . . . . . . . .
73
B. DETAILS OF VARIOUS EQUATIONS OF STATE . . . . . . .
77
C. PHYSICAL PROPERTIES OF WATER AND HEAVY WATER
89
D. SURFACE TENSION AND WORK OF FORMATION IN DFT
91
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
viii
LIST OF ILLUSTRATIONS
Figures
Page
1.1. Schematic pressure — volume phase diagram for a pure substance. The
green solid line is a true isotherm whose intersections (e) with the binodal dome give the equilibrium pressure and volumes of the coexising
vapor-liquid phases. Binodal: solid heavy curve; spinodal: red dashed
curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2. The contributions of the surface and volume terms of the free energy
of formation versus the cluster size. The free energy of formation has
a maximum at the critical size. . . . . . . . . . . . . . . . . . . . . .
4
1.3. Experimental data for water from Ref.[18] illustrating the inadequate
temperature dependence predicted by the classical Becker-Doering theory[4], labeled S-form in the figure. . . . . . . . . . . . . . . . . . . .
6
2.1. Same as Figure 1.2 but the free energy of formation is plotted as a
function of the radius of the cluster. . . . . . . . . . . . . . . . . . . .
13
2.2. The concept of the reference liquid state using a pressure-density isotherm
for a pure fluid. The full circles represent the equilibrium vapor-liquid
states, while the diamonds mark the metastable vapor phase and the
reference liquid phase. . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1. The work of formation for water droplets using the IAPWS-95 EOS
with the three forms of CNT at T=240, 250, and 260 K. . . . . . . .
25
4.2. Comparison of the experimental rates of Woelk and Strey (open circles)
for water with two versions of CNT based on the IAPWS-95 EOS; Pform and S-form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.3. Comparison of the experimental rates of Woelk and Strey (open circles)
for water down to T=220 K with two versions of CNT based on the
CREOS-01 and with the scaled model. . . . . . . . . . . . . . . . . .
27
4.4. The number of water molecules in the critical cluster as predicted by
the nucleation theorem and the P-form calculations. The dashed-line
shows the full agreement with the Gibbs-Thomson equation. . . . . .
28
ix
4.5. The experimantal rates of heavy water by Woelk and Strey down to
T=220 K with the predictions of the P-form of the CREOS-02. . . . .
30
4.6. The P-form results using CREOS-02 at high S compared with two
different sets of supersonic nozzle experiments. The scaled model and
the empirical function also shown at T=237.5, 230, 222, 215, and 208.8
K from left to right. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.7.
As in Figure 4.4 but for heavy water. . . . . . . . . . . . . . . . . . .
32
4.8. The temperature-density isobars of water using the IAPWS-95 EOS
and the CREOS-01 compared to experimental data of Kell and Whalley[75] and Petitet et al.[81]. . . . . . . . . . . . . . . . . . . . . . . .
33
4.9. The work of formation of water at T=240, 250, and 260 K predicted
by the IAPWS-95 and CREOS-01. . . . . . . . . . . . . . . . . . . .
34
4.10. Isothermal compressibility of liquid water at 10 MPa and 190 MPa
calculated from the fit of Kanno and Angell[83]. . . . . . . . . . . . .
35
6.1. The thickness of flat water interfaces at different T using GT, MD
simulations[90], and experimental data[91]. . . . . . . . . . . . . . . .
40
6.2. Density profiles of water droplets predicted by CPHB at different T,
for a supersaturation ratio of 5. . . . . . . . . . . . . . . . . . . . . .
42
6.3. Same as Figure 6.2 but at S=20 . . . . . . . . . . . . . . . . . . . . .
42
6.4. Density profiles of water droplets at T=350 K for different values of
the supersaturation ratio using the CPHB EOS. . . . . . . . . . . . .
43
6.5. Same as Figure 6.4 but using the JA-EOS. . . . . . . . . . . . . . . .
44
x
6.6. Nucleation rate predictions of the CPHB using the P-form and the GT
compared to experimental data of Woelk and Strey[18]. . . . . . . . .
45
6.7. The ratio of the GT work of formation to that of the P-form of CNT
as a function of supersaturation ratio at 260 K. . . . . . . . . . . . .
46
6.8. The number of water molecules in the critical cluster as predicted by
the nucleation theorem and the GT calculations. The dashed line represents full agreement with Gibbs-Thomson equation. . . . . . . . . .
47
6.9. Nucleation rates for GT and two forms of CNT at T=350 K using
different EOSs, as shown in the figure, compared with the MD rate for
TIP4P water and the result of the P-form of CNT using CREOS-01. .
48
6.10. The predictions of different EOSs for the equilibrium liquid density
of water at different T compared to the experimental data generated
using the IAPWS-95. . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
6.11. Density of liquid water using the CPHB EOS (stars) at different P (0.1,
50, 100, 150, 190 MPa) compared to the experimental data calculated
using the IAPWS-95 (circles) . . . . . . . . . . . . . . . . . . . . . .
49
6.12. The predictions of different EOSs for the equilibrium vapor pressure
at different T compared to the experimental data calculated by using
the IAPWS-95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
6.13. Same as for Figure 6.12 except for the equilibrium vapor density. . . .
50
6.14. Experimental nucleation rates of methanol compared to the predictions
of GT and the P-form of CNT with the CPHB EOS. . . . . . . . . .
52
6.15. As in Figure 6.14 but for ethanol. . . . . . . . . . . . . . . . . . . . .
53
6.16. Liquid ethanol density vs. P at different temperatures using the CPHB
EOS (open symbols) and experimental data (solid symbols). . . . . .
53
6.17. Experimental nucleation rates of ethanol compared to calculated rates
using the S-form and the P-form of CNT with the CPHB EOS and the
P-form of CNT using fitted experimental density data[94]. . . . . . .
54
xi
8.1. The total and partial equilibrium vapor pressures of the HSY model
fluid at T=260 K versus mixture composition, x. . . . . . . . . . . . .
62
8.2.
P-x phase diagram of the binary HSY model system. . . . . . . . . .
63
8.3. Surface tension for the pseudo water-ethanol system and measured
values for water-ethanol versus ethanol mole fraction, x. . . . . . . . .
64
8.4. Variation of the partial molecular volume of p-water with composition.
65
8.5. Same as Figure 8.4. but for p-ethanol. . . . . . . . . . . . . . . . . .
65
9.1.
Critical activities of p-water (1) and p-ethanol (2) needed to produce
a constant work of formation of 40 kT. . . . . . . . . . . . . . . . . .
67
9.2. The number of molecules of each component of the critical droplet as
a function of the p-water activity using version 1 and version 2 of the
CNT,as well as the DFT. . . . . . . . . . . . . . . . . . . . . . . . . .
69
9.3. The number of molecules of each component of the critical droplet as
a function of the p-water activity using version 3 of the CNT and the
DFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
9.4. The number of molecules of each component of the critical droplet as
a function of the p-water activity using versions 1, 2, and 3 of the CNT. 70
A.1. Schematic depiction of a spherical critical nucleus in a metastable gas
phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
xii
LIST OF TABLES
Tables
Page
B.1.
The coefficients values of the ideal gas part. . . . . . . . . . . . . . .
78
B.2.
The coefficients and parameters of the residual part. . . . . . . . . .
79
B.3.
The other coefficients and parameters of the residual part. . . . . . .
81
B.4.
The coefficients of the CREOS equation of state. . . . . . . . . . . .
83
B.5.
The coefficients of CREOS-01 and CREOS-02 EOSs. . . . . . . . . .
84
B.6.
The coefficients and parameters of the JA-EOS. . . . . . . . . . . .
85
B.7.
The C parameters for water, ethanol, and methanol of the CPHB EOS. 87
B.8.
The parameters of the CPHB EOS used for water, ethanol, and
methanol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
1. INTRODUCTION
1.1 NUCLEATION PHENOMENOLOGY AND BASIC THEORY
This thesis is primarily concerned with the accuracy of theories of vapor-to-liquid
nucleation based on equations of state of real fluids. Nucleation refers to the kinetic
processes involved in the initiation of first order phase transitions in nonequilibrium
systems. Two phase equilibrium states for a pure substance, e.g. vapor and liquid,
occur at unique pairs of temperature T and pressure P .
For two-phase vapor-
liquid equilibrium, the pressure is referred to as the equilibrium vapor pressure Pve .
Now, if the actual pressure of the vapor Pv is larger than the equilibrium vapor
pressure, the vapor is said to be supersaturated.
This new state of the vapor is
either metastable or unstable. These two types of states are distinguished by their
location with respect to the mean-field spinodal, which is illustrated in Figure 1.1.
nt
oi
lp
ca
iti
cr
P
T6
0
0
Vle
Unstable
e
le
Stab
Meta
Pe
MetaSt
able
T5
T4=Tc
T3
e
Vve
T2
T1
V
Figure 1.1. Schematic pressure — volume phase diagram for a pure substance. The green solid line is a true isotherm whose intersections (e)
with the binodal dome give the equilibrium pressure and volumes of the
coexising vapor-liquid phases. Binodal: solid heavy curve; spinodal: red
dashed curve.
This figure shows a P -V phase diagram for a pure fluid with several isotherms
2
based on a van der Waals equation of state (EOS). The heavy dome-shaped curve
denotes the binodal line, the locus of two-phase, vapor-liquid equilibrium states,
which ends at the critical point. The true isotherms are flat in the two-phase region inside the dome. All mean-field EOSs are similar in that they artificially treat
the fluid as a homogeneous phase with a continuously varying density inside the
two-phase region. This unphysical oversimplification generates the "van der Waals
loops" shown by the isotherms. The spinodal boundary separates mechanically stable regions (metastable states for which (∂P/∂V )T ≤ 0) from mechanically unstable
regions (for which (∂P/∂V )T > 0) as determined by the slope of the isotherms of
the mean-field EOS. If the supersaturated vapor is in contact with its own liquid,
it will condense until the vapor again reaches saturation.
If a container of volume V contains only pure vapor, at a suitably large supersaturation value S = Pv /Pve , droplets will start to form at an appreciable rate as
a result of collisions among vapor molecules. This process of forming a droplet is
known as homogeneous nucleation. If impurities are also present in the container,
the supersaturated vapor will first condense on them in a process referred to as heterogeneous nucleation. Since nucleation plays a key role in many fields ranging from
atmospheric applications to materials science, the study of nucleation has a long history and is currently receiving much attention stimulated by the development of new
experimental and theoretical techniques to measure and predict nucleation rates.
The first comprehensive treatment of the thermodynamics of the nucleation
process is due to Gibbs[1].
Gibbs showed that the reversible work required to
form a nucleus of the new phase consists of two terms: a bulk (volumetric) term
and a surface term. Later, in 1926 Volmer and Weber[2] developed the first nucleation rate expression, by arguing that the nucleation rate should be proportional to
the frequency of collisions between condensable vapor molecules and small droplets
(critical clusters) of the new phase of a size, the critical size, that just permits spontaneous growth. A more detailed kinetic approach for the process of vapor-to-liquid
nucleation was subsequently developed by Farkas[3].
The theory of Volmer, We-
ber, and Farkas was extended a few years later by Becker and Döring[4], Frenkel[5],
Zeldovich[6], and is now known as classical nucleation theory (CNT ).
3
The basic kinetic mechanism assumed by these authors was that small clusters
grow and decay by the absorption or emission of single molecules. In this theory
the clusters are treated as spherical droplets. As in Gibbs’s treatise, the work of
formation of a critical droplet consists of volumetric and surface contributions, but in
the absence of knowledge of the microscopic cluster properties, bulk thermodynamic
properties are used to evaluate the two contributions.
Gibbs’s result for W is
W = Aγ − Vl (Pl − Pv ) ,
4π
W = 4πr2 γ − r3 (Pl − Pv ) ,
3
and it strictly applies to just the droplet of critical size.
(1)
(2)
The surface term Aγ
represents the free energy needed to create a surface. This term always opposes
droplet formation.
The volume term -Vl (Pl − Pv ) represents the stabilizing free
energy obtained by forming a fragment of new phase. The negative sign before the
volume term ensures that new phase formation ultimately lowers the free energy of
the system. In developing the kinetics of nucleation, it is necessary to know the free
energy of formation of droplets of noncritical size. The simplest approximation is
to assume that Gibbs’s result for W applies to all sizes and to rewrite it in terms of
n, the number of molecules in the droplet. In terms of the spherical liquid droplet
model, the surface area and volume are straightforward to rewrite since r ∝ n1/3
for this model.
It is customary to assume that the droplet is an incompressible
liquid and to replace the pressure difference by the difference in chemical potential
between the old and new phases at the same pressure Pv , as explained in more
detail later. It is generally a very good approximation to treat the vapor phase as
ideal, so that the chemical potential difference can then be expressed in terms of the
supersaturation ratio S. One other approximation is needed: the surface tension of
a planar interface is used to evaluate the surface term because the surface tension
of microscopic droplets is unknown. With these simplifications the free energy of
formation of a cluster of n molecules is
∆G γ ∞ A
=
− n ln S = θn2/3 − n ln S ,
kT
kT
(3)
4
where γ ∞ is the surface tension of a planar gas-liquid interface, A is the surface
area of the cluster, which is estimated from the liquid number density ρl , and θ =
−2/3
/kT. The dependence of ∆G on n is illustrated in Figure 1.2.
surface term
Free Energy of Formation
(36π)1/3 γ ∞ ρl
0
n
*
n
0
volume term
Figure 1.2. The contributions of the surface and volume terms of the free
energy of formation versus the cluster size. The free energy of formation
has a maximum at the critical size .
As seen in the figure, ∆G has a maximum at the value n = n∗ , known as the
critical size. If the cluster size n is less than n∗ , the surface term is dominant. As
a result, the cluster has a higher tendency to shrink, i.e., to reduce its free energy,
than to grow, i.e., to increase its free energy.
On the other hand if n > n∗ , the
volume term is dominant, and the cluster has a higher tendency to grow than to
shrink.
From the extremum condition, [d∆G/dn]n∗ = 0, one obtains the simple
relation for the critical size n∗ ,
(n∗ )1/3 =
2θ
,
3 ln S
(4)
which is equivalent to the Kelvin equation for the critical radius r∗
r∗ =
2γ ∞
.
ρl kT ln S
(5)
5
The barrier height is equal to the work of formation of the critical droplet, ∆G∗ =
W ∗ . Volmer and Weber[2] in 1926 argued that the nucleation rate depends exponentially on the work of formation of the droplet. The nucleation rate expression,
which derives from the work of Becker-Döring[4], Frenkel[5], Zeldovich[6], is often
referred to as Becker-Döring theory.
The expression is given by Abraham[7], for
example, as
JCL
with the pre-exponential factor
¶
µ
W∗
,
= J0 exp −
kT
J0 =
r
2γ ∞
vl
πm
µ
Pv
kT
¶2
,
(6)
(7)
where m is the mass of a condensible vapor molecule, vl (= 1/ρl ) is the molecular
volume, Pv is the vapor pressure, and the barrier height of nucleation is
16πγ 3∞
W =
.
3(kT ρl ln S)2
∗
(8)
As seen from the nucleation expression, all the inputs to the theory are experimental quantities which makes the theory easy and popular to use.
For many
years, it was impossible to measure nucleation rates accurately. Instead, what was
usually determined experimentally was the critical supersaturation at which nucleation was observable at a significant rate, whose value was typically not known to
better than one or two orders of magnitude. (One can see from Eqs.(6) and (8)
that J depends sensitively on S, but that S is rather insensitive to changes in J.)
These critical supersaturation measurements were generally in good agreement with
the predictions of CNT for many substances. Over the past twenty years, the development of new experimental techniques with improved precision has allowed the
accurate measurement of nucleation rates for many substances[8—16]. Comparison
of these results with the predictions of CNT has shown that the theory is usually
in error, giving rates that are too low at low temperatures and too high at high
temperatures[10, 17, 18], as illustrated in Figure 1.3.
6
10
10
260 K
250 K
Expt
S-form
240 K
-3
-1
J (cm s )
9
10
H2O
230 K
8
10
7
T=220 K
10
6
10
5
10
15
20
25
S
Figure 1.3. Experimental data for water from Ref.[18] illustrating the
inadequate temperature dependence predicted by the classical BeckerDoering theory[4], labeled S-form in the figure.
Due to limits of CNT , there has been much effort to improve the classical model,
but most of the newer models take the form of correction factors to CNT . [19, 20].
In addition, the improvements of these models are often substance specific, which
limits their applicability. One of the most successful and most general treatments
of the temperature dependence of nucleation rates is the so-called scaled model of
Hale[21, 22] introduced in 1986. The scaled model is based on CNT , and it yields
a universal dependence of nucleation rate on Tc /T − 1.
The two parameters of
this model are the nearly universal constant Ω, which is interpreted as the excess
surface entropy per molecule, and the constant rate prefactor J0 (≈ 1026 cm−3 s−1 ).
The value of Ω for nonpolar substances is around 2.2, whereas for polar materials
it is about 1.5. For later use, and as an example, Ω is 1.476 for heavy water and
1.470 for water.
The model works well for many substances for which the CNT
fails. In the scaled model, the nucleation rate is given by the expression,
7
Ã
J = J0 exp −
16π 3
Ω
3
µ
!
¶3
Tc
− 1 /(ln S)2 .
T
(9)
The most fundamental approach to improving CNT is through the development
of microscopic theories. The goal of any microscopic theory is to avoid the overly
simplistic use of bulk thermodynamic properties in evaluating the free energy of cluster formation. There are several different types of microscopic approaches, which
will only be mentioned here.
Molecular dynamics (MD) computer simulations
have been used to explore properties of small molecular clusters, e.g. by Gubbins
and coworkers[23, 24] and Tarek and Klein[25], and to directly simulate nucleation,
as in the work of Rao and Berne[26], Yasuoka and Matsumoto[28], and ten Wolde
and Frenkel[29]. Monte Carlo (MC) computer simulations have also been used extensively to calculate free energies of cluster formation, e.g. by Lee et al.[27, 30],
Hale et al.[31, 32], and Oh and Zeng[33], and to examine the subcritical cluster size
distribution directly[34]. Hybrid approaches like those of Weakliem and Reiss[35],
Schaff et al.[36] that combine MC or MD simulation results with analytical theory
have also been developed. A brief review by Reiss[37] discusses other approaches
by many other authors not mentioned here.
Another important approach known as the density functional theory (DF T ) [38—
40] will be discussed later in detail. To briefly summarize here, in DF T the free
energy of the nonuniform system, F [ρ(r)], is written as a functional of the local
density ρ(r) at each position r in the fluid. The presence of the nucleus renders
the fluid inhomogeneous. The inhomogeneity is characterized by the density that
varies continuously from its value at the center of the nucleus to its value in the
metastable mother phase far from the nucleus. The properties of the critical nucleus
are determined by finding the density profile that minimizes the nonuniform fluid’s
free energy.
Cahn and Hilliard[40] were the first who developed a type of DF T for nucleation
theory. They proposed the Helmholtz potential to be
F [ρ (r)] =
Z
³
´
c
dr f0 [ρ (r)] + [∇ρ (r)]2 ,
2
(10)
8
where f0 is the Helmholtz free energy density of the homogenous fluid of density ρ,
∇ρ is the gradient of the density, and c is the so-called influence parameter related
to the intermolecular potential.
Because the Helmholtz potential above depends on the gradient of the density,
minimizing Eq.(10) one obtains a differential Euler-Lagrange equation. This theory
is called gradient theory (GT ), or square gradient theory, because of the form of the
free energy functional. The first GT was actually devised many years earlier by van
der Waals[41] to describe the structure of planar interfaces. To apply the GT , one
needs a well-behaved EOS. It should have the form of a cubic equation, similar in
spirit to the van der Waals EOS, that describes the system as a single homogeneous
phase whose density varies continuously throughout the two-phase region.
A more general form of DF T was developed and applied to nucleation theory
by Oxtoby and coworkers[42, 43]. It is a molecular theory that explicitly uses an
intermolecular potential. The theory uses a hard sphere fluid as a reference state
and treats the attractive intermolecular potential as a perturbation. The theory is
developed in terms of the grand potential Ω, which is written in the perturbation
theory as the following functional of the density,
Z
Z Z
Ω [ρ (r)] = dr (fh [ρ (r)] − µρ (r)) +
drdr0 φatt (|r − r0 |)ρ(r)ρ(r0 ) ,
(11)
Here, fh is the Helmholtz free energy density of the hard sphere fluid, φatt is the
long-range attractive part of the potential, and µ is the chemical potential. The
simpler GT can be derived from the more general DF T by expanding the density
in a Taylor series and retaining only the leading nonzero terms.
Minimization
of Eq.(11) generally leads to an integral Euler-Lagrange equation, which must be
solved for the density profile of the nonuniform system.
1.2 BRIEF OVERVIEW OF BINARY NUCLEATION
Many of the above considerations apply as well to homogeneous nucleation of
binary systems, commonly referred to as binary nucleation, but there is a major
difference as well. In binary nucleation, the initial metastable phase and the final
phase are two component systems.
Thus, the kinetics of nucleation involves the
9
formation of clusters of the new phase that generally contain both components.
To apply CNT to binary systems, the most important quantity needed to predict
nucleation rates is the composition of a critical nucleus. If the surface tension is
known as a function of the composition and if ∆P , the difference in pressure inside
and outside the droplet, is known, then the critical radius can be calculated using
the Laplace formula, ∆P = 2γ/r∗ .
Assuming the droplet is incompressible, the
Gibbs-Thomson equations can be derived: ∆µi = −2γvi /r. The differences in the
chemical potential between liquid and vapor phases are represented by ∆µi , while vi
is the molecular volume of component i in the liquid. From the two Gibbs-Thomson
equations, one can determine the composition and the critical radius of the droplet.
In 1950, Reiss[44] proposed a theory based on kinetic and thermodynamic arguments showing that the binary nucleation rate is determined by the passage over a
saddle point in the two-dimensional droplet size space. Later, Doyle[45] used this
theory to study the sulfuric acid-water system, but the Gibbs-Thomson equations
he found contained a term involving the compositional derivative of the surface tension. Because these terms were small for the sulfuric acid-water system, they had
essentially no effect on the calculated critical cluster compositions. When Doyle’s
equations were subsequently applied to strongly surface active systems, such as
ethanol-water or acetone-water, these terms became very important for water-rich
cluster compositions. As a result, the theoretically calculated vapor compositions
needed to produce experimentally observed nucleation rates were many orders of
magnitude lower in the concentration of the surface active component than the experimental concentrations.
Renninger, Hiller, and Bone[46] argued that Doyle’s
treatment of the Gibbs-Thomson equations was inconsistent.
Wilemski[47] pro-
posed a revised classical theory in which the Gibbs surface adsorption equation was
used to cancel the derivative of the surface tension, thus permitting the conventional
Gibbs-Thomson equations to be recovered. It is interesting to note that the conventional Gibbs-Thomson equations had been used in the original, early work on
binary nucleation by Flood[48] and by Döring and Neumann[49], but had then been
forgotten.
10
The predictions of either version of CNT for ideal binary mixtures are fairly reasonable, but for mixtures with a component that strongly segregates on the droplet
surface, e.g. alcohol-water or acetone-water mixtures, problems arise. Doyle’s version of CNT predicts unrealistic results, as just noted, while the revised binary CNT
gives rise to unphysical behavior that violates the nucleation theorem[50] for binary
systems. In an important step to resolving these difficulties, Laaksonen[51, 52] proposed a so-called explicit cluster model to study water-alcohol systems. The model
makes realistic predictions for the vapor concentrations while predicting physical
behavior for the nucleus composition, in accord with the nucleation theorem.
Compared to unary nucleation, less work on microscopic theories of binary nucleation has been performed. Zeng and Oxtoby[43] extended the DF T to treat binary
nucleation for Lennard-Jones mixtures. Talanquer and Oxtoby have used the GT
to study highly nonideal binary systems with parametrized hard-sphere—van der
Waals EOS[53]. Napari and Laaksonen have recently performed DF T calculations
for a site-site interaction model that simulates systems with a highly surface active
component.[54] Hale and Kathmann have performed Monte Carlo simulations to
calculate the free energies of formation of sulfuric acid-water clusters[55].
1.3 MOTIVATION
The principal goal of this thesis is to test a form of classical nucleation theory closest in spirit to the original pioneering work of Gibbs. The usual forms of
CNT are well-known to provide a poor quantitative description of the temperature
dependence of measured nucleation rates, although the predicted dependence on
supersaturation is generally quite satisfactory. To explore this, Gibbs’s original formula was used to calculate nucleation rates for several different substances: water
and heavy water, methanol, ethanol. Significant improvement in the predicted temperature dependence of the nucleation rate was realized only for water and heavy
water. This appears to be due to the extraordinary isothermal compressibility of
these two substances at the low temperatures where nucleation rates are generally
measured.
The other materials studied are much less compressible at low tem-
peratures, and the customary approximation of an incompressible fluid, universally
used in the classical theory, is valid for these substances. The implementation of
11
Gibbs’s original formula requires the use of an accurate equation of state for the fluid
properties. In the case of water, two different EOS were used, but each accurately
treated the anomalously high compressibility of fluid water.
With various equations of state available, it was possible to test a nonclassical
theory of nucleation known as gradient theory. A second goal of this thesis is to determine whether or not the predicted temperature dependence of the nucleation rate
would be improved by this simplest form of density functional theory. Reasonably
good results were found for water using a so-called CP HB EOS, but the gradient
theory results for , methanol, and ethanol were only slightly improved compared to
the predictions of classical theory.
Finally, the application of Gibbs’s original formula to binary nucleation was explored.
The goal of this aspect of the work is to see whether certain unphysical
aspects of classical binary nucleation theory could be alleviated by using a more
exact formulation of the theory. A key difficulty in carrying out this phase of the
research was that for the most interesting binary systems, such as the ethanol-water
system, there are no accurate EOSs in the temperature range of interest. To surmount this difficulty, a model system was devised with properties resembling those
of the ethanol-water system. The EOS for the model system consists of a binary
hard sphere fluid contribution plus an attractive term of the van der Waals form.
The bulk surface tension was computed as a function of mixture composition using
density functional theory for a planar interface. To facilitate the DF T calculations,
attractive potentials of the Yukawa form were employed. The results showed that
Gibbs’s original formula, with the bulk surface tension, also suffered from the same
unphysical behavior as simpler forms of the classical binary theory.
12
2. EQUATION OF STATE APPROACH FOR CLASSICAL
NUCLEATION THEORY
2.1 THEORY
In this chapter, three different versions of classical nucleation theory (CNT ) are
explored to study nucleation rates of water and heavy water.
For two of these
versions, a novel approach based on different equations of state is used to calculate
the work of formation of a critical droplet, W ∗ , which is then used to evaluate the
nucleation rate. The theoretical predictions are compared with the experimental
rates of water and heavy water[18]. The theoretical results are also compared with
the predictions of the scaled model of Hale[21]. The number of molecules in a critical
cluster are compared with the experimental data using the nucleation theorem[50].
2.1.1 Work of Formation. Consider a volume V containing N molecules of
vapor at a chemical potential µv and pressure Pv .
The Helmholtz free energy of
this vapor is
Fi = Nµv − Pv V .
(12)
Upon forming a droplet with n molecules, if we ignore the very small changes in µv
and Pv , the final Helmholtz free energy of the system is
Ff = (N − n)µv + nµl − (V − Vl )Pv − Vl Pl + Aγ ,
(13)
where µl is the chemical potential of a molecule at the internal pressure Pl of the
droplet, Vl is the volume of the droplet, A is its surface area, and γ is the surface
tension. The difference in the free energy between the initial and final systems is
∆F = Ff − Fi = n(µl − µv ) − (Pl − Pv )Vl + Aγ .
(14)
(It should be noted that Eqs.(13) and (14) are not quite rigorous since they fail to
include the surface excess number of molecules[50]. As shown in Appendix A, the
final results below are, nevertheless, correct.)
This free energy difference has a maximum at a specific radius, r∗ , when µl = µv ,
13
∆F Max = −(Pl − Pv )
4π ∗3
r + 4πr∗2 γ ,
3
(15)
which represents the free energy barrier required to be overcome to form a spherical critical droplet of radius r∗ .
Using results of Appendix A, Eq.(14) can be
approximated as the following sum of a surface and a volume term,
∆F = −
4πr3 ∆µ
+ 4πr2 γ ,
3 vl
(16)
where ∆µ is the difference in chemical potential between the initial metastable phase
and the final stable phase and vl is the molecular volume of the stable phase. Figure
2.1 schematically shows the dependence of this free energy as a function of the
droplet radius.
Free Energy of Formation
surface term
MAX
∆F
0
r
0
*
r
volume term
Figure 2.1. Same as Figure 1.2 but the free energy of formation is plotted
as a function of the radius of the cluster.
Applying the Laplace equation, which governs the pressure drop across a curved
interface, specifically Pl − Pv = 2γ/r∗ (see Appendix A), we then obtain
W∗ =
γ3
16π
,
3 (Pl − Pv )2
(17)
14
where W ∗ is the minimum free energy, i.e., the reversible work, required to form a
critical droplet of radius r∗ .
To apply the above formula, which is known as Gibbs formula, one has to know
the exact surface tension at that radius and the pressure inside the droplet. Lacking knowledge of the exact surface tension, the first approximation is to use the
experimental surface tension of a flat interface, i.e., set γ = γ ∞ to obtain
W∗ =
We call this equation the P − form.
γ 3∞
16π
.
3 (Pl − Pv )2
(18)
Usually the pressure inside the droplet is found approximately by making the
assumption that the droplet is incompressible. In this case, we can replace ∆P =
Pl − Pv with (µv − µl (Pv ))/vl , which follows from the thermodynamic identity
∆µ = µv − µl (Pv ) = µl − µl (Pv ) =
ZPl
vl dP,
(19)
Pv
when the molecular volume vl is assumed to be constant and the condition of unstable equilibrium between the critical droplet and the metastable vapor is used.
Note that this definition of ∆µ is identical to Kashchiev’s[56]. Equation (18) then
becomes
W∗ =
16π γ 3∞ vl2
.
3 (∆µ)2
(20)
We call this equation the µ−f orm. This form is most useful if the chemical potential
difference can be found from an equation of state. However, ∆µ is more commonly
found using simpler, but approximate thermodynamic relations. If we assume the
supersaturated and saturated vapors are ideal gases and that the droplet is a tiny
piece of incompressible bulk liquid, then it is easily shown (e.g. in Appendix A)
that
∆µ = kT ln S − vl (Pv − Pve ) ,
(21)
where k is the Boltzmann constant, T is the absolute temperature, and S is the
15
supersaturation value defined as the ratio of the actual vapor pressure to the equilibrium vapor pressure Pve , i.e., S = Pv /Pve .
It is customary to neglect the P v
term, which is almost always extremely small. Equation (20) then reduces to the
most familiar form used in CNT ,
W∗ =
16π γ 3∞ vl2
.
3 (kT ln S)2
(22)
For simplicity we call this equation the S − form.
Applying the first two forms of W ∗ requires knowledge of the actual pressure and
chemical potential inside the droplet. Usually this information is unavailable, and
for this reason experimentalists compare their results with rates predicted using
the S − f orm because the supersaturation ratio is readily determined from the
experimental data.
A less approximate way to evaluate the P − f orm of W ∗ involves calculating the
internal pressure Pl using the equation
kT ln S =
ZPl
vl dP ,
(23)
Pve
which follows from Eq.(19) and the conditions for stable and unstable equilibrium,
µl (Pve ) = µv (Pve ) ,
(24)
µl = µv (Pv ) ,
(25)
along with the ideal gas approximation,
µv (Pv ) − µv (Pve ) = kT ln S .
(26)
The integral on the right-hand-side of Eq.(23) can be evaluated quite accurately
if the liquid density or, equivalently, the molecular volume is known as a function
of pressure. If the pressure dependence of the density is not available from direct
measurements, it may be calculated using the measured liquid isothermal compressibility, preferably as a function of pressure.
2.1.2 Gibbs’s Reference State. A more comprehensive approach for calculating the pressure and chemical potential differences needed in the P − and
16
µ − forms of W ∗ involves using a complete equation of state (EOS).
A com-
plete EOS consists of a functional representation, either analytical or tabular, of
the Helmholtz free energy F of the substance as a function of density and temperature. From the Helmholtz free energy, the pressure and the chemical potential are
readily calculated from standard thermodynamic identities. Thus, F contains all
the information needed to calculate the work formation of a droplet using the first
two forms.
The calculation of the internal pressure Pl from an EOS follows Gibbs’s[1] original
reasoning[56—58]. Upon forming a droplet within a homogeneous fluid with uniform
chemical potential and temperature, the droplet may be so small that its internal
state may not be homogeneous even at the center of the drop. The meaning of the
internal pressure and density of the droplet is then obscured, and these values are
difficult to determine.
To overcome this difficulty, Gibbs introduced the concept
of the reference state as the thermodynamic state of a bulk phase whose internal
pressure Pref and density ρref are determined by the same conditions that exist
for the new phase and the mother phase, i.e., by assuming that the temperature
and the chemical potential are the same everywhere in the nonuniform system.
In mathematical terms, the pressure inside the droplet is calculated such that the
chemical potentials are equal in both the metastable vapor and reference liquid
phases
µv (ρv ) = µl (ρref ) ,
(27)
where ρv is the density of the supersaturated vapor and ρref is the density of the
reference liquid state. As a practical matter, one always calculates differences in
chemical potential, and because Eq.(27) involves phase densities that generally differ
by many orders of magnitude it is convenient to rewrite this equation as an equality
of chemical potential differences measured from the common equilibrium state, for
which
P (ρve ) = P (ρle ) ,
(28)
µv (ρve ) = µl (ρle ) ,
(29)
where ρve and ρle are the equilibrium vapor and liquid densities, respectively. After
17
subtracting the equilibrium value of µ from both sides of Eq.(27), we obtain
µv (ρv ) − µv (ρve ) = µl (ρref ) − µl (ρle ) .
(30)
The chemical potentials are calculated from µ = (∂f /∂ρ)T , where f is the appropriate Helmholtz free energy density for the EOS. Once ρref has been found by
solving Eq.(30), the reference pressure Pref is straightforward to calculate from the
EOS. Figure 2.2 shows the concept of the reference state.
(ρref,Pref)
Pv=pressure of metastable region
Pref=pressure at which µl(ρref) = µv(ρv)
P
(ρv,Pv)
Peq
ρ
Figure 2.2. The concept of the reference liquid state using a pressuredensity isotherm for a pure fluid. The full circles represent the equilibrium vapor-liquid states, while the diamonds mark the metastable vapor
phase and the reference liquid phase.
Once W ∗ has been evaluated, the nucleation rate can be calculated using Eq.(6).
Comparisons of the calculated rates with experimental values will be made in later
sections for various substances.
2.1.3 Number of Molecules in the Critical Nucleus. In addition to the
nucleation rate, another physical quantity of interest is the size of the critical nucleus, which is experimentally determinable from measured nucleation rates using
18
the nucleation theorem in the approximate form[50, 59],
n∗ ≈
∂ ln J
.
∂ ln S
(31)
The experimentally determined values of n∗ can be compared with the theoretical
values based on the different forms of W ∗ using the rigorous form of the nucleation
theorem[56]
∂W ∗
= −∆n∗ / (1 − ρv /ρl ) .
∂∆µ
(32)
For the formation of liquid droplets in a dilute vapor, Eq.(32) reduces to
∂W ∗
= −n∗ .
∂∆µ
(33)
The critical number n∗ can also be computed from more classical considerations.
Since the volume of a spherical critical nucleus is V ∗ = 4πr∗3 /3, one can calculate
the number of molecules in the nucleus from the relation n∗ vl = Vl . Applying the
Gibbs-Thomson or Kelvin equation, Eq.(5), for r∗ , one finds
n∗ =
32πvl2 γ 3∞
,
3(kT ln S)3
(34)
which is equivalent to Eq.(4).
To implement the approach outlined above, there is clearly a need for a satisfactory EOS.
There are many possible candidates in the literature.
Not all of
these are suitable for use in the EOS approach because they are not sufficiently
accurate. Curiously, these less accurate EOSs are actually the only ones suitable
for the gradient theory calculations presented later. For completeness all of the
EOS used in this thesis are presented in the next chapter.
19
3. EQUATIONS OF STATE FOR UNARY SYSTEMS
3.1 WATER
Five EOSs for water were used in the different phases of this thesis work.
3.1.1 IAPWS − 95. This EOS was published by the International Association for the Properties of Water and Steam [60, 61]. It is an analytical equation
based on a multiparameter fit of all the experimental data available at temperatures
above 234 K. It is very accurate and therefore highly suitable for use in the EOS
approach, but only for T ≥ 234 K. This limitation strictly applies to the low T —
low P vapor-liquid equilibrium states. Liquid densities at high P and low T are in
good agreement with the few experimental data available. The low T — low P vapor
behavior also is reasonable. This EOS has one other significant drawback. It fails
to provide a continuous representation of single phase fluid states in the metastable
and unstable regions of the phase diagram, and is, therefore, unsuitable for use in
gradient theory calculations.
In the IAP W S − 95 EOS, the specific Helmholtz free energy f is represented in
dimensionless form as φ = f/RT , and φ is separated into an ideal part, φ0 and a
residual part φr , i.e,
f
= φ = φ0 (δ, τ ) + φr (δ, τ ) ,
RT
where δ = ρ/ρc , τ = Tc /T
0.46151805 kJ/(kg K).
(35)
with Tc = 647.096 K, ρc = 322 kg/m3 , and R =
The subscript c designates a value at the critical point.
We also have the following definitions
0
φ = ln(δ) +
n01
+
n02 τ
+
n03
ln(τ ) +
8
X
i=4
φr =
7
X
ni δ di τ ti +
i=1
+
56
X
i=55
51
X
i=8
ni ∆bi δΨ,
ci
ni δ di τ ti e−δ +
54
X
i=52
0
n0i ln(1 − e−γ i τ ),
(36)
¡
¢
ni δ di τ ti exp −αi (δ − εi )2 − β i (τ − γ i )2
(37)
20
with
∆ = θ2 + Bi [(δ − 1)2 ]ai ,
(38)
1
θ = (1 − τ ) + Ai [(δ − 1)2 ] 2βi ,
(39)
¢
¡
Ψ = exp −Ci (δ − 1)2 − Di (τ − 1)2 .
(40)
All the values of coefficients and parameters of φ0 and φr are listed in Appendix B.
3.1.2. Cross Over Equation of State (CREOS − 01).
Most EOSs
are attempts to improve the van der Waals EOS to give better representations
of the properties of real systems, but these equations generally fail to reproduce
the singular behavior observed at the critical point.
This failure stimulated a
search for a new type of EOS that could describe classical mean-field behavior
far away from the critical region and smoothly cross over to the singular behavior
near the critical point. New equations with this capability have been developed
by Kiselev and Ely for water, which they termed CREOS − 01[62]. Since the
concept behind the crossover EOS is to get the right behavior near a critical point,
to make this equation work at low temperatures, the scenario of a second critical
point at low temperature[67] was exploited by Kiselev and Ely[62]. Even though
the CREOS − 01 equation is a cubic equation, it describes only the liquid states of
the system. Since it does not provide any representation of the vapor states, it is
unsuitable for use in GT calculations for vapor-to-liquid nucleation.
In the CREOS equation, the Helmholtz free energy of the system is cast in terms
of Landau theory[63] as
A(T, ρ) = ∆A(τ , ∆η) +
ρ
µ (T ) + Ao (T ) ,
ρc o
(41)
where A is the dimensionless Helmholtz free energy, A = ρA/ρc RTc , and where
τ = T /Tc − 1, ∆η = ∆ρ = ρ/ρc − 1, and µo (T ), and Ao (T ) are analytical functions
of T .
21
∆A(r, θ) = kr
2−α
α
Ã
2
R (q) aψ0 (θ ) +
5
X
∆i
fi
∆
!
ci r R (q)ψi (θ)
i=1
,
τ = r(1 − b2 θ2 ) ,
1
∆ρ = krβ R−β+ 2 (q) + d1 τ .
(42)
(43)
(44)
All the coefficients and parameters of CREOS − 01 are given in Appendix B.
3.1.3 Jeffery and Austin EOS (JA − EOS). Jeffery and Austin[64] have
developed an analytical equation of state to describe water. It has several interesting
properties, but also an important drawback. Similar to the CREOS equation, it
predicts a low temperature critical point associated with two metastable phases of
supercooled water. It also provides a continuous description of single-phase states in
the two phase region, similar to the van der Waals and other cubic EOSs. However,
since it does not accurately predict the low temperature vapor-liquid binodal line, it
is suitable for quantitative use in gradient theory calculations only for a small range
of higher temperatures.
The JA − EOS consists of three parts. The first part, developed by Song and
Mason[65, 66], is a generalized van der Waals EOS of the specific form,
·
¸
³
PSM
a ´
1
∗
=1+ α−b −
ρ + αρ
−1 .
(45)
ρRT
RT
1 − λbρ
Here, a is the van der Waals constant, (a = 27R2 Tc2 /64Pc = 0.5542 P a m6 mol−2 ), λ
is a constant equal to 0.3241, b∗ and α are related to the Boyle volume, vb , through
α = 2.145vb , b∗ = 1.0823vb , and b is a function of temperature given by
¶
µ
¶
µ
b(T )
T
T
+ b2 ,
= 0.2 exp −b3 ( + b4 ) − b1 exp b5
vb
Tb
Tb
where Tb is the Boyle temperature.
(46)
The second part of the JA − EOS incorporates the effects of hydrogen bonding.
This effect was first treated approximately by Poole et al.[67]. Jeffery and Austin
modified the results of Poole et al. to get this part of the Helmholtz free energy as
³
³
´´
HB
FHB = −fRT ln Ω0 + ΩHB exp −
(47)
− (1 − f )RT ln(Ω0 + ΩHB ) ,
RT
22
where Ω0 , and ΩHB are the numbers of configurations of weak hydrogen bonds
with energy 0 and of strong hydrogen bonds, respectively. These are given by
Ω0 = exp(−S0 /R), and ΩHB = exp(−SHB /R), where S0 and SHB are the entropies
of formation of a mole of weak and strong hydrogen bonds respectively, and
HB
is
the hydrogen bond energy. In this term f is a function of temperature and density
through the following relation
Ã
µ ¶8 !
1 + C1
T
f=
,
exp −C2
exp((ρ − ρHB )/σ) + C1
Tf
(48)
where σ is the width of the region where hydrogen bonds are able to form, ρHB is
the hydrogen bond density, C1 and C2 are constants and Tf is the normal freezing
temperature.
The final part of the equation is called the vapor correction term, and it reads
Pcorr = I1 ρ2 RT ,
(49)
where the correction function I1 is given by
I1 = (α − B)ξ(T )φ(ρ) .
(50)
The auxiliary functions ξ and φ are defined as
µ
¶
Tc 6 (T − κTc )2 + A2
,
ξ(T ) = A1 −A5 exp( )
T
TC2
µ ³ ´ ¶
6.7
exp A4 ρρ
c
φ(ρ) =
³ ´3.2 ,
1 + A3 ρρ
(51)
(52)
c
where A1 , A2 , A3 , A4 , κ are constants, B is the second virial coefficient, defined as
B = α − b∗ − a/RT , Tc and ρc are the critical temperature and density. All of the
constants are evaluated in Appendix B. The total pressure for the JA − EOS is
given by
P = PSM + 2PHB + Pcorr .
3.1.4 Cubic Perturbed Hard Body (CPHB). This EOS was developed
by Chen et al[68, 69] to study the vapor-liquid equilibrium of nonpolar and polar
23
fluids. It employs a generalized hard sphere EOS to treat the hard-core repulsions
between molecules and uses a simple, modified van der Waals term to treat the effects
of attractive forces. The hard-body compressibility factor of Walsh and Gubbins[70]
is used. Walsh and Gubbins modified the well-known Carnahan-Starling[71] EOS,
which is based on hard sphere simulation data. The Walsh and Gubbins modification
covers all the shapes of the molecules from a single sphere to a chainlike molecule
through the use of a nonspherical factor, α. The compressibility factor of Walsh
and Gubbins was simplified to the form
Z rep =
v + k1 b
,
v − k2 b
(53)
where b is the hard core molar volume, v is the molar volume, and k1 , k2 are given
in Appendix B. After adding an empirical attractive term to Eq.(53), the CP HB
EOS reads as
P =
a
RT (1 + k1 b/v)
−
.
v − k2 b
v(v + c)
(54)
The parameters a, b, and c are determined from the critical properties of the fluid.
Details of the CP HB EOS are given in Appendix B.
3.1.5 Peng-Robinson (PR). Van der Waals introduced the first mean-field
theory to study phase behavior in real systems. His EOS was a qualitative breakthrough in understanding, but it lacks quantitative accuracy, particularly with respect to the vapor-liquid equilibrium states.
Peng and Robinson[72] proposed
several modifications to overcome these shortcomings for nonpolar fluids. The P R
equation gives good results in describing nonpolar fluid behavior, but it is moderately successful for polar fluids as well. The P R equation takes the form
P =
RT
a(T )
−
.
v − b v(v + b) + b(v − b)
At the critical point, the following relations are satisfied:
(55)
24
R2 Tc2
,
Pc
RTc
,
b(Tc ) = .0778
Pc
Zc = 0.307 .
a(Tc ) = .45724
(56)
(57)
(58)
At any other T , Peng and Robinson assumed
a(T ) = α(Tr , ω)a(Tc ) ,
(59)
b(T ) = b(Tc ) ,
where Tr = T /Tc ,
1
1
α 2 = 1 + k(1 − Tr2 ) ,
(60)
and k is a substance-specific constant. This constant was correlated to the acentric
factor, ω, and the result was:
k = 0.37464 + 1.54266ω − 0.26992ω 2 .
(61)
3.2 HEAVY WATER: CREOS − 02
CREOS − 02 has the same functional form as CREOS − 01, but with different
parameter values[73]. The parameters and coefficients of this equation are given in
Appendix B.
3.3 METHANOL AND ETHANOL: CPHB
The original CP HB equation of state was developed for non-polar fluids.
An
extension has been made to cover many polar fluids including alcohols. This equation is very sensitive to the parameter values.
To use this equation successfully,
very careful attention should be paid to the values of the critical properties, i.e.,
they should be the same as used by Chen et al.[74] Refer to Appendix B for the
parameters.
25
4. RESULTS OF EOS APPROACH FOR UNARY SYSTEMS
4.1 WATER
Before applying the different equations of state to calculate nucleation rates,
differences in the critical work of formation, W ∗ , for the various forms of CNT were
examined. Figure 4.1 shows W ∗ of water droplets using the IAP W S − 95[60] at
T = 240, 250, and 260K. As can be seen from the graph, the results for the µ−form
and for the S − form are close to each other at low S and start to deviate slightly
at high S. The maximum deviation is of order kT , which will give a difference in
nucleation rates of only a factor of three and is, thus, inconsequential. It is clear
from this figure that the P − form gives significantly different results.
for the P − f orm is much lower than for the other forms.
The W ∗
Since the nucleation
rate depends exponentially on (−W ∗ ), higher nucleation rates will result for the
P − form. An important point to note is that the gap between the P − form
and other versions grows as T decreases, so the predicted temperature dependence
should also be greatly improved.
48 T=260 K
H2O
µ-form
44
W*/kT
240 K
S-form
P-form
250 K
40
36
32
28
6
7
8
9
10
11
12
13
14
S
Figure 4.1. The work of formation for water droplets using the IAPWS-95
EOS with the three forms of CNT at T=240, 250, and 260 K.
26
The nucleation rates of water using the IAP W S − 95 EOS[60] and applying the
different versions of CN T are shown in Figure 4.2. Since the calculated nucleation
rates using the P − f orm are higher, they were divided by 200 for the figure. The
figure shows that the P − f orm performed excellently regarding both the temperature dependence and the supersaturation dependence. Because the predictions of
S − form and µ − f orm are so close to each other, only the results of the S − form
are plotted.
10
10
260 K
H2O
250 K
T=240 K
9
-3
J cm s
-1
10
8
10
7
10
S-form
Expt
P-form/200
6
10
6
7
8
9
10
11
12
13
14
S
Figure 4.2. Comparison of the experimental rates of Woelk and Strey
(open circles) for water with two versions of CNT based on the IAPWS95 EOS; P-form and S-form.
The other EOS used to describe water at low temperature is the CREOS − 01.
Because it fails to describe the vapor states of the fluid, the CREOS − 01 was
used only for the liquid states, while the JA − EOS was used for the vapor, in the
following way. To calculate the equilibrium vapor density, ρve , and liquid density,
ρle , one solves, respectively, the two equations,
exp
(T ) = PJA (ρve ) ,
Pve
(62)
27
exp
(T ) = PCREOS−01 (ρle ) ,
Pve
(63)
where Pve is the experimental equilibrium vapor pressure[18]. Then, to find ρref the
JA − EOS and the CREOS − 01 were combined in the following equation
µJA (ρv ) − µJA (ρve ) = µCREOS−01 (ρref ) − µCREOS−01 (ρle ) .
(64)
The rationale for this procedure is that the JA − EOS is expected to be accurate
for densities and chemical potential differences of vapor states, while the same thing
is true of the CREOS − 01 for the liquid states.
With CREOS − 01[62] results can be calculated over a wider range of temperatures down to T = 220 K, as shown in Figure 4.3. The P − form results are again
divided by the factor of 200. The figure also shows the predictions of the scaled
model[31].
10
10
260 K
Expt
S-form
Scaled
P-form/200
250 K
H2O
240 K
-1
J (cm s )
9
10
-3
230 K
8
10
T=220 K
7
10
6
10
5
10
15
S
20
25
Figure 4.3. Comparison of the experimental rates of Woelk and Strey
(open circles) for water down to T=220 K with two versions of CNT
based on the CREOS-01 and with the scaled model.
28
Both the P − form and the scaled model results describe the data well. The
classical Becker-Döring result, based on the S −form gives a clearly inferior account
of the temperature dependence.
From the experimental rates and the nucleation theorem, the number of molecules
in the critical droplet, n∗ , can be determined. Figure 4.4 shows the experimental
values[18] and the values derived from the P − f orm of W ∗ versus the predictions
of the Gibbs-Thomson formula, Eq.(34), at the different temperatures. Only the
CREOS − 01 EOS was used to calculate n∗ using the formula
32πγ 3∞
ρ ,
n =
3(Pref − Pve )3 ref
∗
(65)
which is readily found from Eqs.(18) and (32). The experimental data were found
by Wölk and Strey[18] using the equation
n∗ =
∂ ln J
−2 .
∂ ln S
(66)
50
H2O
40
Gibbs-Thomsom
Expt
P-form
30
n
*
20
10
0
0
10
*
20
30
40
50
n Gibbs-Thomson
Figure 4.4. The number of water molecules in the critical cluster as
predicted by the nucleation theorem and the P-form calculations. The
dashed-line shows the full agreement with the Gibbs-Thomson equation.
29
The calculated n∗ values using the P − form of the CNT show excellent agreement with the measured ones. This result is not unexpected since the P − form of
the CNT gives the right T and S dependence, and since n∗ is essentially equal to
the derivative of ln J with ln S.
4.2 HEAVY WATER
The only EOS valid at low T to describe D2 O is the CREOS − 02[73].
As
for CREOS − 01, this equation also describes only liquid states, and there is no
other EOS to describe the vapor states. Consequently, to evaluate the chemical
potential of the metastable vapor, the assumption that the vapor is ideal has been
used, i.e., µ(ρv ) − µ(ρve ) = kT ln S. To calculate the equilibrium liquid density, ρle ,
the experimental equilibrium vapor pressure[18], Pve (T ), has been equated with the
CREOS − 02 pressure at the equilibrium liquid density
Pve (T ) = PCREOS−02 (ρle ) .
(67)
To find ρref the ideal vapor assumption was used to obtain
kT ln S = µCREOS−02 (ρref ) − µCREOS−02 (ρle ) .
(68)
The reference pressure is then obtained as Pref = PCREOS−02 (ρref ) after the solution
to Eq.(68) is found.
Figure 4.5 shows the rates, divided by a factor of 100, predicted by the P − form
using the CREOS − 02[73] equation. The results show good agreement with the
experimental T and S dependence.
30
260 K
9
250 K
10
Expt
P-form/100
S-form
D2O
240 K
8
10
-3
J cm s
-1
230 K
7
10
T=220 K
6
10
5
10
15
S
20
25
Figure 4.5. The experimantal rates of heavy water by Woelk and Strey
down to T=220 K with the predictions of the P-form of the CREOS-02.
All the aforementioned experimental data has been taken by Wölk and Strey[18]
using a pulse chamber.
Other interesting experimental data have been taken by
Heath[76, 77], Khan[78], and Kim[79] using a supersonic nozzle technique.
This
technique yields a very high nucleation rate at high supersaturation values. The
results predicted by the P − f orm with CREOS − 02 have been compared with
both the scaled model and an empirical function by Wölk et al[80]. The empirical
function was developed by fitting all the nucleation rate data of Wölk and Strey at
low S. Figure 4.6 shows all the results.
31
18
-3
J cm s
-1
10
Khan et al
Kim et al
P-form
Empirical
scaled model
D2O
17
10
16
10
20
40
60
80
S
100
120
140
160
Figure 4.6. The P-form results using CREOS-02 at high S compared with
two different sets of supersonic nozzle experiments. The scaled model and
the empirical function also shown at T=237.5, 230, 222, 215, and 208.8
K from left to right.
From the above figure, we notice that the scaled model gives very good results at
these high supersaturation values, while the P −form results based on CREOS −02
lie within an order of magnitude of the measured values, but do not reproduce the
T dependence quite as well as for the low S pulse chamber data.
The following graph (Figure 4.7) shows the number of molecules in the critical
droplet calculated from the experimental data[18] and the P − form of W ∗ using
the nucleation theorem plotted versus the number of molecules predicted by using
the Gibbs-Thomson formula at the different temperatures.
32
50
D 2O
40
Gibbs-Thomsom
Expt
P-form
30
n
*
20
10
0
0
10
*
20
30
40
50
n Gibbs-Thomson
Figure 4.7. As in Figure 4.4 but for heavy water.
As for ordinary water, n∗ calculated from the P −form of the CNT is in excellent
agreement with the measured values. Again, since the P − f orm of the CNT
reproduces the experimental T and S dependence of J and since n∗ is essentially
the slope of the lnJ —ln S curve, this good agreement is not surprising.
4.3 DISCUSSION OF WATER RESULTS
The results show a clear advantage of using the P −f orm over the other versions.
Note that the µ− and S − f orms, which were based on the assumption of liquid
incompressibility, give poor results when compared with the experimental data.
This is strong evidence for the invalidity of the assumption that liquid water is
incompressible. Figure 4.8 shows the liquid density as a function of temperature
at different pressures as calculated from IAP W S − 95 and CREOS − 01, which
are excellent agreement with each other and with experiment[75, 81] over wide
ranges of pressure and temperature.
From this figure, one can see that at all
temperatures the density of liquid water depends strongly on the pressure. This
means that liquid water is very compressible, especially at the lower temperatures.
Also note that at a pressure between 190 and 300 MPa, the densities predicted
33
by CREOS − 01 and IAP W S − 95 equations start to differ qualitatively.
The
CREOS − 01 equation predicts that at the higher pressures the well-known density
maximum of water no longer occurs. This is in accord with the experimental density
measurements of Petitet, Tufeu, and Le Neindre[81] that show no density maximum
for P ≥ 200 MP a down to T = 251.15 K.
The disappearance of the density
maximum is also consistent with the observation that water’s viscosity decreases
and its diffusivity increases with increasing pressure up to a pressure of about 200
MPa. At higher pressures, these anomalies in water’s transport coefficients vanish,
and water behaves more normally with further increases in pressure[82]. In contrast,
the IAP W S − 95 equation continues to predict this feature.
This suggests that
nucleation rates calculated using the IAP W S − 95 equation at low T would differ,
perhaps substantially, from those found here using CREOS − 01. This conjecture
awaits a means of using the IAP W S − 95 equation at low T before it can be tested.
It should be noted that for T ≥ 240 K, there is essentially no difference between
the W ∗ (P − f orm) predictions of these two EOSs, as can be seen in Figure 4.9.
300
280
T (K)
260
240
H2O
Expt (Petitet)
Expt (Kell)
IAPWS-95
CREOS-01
500 MPa
0.1 MPa
200
220
200
180
800
50
100
150
900
400
300
190
1000
ρ
1100
3
1200
1300
(kg/m )
Figure 4.8. The temperature-density isobars of water using the IAPWS95 EOS and the CREOS-01 compared to experimental data of Kell and
Whalley[75] and Petitet et al.[81].
34
40
260 K
38
CREOS
IAPWS-95
250 K
T=240 K
W∆P/(kT)
36
34
32
30
28
26
6
7
8
9
10
11
12
13
14
S
Figure 4.9. The work of formation of water at T=240, 250, and 260 K
predicted by the IAPWS-95 and CREOS-01.
Figure 4.10 shows the isothermal compressibility as a function of temperature at
10 MPa (the differences in the isothermal compressibility between 1 atm and 10 MPa
are small) and at 190 MPa, calculated using the fit of Kanno and Angell[83]. From
this figure, it is clear that the isothermal compressibility decreases sharply when the
pressure is increased to values typical of critical nuclei. It should be kept in mind
that the reference pressure for critical droplets can reach very high values, up to
400 MPa or higher, and so the high pressure behavior of the EOS is of considerable
importance in calculating nucleation rates using the P − form of CNT .
35
Kt*10
12
-1
(Pa )
1000
800
10 MPA
600
400
200
230
190 MPA
240
250
260
270
280
290
T (K)
Figure 4.10. Isothermal compressibility of liquid water at 10 MPa and
.
190 MPa calculated from the fit of Kanno and Angell[83]
One last point concerns a purely practical matter. In Chapter 2, an alternative
to using a full EOS to do the P − f orm calculations was noted. This method
was tested using accurate fits for the liquid density as a function of pressure and
employing Eq.(23). Results essentially identical to those shown here were obtained.
36
5. GRADIENT THEORY OF UNARY NUCLEATION
5.1 BACKGROUND
Classical nucleation theory is the most frequently used theory to explain the
nucleation process. In its basic form, the theory depends on experimental measurements as the inputs, which makes it convenient to use. With what is known as the
capillarity approximation, the theory treats the droplet as if it has a sharp interface
with a surface tension equal to that of flat equilibrium interface. Furthermore, the
droplet is usually assumed to be incompressible with a density equal to the bulk
liquid density at low pressure. In reality, the interface between the gas and liquid
phases is never as sharp as envisioned in the model, and it is often quite diffuse.
The density of the droplet is not spatially uniform. For this reason many different
theoretical approaches have been adopted to avoid the approximations of CN T , in
particular the assumptions of a homogeneous droplet with a sharp interface.
Among the different approaches to realistically treat the inhomogeneity of the
droplet is the density functional theory (DF T ). The DF T is a rigorous statistical mechanical approach in which the free energy of the system is expressed as a
functional of the density profile of the entire system. Because DF T involves the
use of realistic intermolecular potentials, it is considerably more difficult to use than
the CNT . Moreover, these potentials are often quite complicated and not so easy
to develop. Gradient theory (GT ) is a methodologically less demanding, but more
approximate relative of DF T . Whereas DF T depends on the intermolecular potential as its main ingredient, the GT , instead, requires a well-behaved mean-field
EOS. In this chapter, GT and the P − form of CNT are applied to study the
nucleation of water, methanol, and ethanol using the CP HB EOS.
The results
are compared to experimental measurements and to the predictions of the S − form
of CNT . Using the CP HB EOS and the JA − EOS, the GT and the P − form
of CNT are also applied to study the nucleation of T IP 4P water at T = 350 K,
which was first simulated by Yasuoka and Matsumoto[84] using molecular dynamics
techniques.
37
5.2 THEORY
Neglecting all the external fields, the GT expression for the Helmholtz free energy,
F , of a pure component is
Z ³
´
c
F =
f0 (ρ) + (∇ρ)2 dV ,
2
(69)
where f0 is the Helmholtz free energy density of the homogenous fluid of density ρ, c
is the so-called influence parameter, which is related to the intermolecular potential,
and ∇ρ is the gradient of the density. The analysis of GT is greatly simplified by
noting[85] that c is only a weak function of density. Then the influence parameter
can be assumed to be constant at constant T .
This parameter characterizes the
inhomogeneity of the fluid. The density distribution that makes F an extremum is
determined by the Euler-Lagrange equation,
µ = µ0 − c∇2 ρ ,
(70)
where µ is the constant chemical potential of the inhomogeneous fluid, and µ0 is the
chemical potential of the homogeneous fluid at density ρ.
For a planar interface, we have a one-dimensional problem, and the above equation can be written as
d2 ρ
1
1 ∂
(f0 − ρµ) .
= (µ0 − µ) =
2
dx
c
c ∂ρ
(71)
Let ω(ρ) = f0 − ρµ, multiply the above equation by 12 dρ/dx, and integrate from
−∞ to ∞. Then apply the boundary conditions, i.e., ρ(∞) −→ ρve and ρ(−∞) −→
ρle , where ρve and ρle are the equilibrium vapor and liquid densities, respectively.
The above equation then reduces to
r
dρ
c
p
= −dx ,
2 ∆ω(ρ)
(72)
where ∆ω(ρ) = ω(ρ) − ω(ρe ), ρe is the equilibrium density of either bulk phase, and
the negative sign indicates that the bulk liquid is located at −∞. To apply the above
equation to a real system, one needs to know the value of the influence parameter.
38
This can be established by using the experimental surface tension of the planar
interface, γ ∞ , to calculate c from the GT expression for the surface tension[86, 87]
ρle
Z
p
=
2c[ω(ρ) − ω(ρve )]dρ .
γ∞
(73)
ρve
The solution to Eq.(72) is the density profile of a flat interface.
Equation (72)
can also be used to determine the thickness of the interfacial region by integrating
between fixed density limits.
A common definition uses ρ− = 0.1ρl + 0.9ρv and
ρ+ = 0.1ρv + 0.9ρl as the lower and upper limits, respectively, so that the "10—90"
interfacial thickness is defined as
r Zρ+
c
dρ
p
t=
.
2
∆ω(ρ)
(74)
ρ−
The primary use for the flat density profile is as the initially guessed profile to
solve Eq.(70) for the radial profile at some initial supersaturation value, S ' 1.
Assuming the droplet is spherical, Eq.(70) can be written as
d2 ρ 2 dρ
1
= (µ0 − µ) ,
+
2
dr
r dr
c
(75)
with the boundary conditions
dρ
−→ 0 as r −→ 0 and ρ −→ ρb as r −→ ∞ ,
dr
(76)
where ρb is the density of the initially uniform metastable phase.
Solving Eq.(75) under the conditions of Eq.(76), enables one to determine the
density profiles of droplets at different values of S. The numerical procedure used
to solve Eq.(75) is based on an iterative finite difference scheme very similar to the
one already described by Li and Wilemski[88]. In one scheme S is slowly increased,
and the profile from the previous value of S is used as an initial guess. The true
profile is obtained by iterating until a predetermined convergence criterion is satisfied at each point. A significant limitation of this scheme is that, particularly at low
temperature, S can be increased by only very small amounts to ensure convergence.
39
An alternative scheme is to apply the foregoing scheme at some relatively high temperature, where convergence is fast. From the high temperature profiles generated
out to an appropriately large value of S, say SM , one lowers the temperature (by 1
K at high T and by 0.5 K at low T ) at the constant SM value thereby generating
a complete set of profiles over the entire temperature range of interest. Each high
S profile corresponds to a small droplet size, and this profile serves as the initial
guess for the next higher or lower value of S at the given temperature. The most
difficult aspect of these calculations is finding converged profiles when the droplet
size is relatively large, i.e., for S values not much larger than 1. This calculation
is easier at high temperatures where the profiles are much more diffuse, as are the
profiles at high S at any T . Converged profiles are found more easily at high S
values.
With the density profile, all other thermodynamic properties of the droplet can be
calculated: Of particular interest are the excess number of molecules in the droplet
and the work of formation of the droplet.
The reversible work is defined as the
difference in the free energy of the system containing the droplet and that of the
homogeneous system:
∗
W =
Z
[f (ρ) − f0 (ρ0 )]dV ,
(77)
where f (ρ) = f0 (ρ) + (c/2)(∇ρ)2 , and f0 (ρ0 ) = ρ0 µ0 − P 0 is the Helmholtz free
energy density of the uniform gas phase.
This result is easily shown[88] to be
equivalent to the original result of Cahn and Hilliard[40]
W
GT
=
Z
[∆ω +
c
|∇ρ|2 ]dV .
2
(78)
The excess number of the molecules in the droplet can be calculated using the
following definition
∆n = 4π
Z∞
[ρ(r) − ρb ]r2 dr ,
0
which is quite general, except for being restricted to spherical droplets.
(79)
40
6. RESULTS OF GRADIENT THEORY FOR UNARY NUCLEATION
6.1 WATER AND T IP 4P WATER
6.1.1 Planar and Droplet Density Profiles from GT . To get a feel for
how well these EOSs can describe interfacial properties, the CP HB EOS and the
JA − EOS were first used with GT to calculate the thickness of planar liquidvapor interfaces using Eq.(74) at different temperatures. Among others, Alejandre,
Tildesley, and Chapela (AT C)[90] have simulated these interfaces using molecular
dynamics (MD) with the SP C/E potential and full Ewald summation. The AT C
simulations provided excellent estimates for the surface tension of water. For this
reason, their results were chosen for comparison with the present GT results. From
their simulation results, they determined the "10—90" thickness of the interface
at various temperatures. Figure 6.1 shows the comparison of the GT results, the
MD simulation results, and a few experimental values determined from ellipsometry
data[91]. The GT calculations were done using the experimental surface tension to
evaluate the influence parameter c.
11
Expt
GT/CPHB
GT/JA
MD-SPC/E
10
9
o
t (A )
8
7
6
5
4
3
300
350
400
450
500
550
T (K)
Figure 6.1. The thickness of flat water interfaces at different T using GT,
MD simulations[90], and experimental data[91].
41
The results clearly depend on the EOS used, but roughly follow the trend of
both the simulations and the experiments. Although the GT results are closer
to the experimental values than are the simulation results, absolute agreement is
lacking.
The larger experimental thicknesses have been attributed to the effects
of capillary waves, which are not present in either the GT calculations or the MD
simulations.
The lack of agreement between the GT results and the simulation
results is not surprising since neither EOS precisely represents the model fluid based
on the SP C/E potential.
Turning now to the droplet calculations, Figures 6.2 and 6.3 show how the density profiles for the CP HB EOS change with temperature. The influence parameter
used in these calculations was evaluated from the experimental surface tension at
each temperature. Results are given at S = 5 and 20, respectively, representative
low and high supersaturation values. In Figure 6.2, for low S at low T , the droplets
have a distinct core where the density is essentially constant. As T increases, the
extent of the uniform core shrinks steadily until the droplet is practically all interface at 350 K. The steady decrease in size with increasing T can be understood
classically in terms of the Gibbs-Thomson (or Kelvin) and Laplace equations. As
T increases at fixed S, the increase in the chemical potential difference between the
equilibrium and metastable phases is given by ∆µ = kT ln S. Then, the thermodynamic relation, (∂µ/∂P )T = 1/ρ, shows that the chemical potential in the droplet
can be increased by raising the droplet’s internal pressure, which is governed by the
Laplace equation, ∆P = 2γ/r. Since the surface tension decreases with increasing
T , r must necessarily decrease to provide the required pressure increase.
Simi-
lar behavior is seen at S = 20, but now the droplets are quite diffuse even at the
lowest value of T . The increase in diffuseness of the droplets at higher S is a consequence of increased proximity to the spinodal. In mean-field theory, at the spinodal
the nucleus and the mother phase are indistinguishable. Thus, as the spinodal is
approached, the nucleus becomes more diffuse with a decreasing core density.
42
S=5
T(K)
220
250
280
320
350
0.05
3
ρ (mol/cm )
0.06
0.04
0.03
0.02
0.01
0.00
0.0
0.5
1.0
1.5
r (nm)
Figure. 6.2. Density profiles of water droplets predicted by CPHB at
different T, for a supersaturation ratio of 5
S=20
T(K)
220
250
280
320
350
0.06
3
ρ (mol/cm )
0.05
0.04
0.03
0.02
0.01
0.00
0.0
0.5
1.0
r (nm)
Figure. 6.3. Same as Figure 6.2 but at S=20
1.5
43
Figures 6.4 and 6.5 show density profiles for the CP HB and JA − EOS,
respectively, at T = 350 K for several supersaturation ratios.
In contrast to
the preceding results, these calculations were made with the influence parameter
evaluated from the T IP 4P surface tension so they could be used for comparison
with the MD results for water nucleation[84]. The JA − EOS produces smaller
droplets with higher density cores, whereas the CP HB predicts somewhat larger
droplets with less dense cores. In each case, these high T profiles are quite diffuse,
i.e., the the flat region in the droplet core is very small. The droplets are practically
all interface.
0.06
density profiles of CPHB water at T=350K
S=5
S=10
S=15
S=20
0.04
3
ρ (mol/cm )
0.05
0.03
0.02
0.01
0.00
0.0
0.2
r (nm)
0.4
0.6
Figure 6.4. Density profiles of water droplets at T=350 K for different
values of the supersaturation ratio using the CPHB EOS.
44
0.06
density profiles of JA water at T=350K
3
ρ (mol/cm )
0.05
S=5
S=10
S=15
S=20
0.04
0.03
0.02
0.01
0.00
0.0
0.2
r (nm)
0.4
0.6
Figure 6.5. Same as Figure 6.4 but using the JA-EOS.
6.1.2 Water Nucleation Rates. Both the GT and the P − form of the CNT
have been used with CP HB EOS to predict nucleation rates of water. The results
were compared with the experimental measurements of Wölk and Strey[18]. Figure
6.6 shows the calculated nucleation rates compared with the experimental data.
The results of GT have been multiplied by a factor of 100 while those of CNT were
divided by 100. As seen in the figure, GT gives a better temperature dependence for
the rate than does the P −form of the CNT . Also, GT gives the right S dependence
at low supersaturation ratios at any T , but it starts to deviate as S increases. It is
also seen that the P −f orm of CNT gives the right S dependence, but the predicted
temperature dependence is hardly different from that of the usual S −form of CNT ,
shown in Figure 1.3, for example. This behavior is not surprising because, as shown
below, the CPHB EOS does not reproduce the density of supercooled liquid water
very accurately.
Recall that CREOS-01 and IAPWS-95 were successful because
they accurately treated the anomalous compressibility of supercooled liquid water.
45
10
10
250K
260K
240K
9
230K
8
10
-3
-1
J (cm s )
10
EXP (Wolk-Strey)
GT*100
P-form/100
T=220K
7
10
6
10
H2O
5
10
5
10
15
20
25
S
Figure. 6.6. Nucleation rate predictions of the CPHB using the P-form
and the GT compared to experimental data of Woelk and Strey[18].
It is also curious that the GT rates are roughly a factor of 104 times smaller
than the classical values. Typically, DF T or GT rates are higher than classical
rates because the nonclassical work vanishes as the spinodal is approached while the
classical W does not. Hence at supersaturations close enough to the spinodal nonclassical rates are higher. But far away from the spinodal, this behavior is reversed:
nonclassical W’s are larger than classical values, and the rates are correspondingly
lower. This behavior is implicit in the recent work of Koga and Zeng[92]. Another
way of expressing their results is shown in the following figure. The Figure shows
that the free energy of formation using the GT is bigger than the CNT result for
S / 69.6, and smaller at larger S. At 220 K, the maximum shifts to S = 21.5 and
the point of equality now lies at S > 100. This behavior indicates that the spinodal
boundary for the CP HB EOS must be very far away from the region of S relevant
to the nucleation experiments. It should be noted that the spinodal boundary is
dependent on the specific EOS used and is not unique.
46
1.25
T=260 K (CPHB)
WGT/W∆P
1.20
1.15
1.10
1.05
1.00
0.95
0.90
0
20
40
60
80
100
S
Figure 6.7. The ratio of the GT work of formation to that of the P-form
of CNT as a function of supersaturation ratio at 260 K.
Figure 6.8 shows the excess number of molecules found from GT with the CP HB
EOS using Eq.(79) and the experimental number of molecules using Eq.(66) from
the Wölk and Strey data. Although the GT predicts the right T dependence of the
rates, its predicted S dependence is not as good as that of the P-form. From the
nucleation theorem, one can see that the bigger the isothermal slope of ln J vs. ln S
is, the higher the number of molecules is predicted. From Figure 6.6 it is clear that
the slope of nucleation rates against S is bigger for GT than for the experimental
results. This behavior is reflected in the results shown in Figure 6.8. The deficiency
of GT in predicting the S dependence is not unexpected. Li and Wilemski found
similarly poor behavior for GT in their study of a hard sphere Yukawa fluid[88].
47
50
H2O
40
Expt
GT-CPHB
Gibbs-Thomson
30
n
*
20
10
0
0
10
20
30
40
50
*
n Gibbs-Thomson
Figure 6.8. The number of water molecules in the critical cluster as predicted by the nucleation theorem and the GT calculations. The dashed
line represents full agreement with Gibbs-Thomson equation.
6.1.3 T IP 4P Water Nucleation. Yasuoka and Matsumoto (Y M)[84] investigated
the homogeneous nucleation of water using the molecular dynamics technique and
the T IP 4P water potential at T = 350 K. They obtained one simulated nucleation
rate with a value of 9.62 × 1026 cm−3 s−1 at a supersaturation ratio of 7.3. It is of
interest to see how this value compares with other theoretical estimates. Thus, GT
was used to calculate water nucleation rates with both the CP HB EOS and the
JA − EOS. Rate calculations have also been made with the P − f orm of CNT
using the CP HB EOS, the JA−EOS, the Peng-Robinson EOS, and CREOS −01
EOS. The last EOS used only at S = 7.3 and yielded the rate 1.83 × 1028 cm−3
s−1 . In all of these calculations, the value of the surface tension reported by Y M
for the T IP 4P potential, γ = 39 erg/cm2 , was used as needed. The calculated
results are compared with the molecular dynamics simulation result in Figure 6.9.
From the figure one can see that there is no advantage in using the GT over the
P − form of CNT or vice versa.
The CP HB predicts higher rates than either
the JA − EOS or the Peng-Robinson (P R) EOS. Since, none of the three EOSs
48
was developed specifically for T IP 4P water, the results are strictly not comparable;
nevertheless, it is satisfying to see that the CP HB, P R and JA − EOS results
bracket the simulated result of the molecular dynamics.
10
TIP4P-water at T=350K
29
CREOS-01
CPHB EOS
Peng-Robinson EOS
-3
-1
J (cm s )
10
28
10
27
MD
10
JA EOS
26
5
6
7
S
8
S-Form
P-Form
GT
MD
CREOS-01
9
10
Figure 6.9. Nucleation rates for GT and two forms of CNT at T=350 K
using different EOSs, as shown in the figure, compared with the MD rate
for TIP4P water and the result of the P-form of CNT using CREOS-01.
6.2 COMPARISON OF THE WATER EOS
To better understand the reasons for the success or failure of the preceding nucleation rate calculations, it is interesting to see to what degree these four different models of water agree or disagree in predicting the actual properties of water.
For T IP 4P , the calculated saturated vapor and liquid densities are 4.66 × 10−4
g/cm3 and 0.9356 g/cm3 , respectively, and the pressure at vapor-liquid equilibrium
is 0.0753 MP a.
Figures 6.10, 6.11, 6.12, and 6.13 compare the various equations
with the single T IP 4P datum. Note that the results of the IAP W S − 95 EOS
may be regarded as the experimental values since this equation describes real water
to high precision.
49
1.00
3
ρle (g/cm )
0.95
0.90
CPHB
IAPWS-95
JA
PR
MD
0.85
0.80
0.75
200
250
300
350
400
450
T (K)
Figure 6.10. The predictions of different EOSs for the equilibrium liquid density of water at different T compared to the experimental data
generated using the IAPWS-95.
300
P=0.1 MPa
190 MPa
T (K)
280
260
240
220
960
1000
1040
1080
3
ρ (kg/m )
Figure 6.11. Density of liquid water using the CPHB EOS (stars) at
different P (0.1, 50, 100, 150, 190 MPa) compared to the experimental
data calculated using the IAPWS-95 (circles)
50
5
1.0x10
IAPWS-95
CPHB
JA
PR
MD
4
8.0x10
P (Pa)
4
6.0x10
4
4.0x10
4
2.0x10
0.0
275
300
325
350
375
T (K)
Figure 6.12. The predictions of different EOSs for the equilibrium vapor
pressure at different T compared to the experimental data calculated by
using the IAPWS-95
-3
5.0x10
IAPWS-95
CPHB
JA
PR
MD
-6
10
-7
-3
4.0x10
10
-8
3
ρve (g/cm )
10
-3
3.0x10
-9
10
-10
-3
2.0x10
10
-11
10
220
230
240
250
260
-3
1.0x10
0.0
200
250
300
350
400
450
T (K)
Figure 6.13. Same as for Figure 6.12 except for the equilibrium vapor
density.
51
Starting with the P R − EOS, one can see that this equation predicts fairly accurately the equilibrium vapor pressure and vapor density, but it is severely deficient
in predicting the equilibrium liquid density. One shouldn’t be too critical of the
P R − EOS, because as noted earlier, this equation was developed to predict the
properties of non-polar fluids. For this reason, the P R − EOS was not used with
the GT to predict nucleation rates of water.
The JA − EOS gives generally poor predictions of all properties on the binodal
except for the equilibrium liquid density. It is worth noting that the JA − EOS is
capable of accurate predictions of the equilibrium vapor pressure if the either the
correct equilibrium vapor or liquid density is supplied independently. It is when the
simultaneous calculation of the vapor and liquid binodal densities is attempted that
the JA − EOS fails. The JA − EOS binodal vapor densities are many orders of
magnitude too low (note particularly the inset of Figure 6.13), and their use would
lead to a gross overestimate of the extent of the metastable vapor phase. For this
reason, we avoided using this equation with GT to calculate the nucleation rates of
water at low T .
The CP HB equation shows good agreement with the experimental values of the
equilibrium properties over most of the temperature range considered, but it fails
to show the liquid density maximum at T ' 4 ◦ C. Moreover, it incorrectly predicts
a monotonically increasing density with decreasing T at low P . It shows the opposite behavior at high P and is generally in poor agreement with the experimental
values. These major flaws disfavor the use of this equation at low T . Despite these
flaws, this equation still displays the appropriate mean-field behavior needed for GT
calculations, and its quantitative predictions of other water properties are generally
acceptable. As a result, it was used for both the low T and high T rate calculations.
In conclusion, it should be noted that none of the equations provides a particularly
accurate description of real water. This failing is clearly shared by T IP 4P water as
the figures also show.
52
6.3 RESULTS FOR METHANOL AND ETHANOL
Both the GT and the P − form of the CNT have been used with the CP HB
EOS to predict nucleation rates of methanol and ethanol. In each case the calculated
rates exceed the experimental values by many orders of magnitude. In the case of
methanol, Figure 6.14 shows that the GT results are better than the results of
the P − form of the CNT by about a factor of 300 when compared with the
experimental results of Strey, Wagner, and Schmeling[93]. Similar trends are seen
for the ethanol results in Figure 6.15, but here the improvement is by a factor of
500. In each figure, the scaling factors that reduce the calculated rates were chosen
to force rough agreement at the higher temperature.
Methanol
9
10
T=272 K
T=257 K
8
-3 -1
J (cm s )
10
7
10
10
6
10
JP/10
JExpt
7
JGT/3*10
5
10
2.2
2.4
2.6
S
2.8
3.0
3.2
Figure 6.14. Experimental nucleation rates of methanol compared to the
predictions of GT and the P-form of CNT with the CPHB EOS.
It should be noted that neither theory reproduces the experimental temperature
dependence, and the reason might be related to the inaccurate predictions of the
CP HB EOS for the density at different temperatures and high pressures. Figure
6.16 shows such predictions for the CP HB EOS as a function of temperature
compared to experimental data[94] at different pressures.
53
Ethanol
10
10
T=286 K
T=293 K
9
-3 -1
J (cm s )
10
8
10
7
10
Jexpt
7
JP/5*10
6
10
5
JGT/10
2.4
2.5
2.6
2.7
2.8
2.9
3.0
S
Figure 6.15. As in Figure 6.14 but for ethanol.
Ethanol
0.88
T=273 K
T=283 K
T=293 K
T=303 K
T=313 K
3
ρ (g/cm )
0.86
0.84
0.82
0.80
0.78
-5
0
5
10
15
20
25
30
35
40
P (MPa)
Figure 6.16. Liquid ethanol density vs. P at different temperatures using
the CPHB EOS (open symbols) and experimental data (solid symbols).
54
The experimental results suggest that the incompressible droplet approximation
might not be so unreasonable for ethanol, although the density profiles for ethanol
(and methanol) droplets under these conditions look qualitatively similar to those
of water droplets. The glaring inconsistency between the CP HB and experimental ethanol densities suggests that the P − f orm calculations based on fits to the
experimental density and Eq.(23) would not be in good agreement with the CP HB
results. This is confirmed in Figure 6.17 which compares the two sets of P − form
results. It is not clear why the S − f orm results, which assume an incompressible
droplet, have the same T dependence as the CP HB P − form results. It is most
likely accidental agreement in the small T range examined.
Ethanol
10
10
T=293 K
T=286 K
9
-3 -1
J (cm s )
10
8
10
7
10
7
P-form/5*10 (CPHB)
6
S-form/5*10
7
P-form/10 (fitting)
Jexpt
6
10
5
10
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
S
Figure 6.17. Experimental nucleation rates of ethanol compared to calculated rates using the S-form and the P-form of CNT with the CPHB
EOS and the P-form of CNT using fitted experimental density data[94].
In contrast, for methanol the P − form results based on fits to the experimental
density and Eq.(23) were identical to those calculated using the CP HB EOS. This
was expected because liquid methanol densities given by the CP HB EOS as a
function of pressure agree well with the experimental values[95].
55
7. BINARY NUCLEATION THEORY
7.1 CLASSICAL NUCLEATION THEORY
Reiss generalized the theory of binary nucleation by including a kinetic mechanism that allows for cluster growth and decay in the two-dimensional space comprising the number of molecules, n1 and n2 , of each species in the cluster[44]. The major
difference between unary and binary nucleation is that in the latter the critical clusP
ter is distinguished not only by its size, but also by its composition (xi = ni /( ni )).
For the Gibbs free energy of a binary cluster, Reiss used the capillarity approximation and found the following approximate result,
∆G = n1 ∆µ1 + n2 ∆µ2 + 4πr2 γ ,
(80)
where ∆µi is the chemical potential difference for a molecule of species i in the vapor
and liquid phases at the same pressure. The free energy surface represented by this
∆G contains a saddle point (sometimes more than one), which functions much as
a pass through a ridge of mountains. That is, the saddle point is the lowest point
on a free energy ridge that separates small, unstable clusters from larger, stable
(actually, growing) fragments of the new phase.
The critical droplet is located at the saddle point, i.e., where the first derivatives
of ∆G with respect to n1 and n2 are zero. In this case, Doyle obtained the following
forms for the Gibbs-Thomson equations[45]
2γv1 3xvm dγ
−
= 0,
r
r dx
2γv2 3(1 − x)vm dγ
+
= 0,
∆µ2 +
r
r
dx
∆µ1 +
(81a)
(81b)
which must be solved for x and r. Here, x is the mole (or number) fraction of the
second component of the fluid, and vm is the mean molecular volume. The mean
molecular volume is related to the partial molecular volume, vi , of each component
through the relation vm = (1 − x)v1 + xv2 .
Note that, in general, all of the
thermodynamic properties of binary mixtures depend on the composition of the
mixture.
56
Under the assumption that the liquid is incompressible, the exact thermodynamic
relation of the pressure and the chemical potential,
µ
¶
∂µi
= vi ,
i = 1, 2
∂P T,ni
(82)
readily integrates to the result
∆µi = vi (Pl − Pv ) .
(83)
Combining Eqs.(81) and (83) with the Laplace equation for the pressure difference,
one can show that vm dγ/dx = 0 must be true[96].
Since, in general the surface
tension depends on composition, this result must be invalid, and it suggests that
Doyle’s versions of the binary Gibbs-Thomson equations are wrong. Following another line of reasoning stimulated by the work of Renninger, Hiller, and Bone[46],
Wilemski[97] proposed a revised thermodynamic cluster model that led to the classical Gibbs-Thomson equations, in which the surface tension derivatives are missing
2γv1
= 0 ,
r
2γv2
= 0 .
∆µ2 +
r
∆µ1 +
(84a)
(84b)
These two equations can be combined to obtain the following relation
∆µ1
∆µ2
=
,
(85)
v1
v2
whose solution yields the so-called bulk, or interior, composition of the cluster. All
of the properties of the critical nucleus can be evaluated using this composition[47].
For many binary systems the composition dependence of the surface tension is
very weak and dγ/dx ≈ 0. These two different methods for finding the critical
nucleus composition then produce very similar results. However, for systems whose
surface tension varies strongly with x, such as water-alcohol systems, the two methods give very different results, and each approach suffers from a serious deficiency,
as described in Section 1.2.
A third approach is possible if one evaluates the critical composition and critical
work of formation using Gibbs’s fundamental conditions of (unstable) equilibrium.
57
Assume the metastable binary system is at a total vapor pressure Pv and temperature T , with a vapor mole fraction yi . Then the properties of the liquid reference
phase that represents the critical nucleus are determined by solving the following
equations simultaneously
µiv (T, Pv , yi ) = µil (T, Pl , xi ) ,
(86)
where µiv and µil are the vapor and liquid chemical potentials of component i, and
yi and xi are, respectively, the vapor and liquid mole fractions of component i. Assuming that the surface tension of the droplet is the same as that of a macroscopic
liquid mixture with composition xi , the radius of the critical droplet can be determined from the Laplace equation (see Appendix A.2) and the pressure difference
Pl − Pv .
As in unary nucleation, if an EOS (introduced in the next chapter) is known for
the system, one can apply Eq.(86) to determine the internal reference pressure Pl .
The free energy of formation at the critical radius can then be evaluated from the
relation
∆G∗ = W ∗ =
16πγ ∗3
,
3(Pl − Pv )2
(87)
which is formally identical to the unary result. We call this version 1.
Applying Eqs.(84) at the critical radius, it can be shown easily that
r∗ = −
∗
2γ ∗ vm
,
(1 − x∗ )∆µ1 + x∗ ∆µ1
(88)
where ∗ denotes a property evaluated at the critical nucleus composition. The
critical free energy of formation, then follows from Eq.(80) as
W∗ =
∗2
16πγ ∗3 vm
.
3((1 − x∗ )∆µ1 + x∗ ∆µ2 )2
(89)
We will call this version 2 when x∗ is determined using the classical Gibbs-Thomson
equations of Wilemski’s revised model, and we call it version 3 when Doyle’s form
of the Gibbs-Thomson equations, Eqs.(81) are used.
In experiments, the main independent variable is known as the vapor activity, ai ,
which is defined as the ratio of the partial vapor pressure of component i, Piv , to
58
the equilibrium vapor pressure of its pure liquid, Pi0 , i.e., ai = Piv /Pi0 . One can also
define the supersaturation ratio of component i as Si (x) = Piv /Pieq (x), where Pieq (x)
is the equilibrium partial vapor pressure of component i over the binary solution of
composition x.
If the vapor is assumed to be ideal, one can write ∆µi = kT ln(ai /ail ) = kT ln Si ,
where ail = Pieq (x)/Pi0 is the liquid activity of component i. Then the work of
formation can be written as
W∗ =
∗2
16πγ ∗3 vm
,
3(kT ln S ∗ )2
(90)
with ln S ∗ = (1 − x∗ ) ln S1 (x) + x∗ ln S2 (x). This form is very popular because it is
expressed in terms of conveniently measured properties.
In Chapter 9, results of these three classical versions are compared for a model
binary fluid whose properties resemble those of an ethanol-water mixture. The
model fluid approach was used because there are no accurate EOSs for the most
interesting binary systems that show surface segregation or enrichment. Since the
results are strictly not comparable with experiment, density functional theory was
used to calculate the most important thermodynamic property of the model fluid,
namely, its surface tension. The other properties were obtained from the mean field
EOS for the system. A binary hard sphere—Yukawa mixture was used as the basis
for the model fluid.
7.2 DENSITY FUNCTIONAL THEORY (DFT)
Zeng and Oxtoby[43] developed an approximate density functional theory for
binary nucleation in which the functional for the Helmholtz free energy takes the
form
F [ρ1 (r), ρ2 (r)] =
Z
2 Z Z
1X
drfh [ρ1 (r), ρ2 (r)] +
drdr0 φij (|r − r0 |)ρi (r)ρj (r0 ) .
2 i,j=1
(91)
where fh is the Helmholtz free energy density of a uniform hard sphere mixture and
φij is the perturbed attractive part of the potential. The grand potential, Ω, of the
system is
Ω [ρ1 (r), ρ2 (r)] = F [ρ1 (r), ρ2 (r)] −
2
X
i=1
µi
Z
drρi (r) ,
(92)
59
where µi is the (constant) chemical potential of the ith component in the system.
The equilibrium droplet density profiles can be generated by applying the conditions
∂Ω/∂ρi = 0, or ∂F/∂ρi = µi , and solving the resulting Euler-Lagrange equations
that read
µi = µih [ρ1 (r), ρ2 (r)] +
2 Z
X
j=1
dr0 φij (|r − r0 |)ρj (r0 ) ,
(93)
where µih is the hard sphere chemical potential.
The free energy density of the homogeneous fluid can be derived from Eq.(91) by
taking the limit of uniform densities. This yields
where
2
1X
f (ρ1 , ρ2 ) = fh (ρ1 , ρ2 ) −
αij ρi ρj ,
2 i,j=1
αij = −
Z
drφij (r) .
(94)
(95)
The pressure and the chemical potential of the homogeneous fluid are then readily
derived from Eq.(94), and they are
and
2
1X
αij ρi ρj ,
P = Ph (ρ1 , ρ2 ) −
2 i,j=1
µi0 = µih (ρ1 , ρ2 ) −
2
X
αij ρj ,
(96)
(97)
j=1
where Ph is the pressure of the binary hard sphere mixture.
7.3 SURFACE TENSION AND REVERSIBLE WORK
An expression for the surface tension of a planar interface can be obtained by
using the definition of the grand potential, i.e., Ω = −P V + γA, where V is the
system volume and A is the area of the interface. If we substitute Eq.(97) into
Eq.(92), it is shown in Appendix D that
¾
Z ½
1
1
γ=
ρ (x) [µ1h (ρ1, ρ2) − µ1 ] + ρ2 (x) [µ2h (ρ1 , ρ2 ) − µ2 ] + P − Ph dx . (98)
2 1
2
Another important relation is the work of formation of a spherical droplet, which
is derived in Appendix D as
¾
Z ½
1
1
Wrev = 4π
ρ1 (r) [µ1h (ρ1, ρ2) − µ1 ] + ρ2 (r) [µ2h (ρ1 , ρ2 ) − µ2 ] + P − Ph r2 dr .
2
2
(99)
60
7.4 DFT FOR THE HARD SPHERE—YUKAWA FLUID
In this thesis, the model fluid is a binary hard sphere—Yukawa mixture.
The
Yukawa potential,
αij λ3 exp(−λr)
φij = −
,
(100)
4πλr
has been chosen as the attractive part of the potential. There is a major advantage
of using this particular potential that arises from its status as the Green’s function of
the Helmholtz equation. By acting with ∇2 on the coupled integral Euler-Lagrange
equations, Eq.(93), we transform them into two coupled differential equations,
"
#
2
X
d2 µih 2 dµih
(101)
+
αij ρj ,
= λ2 µih (ρ1 , ρ2 ) − µi −
dr2
r dr
j=1
that are much simpler to solve numerically.
Further simplifications are possible. For the particular choice, α12 =
√
α11 α22 ,
the so-called Bertholet mixing rule, one can combine the equations for the chemical
potentials, Eq.(93), in a single formula,
µ1h (ρ1 , ρ2 ) − µ1
µ (ρ1 , ρ2 ) − µ2
= 2h √
.
√
α11
α22
(102)
With this expression, Eq.(98) for the surface tension simplifies to
µ
¶
Z ρil
¤1/2 dµih
£
1
2
dρi , i = 1 or 2 ,
γ=
(µih (ρ1 , ρ2 ) − µ1 ) − 2αii (Ph − P )
λαii ρiv
dρi
(103)
where ρiv and ρil are the equilibrium vapor and liquid densities of component i.
Note that this equation does not require the actual density profile in contrast to
the earlier Eq.(98), although one should be careful about choosing which component to integrate over. The smart choice is to pick the component whose density
varies monotonically through the interface[99]. A derivation of Eq.(103) is given in
Appendix D.
Since Eq.(102) defines ρ1 as a function of ρ2 everywhere in the system, only one of
the differential Euler-Lagrange equations, Eq.(101), needs to be solved. This further
reduces the amount of computational work required.
Please note that all the above equations will reduce to those of a unary fluid by
setting either ρ1 or ρ2 to zero.
61
8. PROPERTIES OF THE MODEL BINARY HARD-SPHERE
YUKAWA (HSY) FLUID
8.1 EQUATION OF STATE
In order to approximate the properties of real fluids using the hard sphere—
Yukawa (HSY ) fluid, one has to evaluate the hard core diameters σi and the
αij parameters to produce the right surface tensions and vapor pressures. This
can be achieved by choosing carefully the critical temperatures of each component,
this choice is implemented using the scaled α
e ii parameters, α
e ii = αii /(kT σ 3i ) =
11.1016Tci /T , where Tci is the critical temperature of component i. This choice of
αii has been used previously[42, 100]. All the parameters in this chapter were scaled
using the following rules:
γ = γσ 21 /kT , ρei = ρi σ 31
µ
ei = µi /kT , Pe = P σ 31 /kT , e
fe = fσ 31 /kT, σ = σ 2 /σ 1 , vei = vi /σ 31 .
(104)
(105)
All the lengths are scaled with respect to σ 1 . To optimize the HSY fluid properties
to resemble those of the water—ethanol system, the following parameters have been
estimated to produce the right surface tension of the pure components: λ = 0.709
◦ −1
◦
◦
A , σ 1 = 3A, σ 2 = 4A, Tc1 = 610 K, Tc2 = 544 K, where 1 and 2 refer to water
and ethanol, respectively.
According to the above scaling rules, the EOS , i.e., Eq.(96) appears in dimensionless variables as
where
2
1X
e
e
P = Ph (e
ρ1 , e
ρ2 ) −
α
e ij e
ρie
ρj ,
2 i,j=1
¸
·
3
3
ξ
+
ξ
6
3ξ
2ξ
ξ
0
1
2
2
2
Peh =
,
+
+
π 1 − ξ3
(1 − ξ 3 )2
(1 − ξ 3 )3
(106)
(107)
with ξ i = π (e
ρ1 + e
ρ2 (σ 2 /σ 1 )i ) /6. The chemical potential of each component can be
derived from the free energy density using µi = ∂f /∂ρi , where
2
1X
e
e
α
e ij e
ρie
ρj ,
f = fh −
2 i,j=1
(108)
62
and the Helmholtz free energy density of a hard sphere mixture,
µ 3
¶
ξ 32
ξ
3ξ 1 ξ 2
2
e
fh = e
+
ρ1 ln e
ρ1 + e
ρ2 ln e
ρ2 +
− 1 ln(1 − ξ 3 ) +
, (109)
ξ 0 (1 − ξ 3 ) ξ 0 ξ 3 (1 − ξ 3 )2
ξ 0 ξ 23
is given by the binary Carnahan-Starling equation of Mansoori et al.[101]
At a given temperature T and liquid composition x, the coexisting vapor liquid
densities are determined by solving the following equations simultaneously
µ1v (ρ1 , ρ2 ) = µ1l (ρ1 , ρ2 ) ,
(110)
µ2v (ρ1 , ρ2 ) = µ2l (ρ1 , ρ2 ) ,
(111)
Pv (ρ1 , ρ2 ) = Pl (ρ1 , ρ2 ) ,
(112)
x2 = ρ2 /(ρ1 + ρ2 ) .
(113)
After solving the above equations, one can produce the whole equilibrium phase
diagram. The physical properties shown in Figures 8.1-8.5 were also generated,
because they are needed to calculate the work of formation using CN T .
The
components of the model fluid are referred to as p-water and p-ethanol, where "p"
stands for pseudo, because of their resemblance to real water and ethanol.
T=260 K
-4
4.0x10
-4
3
Pσ1 /kT
3.0x10
-4
2.0x10
-4
Pve
P1
P2
1.0x10
0.0
0.0
0.2
0.4
x
0.6
0.8
1.0
Figure 8.1. The total and partial equilibrium vapor pressures of the HSY
model fluid at T=260 K versus mixture composition, x.
63
Figure 8.1 shows how the equilibrium partial vapor pressures vary with composition, and it also shows the total vapor pressure. Although the absolute magnitudes
of the pure vapor pressures are too high by factors of 120 and 70 for water and
ethanol, respectively, the qualitative behavior is quite similar to that of the water—
ethanol system. Also, the ratio of the calculated equilibrium vapor pressures of the
pure components (p-water to p-ethanol) is 0.64 compared to 0.495 for the real system. Figure 8.2 shows the vapor-liquid equilibrium phase diagram as a function of
the p-ethanol composition. The azetropic composition is realistic for water-alcohol
systems.
4.0x10
T=260 K
-4
3.5x10
-4
3.0x10
-4
2.5x10
-4
2.0x10
-4
3
Pveσ1 /kT
liquid
0.0
vapor
0.2
0.4
0.6
0.8
1.0
x
Figure 8.2. P-x phase diagram of the binary HSY model system.
We have seen earlier that the composition dependence of the bulk surface tension is a key ingredient in classical binary nucleation theory. Figure 8.3 shows the
variation of the calculated surface tension using Eq.(103) with the p-ethanol composition compared to the measured surface tension by Viisanen et al.[102].
The
calculated values (in mN/m) of the pure components, 74.42 for p-water and 25.08
for p-ethanol, are close to the experimental values, 77.45 and 25.04, and the trend
64
captures the desired behavior, the steep decline at small x2 , quite nicely.
T=260
80
DFT
Expt
γ (mN/m)
70
60
50
40
30
20
0.0
0.2
0.4
0.6
0.8
1.0
x
Figure 8.3. Surface tension for the pseudo water-ethanol system and
measured values for water-ethanol versus ethanol mole fraction, x.
Figures 8.4 and 8.5 show the variation of the partial molecular volumes with
the p-ethanol composition. The partial molecular volumes have been evaluated at
the usual constant pressure, 1 atm, using the definitions (∂V /∂ni )P,nj = vi , and
ρi = ni /V , where ni is the number of molecules of component i and V = n1 v1 + n2 v2
is the total volume. The following expression was derived from Eq.(106):
vi =
∂Ph /∂ρi − (α1i ρ1 + αi2 ρ2 )
.
2(P − Ph ) + ρ1 ∂Ph /∂ρ1 + ρ2 ∂Ph /∂ρ2
(114)
The partial molecular volumes show virtually no dependence on composition, in
contrast with the behavior of real water-alcohol systems.
65
T=260 K
1.070
v1/σ1
3
1.065
1.060
1.055
1.050
1.045
0.0
0.2
0.4
0.6
0.8
1.0
x
Figure 8.4. Variation of the partial molecular volume of p-water with
composition.
T=260 K
2.67
v2/σ1
3
2.66
2.65
2.64
2.63
2.62
2.61
0.0
0.2
0.4
0.6
x
Figure 8.5. Same as Figure 8.4. but for p-ethanol.
0.8
1.0
66
8.2 FITTED PROPERTY VALUES
In order to calculate the work of formation at the critical cluster using version 1
from Eq.(87), one needs the use of the EOS and the value of the surface tension at
the critical composition x∗ . An EOS is not used to predict the work of formation
using version 2, and version 3. In these versions, one solves either the Eqs.(84)
for version 2 or Eqs.(81) for version 3. In order to find x∗ and to evaluate W ∗ in
the traditional manner of binary CNT , i.e., without using a full EOS, one needs
to have all the physical properties, γ, v, P1eq , P2eq , as functions of composition. ln γ
e
was correlated by fitting ln γ to a polynomial of seventh order as a function of
7x/(1 + 6x). The molecular volume, v = (1 − x)v1 + xv2 was fitted such that both v1
and v2 are polynomials of fourth order in x, while P1eq , P2eq are written as a function
of the activity coefficients γ i as
Pieq = Pi0 xi γ i .
(115)
In this form the γ i are chosen to follow the van Laar equations:
A
,
(1 + A(1 − x)/Bx)2
B
.
ln γ 2 =
(1 + Bx/A(1 − x))2
ln γ 1 =
(116)
(117)
For the pseudo water-ethanol system, the fitting parameters A and B are 1.0481
and 2.68438, respectively. The corresponding experimental values at 260 K are 0.92
and 1.47 for water-ethanol[102] and 1.313 and 2.3652 for water-propanol[103]. The
fitted equations of v1 , v2 , and γ are in dimensionless units
ve1 = 1.06717 − 0.00263x − 0.08125x2 + 0.09417x3 − 0.03174x4
ve2 = 2.61886 + 0.19638x − 0.29593x2 + 0.21051x3 − 0.05972x4
ln γ
e = 1.40523 − 3.71918y + 15.09805y 2 − 54.43205y 3 + 114.48491y 4
−128.96306y 5 + 74.21279y 6 − 17.8y 9
where y = 7x/(1 + 6x).
(118)
67
9. RESULTS OF THE HSY BINARY FLUID
9.1 CRITICAL ACTIVITIES AT CONSTANT W∗
In this thesis, we compare the various theoretical results in the form of a critical
vapor activity plot at T = 260 K for a constant value of W ∗ , W ∗ /kT = 40 . This
value will produce a nucleation rate of the order 107 cm−3 s−1 .
Figure 9.1 shows
how a2 varies with a1 for the different versions of binary CNT and also includes the
DF T results.
4.0
DFT
Version 1
Version 2
Version 3
*
W /kT=40
T=260 K
3.5
3.0
a2
2.5
2.0
1.5
1.0
0.5
0.0
0
1
2
3
4
5
6
7
a1
Figure 9.1. Critical activities of p-water (1) and p-ethanol (2) needed to
produce a constant work of formation of 40 kT.
It is well known that for binary systems with a highly surface active component,
the critical activity curves calculated using the CNT (version 2) exhibit unphysical
behavior[47, 104, 105]. This version predicts that at the same activity a2 , different
activities of a1 will produce the same value of the work of formation. Such behavior has been produced again in the pseudo water-ethanol system. Version1 also
predicts the same behavior as the standard version 2 calculations, but it lowers
the magnitude of the discrepancy slightly. The close correspondence of version 1
and version 2 is not surprising, because the classical Kelvin equations, Eqs.(84), are
68
easily derived from the fundamental Gibbs condition, Eq.(86), using the assumption
of an incompressible fluid. The predictions of both version 1 and version 2 show a
contradiction with the nucleation theorem[106], which states in mathematical form
that the work of formation has to have a negative slope with increasing activity of
either component. As shown earlier in the thesis, the P −form, or what we called it
here the version 1 is an exact formula except for the approximation of the droplet
surface tension as that of the flat interface.
The major reason for the failure of
version 1 and version 2 is probably the use of the bulk surface tension which forces
the highly curved droplet surface to implicitly assume the composition of a flat interface. Models such as the explicit cluster model[107] that relax this requirement,
but otherwise use the same ingredients as the classical CNT , produce physically
realistic behavior in reasonable agreement with experiment. A second contributing
factor might refer to nature of the EOS, in that this equation probably does not
realistically capture the isothermal compressibility behavior of a real alcohol-water
mixture.
The behavior of version 3 is quite similar to that predicted for real water—alcohol
systems. While no unphysical behavior is predicted, the p-ethanol activities rapidly
drop to unrealistically low values as the p-water activity increases to rather modest
values. It is interesting that the drop-off region for version 3 mirrors the overshoot
region of version 1 and 2.
The figure also shows several points generated by using DF T to predict the work
of formation. It is quite obvious that the DF T improves the results significantly. It
is not unexpected that the results of the DF T will do a better a job over the CNT
regarding many aspects as the temperature dependence (which is not shown here)
and activity dependence. Also note that the results of the DF T show systematic
agreement with the nucleation theorem, which is another of the major advantages
of using the DF T .
Similar results have been found previously by Napari and
Laaksonen using DF T with models based on site-site Lennard-Jones potentials[54,
108].
69
9.2 NUMBER OF MOLECULES IN THE CRITICAL DROPLET
Because it gives information about the composition and size of the critical nucleus, a very interesting piece of information to evaluate is the number of molecules
of each component in the critical droplet. Figures 9.2 and 9.3 show how the numbers of molecules vary as a function of the p-water activity. Figure 9.2 shows the
predictions of version 1 and version 2 of CNT , which are in very close agreement,
and also shows the DF T calculations. Figure 9.3 shows the predictions of version
3 with the predictions of the DF T .
Figure 9.4 compares the results for versions 1 and 2, which are essentially identical, with those of version 3 in the region a1 < 1, where the critical activity curves
for the different versions do not deviate greatly. It should be noted that the results
are not directly comparable because version 3 gives the total numbers of each type
of molecule in the droplet, whereas, versions 1 and 2 give only the numbers in the
bulk core of the droplet. For this highly surface enriched system, one would expect
differences between the total numbers and the core numbers. Thus, the differences
between the different versions are not too surprising.
100
version 1
version 2
DFT
80
60
n
n1
40
20
0
n2
0
1
2
3
4
5
6
a1
Figure 9.2. The number of molecules of each component of the critical
droplet as a function of the p-water activity using version 1 and version
2 of the CNT,as well as the DFT.
70
T=260 K
100
80
60
n
n1
n 1 (version 3)
n 2 (version 3)
DFT
*
40
20
n2
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
a1
Figure 9.3. The number of molecules of each component of the critical
droplet as a function of the p-water activity using version 3 of the CNT
and the DFT.
T=260 K
n2
100
80
n
*
version 1
version 2
version 3
60
40
20
n1
0
0
1
a1
Figure 9.4. The number of molecules of each component of the critical
droplet as a function of the p-water activity using versions 1, 2, and 3 of
the CNT.
71
10. CONCLUSIONS
Generally, in assessing how well the CNT predicts the experimental results for
unary systems, the P-form was ignored in all previous studies due to lack of suitable
EOSs. Instead, the assumption of the incompressibility of the liquid droplet was
made to simplify the analysis. In this thesis, the P-form of CNT has been studied
on several different substances for the first time. This version of CNT proved fairly
successful inpredicting the nucleation rates of water and heavy water using accurate
EOSs. Compared to the more approximate versions of CNT, the P-form shows major
improvements regarding the T dependence, as well as the S dependence, of the rates
for water and heavy water. The highly compressible nature of supercooled water is
clearly the most important factor in understanding water behavior, and this feature
is accurately described in the EOSs that gave the most successful predictions.
On the basis of the water results, the P-form of CNT was expected to predict
better results than the other CNT forms for the nucleation of alcohols. Unfortunately, this expectation was not realized. The disappointing results for ethanol and
methanol might be due to the inaccurate CPHB EOS used to describe alcohols,
or because of inaccurate experimental density measurements, or simply because
alcohols are much less compressible than water or heavy water.
Water nucleation has been also studied for the first time using gradient theory
with the CPHB EOS. The results of this theory show an improvement over the
standard CNT regarding the T dependence. However, these GT results also show
a somewhat poorer prediction for the S behavior at high values of S. Also, it
was found that the predicted rates for GT are lower than the predictions of the
CNT, contrary to what is usually found for a nonclassical theory. This behavior
is understandable in terms of the spinodal location predicted by the EOS: if the
relevant supersaturations are far away from the spinodal, the nonclassical work of
formation is actually larger than the classical value. The GT, with the CPHB EOS,
also gave improved results compared to the CNT forms for the nucleation rates of
ethanol and methanol.
A final area of study concerned the novel application of the P-form of CNT
to binary nucleation theory. The goal of this study was to see if using an EOS
72
could eliminate nonphysical behavior found to arise in one of the standard versions
of the theory. To carry out this study, a simple model of a highly surface active
system, with properties similar to ethanol-water or propanol-water mixtures, was
devised and analyzed using DFT. The properties of this model fluid were employed
in conventional calculations with the different version of classical binary theory.
The P-form of the binary CNT failed to significantly improve the predicted critical
activity curve. The other (Doyle) version gave unrealistic results similar to those
found for alcohol-water systems. The DFT results were both realistic and physically
sound, and they obviously constitute a significant improvement over classical binary
nucleation theory.
73
APPENDIX A
IMPORTANT THERMODYNAMIC RELATIONS
74
APPENDIX A
IMPORTANT THERMODYNAMIC RELATIONS
A.1. CHEMICAL POTENTIAL DIFFERENCE IN THE IDEAL GAS
LIMIT
To obtain Eq.(21) in Sec. 2.1, start from the general thermodynamic relation
∂µl
= vl ,
∂P
(119)
where vl is the partial molecular volume. Then apply this relation to an incompressible liquid droplet for which vl is independent of pressure, and integrate Eq.(119)
over the pressure range Pve to Pv to get
µl (Pv ) = µl (Pve ) + vl (Pv − Pve ) .
(120)
∆µ = µl (Pl ) − µl (Pv ) ,
(121)
The definition of ∆µ is
and with the substitution of Eq.(120) this becomes
∆µ = µl (Pl ) − µl (Pve ) − vl (Pv − Pve ) ,
(122)
At unstable equilibrium, we have µl (Pl ) = µv (Pv ), and at bulk two-phase equilibrium, we have µl (Pve ) = µv (Pve ). With these two identities, Eq.(122) can be written
as
∆µ = µv (Pv ) − µv (Pve ) − vl (Pv − Pve ) .
(123)
In the ideal gas limit µv (Pv ) − µv (Pve ) = kT ln S , then ∆µ finally reduces to
∆µ = kT ln S − vl (Pv − Pve ) .
(124)
The term vl (Pv − Pve ) is negligible except at extremely high supersaturation values
S ≥ 106 , which are virtually unattainable.
A.2. THE LAPLACE EQUATION
Assuming a container contains a gas of total volume V , and total number of
molecules N. Assume also a spherical droplet has been formed of volume Vl with
75
radius r with number of molecules nl . The gas will have a chemical potential µv ,
volume Vv = V − Vl , and number of molecules nv = N − nl − ns , where ns is the
surface excess number of molecules.
For simplicity, the Gibbs surface of tension
dividing surface will be adopted[109]. The droplet molecules will have a chemical
potential µl . The total Helmholtz free energy of the system containing the droplet
is
F = −Pv (V − Vl ) − Pl Vl + γA + (N − nl − ns )µv + nl µl + ns µs ,
(125)
at equilibrium, dF = 0, then
−Pv (dV − dVl ) − Pl dVl + γdA + µv (dN − dnl − dns ) + µl dnl + µs dns
−(V − Vl )dPv − Vl dPl + Adγ + (N − nl − ns )dµv + nl dµl + ns dµs = 0
(126)
Using the constraints of constant total volume and total number of molecules, i.e.,
dV = dN = 0, and employing the constant T Gibbs-Duhem identities (Vv dPv =
nv dµv , Vl dPl = nl dµl , Adγ = −ns dµs ), the above equation becomes
(Pv − Pl )dVl + γdA + (µv − µl )dnl + (µv − µs )dns = 0 .
(127)
To maintain equilibrium for an arbitrary virtual variations dnl and dns , the equilibrium condition, µv = µl = µs , must be satisfied. This leaves the remaining equation
(Pv − Pl )dVl + γdA = 0 .
(128)
Assuming the droplet is spherical, we have dVl = 4πr2 dr and dA = 8πrdr, then the
above equation reduces to
Pl − Pv =
2γ
,
r
(129)
which is known as the Laplace equation.
A.3. THE WORK OF FORMATION
Subtract Eq.(12) from Eq.(125) to obtain
∆F = −(Pl − Pv )Vl + γA + nl (µl − µv ) + ns (µs − µv ) ,
(130)
and apply the equilibrium condition, µv = µl = µs , to obtain
∆F Max = −(Pl − Pv )Vl + γA ,
(131)
76
which equals Eq.(15) for a spherical droplet.
A, µs,ns
Liquid
gas
r
vl,µl,nl
V-Vl, µv, N-nl-ns
0
0
Figure A.1. Schematic depiction of a spherical critical nucleus in a
metastable gas phase.
A.4. THE GIBBS-THOMSON EQUATION
For a quick route to the Gibbs-Thomson equation, note that if the droplet is
incompressible then Eq.(119) integrates to
∆µ = µl (Pl ) − µl (Pv ) =
ZPl
vl dP = (Pl − Pv )vl .
(132)
Pv
As we showed in Appendix A.1.,
∆µ = kT ln S
(133)
to a very good approximation. Then if ∆P = Pl − Pv is replaced with Laplace’s
equation, Eq.(129), we get
∆µ = kT ln S =
2γvl
,
r
(134)
which is known as Gibbs-Thomson or Kelvin equation. More general derivations of
Laplace’s formula and Gibbs-Thomson equation are available[109, 110].
77
APPENDIX B
DETAILS OF VARIOUS EQUATIONS OF STATE
78
APPENDIX B
DETAILS OF VARIOUS EQUATIONS OF STATE
B.1. IAPWS-95
The equation formulated by Wagner and Pruss[61] was adopted by The International Association for Properties of Water and Steam, who released it in 1996. Here,
it will be referred to as IAP W S − 95. The equation represents all the thermodynamic properties for water over the range of available experimental data down to
T = 234K. More information about this equation is available at www.IAPWS.org..
Table B.1 gives all the numerical values of the coefficients in the ideal gas part, while
Tables B.2 and B.3 show all the coefficients and parameters of the residual part.
Table B.1. The coefficients values of the ideal gas part.
γ 0i
i
1 −8.32044648201
0.0
5 0.97315 3.53734222
2 6.6832105268
0.0
6 1.27950 7.74073708
3
3.00632
0.0
7 0.96956 9.24437796
4
0.012436
i
n0i
n0i
γ 0i
1.28728967 8 0.24873 27.5075105
79
Table B.2. The coefficients and parameters of the residual part.
i
ci di
ti
ni
1
0 1 −0.5
2
0 1 0.875
3
0 1
1
−0.87803203303561 × 101
4
0 2
0.5
0.31802509345418
5
0 2
0.75
−0.26145533859358
6
0 3 0.375 −0.78199751687981 × 10−2
7
0 4
1
0.88089493102134 × 10−2
8
1 1
4
−0.66856572307965
9
1 1
6
0.20433810950965
10
1 1
12
−0.66212605039687 × 10−4
11
1 2
1
−0.19232721156002
12
1 2
5
−0.25709043003438
13
1 3
4
0.16074868486251
14
1 4
2
−0.40092828925807 × 10−1
15
1 4
13
16
1 5
9
17
1 7
3
18
1 9
4
19
1 10
11
20
1 11
4
21
1 13
13
22
1 15
1
−0.62639586912454 × 10−9
23
2 1
7
−0.10793600908932
24
2 2
1
0.17611491008752 × 10−1
25
2 2
9
0.22132295167546
0.12533547935523 × 10−1
0.78957634722828 × 101
0.39343422603254 × 10−6
−0.75941377088144 × 10−5
0.56250979351888 × 10−3
−0.15608652257135 × 10−4
0.11537996422951 × 10−8
0.36582165144204 × 10−6
−0.13251180074668 × 10−11
80
Table B.2. continued
i
ci di ti
ni
26
2 2 10
−0.40247669763528
27
2 3 10
0.58083399985759
28
2 4
3
29
2 4
7 −0.31358700712549 × 10−1
30
2 4 10
−0.74315929710341
31
2 5 10
0.47807329915480
32
2 6
33
2 6 10
34
2 7 10
35
2 9
1
36
2 9
2 −0.29052336009585 × 10−1
37
2 9
3
38
2 9
4 −0.20393486513704 × 10−1
39
2 9
40
2 10 6
41
2 10 9
42
2 12 8 −0.16388568342530 × 10−4
43
3 3 16
44
3 4 22
45
3 4 23 −0.76788197844621 × 10−1
46
3 5 23
47
4 14 10 −0.62689710414685 × 10−4
6
0.49969146990806 × 10−2
0.20527940895948 × 10−1
−0.13636435110343
0.14180634400617 × 10−1
0.83326504880713 × 10−2
0.38615085574206 × 10−1
8 −0.16554050063734 × 10−2
0.19955571979541 × 10−2
0.15870308324157 × 10−3
0.43613615723811 × 10−1
0.34994005463765 × 10−1
0.22446277332006 × 10−1
48
6 3 50 −0.55711118565645 × 10−9
49
6 6 44
−0.19905718354408
50
6 6 46
0.31777497330738
51
6 6 50
−0.11841182425981
81
Table B.3. The other coefficients and parameters of the residual part.
i
ci
di
ti
ni
52
0.0 3
0
−0.31306260323435 × 102
53
0.0 3
1
54
0.0 3
4
i
ai
αi β i
γi
εi
20 150 1.21 1
0.31546140237781 × 102
20 150 1.21 1
−0.25213154341695 × 104
20 250 1.25 1
ni
Ci Di Ai β i
bi Bi
55
3.5 0.85 0.2
−0.14874640856724
28 700 0.32 0.3
56
3.5 0.95 0.2
0.31806110878444
32 800 0.32 0.3
Some important relations
2
P =ρ
µ
∂f
∂ρ
¶
,
(135)
T
P (δ, τ )
= 1 + δφrδ ,
ρRT
(136)
where δ = ρ/ρc and
φrδ
=
·
∂φr
∂δ
+
+
¸
54
X
i=52
56
X
i=55
=
τ
7
X
ni di δ
di −1 ti
τ +
i=1
51
X
i=8
di ti −αi (δ−εi )2 −β i (τ −γ i )2
ni δ τ e
ni
·
∂δ∆bi ψ
∂δ
¸
ni e−δ
·
ci
¤
£ di ti
δ τ (di − ci δ ci )
¸
di
− 2αi (δ − εi )
δ
,
(137)
with
£
¤ai
,
∆ = θ2 + Bi (δ − 1)2
£
¤ 1
θ = (1 − τ ) + Ai (δ − 1)2 2βi ,
2
ψ = e−Ci (δ−1)
−Di (τ −1)2
.
(138)
(139)
(140)
82
B.2. CREOS-01, -02
The parameters β, α, and ∆i in the CREOS − 01, −02 are the universal criti-
cal exponents, b2 is a universal constant parameter, the scaled functions ψi (θ) are
universal analytical functions of the parametric variable θ, as defined in Section 3.2.
The other parameters, k, d1 , a, and ci are characteristic parameters of the system
of interest. The universal functions, ψi , are given by
Ψ0 (θ)
Ψ1 (θ)
Ψ2 (θ)
Ψ3 (θ)
Ψ4 (θ)
Ψ5 (θ)
·
¸
b2 − 1
2γ − 1
1
1 − 2β
2 2
2 2 2
+ 2β
(1 − b θ ) −
(1 − b θ ) ,
= 4 2β
2b
2−α
γ(1 − α)
α
¸
¸·
·
γ + ∆1
1
2 2
=
− (1 − 2β)b θ ,
2b2 (1 − α + ∆1 ) 2 − α + ∆1
¸
·
¸·
1
γ + ∆2
2 2
=
− (1 − 2β)b θ ,
2b2 (1 − α + ∆2 ) 2 − α + ∆2
¤
1 £
= θ 3 − 2(e0 − β)b2 θ2 + e1 (1 − 2β)b4 θ4 ,
3
¤
1 2 3£
= b θ 1 − e2 (1 − 2β)b2 θ2 ,
3
¤
1 2 3£
= b θ 1 − e4 (1 − 2β)b2 θ2 .
(141)
3
The crossover function, R(q), is defined in the following expression
¶2
µ
q2
,
R(q) = 1 +
1+q
(142)
where the crossover variable, q, is related to the parameter r through
q=
√
rq ,
(143)
with
τ = r(1 − b2 θ2 ) ,
(144)
where τ = T /Tc − 1. The µo (T ), Ao (T ) are analytical functions of temperature and
are given by
83
µo (T ) =
3
X
mj τ j ,
(145)
j=1
Ao (T ) = −Zc +
3
X
Aj τ j ,
(146)
j=1
where Zc is the critical compressibility given by Pc /ρc RTc . Table B.4. shows all
the universal constants, while Table B.5. shows the system dependent parameters
for H2 O and D2 O.
Table B.4. The coefficients of the CREOS equation of state.
α = 0.11
β = 0.325
γ = 2 − α − 2β = 1.24
b2 = γ−2β ∼
= 1.359
γ(1−2β)
f1 = 0.51
∆1 = ∆
f2 = 2∆1 = 1.02
∆2 = ∆
∆3 = ∆4 = γ + β − 1 = 0.565
∆5 = 1.19
f3 = ∆
f4 = ∆3 − 1 = 0.065
∆
2
1
f
∆5 = ∆5 − = 0.69
2
e0 = 2γ + 3β − 1 = 2.455
e1 = (e0 − β) (2e0 − 3) / (e0 − 5β) ∼
= 4.9
e2 = (e0 − 3β) / (e0 − 5β) ∼
= 1.773
e3 = 2 − α + ∆5 = 3.08
e4 = (e3 − 3β) / (e3 − 5β) ∼
= 1.446
84
Table B.5. The coefficients of CREOS-01 and CREOS-02 EOSs.
P arameter
H2 Oa
D2 Ob
k
0.372389
0.319254
d1
0.171848
0.200011
a
192.657
211.969
c1
86.1386
379.046
c2
−2116.6
−265.477
c3
180.877
407.576
c4
−298.053
−458.119
g
9.71434
13.9632
A1
−0.873229 −0.712249
A2
173.177
215.237
A3
32.5782
50.5706
m1
0.88195
1.91944
m2
−110.191
−155.576
m3
−10.2527
−14.7238
The data in the second column have been taken from Kiselev and Ely[62], while
column three has been taken from the same authors[73]. Note that in the second
paper[73], there are misprints in the signs of c1 ,
C2 ,
and c3 .
The critical parameters of the second critical point in the supercooled water are:
Tc = 188K, Pc = 230.MP a, ρc = 1100 Kg.m−3 , while for supercooled heavy water
they are: Tc = 195K, Pc = 230 MP a, ρc = 1220 Kg.m−3 .
85
B.3. JA-EOS
Refer to Sec. 3.3 for the defining equations.
Table B.6. The coefficients and parameters of the JA-EOS.
first term
second term
third term
Pc = 220.5 MP a
Tc = 647.3 K
Tb = 1408.4 K
3
Tf = 273.15K
3
vb = 41.782 cm /mol
ρHB = 0.8447g/cm
λ = 0.3241
C1 = 0.714024
α = 2.145 × vb
C2 = 0.18
b = 1.0823 × vb
σ = 0.168695ρHB
b1 = 0.250810
S0 = −61.468 J/mol.K
b2 = 0.998586
SHB = −5.1278 J/mol.K
∗
b3 = 21.4
b4 = 0.0445238
b5 = 1.01603
HB
= −11490 KJ/mol.
ρc = 1./56 g/cm3
κ = 0.836575
A1 = −12.1637
A2 = 0.228358 × 105
A3 = 13.3273
A4 = −0.0610028
A5 = 1.87317
86
B.4. CPHB EOS
Chen et al.[69] applied the Walsh-Gubbins EOS and simplified it to
Zrep =
v + k1 b
,
v − k2 b
(147)
where b is the hard core volume, and k1 , k2 were correlated to the nonspherical
factor α as
k1 = 4.8319α − 1.5515 ,
(148)
k2 = 1.8177 − .1778α−1.3683 .
(149)
Note that when α = 1, then Eq.(147) gives numerical results equivalent to the
Carnahan-Starling expression[69].
After adding an empirical attractive term to Eq.(147), the CP HB EOS is
RT (1 + k1 b/v)
a
−
,
(150)
v − k2 b
v(v + c)
where a, b, and c are calculated from the conditions defining the critical point
¶
µ
∂P
= 0,
(151)
∂v c
µ 2 ¶
∂ P
= 0,
(152)
∂v 2 c
P =
Pc vc
= Zc .
RTc
From these conditions, a, b, and c are defined as
Ωac R2 Tc2
F1 (T ) ,
Pc
Ωbc RTc
F2 (T ) ,
b =
Pc
Ωcc RTc
,
c =
Pc
a =
(153)
(154)
(155)
(156)
where Ωac , Ωbc , Ωcc are determined through the following equations
Ωcc = 1 + k2 Ωbc − 3ζ c ,
ζ 3 − k1 Ωbc Ωcc
.
Ωac = c
k2 Ωbc
(157)
(158)
87
k2 Ωbc 3+(2k1 k2 +2k22 −3k22 ζ c )Ω2bc +(k1 +k2 −3k1 ζ c −3k2 ζ c +3k2 ζ 2c )Ωbc −ζ 3c = 0 . (159)
The last term of Eq.(159) in the original paper has been mistyped as 3ζ c instead
of ζ 3c . On solving this equation the lowest positive root is used.
The nonspherical factor is given by
α = 1.0003−0.2719M +3.731M 2 −1.0827M 3 +0.1144M 4 −4.1276×10−3 M 5 , (160)
where M = mw ω/39.948 , mw is the molecular weight.
The parameters ζ c , F1 , and F2 are calculated from the following equations
³
h
i2
p ´
(161)
F1 = 1 + C1 1 − TR + C2 (1 − TR ) ,
·
³
´
³
´2
³
´3 ¸2
2/3
2/3
2/3
F2 = 1 + C3 1 − TR
+ C4 1 − TR
+ C5 1 − TR
,
(162)
where C2 and C5 depend on the fluid properties through
¶
µ
¶2
µ
p
Tc
Tc
− 2.0005
+ 5.2614 ωZc ,
(163)
C2 = −1.4671 + 3.6889
αTb
αTb
(164)
.
C5 = 7.9885 − 4.3604eω + 1.4554mw ω 3.063 − 21.395α(ζ c − Zc ) − 4.0692Zc α1.667
The parameter ζ c was correlated through
−4
ζ c = 0.2974 + 0.1123ω − 0.9585ωZc + 7.7731 × 10 mw ω
µ
Tb
Tc
¶
,
(165)
where Tb , Tc , TR (= T /Tc ) are the boiling, critical and reduced temperatures, respectively. Equation (153) defines Zc , the critical compressibility factor. The C1 , C3 ,
and C4 coefficients are given in the following table for water, methanol, and ethanol.
Table B.7. The C parameters for water, ethanol, and methanol of the
CPHB EOS.
Substance
Water
C1
C3
C4
0.28111 2.18987 -2.03823
Methanol -1.73089 4.55337 -7.42214
Ethanol -1.20047 7.36221 -13.2154
Note that in the table in the original paper, C3 , and C4 were mistyped as C2 , and
C3 .
88
The following table contains different properties of water, ethanol, and methanol.
As mentioned in Chapter 3, it is important to use the same critical temperature as
Chen et al. to get reasonable results.
Table B.8. The parameters of the CPHB EOS used for water, ethanol,
and methanol.
Substance Tc (K)
Tb (K) Pc (MP a) mw (g/mol)
ω
647.3
373.2
220.5
18.015
.348
Methanol 512.6
337.8
80.972
32.042
0.559
61.37
46.069
0.6436
Water
Ethanol 512.93 351.443
89
APPENDIX C
PHYSICAL PROPERTIES OF WATER AND HEAVY WATER
90
APPENDIX C
PHYSICAL PROPERTIES OF WATER AND HEAVY WATER
C.1 WATER
Correlations for the surface tension (mN/m) and equilibrium liquid density
(g/cm3 ) are
γ = 93.6635 + 0.009133T − 0.000275T 2 ,
(166)
ρle = 0.08 tanh x + 0.7415t0.33
+ 0.32 ,
r
(167)
where tr = (Tc − T )/Tc is the reduced temperature, and x = (T − 225)/46.2, and Tc
(647.15 K) is the critical temperature of water.
The experimental equilibrium vapor pressure (P a) of water[75], Pve (T ) can be
evaluated from
exp
Pve
(T ) = exp(77.34491 − 7235.42465/T − 8.2 ln T + .0057113T ) .
(168)
C.2 HEAVY WATER
Correlations for the surface tension and equilibrium liquid density are
0
γ = 93.6635 + 0.009133T − 0.000275T
02
,
ρle = 0.09 tanh x + 0.84t0.33
+ 0.338 ,
r
(169)
(170)
where T 0 = 1.022T , tr = (Tc − T )/Tc is the reduced temperature, with x = (T −
231)/51.5, and Tc (643.89K) is the critical temperature of heavy water. Please note
that in the original paper of Wölk and Strey[18], a misprint is found in the formula
of the surface tension for heavy water.
The experimental equilibrium vapor pressure[75], of heavy water can be evaluated
from
exp
Pve
(T )
where
µ
¶
Tc
1.9
2
5.5
10
(α1 τ + α2 τ + α3 τ + α4 τ + α5 τ ) ,
= Pc exp
T
(171)
α1 = −7.81583, α2 = 17.6012, α3 = −18.1747, α4 = −3.92488, α5 = 4.19174, Tc =
643.89, Pc = 21.66MP a, and τ = 1 − T /Tc .
91
APPENDIX D
SURFACE TENSION AND WORK OF FORMATION IN DFT
92
APPENDIX D
SURFACE TENSION AND WORK OF FORMATION IN DFT
D.1 SURFACE TENSION
Start with the thermodynamic definition of the grand potential for a two-phase
system at pressure P , with volume V , and interfacial area A,
Ω = −P V + γA .
(172)
From Eq.(172) with Eqs.(92) and (91), the Aγ(= Ω + P V ) term can be expressed
as
Aγ =
Z
1X
fh (ρ1 , ρ2 )dr+
2
Z
Z
X Z
drdr φij (|r − r |)ρi (r)ρj (r )−
µi ρi (r)dr+ P dr .
0
0
0
(173)
The term involving the attractive potential φij can be eliminated with the help of
Eq.(93). The resulting equation can then be simplified by noting that the density
varies solely in the x direction, perpendicular to the planar interface. The resulting
expression for the surface tension, then reads as
Z
Z
Z
X Z
1X
γ = fh (ρ1 , ρ2 )dx +
ρi (µi − µih )dx −
µi ρi (x)dx + P dx . (174)
2
The fh term can be eliminated using the definition
fh =
X
ρi µih − Ph .
Then, we obtain the somewhat simpler result,
¾
Z ½
1
1
ρ (µ − µ1 ) + ρ2 (µ2h − µ2 ) + (P − Ph ) dx .
γ=
2 1 1h
2
(175)
(176)
Now, for our HSY model fluid, writing Eq.(93) for the flat interface we obtain
d2 µ1h
= λ2 (µ1h (ρ1 , ρ2 ) − µ1 − α11 ρ1 − α12 ρ2 ) ,
dx2
(177)
d2 µ2h
= λ2 (µ2h (ρ1 , ρ2 ) − µ2 − α21 ρ1 − α22 ρ2 ) .
dx2
(178)
and
93
Equations (177) and (178) can be simplified further by using the so-called Bertholet
√
mixing rule (α12 = α11 α22 ). Note that by using this mixing rule, the pressure
EOS, Eq.(96) in Chapter 7, can also be simplified to
√
1 √
P = Ph − ( α11 ρ1 + α22 ρ2 )2 ,
2
(179)
which will be used to replace the terms α11 ρ1 + α12 ρ2 or α21 ρ1 + α22 ρ2 in Eqs.(177)
and (178).
Now multiply Eq.(177) by dµ1h /dx and Eq.(178) by dµ2h /dx and note that
µ
¶2
dµ d2 µih
1 d dµih
= ih
.
(180)
2 dx dx
dx dx2
Equations (177) and (178) can then be written as
µ
¶2
1 d dµ1h
dµ
= λ2 (µ1h (ρ1 , ρ2 ) − µ1 − α11 ρ1 − α12 ρ2 ) 1h ,
2 dx
dx
dx
and
1 d
2 dx
µ
dµ2h
dx
¶2
= λ2 (µ2h (ρ1 , ρ2 ) − µ2 − α21 ρ1 − α22 ρ2 )
dµ2h
.
dx
(181)
(182)
With the help of the Gibbs-Duhem identity
dPh =
X
(183)
ρi dµih ,
Eq.(179), and the differential of the Eq.(102),
√
√
α11 dµ2h = α22 dµ1h ,
(184)
it is easily seen that after integrating both sides of Eq.(181), we obtain
µ
¶2
¤
£
dµ1h
= λ2 (µ1h − µ1 )2 − 2α11 (Ph − P ) .
dx
(185)
Now substitute Eq.(102) into Eq.(176), and use Eq.(177) to further simplify this
equation. Then the expression for the surface tension can finally be written as
Z
ª
©
1
γ=
(186)
(µ1h − µ)2 − 2α11 (Ph − P ) dx ,
α11
or even more simply as,
1
γ= 2
λ α11
Z µ
dµ1h
dx
¶2
1
dx = 2
λ α22
Z µ
dµ2h
dx
¶2
dx ,
(187)
94
with the use of Eq.(185), or the similar pair of equations with i = 2. These specific
forms still require detailed knowledge of the structure of the interface to complete
their evaluation.
To avoid this, Eq.(187) can be transformed by changing the
independent variable from x to one of the densities by noting that
µ
¶2
¶
¶
µ
µ
dµ1h
dµ1h
dµ1h dµ1
dµ1 =
dx =
dρ .
dx
dx
dx
dρ1 1
(188)
The final formula may then be cast in terms of either component 1 or 2 as
µ
¶
Z ρil
¤1/2 dµih
£
1
2
dρi , i = 1 or 2 .
(µih (ρ1 , ρ2 ) − µi ) − 2αii (Ph − P )
γ=
λαii ρiv
dρi
(189)
D.2 WORK OF FORMATION
The reversible work is defined as the difference in the grand potential between
the initial uniform system and the final system containing a droplet,
Wrev = Ω (ρ1 , ρ2 ) − Ω0 (ρ1b , ρ2b ) ,
(190)
where ρ1b , ρ2b are the densities of the uniform system. For the initial system, we
have Ω0 (ρ1b , ρ2b ) = −P V . For the nonuniform system, we have
¾
½
Z
X
1X
ρi (µi − µih ) ,
µi ρi +
Ω = dr fh (ρ1 , ρ2 ) −
2
where, as usual,
fh =
X
ρi µih − Ph .
(191)
(192)
Substitute Eq.(192) into Eq.(191), and assume the droplet is spherical, then Eq.(190)
becomes
Wrev = 4π
Z ½
¾
1
1
ρ (r) [µ1h (ρ1, ρ2) − µ1 ] + ρ2 (r) [µ2h (ρ1 , ρ2 ) − µ2 ] + P − Ph r2 dr .
2 1
2
(193)
95
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100
VITA
My name is Abdalla Ahmad Obeidat; I was born on December 4, 1970 in Youbla,
Jordan. I graduated from Koforsoom High School in 1988. I received my Bachelor
degree in Physics with minor in Mathematics in May of 1992 from Yarmouk University. In August of 1992, I began my graduate work in the same university, and I
received my Masters in the field of Magnetism from Physics department by October
of 1994.
In January of 1995, I start teaching undergraduate labs at Jordan University of
Science and Technology till the end of 1997.
By August of 1998, I started working toward my Doctor of Philosophy degree in
Physics at University of Missouri-Rolla. I worked one year as a teaching assistant,
while I was working as a research assistant for Dr. Wilemski that involves Unary
and Binary Nucleation. In December of 2003, I was awarded a Doctor of Philosophy
degree in Physics.