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Surface tension and computer simulation of polyatomic fluids

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Surface tension and computer simulation of polyatomic fluids
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Haile, James Mitchell, 1946-
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English
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xxxi, 376 leaves : ill. ; 28 cm.

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Fluids ( jstor )
Interfacial tension ( jstor )
Liquids ( jstor )
Mathematical tables ( jstor )
Molecular dynamics ( jstor )
Molecules ( jstor )
Perturbation theory ( jstor )
Quadrupoles ( jstor )
Simulations ( jstor )
Spherical harmonics ( jstor )
Liquids ( lcsh )
Surface tension ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis--University of Florida.
Bibliography:
Includes bibliographical references (leaves 357-375).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by James Mitchell Haile.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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SURFACE TENSION AND COMPUTER SIMULATION
OF POLYATOMIC FLUIDS










By

JAMES MITCHELL HAILE


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY







UNIVERSITY OF FLORIDA


1976




SURFACE TENSION AND COMPUTER SIMULATION
OF POLYATOMIC FLUIDS
By
JAMES MITCHELL HAILE
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1976


ACKNOWLEDGEMENTS
It is a pleasure to express my gratitude to those who have freely
contributed to this work through their instruction, guidance, and advice.
Keith Gubbins initiated the research reported herein and enthu
siastically stimulated and supported its development, as well as my own
professional growth. Dr. Gubbins consistently provided an extraordinary
environment for learning, provided a strong example of the scientific
method, and exposed me to numerous knowledgeable scientists and engineers
on both sides of the Atlantic. In addition, he expended considerable
effort in obtaining the financial support, travel and computer funds
which made this work possible.
Bill Streett generously allowed me to spend several months in
his laboratory at the U.S. Military Academy and patiently taught me
molecular dynamics. He obtained the copious amounts of computer time
used in the molecular dynamics work reported here and kept the program
running in my absence. Many of the ideas for presenting the molecular
dynamics results came to light in discussions with Colonel Streett.
Further, I am grateful to Colonel and Mrs. Streett for the hospitality
extended to me during my visits to West Point.
John O'Connell, University of Florida, continually inspired me
through open-ended questioning concerning classical and statistical
thermodynamics, science, engineering, and, most importantly, the
character of life.
ii


Chris Gray, University of Guelph, instructed me in spherical
trigonometry, spherical harmonic expansions, Racah algebra, etc.,
thereby developing in me a healthy respect for the physicist's view
of applied science.
I have benefited greatly from countless discussions with my
colleague Chorng-Horng Twu on various aspects of thermodynamics,
statistical mechanics, numerical methods, and Chinese cooking.
S5ren Toxvaerd, University of Copenhagen, contributed much
valuable advice on the theory and associated calculations for fluid
interfaces. Thanks are also due Dr. Toxvaerd for providing a copy
of his computer program for calculating the vapor-liquid interfacial
density profile for Lennard-Jones fluids.
Peter Egelstaff allowed me to spend several months in the
stimulating atmosphere of the Physics Department at the University
of Guelph. I am grateful to the faculty and staff for their hospi
tality and for the large amount of NOVA 2 computer time made available
to me. I am especially thankful to Dan Litchinsky for useful advice
on NOVA 2 software and to Ross McPherson for timely hardware support
on the NOVA. I am also indebted to Shien-Shion Wang for spending many
hours in teaching me the Monte Carlo method.
Dick Dale and Ron Franklin of the Engineering Information Office,
University of Florida, gave timely and enthusiastic photographic tech
nical assistance in producing the filmed animation of molecular dynamics
simulations. Larry Mixon in the Northeast Regional Data Center, Univer
sity of Florida, provided valuable software support in developing the
filmed animation technique.
iii


I am grateful to Dr. J. W. Dufty for serving on the supervisory
committee. I would also like to remember Dr. T. M. Reed who was an
original member of the committee and who strongly encouraged me in the
initial phases of the research reported here.
P.S.Y. Cheung, T. Keyes, C. G. Gray and R. L. Henderson, W. B. Streett,
and S. Toxvaerd kindly provided manuscripts of their work prior to publica
tion.
Mrs. J. Ojeda, University of Florida, performed the remarkably
excellent typing of the manuscript.
Finally, I am grateful to Tricia who, in addition to all the
\
usual annoyances with which wives of Ph.D. students are plagued,
quietly endured our being separated for the greater part of the last
year and a half of this work.
The three dimensional drawings presented in Chapter 7 were done
on a Gould 5100 electrostatic plotter driven by the IBM 370/165 at the
Northeast Regional Data Center, University of Florida. The associated
software was the SYMVU Computer Graphics Program, Version 1.0, of the
Laboratory for Computer Graphics and Spatial Analysis, Harvard University.
I thank the Petroleum Research Fund (administered by the American
Chemical Society) and the National Science Foundation for financial
support of this study.
iv


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
LIST OF TABLES ix
LIST OF FIGURES xv
KEY TO SYMBOLS xxii
ABSTRACT xxix
CHAPTERS:
1 INTRODUCTION 1
1.1 Theory of Surface Properties 6
1.2 Computer Simulation Methods 9
1.3 Outline of Dissertation 10
2 THEORY OF SURFACE TENSION 13
2.1 General Expressions for Surface Tension of
Polyatomic Fluids 13
2.2 General First Order Perturbation Theory for
Surface Tension 18
2.3 Perturbation Theory for Surface Tension using
a Pople Reference 23
2.4 Fowler Model Expressions for Perturbation Terms
Y2A Y2B Y3A Y3B in PPle ExPansion 29
2.5 Superficial Excess Internal Energy from the
Pad Perturbation Theory for Surface Tension 32
3 NUMERICAL CALCULATIONS OF SURFACE TENSION 34
F F F F
3.1 Evaluation of Y2A, Y3g> Y3A> and y3b 35
3.2 Surface Tension Calculations for Model Fluids.... 40
v


TABLE OF CONTENTS (Continued)
CHAPTERS: Page
3.3 Calculation of the Superficial Excess Internal
Energy for Model Fluids 46
3.4 Surface Tension Calculations for Real Fluids 48
3.5 Correlation of Surface Tension for Pure Poly
atomic Liquids.... 58
4 VAPOR-LIQUID DENSITY-ORIENTATION PROFILES 72
4.1 First Order Perturbation Theory for p(z.U)^) 72
4.2 Calculations of p(z^co^) for Overlap and Dis
persion 78
5
6
MONTE CARLO SIMULATION OF MOLECULAR FLUIDS ON. A
MINICOMPUTER 94
5.1 Introduction 94
5.2 Monte Carlo Method for Nonspherical Molecules 96
5.3 Description of the Minicomputer System 101
5.4 Monte Carlo Program for the NOVA 102
5.5 Comparison of NOVA Results with Full-Size
Computer Results 107
5.6 Conclusions '. 112
MOLECULAR DYNAMICS METHOD FOR AXIALLY SYMMETRIC
MOLECULES 114
6.1Introduction 114
6.2Expressions for the Force and Torque for Axially
Symmetric Molecules 117
6.3 Method of Solution of the Equations of Motion
and the Molecular Dynamics Algorithm 129
6.4 Evaluation of Pair Correlation Functions 136
6.5Equilibrium Properties from the g^ ^ n/r12^
146


TABLE OF CONTENTS (Continued)
CHAPTERS: Page
7 MOLECULAR DYNAMICS RESULTS 158
7.1 Potential Models 158
7.2 Equilibrium Properties 171
7.3 Spherical Harmonic Coefficients, g (r,) 191
7.4 Angular Pair Correlation Function 215
7.5 Site-Site Pair Correlation Functions 238
7.6 Filmed Animation of Molecular Motions 250
8 CONCLUSIONS 259
8.1 Theory for Surface Tension of Polyatomic Fluids... 259
8.2 Theory for the Interfacial Density-Orientation
Profile of Polyatomic Fluids 261
8.3Computer Simulation of Polyatomic Fluids 262
APPENDICES:
A EXPRESSIONS FOR THE ANGLE AVERAGES IN EQUATIONS (3-4)
TO (3-7) 267
B COORDINATE TRANSFORMATION AND INTEGRATION OVER EULER
ANGLES TO OBTAIN EQUATIONS (2-89) AND (2-90) 270
B.l Choice of Euler Angles 270
B.2 Evaluation of Integral I 273
C MODELS FOR ANISOTROPIC POTENTIALS OF LINEAR MOLECULES... 277
D EXPRESSIONS FOR y^, Y^g. AND y^g FOR VARIOUS
ANISOTROPIC POTENTIALS FOR AXIALLY SYMMETRIC MOLECULES.. 283
E THE INTEGRALS KY(U';nn'n") AND LY(£;nn') 288
F EXPRESSIONS FOR THE SPHERICAL HARMONIC COEFFICIENTS
g£ £ m(r12> IN EQUATION (6-77) 292
G THE INTEGRAL I USED TO CALCULATE THE ANGULAR COR
RELATION PARAMETER G2 FOR QUADRUPOLES 294
vii


TABLE OF CONTENTS (Continued)
APPENDICES: Page
H VALUES FOR THE g (r. J COEFFICIENTS 297
1 zm ^
I VALUES FOR THE J INTEGRALS 328
n
J VALUES OF THE SITE-SITE CORRELATION FUNCTIONS 349
K VALUES OF THE INTEGRAL H^2,6^ 335
LITERATURE CITED 357
BIBLIOGRAPHY 365
BIOGRAPHICAL SKETCH 376
viii


2
3
4
5
6
7
8
9
10
11
12
13
Page
5
47
57
62
100
106
109
150
151
152
153
154
155
LIST OF TABLES
Examples of Macroscopic and Microscopic Interfacial
Properties Related by Equations of the Form (1-1)
Test of the Gibbs-Helmholtz Equation in the Fowler
Model Perturbation Theory for Lennard-Jones plus
Quadrupole Fluids
Potential Parameter Values used in Calculating Surface
Tension.
Values for the Parameters a^ and a2 in the Surface
Tension Correlation of Equation (3-61)
Equilibrium Properties in the Form of Ensemble Averages...
Approximate Number of Monte Carlo Configurations
Generated per Hour on the NOVA 2
Comparison of NOVA and CDC Results for Property Values
of Lennard-Jones + Quadrupole Model Fluid. kT/e = 0.719,
pa3 = 0.80, Q/(ea5)!/2 = 1
Expressions for in Terms of g (r,J
a J^a)^ ^1 zm 1Z
for Various Model Potentials
Expressions
8l1H2(r12)
for the Configurational Energy in Terms
for Various Model Potentials
Expressions for the Pressure in Terms of g^ ^ (r-^)
Various Model Potentials 1.2
of
for
Expressions for the Fowler Model Surface Tension in
Terms of g0 (r,) for Various Model Potentials
Expressions for the Fowler Model Surface Excess Internal
Energy in Terms of g (r ) for Various Model
Potentials 1.2?
Expressions for the Mean Squared Force in Terms of
g (r,) for Various Model Potentials
a/. JO-ID J.Z


Table
Page
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Expressions for the Mean Squared Torque in Terms of
g £ m^r12^ ^r ^ar*ous Mdel Potentials 156
Expressions for the Angular Correlation Functions in
Terms of g^ ^ m^r12^ ^or ^ar^ous Model Potentials 157
Primary Orientations for Pairs of Linear Molecules 162
Property Values of a Lennard-Jones plus Quadrupole
Fluid Obtained in this Work and Compared with those
given by Berne and Harp 177
Equilibrium Properties for Lennard-Jones plus Quadru
pole Fluid at pa3 = 0.85, Q/(ea3)-*-/2 = 2/2 181
Equilibrium Properties for Lennard-Jones plus Quadru
pole Fluid at po3 = .931, Q/(ea5)1/2 = 0.707 182
Equilibrium Properties for Lennard-Jones plus Quadru
pole Fluid at po3 = 0.85, Q/(ea5)1/2 = 1.0 183
Anisotropic Contributions to Equilibrium Properties for
Lennard-Jones plus Quadrupole Fluid at pa3 = 0.85,
0 /(ea5)1/2 = 1/2 188
Anisotropic Contributions to Equilibrium Properties for
Lennard-Jones plus Quadrupole Fluid at pa3 = .931,
Q/iec5)1/2 = 0.707 189
Anisotropic Contributions to Equilibrium Properties for
Lennard-Jones plus Quadrupole Fluid at pa3 = 0.85,
Q/( ea5)1/2 = 1.0 190
Equilibrium Properties for Lennard-Jones plus Overlap
Fluid at pa3 = 0.85, = 0.10 192
Equilibrium Properties for Lennard-Jones plus Overlap
Fluid at pa3 = 0.85, = 0.30 193
Effect of Potential Model and State Condition on the
Fii'st Peak Height of the g Coefficients 201
l zm
Comparison of Molecular Dynamics Results for Equilibrium
Properties with Values Obtained from
£^£^1
J Integrals for Lennard-Jones plus Quadrupole Fluid
with po3 = 0.85, kT/e = 1.277 Q/(ea5)]/2 = 0.5 210
x


Table
Page
28 Comparison of Molecular Dynamics Results for Equilib
rium Properties with Values Obtained from
J Integrals for Lennard-Jones plus Quadrupole
Fluid with pa3 = 0.85, kT/e = 1.294, Q/(ea5)1^ = 1.0... 211
29 Comparison of Molecular Dynamics Results for Equilib
rium Properties with Values Obtained from
l^m
J Integrals for Lennard-Jones plus Overlap Fluid
with po3 = 0.85, kT/e = 1.291, 6 = 0.10 212
30 Comparison of Molecular Dynamics Results for Equilib
rium Properties with Values Obtained from
J Integrals for Lennard-Jones plus Overlap Fluid
with po3 = 0.85, kT/e = 1.287, 6 = 0.30 213
31 Range of Values for Orientational Contributions to
Property Integrands for Quadrupole and Overlap Fluids... 243
Cl Expressions for the Expansion Coefficients E for
Various Interaction Potentials for Linear Molecules 279
C2
C3
C4
D1
D2
D3
D4
El
E2
Expressions for Anisotropic Potential Models in the
Intermolecular Frame of Figure 32 280
Expressions for Anisotropic Potential Models in the
Intermolecular Frame, using y rather than
Derivatives of Various Anisotropic Potentials for
Evaluating the Force and Torque from Equations (6-25)
and (6-34)
281
282
y
Expressions for yj: for Various Anisotropic Potentials
for Linear Molecules 284
F
Expressions for y^ for Multipole Potentials for
Linear Molecules 285
F
Expressions for yB for Various Anisotropic Potentials
for Axially Symmetric Molecules 286
F
Expressions for y^B for Multipole Potentials for
Linear Molecules 287
The Integrals K^(££'£";nn'n") for Pure Fluids 289
The Integrals L^(£;nn') for Pure Fluids 290
xi


Table
Page
E3 The Constants in Equation El 291
G1 The Integral I ^ for Pure Fluids 296
HI Values of goOO^r12^ ~ 400^r12^ fr Fennard-dones plus
Quadrupole Fluid at kT/e = 1.277, pPd = 0.85,
Q/( ea5)1/2 = 0.5 298
H2 Values of g420^r12^ ~ §442^r12^ for the Fluid of
Table HI 300
H3 Values of 8443(1^) g660^r12^ for the Fluid of
Table HI 302
H4 Values of g000^r12^ 4Q0^r12^ fr Lennard-Jones plus
Quadrupole Fluid with kT/e = 0.765, pad = 0.931, and
Q/(eo5 d/2 = 0.707 304
H5 Values of g420^r12^ g442^r12^ for the Fluid of
Table H4 306
H6 Values of g^Cr^) 8660^12^ for the Fluid of
Table H4 308
H7 Values of g000^r12^ g400^r12^ fr Lennard-Jones plus
Quadrupole Fluid with kT/e = 1.294, pad = 0.85, and
Q/(eo5)1/2 = 1.0 310
H8 Values of g420^r12^ g442^r12^ fr t*ie Fdudd f
Table H7 312
H9 Values of 8443(^2) 8640^12^ for the Fluid of
Table H7 314
H10 Values of goo0^r12^ g400^r12^ fr Lennard-Jones plus
Anisotropic Overlap Fluid with kT/e = 1.291, pa^ = 0.85,
and = 0.10 316
Hll Values of g420^r12^ g442^r12^ fr t^ie Fdudd f
Table H10 318
H12 Values of 8443(^2) g660^r12^ for the Fluid of
Table H10 320
H13 Values of gQ00^r12^ ~ g400^r12^ fr Lennard-Jones plus
Anisotropic Overlap Fluid with kT/e = 1.287, pad = 0.85,
and 6 = 0.30 322
xii


Table
Page
H14
Values of g49n(r19)
Table H13........7.
- §442^r12^ for t*ie FFuid oF
324
H15
Values of g,,.(r J
Table H13........7.
- §660^12^ for the Fluid of
11
The Integrals J^
Quadrupole Fluid.
Q/Ceo5)1/2 0.5...
222
- J for a Lennard-Jones plus
pa3n= 0.85, kT/e = 1.277,
12
The Integrals J^^
n
440
- J for the Fluid of Table 11...
n
330
13
441
The Integrals J
n
- J^O for the Fluid of Table 11...
n
331
14
620
The Integrals J^
- J^ 0 for the Fluid of Table 11...
n
332
15
The Integrals J^^
Quadrupole Fluid.
QCea5)1/2 0.707..
222
- J for a Lennard-Jones plus
pa3n= 0.931, kT/e = 0.765,
16 The Integrals J^*3 for the Fluid of Table 15 334
n n
17 The Integrals J^^ J^^ for the Fluid of Table 15 335
n n
18 The Integrals J^ for the Fluid of Table 15 336
n n
000 222
19 The Integrals J J for a Lennard-Jones plus
Quadrupole Flui§. pa^n= 0.85, kT/e = 1.294,
Q/Ceo5)1/2 = 1.0 337
400 440
110 The Integrals J J for the Fluid of Table 19 338
n n
111 The Integrals for the Fluid of Table 19 339
n n
112 The Integrals J^ for the Fluid of Table 19 340
n n
000 222
113 The Integrals J J for a Lennard-Jones plus
Anisotropic Overlap Fluid. pp3 = 0.85, kT/e = 1.291,
6 = 0.10 341
400 440
114 The Integrals J J for the Fluid of Table 113.... 342
n n
115 The Integrals J^^ for the Fluid of Table 113.... 343
n n
116 The Integrals J^ for the Fluid of Table 113.... 344
0 n n
000 222
117 The Integrals J J for a Lennard-Jones plus
Anisotropic Overlap Fluid. p3 = 0.85, kT/e = 1.287,
= 0.30 345
xiii


Table
Page
400 440
118 The Integrals J J for the Fluid of Table 117.... 346
n n
119 The Integrals J441 J6 for the Fluid of Table 117.... 347
n n
120 The Integrals J^ j^O for Fluid Qf Table 117.... 348
n n
J1 Site-Site Correlation Function for Lennard-Jones plus
Quadrupole Fluid with Z/o = 0.3292, kT/e = 1.277,
pa3 = 0.85, Q/(CO5)1/2 = 0.5 350
J2 Site-Site Correlation Function for Lennard-Jones plus
Quadrupole Fluid with i/a = 0.2955, kT/e = 0.765,
pa3 = 0.931, Q/(eo5)l/2 = 0.707 351
J3 Site-Site Correlation Function for Lennard-Jones plus
Quadrupole Fluid with i/o = 0.3292, kT/e = 1.294,
pa3 = 0.85, Q/(ea3)l/2 = i.o 352
J4 Site-Site Correlation Function for Lennard-Jones plus
Anisotropic Overlap Fluid with £/o = 0.3292, kT/e = 1.287,
pa3 = 0.85, 6 = 0.10 353
J5 Site-Site Correlation Function for Lennard-Jones plus
Anisotropic Overlap Fluid with i/o 0.3292, kT/e = 1.287,
po3 = 0.85, = 0.30 354
K1 The Integral for pure Fluids 356
xiv


LIST OF FIGURES
Figure Page
1 Relation of Theory, Experiment, and Computer Simula
tion in the Study of Liquids 2
2 Variation of Fluid Density with Position through a
Planar Vapor-Liquid Interface 4
3 Two Possible Values for the Maximum Zj Value for a Pair
of Molecules in the Fowler Model Interface 22
4 Fowler Model Surface Tension for a Fluid of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Dipole Model Potential. pa^ = 0.85,
kT/e = 1.273 42
5 Fowler Model Surface Tension for a Fluid of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Dipole Model Potential. pa^ = 0.45,
kT/e = 2.934 43
6 Fowler Model Surface Tension for a Fluid of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Quadrupole Model Potential. pa^ = 0.85,
kT/e = 1.273 44
7 Fowler Model Surface Tension for Fluids of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Various Anisotropic Potentials. pc?3 = 0.85,
kT/e = 1.273 45
8 Corresponding States Plot for Surface Tension of
Simple Liquids 52
9 Surface Tension for CO2 Comparing Perturbation Theory
Calculations with Experimental Values 55
10 Surface Tensions for C2H2 and HBr Comparing Perturba
tion Theory Calculations with Experimental Values 56
11 Test of Surface Tension Correlation for CO2 64
12 Test of Surface Tension Correlation for Acetic Acid 65
13 Test of Surface Tension Correlation for Methanol 66
xv


Figure Page
14 Comparison of Surface Tensions Calculated from the
Correlation with Experimental Values for Several
Polyatomic Liquids 68
15 Comparison of Surface Tensions Calculated from the
Correlation with Experimental Values for Several
Polyatomic Liquids 69
16 Test of Surface Tension Correlation for n-Hexane and
n-Octane 70
17 Comparison of Surface Tensions Calculated from the
Correlation with Experimental Values for Several
Hydrocarbons 71
18 Interfacial Density Profile for Lennard-Jones Fluid 80
19 Interfacial Density-Orientation Surface for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Dispersion Model Potential. kT/e = 0.85,
K = 0.25 83
20 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Dispersion Model Potential. kT/e = 0.85,
K = 0.25 84
21 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Dispersion Model Potential. kT/e = 0.85,
K = 0.25 85
22 Difference in Normal and Tangential Components of Stress
Tensor for Lennard-Jones Fluid 87
23 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Anisotropic Overlap Model Potential.
kT/e = 0.85, = 0.10 89
24 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Anisotropic Overlap Model Potential.
kT/e = 0.85, 6 = 0.10 90
25 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Anisotropic Overlap Model Potential.
kT/e = 0.85, 6 = -0.10 91
xv i


Figure Page
26 Interfacial Density-Orientation Profiles for a Fluid
of Axially Symmetric Molecules Interacting with Lennard-
Jones plus Anisotropic Overlap Model Potential.
kT/e = 0.85, 6 = -0.10 92
27 Simplified Schematic Flow Diagram of Fortran Monte
Carlo Program Developed for NOVA 2 2.05
28 Comparison of CDC and NOVA 2 Monte Carlo Results for
the Center-Center Pair Correlation Function for a
Lennard-Jones plus Quadrupole Fluid 210
29 Comparison of CDC and NOVA 2 Monte Carlo Results for
the Angular Pair Correlation Function for a Lennard-
Jones plus Quadrupole Fluid for Molecular Pairs in
the Tee Orientation Ill
30 Methods of Specifying the Orientation of an Axially
Symmetric Molecule 119
31 Orientation Angles for Axially Symmetric Molecules in
an Arbitrary Space Fixed Frame 121
32 Orientation Angles for Axially Symmetric Molecules in
the Intermolecular Frame 122
33 Geometry of a Pair of Diatomic Molecules 142
34 Pair Potential for Lennard-Jones plus Dipole Model
Fluid at Primary Pair Orientations 161
35 Pair Potential for Lennard-Jones plus Quadrupole Model
Fluid at Primary Pair Orientations 165
36 Surface of the Lennard-Jones plus Quadrupole Pair
Potential for the Tee Orientation as a Function of
the Quadrupole Strength 1 166
37 Pair Potential for Lennard-Jones plus Dipole, Dipole-
Quadrupole, and Quadrupole Model Fluid at Primary Pair
Orientations. p/(e3)l/2 = 1.0, Q/(ea3)!/2 = 2.75 167
38 Pair Potential for Lennard-Jones plus Dipole, Dipole-
Quadrupole, and Quadrupole Model Fluid at Primary Pair
Orientations. y/(ea3)l/2 = 1.75, Q/(ea3)1/2 = 1.0 168
39 Pair Potential for Lennard-Jones plus Anisotropic
Overlap Model Fluid at Primary Pair Orientations 270
xvii


Figure
Page
40
41
42
43
44
45
46
47
48
49
50
51
52
53
Mean-Squared Displacement of Molecular Centers of
Mass for Lennard-Jones plus Quadrupole Fluid
Fluctuation in Temperature for Lennard-Jones plus
Quadrupole Fluid
Fluctuation in the Ratio of Translational to Rotational
Kinetic Energy for Lennard-Jones plus Quadrupole Fluid 176
Effect of Quadrupole Moment on the Center-Center Pair
Correlation Function at pa3 = 0.85 194
Spherical Harmonic Coefficients g for Lennard-Jones
q y^2m
plus Quadrupole Fluid at po = 0.85, kT/e = 1.294,
Q/(ea5)1/2 = 1.0 196
Spherical Harmonic Coefficients g/n for the Fluid of
. . 4 X/ oITl
Figure 44 4 197
Spherical Harmonic Coefficients g,. for the Fluid of
,, 44m
Figure 44 198
Spherical Harmonic Coefficients g for the Fluid of
Figure 44 2 199
Effect of Anisotropic Overlap Parameter on the Center-
Center Pair Correlation Function at pa3 = 0.85 203
Spherical Harmonic Coefficients gn for Lennard-Jones
2.)
plus Anisotropic Overlap Fluid at pa3 = 0.85, kT/e = 1.287,
6 = 0.30 204
Spherical Harmonic Coefficients g. for the Fluid of
Figure 49 ?T 205
Spherical Harmonic Coefficients g,, for the Fluid of
Figure 49 T 206
Spherical Harmonic Coefficients g,. for the Fluid of
Figure 49 .2 207
2
Integrands tg22C)(r) ~ 2g221^ + 28222^r^r and
[g220^ + 4/38221^ + 1/38222^r^r_3 for G2 and Ua
respectively, for Lennard-Jones plus Quadrupole Fluid 214
xviii


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56
57
58
59
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61
62
63
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65
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217
218
220
222
223
226
227
228
229
231
232
234
Angular Pair Correlation Function for the Lennard-Jones
plus Quadrupole Fluid of Figure 44 for the Tee Orienta
tion (0^ = 90, 02 = 0,

Angular Pair Correlation Function for the Lennard-Jones
plus Quadrupole Fluid of Figure 44 for the Cross and
Parallel Orientations (0^ = = 90)
Angular Pair Correlation Function for the Lennard-Jones
plus Quadrupole Fluid of Figure 44 for a Skewed Orienta
tion (0^ = @2 = Angular Pair Correlation Function for the Lennard-Jones
plus Anisotropic Overlap Fluid of Figure 49 for the Tee
Orientation (0^ = 90, 2 = 0,

Angular Pair Correlation Function for the Lennard-Jones
plus Anisotropic Overlap Fluid of Figure 49 for the Endon
Orientation (0^ = @2 =

Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 0 = 90, = 0,
with Q* = 0.5, T* = 1.277, p* = .85 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 0 = 90,

with Q* = 1.0, T* = 1.294, p* = .85 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones .plus Quadrupole Fluid for 0.. = 90, 0 = 90,
with Q* = 0.5, T* = 1.277, p* = .85 7 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 01 = 90, 0 = 90,
with Q* = 1.0, T* = 1.294, p* = .85 7 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 0, = 45, 0 = 45,
with Q* = 0.5, T* = 1.277, p* = .85 7 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 0, =45, 0 = 45,
with Q* = 1.0, T* = 1.294, p* = .85 7 7
Comparison of Peak Heights in the Angular Pair Correla
tion Function with Well Depths in the Pair Potential
for the Lennard-Jones plus Quadrupole Fluid with
Q/Cea5)1/2 = 1.0
XIX


Figure Page
66 Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Anisotropic Overlap Fluid for
0X 90, 67 Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Anisotropic Overlap Fluid for
= 90,

o
68 Integrand for Internal Energy, r u(12)g(12), for
Lennard-Jones plus Quadrupole Fluids for the Tee
Orientation 239
69 Integrand for Pressure, r^ g(12), for Lennard-
Jones plus Quadrupole Fluids for the Tee Orientation 240
2
70 Integrand for Internal Energy, r u(12)g(12), for Lennard-
Jones plus Anisotropic Overlap Fluids for Parallel
Orientation 241
71Integrand for Pressure, r^ ^ g(12), for Lennard-
Jones plus Anisotropic Overlap Fluids for Parallel
Orientation 242
72 Site-Site Pair Correlation Function for Lennard-Jones
plus Quadrupole Fluids 245
73 Possible Square Packing of Lennard-Jones plus Quadru
pole Molecules for Interpreting gag(r) 247
74 Site-Site Pair Correlation Function for Lennard-Jones
plus Anisotropic Overlap Fluids 249
75Box Representing the Molecular Dynamics System with the
Volume Element Sampled for the Filmed Animation Indicated. 253
76 Initial FCC Lattice Configuration of Lennard-Jones
Molecules in the Volume Element Sampled in the Filmed
Animation 256
77 Frame from the Filmed Animation of Lennard-Jones Mole
cules Corresponding to the Sixth Time-Step in the
Molecular Dynamics Calculation 257
78 Frame from the Filmed Animation of Lennard-Jones Mole
cules Corresponding to the 101st Time-Step in the
Molecular Dynamics Calculation 258
xx


Page
Figure
B1
Rotations
Defining the Euler Angles
40x1
271
B2
Rotations
Interface
in the Triangle 123 in the
to Define Values for z
max
Fowler Model
... 274
xxi


KEY TO SYMBOLS
c(£1
A
. c
A.
x
A(ziz12)
B(ZlZl2)
D
£
i
mn
£*
mn
Fi
F2A,F2B
F3AF3B
n
Roman Upper Case
Helmholtz free energy
Configurational Helmholtz free energy
The ith term in the perturbation expansion for
Helmholtz free energy
Function defined in Equation (4-26)
Function defined in Equation (4-27)
Residual contribution to constant volume heat capacity
Clebsch-Gordan coefficient
Diffusion coefficient
Representation coefficient
Complex conjugate of representation coefficient
Force on molecule 1
Function defined in Equation (2-32)
Functions defined in Equation (2-62) and (2-63),
respectively
Functions defined in Equations (2-77) and (2-78),
respectively
Angular correlation parameter
Integral defined in Equation (3-34)
xxii


I =
I.(r)
^nn' Z
K =
K
KY
L =
L =
LY =
N =
"
P =
P.
l
P(x)
o(i)
Q
*
T
Moment of inertia
Functions defined in Equations (6-65) and (6-69)
Integral defined in Equations (2-40) and (B7)
Integral defined in Equation (Gl)
Integral defined in Equation (3-16)
Integral defined in Equation (6-89)
Coefficient of ellipticity for plane polarized light
Integral defined in Equation (3-25)
Integral defined in Equation (3-18)
Functions defined in Equations (4-30) and (4-31),
respectively
Angular momentum operator
Integral defined in Equation (3-24)
Integral defined in Equation (3-17)
Number of molecules
Avogadro's number
Pressure
t h
The i component of the local polarization vector
Legendre polynomial of order i
Probability of the i^ state occurring, defined in
Equation (5-8)
Quadrupole moment
5 1/2
Reduced quadrupole moment = Q/(ea )
General multipole moment
Temperature
Critical temperature
xxiii


tr
*
T
= Reduced temperature, T/T^
= Reduced temperature, kT/e
U
s
= Superficial excess internal energy
V
= Volume
V
c
= Critical volume
Y£m
*
Y*m :
= Spherical harmonic
= Complex conjugate of
Z
= Configurational integral
Roman Lower Case
a. =
i
= Semiempirical parameters defined in Equations (3-50)
to (3-53)
c =
= Constant defined in Equations (4-21) and (4-22)
c =
= Cosine of angle d). .
ij
c. =
1
= Cosine of angle 0^
c(Y) =
= Cosine of angle y
c(zlz2r12) =
= Interfacial direct correlation function
d =
5 Hard sphere diameter defined in Equation (3-39)
d,dft
5 Maximum allowable step lengths for translational and
rotational motion, respectively, of molecules in one
Monte Carlo generated configuration
£ '
Interfacial pair distribution function for spherical
molecules
f(zl£l2£l3)
Interfacial triplet distribution function for spherical
molecules
g(r12) =
Radial distribution function
xxiv


gc(rl2)
= Center-center pair correlation function
gaB^r12)
= Site-site pair correlation function
g(zlz2r12)
= Interfacial pair correlation function
g(r12wla)2)
= Angular pair correlation function
g(12)
= g(r12J1t02)
gi.1i,2m(rl2)
= Coefficients in the expansion of g(12) in spherical
harmonics of the molecular orientations
fi. =
1
= Unit vector aligned along the axis of linear molecule i
k =
= Boltzmann's constant
i =
= Molecular bond length between atoms
m :
= Molecular mass
n :
= Index of refraction
n =
= Constant, values given in Equations (4-21) and (4-22)
n =
= Exponent of r in repulsive part of Mie potential,
Equation (3-32)
n =
s
= Power of r in various model potentials
PN
= Normal component of the stress tensor
pt =
= Tangential component of the stress tensor
r. =
i
= Vector location of center of mass of molecule i
-12
= Vector separation of centers of mass of molecules 1 and 2
rl2
*
r12
: Magnitude of r^2
Reduced distance, r^^/o
r =
m
: Value of r where pair potential is a minimum
s. =
i
Sine of angle 0^
t =
Time
u(ri2) =
Spherically symmetric intermolecular pair potential
XXV


u(r12lu2)
= Orientation dependent intermolecular pair potential
u (12)
= u(r12c10)2)
ao{z12>
= Isotropic, reference contribution to u(12)
ua(12)
"*(r12)
= Anisotropic contribution to u(12)
= Reduced potential, u(r^2)/e
u
s
= Superficial excess internal energy per unit of surface area
V
= Molecular translational velocity
X12
= x-component of r^2
y12 '
= y-component of r^2
y(ri2) ;
= Function defined in Equation (3-38)
Z1 :
= z-coordinate location of molecule 1
Z12 :
= z-component of r^2
z =
max
*
zi =
= Maximum value of z-, defined in Equations (2-42) and (B6)
= Reduced coordinate, z^/o
5 =
Roman Script
= Interfacial area
U =
= Total intermolecular potential
E =
= Coefficients in the spherical harmonic expansion of the
anisotropic pair potential
Greek Upper Case
r. =
i
: Surface adsorption of component i
A =
Triplet of indices, for example
52 =
Volume in angle space
xxvi


2 = Angular velocity
= The l^k component of Q_
a
a.
i
a. ,B.
3
Y
Y
Y
F
*
Y
6
. .
ij
e
0(x)
0.
Greek Lower Case
= Adjustable parameter in Equations (4-26) and (4-27)
= Interior angle in the triangle formed by r^2 ri3 r23
at molecule i
= Constants in the predictor-corrector algorithm; values
given in Equations (6-45) and (6-46), respectively
= Azimuthal and polar angles, respectively, for molecular
orientation in the space fixed frame
= 1/kT
= Angle between the axes of a pair of linear molecules
= Surface tension
= Fowler model surface tension
~ The ith term in the perturbation expansion for surface
tension
2
= Surface tension reduced by potential parameters, yO /e
= Surface tension reduced by critical constants,
Y(Vc/NA)2/3(kTc)"1
= Dimensionless anisotropic overlap parameter
= Kronecker delta
= Interinolecular potential energy parameter
= Unit step function
- Polar angle for molecular orientation in the inter-
molecular frame
xxvii


Angular displacement vector
Polar angle for r^ in spherical coordinates
Dimensionless anisotropic polarizability
Wavelength of incident light in light scattering
experiments
Perturbation parameter
Dipole moment
3 1/2
Reduced dipole moment, y/(eo )
Random numbers
Fluid number density
Interfacial number density profile
Interfacial number density-orientation profile
First order term in perturbation expansion for p(z^OJ^)
Intermolecular potential distance parameter
Standard deviation of property i
Torque on molecule 1
Azimuthal angle for molecular orientation in the inter
molecular frame
Azimuthal angle for in spherical coordinates
Function defined in Equation (3-19)
Set of variables specifying the orientation of molecule i
xxviii


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
SURFACE TENSION AND COMPUTER SIMULATION
OF POLYATOMIC FLUIDS
By
James Mitchell Haile
December, 1976
Chairman: Keith E. Gubbins
Major Department: Chemical Engineering
A third order thermodynamic perturbation theory for the surface
tension of polyatomic liquids has been developed using a Pople reference
fluid. The expansion includes terms containing the unknown pair and
triplet interfacial correlation functions, gQand g^Cz^r^X23)
for the reference fluid. These are removed by using the Fowler approxima
tion for the interface. The resulting theory has been tested against
Monte Carlo results for the Fowler model surface tension of a fluid
whose molecules interct with a Lennard-Jones plus dipole potential.
The behavior of the theory compared with similation parallels that
of bulk fluid properties; namely, the second order theory agrees with
the Monte Carlo results for small values of the dipole moment up to
3 1/2
y/(ea ) 0.6. For larger dipole strengths neither the second nor
third order theories agree with the computer simulation results. How
ever, when the third order expansion is recast in the form of a simple
[1,2] Pad approximant, the theory agrees with the Monte Carlo results
up to dipole moments as large as u/(ecr ) = 1.75. This Pad! theory
has been used to calculate the surface tension of pure dipolar and
xx ix


quadrupolar liquids. The theory agrees with experimental values of
surface tension in the neighborhood of the triple point; however, the
Fowler model does not give the correct temperature dependence of the
surface tension. The theory has also been used as a basis for develop
ing useful correlations of surface tensions of pure polyatomic liquids.
A first order perturbation theory has been developed for the
interfacial density-orientation profile p(z^o^) for polyatomic fluids.
Upon introduction of a Pople reference, the first order term
vanishes for multipolar anisotropies, but does not vanish for anisotropic
overlap or dispersion potential models. Calculations of p(z^aj^) for
axially symmetric molecules interacting with each of the latter potentials
have been performed, using a Lennard-Jones reference fluid and the inter
facial pair correlation function model used by Toxvaerd. The calculations
indicate that the axially symmetric molecules have preferred orientations
in the interfacial region.
A method has been devised for using a NOVA 2 minicomputer with
32K words of core and external disc storage to perform Monte Carlo
simulations of 128 nonspherical molecules. The simulations generally
require several days of continuous calculation; however, several
equilibrium property values and values for the angular pair correlation
function gCr^co^t^) at five to seven specific orientations may be obtained
at a fraction of the cost of doing the calculations on a full size machine.
Results from the minicomputer compare within the statistical precision
with results previously obtained on CDC and IBM machines.
The method of molecular dynamics has been used to study systems
of 256 molecules interacting with Lennard-Jones plus quadrupole and
xxx


Lennard-Jones plus anisotropic overlap potentials. The equilibrium
properties determined include configurational internal energy,
pressure, Fowler model surface tension, Fowler model surface excess
internal energy, mean squared force, and mean squared torque. In
addition, the coefficients g^ ^ m^ri2^ :''n an exPansin fr the
angular pair correlation function gCr^^ci^) in terms of products
of spherical harmonics of the molecular orientations are determined.
Site-site pair correlation functions g D(r) are also found. Relations
dp
are developed between the gg (r ) coefficients and the above listed
A/
equilibrium properties for several anisotropic potential models. Study
is made of orientational structure in quadrupolar and overlap fluids
via g(r^2W-^W2^ as obtained from the recombined spherical harmonic
expansion. A method of producing filmed animations of molecular
motions from molecular dynamics data is described.
xxxi


CHAPTER 1
INTRODUCTION
The development of rigorous methods for prediction of properties
of liquids must come from an understanding of how molecules are dis
tributed and interact with one another. Such fundamental understanding
of liquids may be pursued from three directions: theory, experiment,
and computer simulation. Molecular theory is the domain of statistical
mechanics; molecular level experimentation usually involves light,
x-ray, or neutron scattering, while computer simulation embodies the
Monte Carlo and molecular dynamics techniques. Computer simulation
adds a powerful new dimension to the study of matter which in no way
supplants the other two methods of study. As indicated in Figure 1,
simulation complements both theory and experiment. Thus, comparison
of simulation with laboratory experiment provides information about
the molecular interactions in the liquid. On the other hand, com
parison of simulation with theory provides a stringent test of the
theory.
In spite of current gains being made in the study of liquids,
fundamental understanding of fluid-fluid interfacial phenomena has
been slow to develop. The difficulty with study of interfacial
properties arises from the inhomogeneity, nonuniformity and anisotropy
of the interfacial region between homogeneous fluid phases. In general,
microscopic fluid properties vary through the interface from their
values in one bulk phase to their values in the other bulk phase, as
1


THEORY
EXPERIMENT
COMPUTER
SIMULATION
Figure 1. Relation of Theory, Experiment, and Computer
Simulation in the Study of Liquids


3
in Figure 2. These properties include density, refractive index,
dielectric constant, etc. The oft measured (macroscopic) properties
of interfaces are, in many cases, related to integrals over the
microscopic properties [1]:
X
OO
f
M(z)]dz
(1-1)
where X represents a macroscopic property, M is a microscopic property,
subscript B indicates, usually, a bulk phase value and k is a pro
portionality constant. Examples of specific properties having the
general form of (1-1) are given in Table 1. Equation (1-1) indicates
that prediction of macroscopic interfacial properties and comparison
with experiment is not a completely satisfactory test of theory. Of
more value is understanding of the microscopic properties and their
relations to macroscopic properties of interest. In the case of
theoretical study, such an approach must lie in statistical mechanics.
The prediction of macroscopic property values in a variety of
situations, e.g., over a large portion of the phase diagram and in
multicomponent systems, remains a vital engineering concern. In the
case of interfacial problems, the macroscopic property of interest is
the interfacial tension. The ability to predict interfacial tensions
is required in the equipment and process design and operation of numerous
fluid-fluid contacting operations, such as distillation, solvent extrac
tion, liquid membrane separation techniques, and tertiary oil recovery.
\
Interfacial tension is also important in understanding various biological
processes such as blood oxygenation and eye lubrication. Of particular


4
z
Figure 2.
Variation of Fluid Density with Position through
a Planar Vapor-Liquid Interface


5
TABLE 1
Examples of Macroscopic and Microscopic
Interfacial Properties Related by Equations
of the Form (1-1)
X = k
[M M(z)]dz
D
r.
i
[Pi(z) p]dz +
[pi(z) p^]dz
(1-2)
Y =
[pN PT(z)]dz
(1-3)
K =
X + D
00
P (z)
P (z)
z
X
P ()
p ()
CO
z
X
dz
(1-4)
t
Reference [2]
r.
i
pi
PI
pi
Y
N
References [3,4]'
surface adsorption of component i
density of component i
bulk vapor phase density
bulk liquid phase density
surface tension
normal component of stress tensor
tangential component of stress tensor
PT(z)
K
P.
i
n
A
= coefficient of ellipticity for light plane polarized at
45 to the plane of incidence, when incident at Brewster's
angle
= ith component of local polarization vector
= bulk phase index of refraction
= wavelength of incident beam


6
importance is the determination of how molecular characteristics of
the fluid contribute to the interfacial tension. An important goal
in design considerations is the improved efficiency of fluid-fluid
contacting operations by modifying the system (e.g., by introduction
of additives) in order to lower the interfacial tension. This, in
turn, leads to consideration of how molecules absorb and orient
themselves in the interfacial region.
1.1 Theory of Surface Properties
Readily used methods currently available for predicting inter
facial tension have been reviewed [5,6]. These methods include
corresponding states approaches, techniques based on regular solution
theory and scaled particle theory, and more empirical methods. These
methods are generally applicable to spherical and nearly spherical
molecules, nonpolar polyatomics and their mixtures. These methods
fail to accurately predict interfacial tensions for strongly polar,
quadrupolar or associating liquids.
Almost all rigorous theoretical work on interfacial phenomena
has been for fluids with spherically symmetric molecular interactions.
Two rigorous relations have been developed for surface tension. The
Kirkwood-Buff equation relates the surface tension y to the inter
facial density profile p(z^) and the interfacial pair correlation
function g(z1z2r12) [2,7]
Y = T
dz,
d12
du(r12)
dr
12
P S
*12-
J12
12
(1-5)


7
The Kirkwood-Buff equation is valid for spherical molecules and
assumes the intermolecular potential to be a sum of pair potentials.
This relation is intractable as it stands because of the unknown
function gCz^z^r^^)- Consequently, the Kirkwood-Buff relation has
been studied using various simplifying models for the interfacial
region. The simplest such model is due to Fowler [8] and assumes
an abrupt transition from liquid to vapor phase. Further, the vapor
phase density is assumed to be negligible compared to the liquid
density. Thus, the model may be expressed as:
p(z1)p(z2) g(z1z2ri2^
0(-z1)0(-z2) pLgL(r12)
(1-6)
where 0 is the unit step function,
0(x)
1 if x ^ 0
0 if x < 0
(1-7)
subscript L indicates a bulk liquid property and the negative z
direction is into the liquid. The resulting Fowler-Kirkwood-Buff
expression for surface tension is:
0
4 du(r12} ,
rl2 dr12 gL r12
(1-8)
As could be expected, the Fowler-Kirkwood-Buff theory works well near
the fluid triple point but gives increasing errors in surface tension
as the temperature is raised towards the critical point. Recent


8
evidence indicates that the good agreement at the triple point is
due to cancellation of errors [6].
The second rigorous relation for surface tension is a
generalized van der Waals equation which gives the surface tension
in terms of the density gradient dp(z^)/dz^ and the interfacial
direct correlation function cCz^^r^) [9]:
CO oo oo

f
dz..
dz
1
2
CO
-CO
dx
12
dy
dp(z^) dp(z2)
12 dz.
dz,
c(z1z2r12)(x^2+ y^2) (1-9)
This relation is more general than the Kirkwood-Buff expression since
no assumption of pairwise additive potential is made in its derivation.
It has the further advantage that the direct correlation function c(r^2)
is generally of shorter range than the pair correlation function g(r^2).
The generalized van der Waals equation (1-9) has not been as thoroughly
studied as the older Kirkwood-Buff formula.
The Kirkwood-Buff equation has recently been generalized to
nonspherical molecules [10]. From both a practical standpoint and a
desire to gain understanding of interfacial phenomena, there is strong
need for using this new relation as a basis for developing predictive
methods for surface tension of polar and quadrupolar systems.
Evaluation of the interfacial density profile p(z^) is required
in order to determine the surface tension from the rigorous expressions
(1-5) and (1-9). Study of the interfacial density profile is also
of interest per se. Nearly all studies thus far have been for
spherical molecules. A variety of approaches have been used: a)
van der Waals theory [11], b) constant chemical potential through


9
the interface [12,13], c) first Born-Green-Yvon equation [14], d)
constant normal pressure through the interface [15,16], e) minimiza
tion of system free energy obtained by perturbation theory [17,18],
and f) computer simulation [19,20,21]. Determination of surface
tension for nonspherical molecules will require knowledge of the
interfacial density-orientation profile p(z^U)^), where oo^ is a set
of Euler angles specifying the orientation of molecule 1. Further,
there is considerable interest in determining how modification of
molecular orientation affects 'the surface tension.
1.2 Computer Simulation Methods
In the computer simulation approach to the study of liquids
either the Monte Carlo or molecular dynamics technique may be used.
These methods provide detailed information on equilibrium and time
dependent properties and on liquid structure for molecules interacting
with known force laws. In addition to use of simulation results as
a standard for evaluating theories of liquids, there is now serious
interest in using simulation as a vehicle for evolving and evaluating
realistic potential models for liquids. Much of the simulation work
has been performed for simple, spherical molecules. Only in the past
few years has simulation of nonspherical molecules been undertaken.
There is need of exploiting these simulation methods as fully as
possible since they provide a wealth of detailed information for
study.
Much of the effort in the history of computer simulation has
been expended in developing the methods, demonstrating their usefulness
and potential applicability and, in general, making simulation a


10
respectable research tool. There is currently a need to explore
methods for improving the efficiency of these calculations with a
view towards conserving computer resources and making the simulation
techniques accessible to more users. One possible approach is the
use of a minicomputer for performing simulations of liquids.
1.3 Outline of Dissertation
This work is divided into two parts. Part I includes
Chapters 2-4 and is concerned with the theory of vapor-liquid
interfaces for nonspherical molecules. In Chapter 2 a statistical
mechanical perturbation theory is developed for the surface tension
of polyatomic fluids. The general first order perturbation term is
obtained and the second and third order terms are found when a Pople
reference is used. Specific relations for the perturbation terms
are given for several anisotropic potential models: dipole, quadrupole,
overlap and dispersion. The corresponding perturbation terms are also
derived when the Fowler" approximation for the interface is introduced.
In Chapter 3 numerical calculations are reported for surface
tension using the perturbation theory in the Fowler model. The results
are compared with experiment and computer simulation. Semiempirical
methods based on perturbation theory are explored for correlating
surface tensions of polyatomic and polar substances.
Chapter 4 develops a perturbation theory for determining the
vapor-liquid interfacial density-orientation profile of fluids with
anisotropic potentials. Calculations are presented for overlap and
dispersion model interactions. The calculations predict that these
nonspherical molecules exhibit preferred orientations in the inter
facial region.


11
Part II of this work describes computer simulation studies
of linear molecules. Chapter 5 reports development of a method for
performing Monte Carlo calculations for linear molecules on a NOVA 2
minicomputer.
Chapter 6 describes the molecular dynamics method for linear
molecules, including: derivation of expressions for efficient evalua
tion of the force and torque exerted on a molecule due to various
anisotropic potential interactions, the method used to solve Newton's
equations of motion and the molecular dynamics algorithm for linear
molecules, evaluation of the coefficients in a spherical harmonic
expansion of the angular pair correlation function, and development
of relations between these coefficients and various fluid equilibrium
properties.
Chapter 7 reports the results of the molecular dynamics
calculations for equilibrium property values obtained for Lennard-
Jones plus quadrupole and Lennard-Jones plus anisotropic overlap
fluids. Comparisons of the equilibrium property values are made
with perturbation theory predictions in the case of the quadrupole
fluid calculations. Study is made of the local structure in these
fluids using the molecular dynamics determined angular pair cor
relation functions. A method for producing filmed animations of
molecular motions from molecular dynamics calculations is presented.
Chapter 8 draws conclusions from this study.


PART I
THEORETICAL STUDY OF FLUID INTERFACES


CHAPTER 2
THEORY OF SURFACE TENSION
2.1 General Expressions for Surface Tension of Polyatomic Fluids
There are two rigorous expressions for the surface tension of
polyatomic fluids. One is the generalized van der Waals equation (1-9).
The second is a generalization of the Kirkwood-Buff expression (1-5)
which has been previously derived [10,22]. In this section the deriva
tion for the general Kirkwood-Buff equation is summarized and its
simplified form obtained when the Fowler approximation is made.
2.1.1 Generalized Kirkwood-Buff Formula
Consider a planar interface between vapor and liquid phases with
a coordinate system oriented such that the xy plane is in the plane of
the interface and the +z direction points into the vapor. The two phase
system has N molecules .in volume V at temperature T and interfacial area
S. Thermodynamically, the surface tension y is related to the Helmholtz
free energy A of the two phase system by [23]:
Y =
9A
9 S
NVT
(2-1)
c
Since only the configurational part of the free energy A depends on
the interfacial area, (2-1) may be written as:
Y = kT
L £n Z
NVT
(2-2)


14
where
A = kT £n Z
(2-3)
is the statistical mechanical definition of the configurational
Helmholtz free energy, k = Boltzmann's constant, and Z is the con
figurational integral. For nonspherical molecules:
Z =
r N N -3(rNwN)
dr dw e
(2-4)
where 3 = 1/kT and U is the full intermolecular potential. For non
spherical molecules U depends on the orientations ) of the molecules,
as well as the positions of their centers of mass _r. The orientations
0) are usually specified by a set of Euler angles OJ = body-fixed reference frame on the molecule and a space-fixed frame
located external to the system. Substituting (2-4) into (2-2):
Y = -
kT
3_

f W \
, N N -3U(r w )
dr dto e -
3 S
.
(2-5)
NVT
The differentiation in (2-5) cannot be done immediately since the
N
integration limits on the integrals over r_ depend on 5. We, therefore,
follow Green [24] and change _r variables using the transformation:
cl/2 ,
x = b x
Ql/2 ,
y = S y
V ,
2 = s 2
(2-6)
Performing the differentiation in (2-5) and changing back to the old
variables gives:


15

f MM
, N N -gU(r u )
dr di) e -
(~lu N N.
8u(r w )

.35
NVT
(2-7)
Assuming the potential to be pairwise additive,
(2-7) becomes
U = j i u (r .co ,io.)
ij i j
i (2-8)
Y
d£.ld£2djldJ2
f (£1£2JiJ2'>
8u (12)]
l 95 J
NVT
(2-9)
where u(12) = uCr^^j^^ and tde de^inition of the angular pair dis
tribution function has been used:
f (£1£2jij2)
N(N-l)
Z
d£3'
diNd3'
dor
ant N Nn
-pu(r to )
e
(2-10)
Integrating (2-9) over x^, and transforming r\, to £^2 §dves
dzl d12 dtaidJ2
f ^Z1-12C1 3u(12)
9S
NVT
(2-11)
P 3u(12)
as
can be evaluated for nonspherical molecules by considering:
3u(12) 9-12 3u(12)
as 6 as ar12
1 3u(12) 3u(12)
= 2 [ri2 177 3z12 -alj
(2-12)
(2-13)
Hence, (2-11) becomes:


16
OO
1
Y = 4
dzl
d£1? dc1dco2 f (z1£1 ?oj1c9)
3u(12) 3u(12)
12 3r10 ~i2 3z,
-OO
12 12
Defining the angle average by:
(2-14)
<>
lU2 tt2
( )dJ1do2
where
fi =
dw
(2-15)
(2-16)
Equation (2-14) can be written:
Y = 7-
dz,
d12
3u(12)
3r
- 3z
3u(12)
12
12
]>
8z12 V2
(2-17)
Equation (2-17) is one form of the general Kirkwood-Buff equation.
Another form may be obtained by transforming (2-17) to spherical
coordinates. Since [25]
3u(12)
3z
12
= cos 0
X12,y12
12
3u(12)
3r
12
sin 0
12
012^12
12
3u(12)
30
12
r12^12
then
(2-18)
3u(12)
r, ^ 3z
3u(12)
12 3r
12
12 3z
12
2P2(cos ei2)r12 9r
+ 3 sin 012 cos 012 9Q
12
3u(12)
12
(2-19)
where P2(x) is the second order Legendre polynomial, P2(x) = y (3x 1)
Substituting (2-19) into (2-17) gives:


17
Y = YA +
(2-20)
ft
2
dz,
dri2 P2(cos 012) r12
V -
3ft
B 4
dz.
d12 sIn 012 cos S12 <£(zl12lw2) <2'22)
Equations (2-20,21,22) give the general Kirkwood-Buff equation. The
equation applies to general shaped molecules, the only assumption being
a pairwise additive potential.
In the case of spherical molecules, the potential u(12) goes to
u(r,) so that the derivative in the Y term vanishes and Y, reduces to
12 B A
the Kirkwood-Buff equation (1-5).
2.1.2 Fowler-Kirkwood-Buff Equation for Nonspherical Molecules
The general expression for the surface tension (2-20) is in
tractable as it stands due to the unknown distribution function
f (z^r^co (02)' Varius simplifying assumptions can be made for f to
enable calculations to be performed. The simplest assumption is that
due to Fowler [8]:
2
f(z1r12(a10)2) = 0(-Zl) 0(-z2) gL(12) (2-23)
where 0 is the unit step function as in (1-7), p is the bulk liquid
1j
density, and g is the bulk liquid angular pair correlation function.
Li
Introducing the Fowler model (2-23) into (2-21) allows the integrations
over z^ and co^2 to be performed giving:


18
ttpt
Ya =
dr
12 12
<8,(12)
3u(12).
9r.oo,co
12 12
(2-24)
where the superscript F indicates Fowler model. Similarly, (2-22)
reduces to:
YB 32
3tr2 2
PL
3 9u(12)
drl2 ri2 gL(12) 90. w.w
(2-25)
0
12
12
Further, the Yr, term can be shown to vanish by symmetry arguments. One
D
such argument is that 9u(12)/90^2 is proportional to the 0^ component
of the force on molecule 2 due to molecule 1. This should vanish by
symmetry when averaged over U)^ and Thus, the Fowler-Kirkwood-Buff
equation for nonspherical molecules is:
TTp
Y =
, 4 9u(12)^
dr. r.
12 12
0
3r12 l2
(2-26)
For the special case of spherical molecules, (2-26) reduces to
the usual FKB expression (1-8).
2.2 General First Order Perturbation Theory
for Surface Tension
2.2.1 Rigorous First Order Term
The difficulty with using the general Kirkwood-Buff expression
(2-20) lies with the, in general, unknown interfacial distribution
function f (z^_r^2u)^lJJ2^ This problem may be avoided by use of per
turbation theory, if the interfacial pair distribution function is
available (or may be approximated) for some reference substance.
Perturbation theory for fluid properties may be developed by con
sidering the anisotropic pair potential u(12) to be the sum of a
(known) reference term uq and a perturbing term u^:


19
u(12;A) = uq(12) + Au (12) (2-27)
where A is a perturbation parameter such that when A = 0, (2-27) gives
the reference potential and when A = 1, (2-27) gives the full potential.
Expanding the two-phase system Helmholtz free energy (2-3) in powers of
A and setting A = 1 gives:
A = A + A. + '
o 1
(2-28)
where
A1 2
dld2
3u(12;A)
9A
(2-29)
A=0
and fQ is the interfacial angular pair distribution function for the
reference system.
The corresponding expansion for surface tension is obtained by
applying (2-1) to (2-28):
Y Y0 + Yi +
where = ~2
9_
9S
-ld-2 Fl(zl£l2)
NVT
and V2!^) E 9u(12;A)
9A
A=0
(2-30)
(2-31)
(2-32)
Applying Green's method as in section (2.1), integrating over x^ and
y^, and transforming _r^ to _r 2 (2-31) becomes:


20
^1 2
dz.
d£l2 S
8F1(z112)
3S
NVT
(2-33)
S SF^ (z^r\^2)/3S may be evaluated by considering:
9Fi(zili?) 3jrn 9 3F (z.,r..9) 3F (z r ) 3z
5^ = s _^£ .KJ- + s i__L_L£__i (2-34)
9 S
3 S
812
3z^
9S
12 3F1(Z1^12)
912
3 9F1(z1-12)
2 Z12 9z12
(2-35)
Thus, (2-33) becomes:
8Fl(z!12>
3z,
dz.
8F1(z1-12) 3 9F1(z1-12)
drl2 Z1 3z, + 2 Z12 3z12
3F(z,r._)
-12
lv 1-12;
912
(2-36)
The second two terms in (2-36) can be integrated by parts and shown
to cancel, leaving:
Y1 = '
d12
dzl Z1
3u(12;A)
3A
A=0
9fo(zA2V2)
3z,
lu2
(2-37)
Equation (2-37) is valid for spherical or nonspherical reference fluids
the only assumption has been a pairwise additive potential. Note that
if the reference fluid is taken to be a Pople reference, defined by
(2-48) below, then y^ vanishes.


21
2.2.2 First Order Term in the Fowler Model
To make (2-37) amenable to calculation, the Fowler approximation
(2-23) may be introduced:
^1 2
2(2
d12
dzl Z1 3i7 9(-zl) 6(-z2)<:
3u(12;A)
3A
n8L^12^>(ja1a)
A=0 12
(2-38)
Substituting (2-15) and changing the order of integration:
2(2
d12
dco^ da>2
3u(12;A)
3A
(2-39)
A=0
where I =
z
dzl Z1 3¡~ 0(_Z1) 0(Z2) goL(12)
(2-40)
The integral I may be evaluated by parts to give:
I
z
g T (12) z
oL max
(2-41)
where
max
-rl2 cos 012
lf 012 > 2
if 012 < I
(2-42)
and 0^2 is the spherical coordinate polar angle for _r^> as shown in
Figure 3. Equation (2-39) becomes, therefore:
2
Yl = +
2(2
dr
12
d(j0^d(jJ2
3u(12;A)
3A
A=0
g T (12) z
oL max
(2-43)


22
z
z
a. 12 < 1T^2
b. 012 > tt/2
Figure 3. Two Possible Values for the Maximum z^ Value (zmax)
for a Pair of Molecules in the Fowler Model Interface


23
Reintroducing the angle average (2-15) and noting that g(12) in the
bulk liquid is independent of (2-43) can be written as:
Performing the integration over U)^ gives:
(2-45)
Equation (2-45) is the general result for the first order perturbation
term in the Fowler approximation.
2.3 Perturbation Theory for Surface Tension
using a Pople Reference
When the general expansion for surface tension (2-30) is in
creased to higher order, the second order term is found to include a
term containing the reference four-body distribution function. Higher
order terms in the expansion contain even higher order multibody terms.
These complicated terms can be made to vanish up to at least the second
order term in the expansion by using the isotropic reference potential
first suggested by Pople [26]:
u0(ri2) =
(2-46)
With this choice of reference, (2-27) becomes:
u(12;A) = u (r ) + Au (12)
o 1/ a
(2-47)


24
(2-46) and (2-47) together give the simplification:
9u(12;A)
9A
>
A=0 u
i2
= = 0
a Ulw2
Then (2-29) and (2-37) have:
(2-48)
Ax = Y1 = 0 (2-49)
and the second and third order terms (A^A^.y^y^) simplify. Thus, in
the Pople expansion, (2-28) and (2-30) become, to third order:
AC = + A2 + A3 (2-50)
Y = Yo + Y2 + Y3 (2-51)
2.3.1 Derivation of Second Order Terms, y01 and y
Zn Z D
The second order term in (2-50) for the bulk phase liquid is [27]:
^ldr2 (2-52)
where g^(12) is the first order term in a perturbation theory expansion
of the angular pair correlation function. When the expansion is about
a Pople reference, g^(12) is given by [28]:
§1(12) = Bua(12) go(r12) 6P
dr [ + ] x
3 a 03^ a (jl)^
go(r12r13r23)
(2-53)


25
where §0(r^2rl3r23^ '*'s t*ie triplet correlation function for the reference.
When the anisotropic potential contains only spherical harmonics of order
it ^ 0, such as multipoles, then the angle averages in the second term in
(2-53) vanish. Such potential models as anisotropic overlap and dispersion
contain L = 0 spherical harmonics, in which case the second term must be
included.
Equation (2-52) may be written, therefore, as
A2 A2A + A2B
(2-54)
where
P2B
2 A
dld2 go(r12) (2-55)
Bp
3 r
2B
dr dr dr, g (r10riarQ)
1 2 3 &o 12 13 23 a a a a W1W2U3
(2-56)
Since _r^, and _r^ are each integrated over in (2-56), the indices are
dummy indices and (2-56) may, hence, be written as:
PB'
2B
dr dr dr g (r r r_) (2-57)
1 2 3 o 12 13 23 a a
For a fluid nonuniform in the z-direction, (2-55) and (2-57)
generalize to:
2A 4
Lld-2 Po(zl)Po(z2> 0)lM2
(2-58)


26
2B 2
dr dr dr p (z,.)p (z)p (z) g (z..r r,^)
1 2 3 o 1 o 2 o 3 o 11213 a a 00^0)2^2
(2-59)
The corresponding terms in the expansion for surface tension (2-51)
may be found by applying (2-1) to (2-58) and (2-59):
Y
2A 4
2B 2
3_
as
a_
as
d-ld-2 F2A(z1£12)
- NVT
dr1dr2dr3 F^z^)
(2-60)
(2-61)
NVT
where
F2A Po(zl)Po(z2) go(zl12) Ul(02
(2-62)
F2B E Po(z1)Po(z2)Po(z3) 8o(zl£l213) Equations (2-60) and (2-61) have the same form as (2-31); thus they may
be evaluated in a similar manner to give, analogous to (2-37):
r -!
2A 4
dr
12
dz, z
3F2A(Z1-El2)
1 1 3z,
(2-64)
Y
2B 2
d12d13
dz, z
9F2B(Z1-12^
1 1 az,
(2-65)
Substituting (2-62) and (2-63) into (2-64) and (2-65), respectively,
and using the relations:


27
f
o
(zlil2> '
p0(zi)p0(z2^ go^Zl-l2^
(2-66)
fo(zl-12-13) Po(z1)Po(z2)Po(z3) go(zl-12-13)
(2-67)
gives,
2A
3
4
dr
12 a w to2
dz, z
3fo(zlil2)
1 1 3z,
(2-68)
2B
dr,dr10
-12 -13 a a oi
dz^ z^
3fo(zlll2ll3)
3z,
(2-69)
The derivation of (2-68) and (2-69) only assumes a pairwise
additive potential and use of a Pople reference fluid. If the aniso
tropic potential contains only £ ^ 0 spherical harmonics, then 2$
vanishes because of (2-53).
2.3.2 Derivation of Third Order Terms, Y_. and Y
j_i-3 A L3 B
The third order term in (2-50) for the bulk phase liquid is:
A
3
d£]dJr2
g9 (12)>
0)^2
(2-70)
where g2 is the second order term in the perturbation theory expansion
for g(12) [28J. If only anisotropic potentials containing £ / 0 spherical
harmonics are considered, g2(12) simplifies. Using arguments anologous
to those for simplifying the AD term in (2-56), and generalizing to a
fluid which is nonuniform in the z-direction, (2-70) becomes (for £ ^ 0
harmonics):


28
A3 A3A + A3B
A3A
B
12 J
dridr0 Pn^l^o^2?) g_(z,r J
I 2 o 1 o 2 o 112 a W1W2
A3B 6
dld2d3 P0(zl)po(z2)po(z3) go(z1^12^13)
M
a a a
(2-71)
(2-72)
(2-73)
Applying (2-1) to (2-71) gives:
Y3 Y3A + Y3B
(2-74)
where
Y3A 12
3B
r
6
9
9S

9
9S

dridr2 F3A(Zl£12)
(2-75)
NVT
dld2d3 F3B(zl1213}
(2-76)
NVT
and
F3A Wl0,2
(2-77)
F (z r r ) = p (z )p (z )p (z ) g (z r r )
3B 11211 o 1 o 2 o 1 o 11211 a a a oi-^co^to^
(2-78)
Again, (2-75) and (2-76) have the same form as (2-31) and,
consequently, they may be evaluated in the same manner to give


29
analogous to (2-37) (the Jacobian of the transformation djr^dj:^dr_3 = J
dr^dr^dr.-^ is unity):
Y3A 10
£
12
dr19
12 a
dz. z
3fo(zlil2)
1 1 3z,
(2-79)
3B
dr dr19
12 13 a a a Ia12W3
dzl Z1
9fo(zl-12-13)
9z,
(2-80)
where f^z^r.^) and fQ (zjjr^JL^) are 8^ven by (2-66) and (2-67), respec
tively.
The derivation of (2-79) and (2-80) assumes: a) pairwise additive
potential, b) use of a Pople reference fluid, and c) the anisotropic
potential contains only terms with spherical harmonics of order £ ^ 0.
2.4 Fowler Model Expressions for Perturbation
Terms y^, Jnb_l_JL3aj_L3d in Pople Expansion
The Fowler approximation for the spherically symmetric Pople
reference may be expressed as:
fo(z1^12} = 0(-zi)0(-z2) pL.8oL(r12) (281)
fo^Zl1213^ = 0(-z1^0(-z2^9(-z3^ PL g0l/r12^ (2-82)
Putting (2-81) into (2-68), (2-69), and (2-79) for Y2A> ^3 and YoA>
respectively, and using (2-82) in (2-80) for y all give terms analo-
gous to (2-39) for y^ in the Fowler model. In each case, the integration
over z^ may be done by parts giving, analogous to (2-44):


30
Bp
2 00
2A
2B
Bp
0
3
dr r2 g (r )
12 12 a wi)2 oL 12
doo.. z (2-83)
12 max
dr12 rl2
dr r..
13 13 a a W1W2W3
dco, 0 dco1 g (r10r _r) z
12 13 oL 12 13 23 max
(2-84)
Y
D2 2 00
F ^ PL
3A 12
0
dr r2 g (r10)
12 12 a 0)^2 6oL 12
dco z (2-85)
12 max
Y
2 3
F 6 PS
3B 6
dr12 rl2
dr rf_
13 13 a a a W1W2W3
dw dco g (r r r ) z
12 13 oL 12 13 23 max
(2-86)
In the case of the two-body terms, Y2A an^ Y^> values fr
Zmax are §lven by (2-42); see Figure 3. Hence, the integration over
co
12 can be performed giving:
Y
2 00
,F *BpL t
2A 4
0
dr 0 r2 g (r )
12 12 &oL 12 a co.^
(2-87)
Y
0 2 2
F 716 PL '
3A 12
dr r2 g (r )
12 12 oL 12 a )i)2
(2-88)
The integrations over un and co, in the three-body terms, Yon
1Z J Zd
and Yon cannot be performed immediately since z and g T (r,,r,,,,r~)
Jij IT13X OL 1Z 1 j Z j
depend on these angles. A portion of this angle dependence may be
integrated out, however, if the angles ^2^12^13^13 are transformed to
a set of angles which include Euler angles specifying the orientation


31
of the triangle whose vertices are the molecules 1, 2 and 3. Details
of the coordinate transformation and integration over the Euler angles
are reserved for Appendix B. The resulting expressions are:
Y
2 3 00
F *>L '
2B 2
dr12 r12
dr13 rl3
r12+ r13
dr23 r23 SoL(r12r13r23)
r12" r13 ^
x (r + r + r__)
a a wiU)2a)3 12 13 23
(2-89)
22 3
* 6 PL '
3B
dr12 r12
dr13 r13
r12+ r13
dr23 r23 goL(r12r13r23>
ri2- r13l
X (r + r + r) (2-90)
a a a 12 13 23
F F F F
Computationally convenient forms for y y yD, and You are
zA 3A zB 3B
derived from (2-87), (2-88), (2-89), and (2-90), respectively, in the
F F
next chapter. Equations for each of these perturbation terms y YOI>,
zA zB
F F
^3A ^3B ^or sPec^^^c anisotropic potentials are tabulated in Appendix D.
In the case of bulk fluid properties, Stell et al. [29] have
obtained improved results over the perturbation theory by resumming
the series (2-50) in the form of a Pad approximant:
(2-91)
The analogous form for the surface tension expansion (2-51) is:


32
Y = Y
o
+
1
Y
2
(2-92)
Calculations based on (2-92) in the Fowler model are given in the next
chapter.
2.5 Superficial Excess Internal Energy from the
Pad Perturbation Theory for Surface Tension
The surface excess internal energy is related to the temperature
dependence of the surface tension by the classical Gibbs-Helmholtz equa
tion:
u
s
= Y T
(2-93)
where u^ is the surface excess internal energy, Us> per unit of surface
area 5:
(2-94)
The Fowler approximation does not predict the correct temperature depen
dence of the surface tension for fluids of spherical molecules. Further,
the Fowler model may be used to obtain an equation for ug in terms of
the pair distribution function for the bulk liquid [2], and this second
equation is inconsistent with (2-93) [30,31]; Freeman and McDonald show
that these two expressions give quite different results for use in the
case of Lennard-Jones liquids [31]. To determine the validity of (2-93)
for the Pad perturbation theroy for surface tension presented above,


33
we combine (2-92) and (2-93) to obtain:
u = u + u (2-95)
s so sa
where u is
so
the isotropic reference contribution,
and
u
sa
T
3Yo 3Yo
_ T
3T 9T
(2-96)


CHAPTER 3
NUMERICAL CALCULATIONS OF SURFACE TENSION
The Fowler model perturbation theory developed in Chapter 2
has been used to calculate surface tensions for pure polyatomic
fluids. The intermolecular potential used for the fluids is of
the form:
u(r12a)lW2) = Uo(r12) + Ua(rl2)lJ2) (3_1)
where uq is the Lennard-Jones potential,
Uo(ri2} = 4e[(a/r12)12 (a/r^)6] (3-2)
and u is the dipole, quadrupole, anisotropic overlap or anisotropic
dispersion potential. Equations for these potentials are given in
Appendix C. In these calculations the superposition approximation
is made for gQ^12*13r23) :
go(rl2rl3r23) So(rl2) go(r23) go(rl3)
(3-3)
and Verlet's molecular dynamics results are used for gQ(r) [32].
In Section 3.1 forms amenable to calculation are presented
In Sections 3.2 and 3.3 calculations
, F F F J F
fr Y2A Y2B Y3A and Y3B*
34


35
of surface tension and surface excess internal energy are presented
for various model fluids which obey (3-1); comparisons with Monte Carlo
calculations are made for surface tension of polar liquids. In Section
3.4 results for real fluids are given and compared with experimental
measurements. In Section 3.5 the perturbation theory is used as a
basis for developing a correlation of surface tension for polyatomic
liquids.
Irl Evaluation of y^, y^A and
Rewriting (2-87,88,89,90) in dimensionless form:
k ? co
F* F a TT PL
Y2A y2A e 4 T*
F* F (3 tt PL
r3A Y3A e 12 t*2
0
*2
dr* r*3 g T(r* )
12 12 oL 12 a 0)^0)^
*
dr. r. g (r )
12 12 6oL 12
F* F o'
k 7 co
2 pT
IT L
Y2B Y2B e 2 *
* *
dr12 r12
* *
dr13 r13
* *
r12+ r13
. *
dr23 r23
* i
r12" r13'
lu2
(3-4)
(3-5)
goL(rl2r13r23) (r12+ rl3+ r23)ru)) (3 6)
12 3
*3
CO
I
oo
a2
-TT2 V
* *
dr12 r12 .
/
JU 4-
dr13 r13
£
6 *2 J
* k
-f- r
12 13
* *
dr23.r23
k k i
I r12~ r13'
* * * * *
X goL rl2r] 3r23^ ^rl2+ r13+ r23^
a)lJ2W3
k
where p
kT/e,
*
u
a
u /e, and r
a
r/a.
(3-7)


36
The angle averages in (3-4,5,6,7) have been evaluated [33] and
are tabulated in Appendix A. Substituting those expressions into
(3-4,5,6,7) gives:
yZ I I, y£(A;ss)
2A
(3-8)
A
ss
y\l- I l yllw-.ss')
AA'
SS
, 2B
(3-9)
Y31= I I Y^(AAA";ss's")
J AA'A" sss" J
(3-10)
F*
r3B I I y**(M'A";ssV)
J AA'A" ss's" J
(3-11)
In each of these equations A = Z^Z^Z; in (3-10) A' = Z'^Z'^Z' A" = ,
while in (3-9) and (3-11) A' = A" = SL^Z^Z". In these equations,
F* i\ r. tzx. t 11 T r
Y2a(A;ss ) (2£,+1)(2£0+l) Jn +n ,-l(P ,T ) ^
*2
L (2£ + 1)
* *
16T '~l-~/ s s'
x Eg(A;n1n2) Eg,(A;n1n2)/e2
nln2
(3-12)
*3
^2B(AA' ss') g T* ni
x l (-) Es(£10£1;n10) Es, (^O^jn^/e (3-13)


37
*2
Yo*(AA,A";ss's") = ,/9 -r7 (2£+l)(2£'+l)(2£"+l)
J 96tt 7 T
x Jn . i(P*>T*)
n +n ,+n
s s s
£" £' £
0 0 0
£J £.[ £
*2 £2 £2
£" £' £
n1n2n1
_ __ M __ M
2nl 2
£"
*1
£1
^2
*2
*2
il
-1'
2
-2
-2-
(3-14)
Y~*(AA'A";ss's")
Jd
Es(A;nin2) E^CA'jnjnp /e~
2 p*3 £_+£'+£"
L / \ Z o r
6 t*2 \£'
x ^ o o "
f(2£+l)(2£'+l)]
1/2
££J (2£1+1)(2£2+1)(2£^+1)
x
£ £' £"
£' 5 £
J3 x,2
K1 (££'£";n n'n")
s s s
n1 +n+nl
x I (-) Es(A;nin2) Es,(A,;n[nJ)
nln2n3
x EslI(A";n2n)/e-
(3-15)
In these equations, is the Kronecker delta, n = -n,
£" £' £
0 0 0
is a 3j symbol,
[ £ £' £
|m m' m
tO
is a 6j symbol [34,35], and


38
£" j?' a
i l l
*2 £2 £2
is a 9j symbol [36]. The Eg are coefficients in a
r v a
spherical harmonic expansion of the anisotropic potential u The
superscript on Eg indicates a complex conjugate. Details of the
expansion and equations for Eg for specific interactions are given in
Appendix C.
£ ^
In (3-12) and (3-14) Jn(p ,T ) is the single integral:
. -k k
Jn(p ,T )
* *-(n-2) .
dr, r,v g(ri2^
12 12
(3-16)
Values of the J integrals are tabulated elsewhere and have been fitted
n
k k
to an empirical equation in p and T [37].
y
In (3-13) L (£^;ngn ,) is the triple integral:
L^(£;nn') =
* *-(n-l)
drl2 rl2
* *-(n'-l)
dr13 rl3
* *
rl2+r13
r12 r13'
X dr23 r23^r12+ rl3+ r23^ goL^r12rl3r23^ P£^cos l^ 17')
t h
Here is the £ order Legendre polynomial and is the interior angle
y
at molecule 1 in the triangle formed by r^j r^3> anc* r23 Values of L
are tabulated in Appendix E and have been fitted to an empirical equation
k k
in p and T .
In (3-15) K1 (££'£";ngns,n^,,) is the triple integral:


39
KY(££,£,,;nn'n")
dr* r*-(n-X)
12 12
* *-(n'-l)
drl3 ri3
* *-(r\"-l) * *
x dr r ^ o (x r r )
23 23 goLv 12 13 23;
* *
ri2+ rl3
* ,
r12_ r13'
(ri2+ r13+ r23) (t1a2a3)
(3-18)
The function ^is given by a spherical harmonic expansion:
^££'£,"(aIa2a3^ = ^ C(££'£";mm'm") Ym(u>12)
mm'm"
* Y£'m'(W13) Y£"m"(W23)
(3-19)
where C(££'£";mm'm") is a Clebsch-Gordan coefficient [33]. The are
spherical harmonics in the convention of Rose [38]. In (3-18) and (3-19)
the cu are the interior angles at molecule i in the triangle formed by
r12 rl3 and r23 Expressions for for multipole interactions
are given in Appendix A.
Note that y!^(AA A"; ss s") given by (3-14) vanishes if (£+£'+£")
is odd or if the molecules are linear and either (£^+£3+£^) or
F*
(£+£'+£") is odd. yOT) (AA';ss ) vanishes unless the anisotropic potential
Z Z Z ZJd
contains £=0 spherical harmonics.
F* F* F* F*
Specific expressions for YA, Y,, Y,,. and Y_ have been evaluated
2A 2B 3A 3B
from (3-12), (3-13), (3-14), and (3-15), respectively, for various aniso
tropic potentials. The results are tabulated in Appendix D.
F
The contributions to y given in (3-12,13,14,15) are simply
related to the corresponding terms in the free energy expansion [33]:


40
^(A;as') - f
n +n .-1
s s
^ n +n ,
s s
A2A(-A;SS'-)
kT
(3-20)
YF2B(M,;ss,) - V
** L,
L (£.,;n n ,)
1 s s' J
A2b(AA';ss')
kT
(3-21)
a a
Y3A(AA'A";sss") = -
n +n ,+n -l
s s s
n +n ,+n '
s s s
A (AA'A";ss's")
-j A
NkT
(3-22)
Y3B(AAA";ss's") = -
p*T* KY(££*r;nsns,nsll) (AA A"; ss s")
8 K (££'£";nsns,ns)
NkT
(3-23)
The L and K integrals in (3-21) and (3-23) are defined by:
L(£;nn') =
H *-(n-l)
d 12 12
* *-(n'-l)
d 13 13
* *
ri2+ rl3
k k
dr23 r23
* a .
rl2_ r13'
a a a
X go(r12r13r23) VC0S al)
(3-24)
K(££'£";nn'n")
dr* r*-(n-1)
12 12
, *-(n'-l)
d 13 13
A A
r + r
12 13
r12 r13'
x dr23 r23 go^rl2r13r23^ ^££'£"^aia2a3^ ^ 25^
3.2 Surface Tension Calculations for Model Fluids
In this section model fluid calculations using the Pad per
turbation theory for molecules obeying potentials of the form (3-1)
are presented. For these calculations, the reference fluid surface
F
tension YQ was obtained from the Fowler model expression for a
Lennard-Jones fluid:


41
YV/e = 3ttP*2[J5 2JU] (3-26)
where the integrals are defined by (3-16) and are tabulated elsewhere
[37]. Figures 4 and 5 show the effect of dipolar forces (A E £ = 112)
on the surface tension as predicted by the second order theory, the third
order theory, and the Pad! theory (2-92). The points on these figures
are Monte Carlo calculations of the Fowler model surface tension for
the Lennard-Jones plus dipole fluids [39]. Figure 6 shows similar
results for quadrupoles (A = 224). The second order and third order
terms used in these calculations were determined from the expressions
given in Appendix D.
The results in Figures 4, 5, and 6 are similar to those found
from the corresponding theories for the Helmholtz free energy [29,40].
Including the third order term extends the range of application of the
expansion somewhat; however, the third order term overcorrects the
second order theory for |j* or Q* > 1. The Pad! theory, on the other
hand, interpolates between the second and third order theories. In
the case of the polar fluids, the Pad agrees well with Monte Carlo
results up to y* 1.75.
In Figure 7 comparison is made of the effects of various aniso
tropies on the surface tension. The dipole and quadrupole curves are
the Pad! results from Figures 4 and 6, respectively. As in the case
of bulk fluid thermodynamic properties, for a given value of the
multipole strength (y or Q ), the quadrupole potential is found to
have a larger effect on surface tension than the dipole potential.


42
Figure 4. Fowler Model Surface Tension for a Fluid of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Dipole Model Potential. pa^ = 0.85, kT/e = 1.273


U)
Figure 5. Fowler Model Surface Tension for a Fluid of Axially Symmetric Molecules Interacting
with Lennard-Jones plus Dipole Model Potential. pa^ = 0.45, kT/e =-* 2,934


44
Q/(cr5)1/2
Figure 6. Fowler Model Surface Tension for a Fluid of
Axially Symmetric Molecules Interacting with
Lennard-Jones plus Quadrupole Model Potential.
pa3 = 0.85, kT/e = 1.273


0-2/e
45
Figure 7. Fowler Model Surface Tension for Fluids of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Various Anisotropic Potentials. p= 0.85,
kT/e = 1.273


46
The anisotropic overlap and dispersion results are from the second order
F F
theory, using expressions for Yo and Yor> given in Appendix D.
ZA ZB
3.3 Calculation of the Superficial Excess Internal
Energy for Model Fluids
To obtain the surface excess internal energy from the Pad
expression (2-96), the temperature derivatives of the anisotropic
contributions to the surface tension are required. Equations (3-12),
(3-13), (3-14), and (3-15) give, respectively:
3y^(A;ss')
* = Yo*(A;ss')
9T 2A
9£n J
n +n ,
s s -1
3T
(3-27)
9T
ZD
9£n L3(£,;n n ,)
1 s s
9T
(3-28)
3Y^a(AA,A";ss's")
= Y~(AA'A";ss's")
3T 3A
9£n J
n +n ,+n ,
s s s -1
9T
(3-29)
9Y^(AA'A";ss's") _
* = Yor (AA' A"; ss s")
9T
9£n K^(££'£";n n ,n ,,
s s s 2
9T
(3-30)
In Table 2 values for ug from (2-95) and (2-96) are compared
with computer simulation results for Lennard-Jones plus quadrupole
model fluids (see Chapters 6 and 7). The reference fluid surface
excess internal energy, was obtained from the Fowler model
expression:


47
TABLE 2
Test of the Gibbs-Helmholtz Equation in the Fowler Model
Perturbation Theory for Lennard-Jones Plus Quadrupole Fluids
Q/(ea5)1/2
pa3
kT/e
Pad
F 2,
u o /e
s
MD
0.5
0.85
1.277
1.913
1.923 .016
0.707
0.931
0.765
2.670
2.656 .012
1.0
0.85
1.294
2.505
2.475 .019


48
UgQcr2/e = 2ttP*2 [j5 Ju] (3-31)
Y
where the J integrals are defined by (3-16). The values for u were
n sa
determined in the simulation by evaluating the ensemble average given
in Chapter 5.
In view of the demonstrated inapplicability of Equation (2-93) in
the Fowler approximation for Lennard-Jones fluids [31], the agreement
between the theory and computer simulation shown in Table 2 is
surprising. Since the inconsistency in the Fowler expressions for y and
u is not limited to spherical potentials [6], the results in Table 2
s
must be fortuitous. The agreement may, in part, be attributed to the
high density state conditions considered, wherein the Fowler approxima
tion is more accurate. Much of the agreement, however, must be due to
cancellation of errors between the y and dy/dT terms in (2-93).
3.4 Surface Tension Calculations for Real Fluids
The Pad perturbation theory developed in Chapter 2 has been used
to predict pure liquid surface tensions for CO^, C^H^, and HBr. In these
calculations the reference fluid was taken to be a Mie (n,6) fluid. The
anisotropies considered were the multipole interactions up through the
quadrupole-quadrupole term, as well as anisotropic dispersion and
overlap contributions.
3.4.1 Mie (n,6) Reference Contribution to Surface Tension
The Mie (n,6) fluid was taken as the reference in the perturbation
theory calculations since Twu has determined values for e, O, and n by


49
fitting perturbation theory calculations of liquid densities and pressures
to experimental values along the orthobaric line for the fluids considered
here [37]. It is felt that the test of the Pad! theory for surface ten
sion is strengthened by using these independently determined potential
parameters.
The Mie (n,6) potential u^n,^(r) is given by [41]:
u(n>6)(r)
ne
(n-6)
6(n-6)
-1
(3-32)
To determine the surface tension for this potential, the Helmholtz
free energy of the nonhomogeneous, two phase, (n,6) fluid is expanded to
first order in powers of n ^ about the Lennard-Jones (12,6) fluid free
energy. The surface tension is obtained by applying the thermodynamic
definition (2-1). Then, the Fowler approximation is made in order to
obtain a form amenable to calculation.
The expansion of the (n,6) free energy about the (12,6) may be
done in two ways. In one method, the values of £ and a are taken to be
the same for the two fluids. The expression for Yq resulting from
this expansion is:
F(n,6) 2 F(12,6) 2
---£ --a Y £ 48,p*2 [. 2U2-6> J<12-6)} + 6
1 1_
n 12
(3-33)
In (3-33) is the single integral:
h26)(p\t*) e
* *-(n'_2) (12,6). *, ...
dr r nr gv 7(r ) (3-34)
0


50
Values of this integral for n' = 11 are tabulated in Appendix K and
* *
have been fitted to an empirical equation in p and T .
In the second method of doing the expansion, the values of £
and r^ are taken to be the same for the (n,6) and (12,6) fluids. Here
r^ is the value of r where u(r) = -£. The expression for
resulting from this second expansion is:
F(n,6) 2
y a
o
F(12,6) 2
Yo 0 *2 r T (12,6)
48fTp [ (l-£n2) *
1_
12
1 .(12,6) , (12,6) .
2 J5 + 6 H11 J
(3-35)
When the (n,6) fluid is used as the reference, the second and third
tion 1
(n, 6)
order terms and y^ in the perturbation theory involve integrals over
the (n,6) pair correlation function g^^/(r). These (n,6) correlation
functions can be related to Lennard-Jones g^^^^(r) functions (for which
there are molecular dynamics results [32]) in the following way [42,43]:
and
g(n,6)(r) s g(12,6),rep = e~0u( n6)rep yHS(d(n,6)}
g0(126)(r) g126)rep e-BuU26)rep yHS(d(12,6)}
(3-36)
(3-37)
where the superscripts rep and HS indicate the repulsive and hard sphere
potential contributions, respectively. The function y is defined by:
, s. 3u(r) s
y (r) = e g (r)
(3-38)


51
In (3-36) and (3-37) the hard sphere diameter d is taken to
be [42]:
d
[1 e-^Wjdr
Further, assuming
(3-39)
yHS(d(n-6)) yHV12-6))
(3-40)
(3-36), (3-37), and (3-40) give:
g(n,6)(r) g(12,6)(r) e-6[u(n-6)>rel> uU2,6),rPj (3-41)
where (u(n-6> rcp ua2,6>,rep vanlshes £or r > r .
in
Using (3-41) with the Mie (n,6) potential in the integrals J^,,
Equation (3-16), and K(££'£";nn'n"), Equation (3-24), Twu has found the
resulting values to be negligibly different from those values obtained
for the Lennard-Jones (12,6) potential [37], at least for values of n
close to 12. Hence, in the calculations reported here, the Lennard-Jones
(12,6) pair correlation function has been used in evaluating the terms y^
and in the surface tension expansion.
In the calculations reported here, Equation (3-35) was used to
determine the reference fluid contribution. Values for the Lennard-Jones
term y^(12,6) were obtained from a corresponding states plot of surface
tension for simple fluids, Figure 8. For the temperature range
0.6 kT/e < 1.24, the curve in Figure 8 obeys:


52
Figure 8. Corresponding States Plot for Surface Tension
of Simple Liquids


53
(12 6) 2. *2 *
Yq a /e = 0.8950T 3.5177T + 3.0166
(3-42)
3.4.2 Anisotropic Contribution to Real Fluid Surface Tension
The calculations presented here include anisotropic dispersion,
overlap, and multipole contributions up to the quadrupole-quadrupole
term:
u =
(202) + u,. (022) 4- Uj. (224) + u (202) + u (022)
dis dis dis over over
+ u (202) + u .. (022) +u, (224)
over-dis over-dis dis-Q
+ u .(112) + u (123) + u .(213) + u .(224)
mul mul mul mul
(3-43)
wherein the subscripts dis, over, Q, and mul refer to dispersion, overlap,
quadrupole, and multipole, respectively. Appropriate parts of this model
have been used by Flytzani-Stephanopoulos e_t ail. [33] and Twu [37] in
studies of bulk fluid thermodynamic properties. The multipole contribu
tions to surface tension for this model potential are obtained by combining
(3-8), (3-9), (3-10), and (3-11) with (3-12), (3-13), (3-14), and (3-15),
respectively:
Y2A Y2A(112) + 2 Y2A(123) + Y2A(224)
Y3A 3 13jU12;112;224) + 6 y!j* (112; 123; 213)
T-* C,&
+ 6 y^A(123; 123; 224) + y^(224 ; 224; 224)
T7& TT&
Y3B = Y3B(H2;112;112) + 3 Y3B(H2;123;123)
+ 3 Y^b(123;123;224) + Y3*(224;224;224)
(3-44)
(3-45)
(3-46)


54
Expressions for the terms on the right side of (3-44), (3-45), and (3-46)
F*
are given in Appendix D. The term is zero for multipoles since only
terms with l 0 spherical harmonics occur in the multipole potentials.
The dispersion and overlap anisotropies in (3-43) have been
included in only the second order term y^ in calculating the surface
tension from the Pad perturbation theory (2-92). The inclusion of
dispersion and overlap contributions to the third order term y^ requires
evaluation of difficult multibody terms. Expressions for the y and
Aci.
y2g terms for anisotropic overlap and dispersion are given in Appendix D.
3.4.3 Results for Real Fluids
Figure 9 compares the Pad! predictions of surface tension for
CC>2 with experiment, while Figure 10 shows a similar comparison for
C2H2 and HBr. The experimental values for surface tension of C02 and
C2H2 were taken from the compilation by Jasper [48]. The corresponding
values of saturated liquid densities for C02 and C2H2 were taken from
Vargaftik [49]. Experimental values of surface tension and saturated
liquid density for HBr were obtained from Pearson and Robinson [50].
Values of the potential parameters for C02> C2H2, and HBr were taken
from Twu [37] and are listed in Table 3.
The deviations in surface tension between theory and experiment
are less than 10% for C02 and less than 12% for C2H2 and HBr for the
temperatures shown in the accompanying figures. The consistent deviations
between theory and experiment, especially for C2H2 and HBr, suggest that
adjustment of the potential parameters would improve the agreement. It
is not a very informative test of the theory, however, to adjust potential


7ct2/
55
0.8 1.0 1.2
kT/e
Figure 9. Surface Tension for CO2 comparing Perturbation
Theory Calculations (points) with Experimental
Values (line)


y o~21$
56
Figure 10. Surface Tension for C2H2 and HBr comparing
Perturbation Theory Calculations (points)
with Experimental Values (lines)


57
TABLE 3
Potential Parameter Values Used in
Calculating Surface Tension
Fluid
e/k
(K)
0
(A)
n
M(1018)
(esu cm)
Q(1024)
, 2.
(esu cm )
6
K
co2
244.31+
3.687
16

-4.30 [51]
-0.1
0.257
c2h2
253.66
3.901
16

5.01 [52]
0.3
0.270
HBr
248.47
3.790
12
0.788 [51]
4.0 [51]


All values for e, o, n, 6 and K are taken from Twu [37].


58
parameters using surface tension in order to calculate surface tension.
The deviations between theory and experiment shown in Figures 9 and 10
do not increase much with temperature, and, in fact, for CC^ the devia
tions decrease. This is in contrast to the results found for simple
Lennard-Jones fluids wherein surface tension calculated in the Fowler
model show rapidly increasing disagreement with experiment as the
temperature is increased [2,31]. Further, the Fowler model values of
surface tension for Lennard-Jones fluids are generally larger than the
experimental values at the same temperature. It may be that, in addi
tion to the questionable adequacy of potential parameter values used
here, the Pade approximant in some way corrects for errors introduced
in using the Fowler model for the interface. There is some evidence
for this from the Pad values for the surface excess internal energy
presented in Section 3.3.
3.5 Correlation of Surface Tension for
Pure Polyatomic Liquids
The perturbation theory for surface tension presented in Chapter 2
has been used as a basis for correlating surface tensions of a large number
of polyatomic liquids. The perturbation theory gives the surface tension
in the Fowler approximation as:
Y Y0 + Y2A + Y2B + Y3A + Y3B (3-47)
A simple correlation for y may be obtained by making the van der Waals-
type assumption that the reference fluid radial distribution function is
a constant:


59
?oL(r12} = C
(3-48)
Using (3-48) in (2-87), (2-88), (2-89), and (2-90), together with a
similar approximation for the triplet correlation function, gQ(r^2r13r23^
(3-47) becomes, in reduced form:
*
Y
, *2 *2 *3 *4
. a p a'p a'p a'p
= y + 1 L + -2- -L- + 3 L +
o * *2 *2
(3-49)
where
a' = I
al 4
=
2 2
a a?
dr r ]f
12 12 a a)lt02
(3-50)
CO
I
CO
* *
A A
dr12 r12
dr13 r13
0 0 | r
A A
A A
-f- r
12 13
AAA A
dr0- r0
A 23a 23 a a W1W2W3
12 rl3'
x (r 4- r 4- r )
k 12 13 23;
(3-51)
a. = -
7T
12
A A3 AT
dr r
12 12 a ^1^2
A A
dr12 rl2
A A
dr13 r13
r* + r*
12 13
A A
,dr23,r23
r r I
12 131
(3-52)
AAA AAA
X (r10 + r..., + r~)
a a' av a)lJ2a>3 12 13 23
(3-53)
Equation (3-49) can be written in an equivalent form using the
critical constants, T^, V and p^ as reducing parameters rather than
the potential parameters, e and O:


60
y +V£ + ^ + !A + V
roR T T 2 2
R R
R R
where
Yr = y(vc/na)2/3 (ut.)'1
(3-54)
(3-55)
PR = P/Pc (3-56)
Tr = T/Tc (3-57)
In transforming (3-49) into (3-54), the proportionality of the potential
parameters to the critical constants is obtained by the usual correspond
ing states method [41]- Here, however, the potential is for polyatomic
fluids and, therefore, contains parameters in addition to the energy and
distance parameters £ and O, e.g., the multipole moments, y, Q, and
anisotropic polarizability, K, and overlap parameters 6. For such an
intermolecular potential, if the usual derivation of corresponding states
theory is followed [41], one finds:
* c *
Tc = ~~ = c^y ,Q ,6,k, )
* 3 *
Pc = pca = c2(y ,Q ,6,k,)
p a3
* C *
~ ^ (y > Q
(3-58)
(3-59)
(3-60)
where c^, c^, and c^ are constants only if the anisotropic potential para-
* *
meters y Q o, K, are kept fixed. However, these "constants" can
be absorbed in the terms a^, a2, etc., as has been done in (3-54). Thus,
2
= a|c2/c^, etc. The transformation from the potential parameters £
and a as reducing parameters to critical constants as reducing parameters
may then be made in the usual way [41].


61
Equation (3-54) has been used as a basis for correlating surface
tension by treating the parameters a^ as semiempirical constants. Various
truncated forms of (3-54), including its Pad form analogous to (2-92),
have been tested against experimental surface tensions for numerous poly
atomic liquids. The form giving the best comparison with experiment was
found to be that terminated after the y term:
ZB
YR YoR + T ^1 + a2PR)
R
(3-61)
Values for a-^ and a^ for use in (3-61) have been determined for numerous
polyatomic liquids by least squares fitting to available experimental
data. The resulting values are listed in Table 4. In these calculations,
the reference contribution was obtained from a fit of reduced surface
tensions of the inert gases and methane, analogous to the curves in
Figure 8.
Y = 2.4724T^ 7.5918TD + 5.0748 (3-62)
OK K K
which applies for 0.4 < T £ 0.95.
R
The validity of the correlation (3-61) may be tested by determining
2
whether experimental data give a linear relation between (y y )T /p
R oR R R
and p as implied in (3-61). Such a test has been conducted for several
K
polyatomic liquids, and results for carbon dioxide, acetic acid, and
methanol are shown in Figures 11, 12, and 13, respectively. These
figures are typical results for small to moderately large polyatomics
and indicate a satisfactory correlation. The corresponding comparisons


62
TABLE 4
Values for the Parameters and a^ in the
Surface Tension Correlation of Equation (3-61)
Substance
Range
a1(10^)
a2(10 )
References
T
r
P Y
Paraffins
Ethane
.43-
.59
6.880
-2.246
49
48
Propane
.50-
.77
14.17
-5.606
53
48
n-Butane
.50-
.57
8.548
-2.469
49
49
i-Butane
.52-
.60
7.749
-2.082
49
49
n-Pentane
.54-
.67
6.504
-1.498
49
49
i-Pentane
.59-
.66
6.250
-1.451
49
49
n-Hexane
.54-
.93
-2.398
1.918
49
49
n-Heptane
.54-
.95
-1.860
1.910
49
49
n-Octane
.48-
.90
1.335
0.7617
49
49
i-Octane
.50-
.67
8.536
-1.942
49
49
n-Nonane
. 46-
.63
7.340
-1.230
49
49
n-Decane
.44-
.60
8.699
-1.576
49
49
n-Dodecane
.41-
.57
11.42
-2.262
49
49
n-Tridecane
.40-
.55
12.06
-2.360
49
49
n-Tetradecane
.41-
.54
12.21
-2.339
49
49
n-Pentadecane
.40-
.53
12.83
-2.452
49
49
n-Hexadecane
.41-
.55
11.59
-1.893
49
48
n-Heptadecane
.41-
.54
11.10
-1.720
49
48
n-Octadecane
.40-
.52
10.25
-1.439
49
48
n-Nonadecane
.41-
.52
11.15
-1.596
49
48
n-Eicosane
.40-
.51
11.15
-1.539
49
48
Cycloparaffins
Cyclopentane
.55-
.61
5.689
-1.339
49
48
Methylcyclo-
.53-
.59
5.973
-1.288
49
48
pentane
Ethylcyclo-
.50-
.55
10.65
-2.832
49
48
pentane
Cyclohexane
.51-
.62
6.249
-1.351
49
48
Methylcyclo-
.48-
.65
2.885
-0.4730
49
49
hexane
t
Reduced temperature
range over
which a.
and a were
fitted.
Sources of experimental data for liquid densities p and surface
tensions y.


63
TABLE 4 (Continued)
Substance
2 2
Range a (10 ) a-UO ) References
Tr P Y
Olefins
Propylene
.53-.67
7.143
-2.173
49
48
1-Butene
.48-.70
4.752
-0.9880
49
48
2-Butene
.51-.65
3.606
-0.8401
49
48
1-Hexene
.54-.64
8.630
-2.142
49
49
1-Octene
.47-.56
8.047
-1.876
49
49
Cyclopentene
.56-.62
5.445
-1.195
49
49
Aromatics
Benzene
.48-.93
0.4634
0.5432
53
11
Toluene
.46-.63
8.130
-2.080
49
49
Ethylbenzene
.44-.60
9.602
-2.384
49
49
Isopropylbenzene
.46-.57
8.175
-1.851
49
49
Alcohols
Methanol
.53-.92
11.60
-4.971
49
49
n-Propanol
.55-.68
25.56
-8.854
49
48
i-Propanol
.56-.60
33.13
-10.98
49
49
n-Butanol
.48-.54
20.35
-6.624
49
49
Organic Halides
Methyl Chloride
.68-.73
1.215
-0.1476
53
48
Ethyl Bromide
.56-.60
15.00
-4.565
53
48
Carbon Tetra-
.49-.89
0.9412
0.4869
49
49
chloride
Chlorobenzene
.43-.88
2.138
-0.000452
49
49
Oxides
Carbon monoxide
.61-.68
8.502
-2.748
49
48
Carbon dioxide
.71-.95
-1.587
1.395
49
49
Water
.44-.58
34.75
-11.90
54
48
Others
Acetic Acid
.49-.86
5.984
-2.906
49
49
Acetone
.54-.69
6.178
-1.783
49
49
Ammonia
.49-.58
6.878
-2.269
53
48
Aniline
Carbon disulfide
.39-.65
.51-.58
13.38
5.968
-3.976
-1.869
49
49
Chlorine
.47-.57
4.103
-1.340
49
48
Diethyl ether
.59-.95
-0.4972
1.227
49
49
Ethyl acetate
.52-.90
0.08686
1.058
49
49


64
Figure 11. Test of Surface Tension Correlation (line)
for CC>2


65
Figure 12. Test of Surface Tension Correlation (line) for
Acetic Acid


66
Figure 13. Test of Surface Tension Correlation (line) for
Methanol


67
of predicted surface tensions with experimental values for these and
several other liquids are shown in Figures 14 and 15.
The linear relation suggested by (3-61) is not obeyed by experi
mental data for long chain hydrocarbons, however. Typical plots are
shown in Figure 16. In spite of the poor correlation for these
substances, the predicted surface tensions using (3-61) were usually
within 3% of the experimental values as shown in Figure 17.
Generally, the correlation of (3-61) reproduced the experimental
data for substances tested here within 3% for values of T < 0.92. In
K
order for the correlation to apply in the critical region, we must have:
C.P. C.P.
0
(3-63)
hence, from (3-61):
(3-64)
Then
yr YoR+ aT (1 V
K
(3-65)
in the critical region. This relation was not obeyed by many of the
liquids in Table 4.


68
Figure 14. Comparison of Surface Tension Calculated from
the Correlation (lines) with Experimental Values
(points) for Several Polyatomic Liquids


Full Text

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PAGE 347

7$%/( + 9DOXHV RI JS44Aa IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG ZLWK N7H SHUr DQG 6 R 0 $ & f f * F F f f f f F F & 4 M nf & f F U & 4 f &2, U? f & F f f  f ff & & E & & & & V U & & fA & n& F & & R & f L A R fR & & E & f & & M G & R F f & E L  f r  f G R M & f 4R 9 F  & S & && F R  2n f 2 A BB U M nV F V F & f G & & & F L A' fY! f & e & & Yr f F & f V F & R & & F & & f & & f 9  & f  & & & E f F M F &2M & e & f L f & O & & & & & f§ n? & & F f R f & & f§ f & f & & & F O R & & E & 8 O f B Q m LB f§ 4 & & & 2 IM f R M ‘ f f§ f  & & & * FO f & r & r F F U f ‘ YM f  R & F G f F & B U UV r B f ?M r B U L Q L Z f n & F & X & ‘ & & F f M && & & L RRFH U! rW f r  & & & U Q A -

PAGE 348

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PAGE 349

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PAGE 350

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PAGE 358

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PAGE 362

1 2 2 X 7$%/( 7KH ,QWHJUDOV 7 F X IRU WKH )OXLG ,, ,, RI 7DEOH L 1O f 1 f -1f -1 f -1 f & T & R & R& & 2& & & R & e & & & A &

PAGE 363

7$%/( 7KH B W ,QWHJUDOV 7 S 8 IRU WKH )OXLG /, ,, RI 7DEOH 1 -Q& -1f -1f L r & r r O & f§ *& f§ r

PAGE 364

7$%/( P 7 W B 7 7KH ,QWHJUDOV IRU D /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG SDA N7H m9 1 M Q RRR f -1 f 1 f -1  f -1f f f f f f f f f f f f f f f f f f &R R D M & f§ R R R f§ & f§ G& &O & R f§ &  f§ A M & WO9 ( -QPf Q

PAGE 365

1 2 2 7$%/( 7 B W W F M 7KH ,QWHJUDOV IRU WKH )OXLG RI 7DEOH MQ! MQf MQ & f& G & & & & 2 2& 22 R 2& -1 2& 2& & &22& -1f & &

PAGE 366

1 2 2 R 7$%/( 7KH ,QWHJUDOV RI 7DEOH IRU WKH )OXLG M Q L f r RR B! f fD & R R 1 f & R O -1-f &r & & -1f & & & R & 1 & f f§

PAGE 367

7$%/( B W W 7KH ,QWHJUDOV RI 7DEOH IRU WKH )OXLG 1 -1  f -1f -1f f§ R f§£ & && f§ f G

PAGE 368

1 2 ,6 7$%/( A 7 7 7 7KH ,QWHJUDOV IRU D /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG SJ N7R 4HRf -O1L2&2f -,1f -1f -1 -1f f f f f f f f f R R f & & & f & & f§ f§ f§ f§ a W R f§ & f§ & & & & & &

PAGE 369

1 2 & 7$%/( 7KH ,QWHJUDOV IRU WKH )OXLG RI 7DEOH -Q G M f V -1f & fA & & -1  f & -1 f f R & & -1&f & & &

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1 2 2 7$%/( ,OO 7KH ,QWHJUDOV U Y IRU WKH )OXLG RI 7DEOH 1 O f -1f -1Lf MQ f n -&1f &R & & &2 & f R & & œ R &  R &2 &2 & f R f • &

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7$%/( 7 W U 7KH ,QWHJUDOV IRU WKH )OXLG RI 7DEOH 1 -1f -1-f & f§  & A f f§ f§

PAGE 372

1 2 7$%/( P 7 W 7 7KH ,QWHJUDOV IRU D /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG SF7 N7H -1242-1-f -1f -1f -1f f f f f & RR & A M f§ & &6 2 2LO && & & &

PAGE 373

7$%/( A B 7KH ,QWHJUDOV 7 & W8 IRU WKH )OXLG RI 7DEOH Q 1 MQ f -1 f -1f -1f -1 f & f§& & G O f R ,G & 2& D & & & f§  & L  & f 2MM O & & & 2222OR  & L R f§ & &

PAGE 374

1 2 D 6 DR DL DD  7$%/( 7KH ,QWHJUDOV IRU WKH )OXLG RI 7DEOH 1L f & R RRRDD R RRRRDR -1RRRLD R RRLDL RRRRD & RRRRD L & R FRRDD -1f f§ R & f§ & f§ &• &22 &G -1  f 22& & RRFRD L R F R R D & & *2 2& -1cf & &

PAGE 375

7$%/( 7KH ,QWHJUDOV r IRU WKH )OXLG /, RI 7DEOH 1 -1f -1f -1L f & R & L L D r f§ & L D & r

PAGE 376

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