ABSTRACT 
 
 
 
 
Title of Document: MESOSCOPIC THERMODYNAMICS IN 
SMOOTH AND CURVED INTERFACES IN 
ASYMMETRIC FLUIDS 
  
 Heather Jane St. Pierre, Doctor of Philosophy, 
2009 
Directed By: Professor Mikhail Anisimov 
Institute for Physical Science and Technology 
and 
Department of Chemical and Biomolecular 
Engineering 
 
 
Phase separation has a significant impact on many chemical engineering applications, 
and since the phase transition of asymmetric one-component fluid or mixture 
interfaces can be gradual, or smooth, further analysis is warranted for fluid separation 
on the mesoscale.  Complete scaling is applied to account for fluctuations in the 
critical region and to model the interfacial profiles of one-component and dilute 
binary mixtures, as well as calculate the curvature correction to the surface of tension 
(or Tolman length).  Well-established symmetric profiles were selected for the order 
parameter and thermal scaling densities for use in complete scaling to model these 
asymmetric fluids.  Real fluid asymmetry was applied to these profiles though the 
scaling coefficients of one-component fluids.  Scaling coefficients for mixtures were 
introduced through the experimental critical parameters of the specific mixture; 
  
characteristics accounted for included consideration of the difference in molar volume 
between solvent and solute, changes in critical temperature and pressure with 
concentration as well as the Krichevskii parameter.  Flory theory was used to 
approximate the scaling coefficients in dilute polymer solutions to determine the 
Tolman length.  These results indicated that Tolman?s length diverges near the critical 
point of separation and with an increasing degree of polymerization.  As an infinite 
degree of polymerization is approached, the Tolman length becomes half of the width 
of the interface.   Since fluid behavior near the critical point of separation is universal, 
many of the theoretical expressions shown in this work can be applied to other 
asymmetric systems.  The results of this analysis showed that a large difference in 
molecular volume led to a higher degree of fluid asymmetry in polymer solutions.  
The concentration of added solute, specifically in dilute n-heptane-ethane solutions 
resulted in increased fluid asymmetry in density profiles, as well as a shift toward the 
density of the added solute.  When considering dilute mixtures of aqueous n-hexane 
and n-heptane-ethane solutions, a slight increase in concentration of solute, coupled 
with an increasing temperature distance to the critical point of separation, yielded the 
greatest increase in fluid asymmetry in concentration profiles. 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
MESOSCOPIC THERMODYNAMICS IN SMOOTH AND CURVED 
INTERFACES IN ASYMMETRIC FLUIDS 
 
 
 
By 
 
 
Heather Jane St. Pierre 
 
 
 
 
 
Dissertation submitted to the Faculty of the Graduate School of the  
University of Maryland, College Park, in partial fulfillment 
of the requirements for the degree of 
Doctor of Philosophy  
2009 
 
 
 
 
 
 
 
 
 
 
Advisory Committee: 
Professor Mikhail A. Anisimov, Dept. of Chemical and Biomolecular Engineering, Chair 
Professor Robert M. Briber, Dept. of Materials Science and Engineering 
Professor Mohammed Al-Sheikhly, Dept. of Materials Science and Engineering 
Professor Raymond A. Adomaitis, Dept. of Chemical and Biomolecular Engineering 
Professor Sheryl H. Ehrman, Dept. of Chemical and Biomolecular Engineering 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
? Copyright by 
Heather Jane St. Pierre 
2009 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ii 
 
Acknowledgements 
 
I would like to take the opportunity to thank the members of our research 
group, as all have provided invaluable thoughts and input.  In particular, I would like 
to thank Chris Bertrand, as well as Daphne Fuentevilla and Deepa Subramanian for 
their incredible support throughout this process.   
I also wish to thank the faculty and staff of the Chemical and Biomolecular 
Engineering Department for their assistance throughout this process.  Specifically, I 
would like to thank my advisor, Professor Mikhail Anisimov for his guidance, 
patience and for keeping me motivated throughout this research, as well as 
Distinguished University Research Professor Jan Sengers for his guidance and review 
during completion of this work.  Without them, this research would not have been 
possible.  
Lastly, I would like to thank my husband, Eric, for his unending love and 
support throughout this research.    
 
 iii 
 
Table of Contents 
 
 
Acknowledgements ...................................................................................................ii 
Table of Contents .....................................................................................................iii 
List of Tables.............................................................................................................v 
List of Figures ..........................................................................................................vi 
Chapter 1: Introduction..............................................................................................1 
1.1   The Smooth Interface.....................................................................................1 
1.2   Surface Tension and Curved Smooth Interfaces .............................................2 
1.3   Asymmetry and Its Effect on Interfacial Profiles............................................4 
1.4   Methodology and Overview ...........................................................................5 
Chapter 2: Mean-Field Approximation of the Smooth Interface .................................7 
2.1  Determining the Local Helmholtz-Energy Density..........................................7 
2.1.1  Landau Expansion and the Landau-Ginsburg Functional ..........................8 
2.2  Approximating the Density Profile near the Interface ....................................14 
2.3  Determining the Width of the Interface .........................................................16 
2.4  Position and Density Dependence of the Local Helmholtz-Energy Density ...19 
2.5  Mean-Field Estimation of Interfacial Tension................................................20 
Chapter 3: Accounting for Fluctuations near the Critical Point.................................24 
3.1  Effects of Critical Fluctuations and Fluid Asymmetry ...................................24 
3.2  Scaling Theory..............................................................................................26 
3.2.1  General Scaling Equations......................................................................27 
3.2.2  Complete Scaling for a One-Component Fluid .......................................30 
3.2.3  Complete Scaling for a Dilute Binary Mixture........................................34 
3.3  The Smooth Interface....................................................................................40 
3.4  Estimations of Surface Tension.....................................................................44 
3.4.1  Renormalization Group Theory Results..................................................44 
3.4.2 Dimensional Scaling Analysis .................................................................45 
3.4.3  Comparison of Methodologies................................................................48 
Chapter 4: Curved Interfaces and Surface Tension...................................................50 
4.1  Tolman?s Length: Curvature Correction to the Surface Tension ....................50 
4.1.1  Young-Laplace Approximation of Surface Tension................................50 
4.1.2  Expression for Tolman?s Length.............................................................51 
4.1.3  Predictions of the Behavior of Tolman?s Length.....................................53 
4.2  Tolman?s Length Near the Critical Point .......................................................55 
4.3  General Thermodynamic Expression for Tolman?s Length............................60 
Chapter 5:  Tolman?s Length for Polymer Solutions and Other Dilute Binary Mixtures
................................................................................................................................63 
5.1  Polymer Solutions: Theory and Background .................................................63 
5.1.1  Phase Coexistence..................................................................................63 
5.1.2  Fluctuations............................................................................................67 
5.2  Tolman?s Length in Near-Critical Polymer Solutions ....................................69 
5.2.1  Approximation of Scaling Coefficients...................................................69 
5.2.2  Approximation of Critical Amplitudes....................................................74 
 
 iv 
 
5.2.3  Tolman?s Length near the Critical Point .................................................75 
5.2.4  Comparison of Terms.............................................................................77 
5.3  Tolman?s Length at High Degrees of Polymerization ....................................80 
5.4  Crossover Expression for Tolman?s Length in Polymer Solutions .................83 
5.4.1  Crossover Correlation Length.................................................................84 
5.4.2  Crossover Tolman-Correlation Length Ratio ..........................................85 
5.4.3  Crossover Tolman?s Length ...................................................................86 
5.5  Tolman?s Length in Other Polymer Applications...........................................90 
5.6  Near-Critical Tolman?s Length in Dilute Binary Mixtures.............................90 
Chapter 6:  Concentration and Density Profiles for Asymmetric Fluids and Binary 
Fluid Mixtures.........................................................................................................93 
6.1  Background and Introduction ........................................................................93 
6.2  Determination of Scaling Densities for Use in Complete Scaling ..................94 
6.2.1  Symmetric Order Parameter ...................................................................95 
6.2.2  Thermal Scaling Density ........................................................................97 
6.3  Modeling Density Profiles for One-Component Fluids..................................99 
6.3.1  Application to Water ............................................................................100 
6.3.2  Application to Ethane...........................................................................102 
6.3.3  Application to n-Heptane......................................................................102 
6.4  Fisher Renormalization: Temperature Correction for Concentration............104 
6.5  Modeling Concentration Profiles for Dilute Binary Solutions......................108 
6.5.1  Application to Dilute Aqueous n-Hexane Solutions..............................110 
6.5.2  Application to Dilute n-Heptane-Ethane Solutions................................113 
6.5.3  Effect of Temperature on Profile Asymmetry .......................................115 
6.5.4  Effect of Concentration on Profile Asymmetry.....................................118 
6.6  Modeling Asymmetric Density Profiles for Dilute Binary Solutions............122 
6.6.1  Application to Dilute Polystyrene-Cyclohexane Solutions....................123 
6.6.2  Application to Dilute Ethane-n-Heptane Solutions................................126 
6.6.2  Effect of Temperature on Asymmetry in Density Profiles.....................127 
6.6.3  Effect of Concentration on Asymmetry in Density Profiles...................129 
6.7  Conclusions ................................................................................................133 
Chapter 7:  Summary and Outlook.........................................................................134 
Appendix A:  Comparison of Mean-Field Surface Tension Calculations................138 
A.1  Comparison with Landau and Lifshitz ........................................................138 
A.2  Comparison with Rowlinson and Widom ...................................................140 
References.............................................................................................................144 
 
 
 
 
 
 
 
 
 
 v 
 
List of Tables 
 
Table 3.1 Comparison of mean-field and scaling methods for the estimation 
of surface tension coefficients??????????????.. 48 
  
Table 6.1 Comparison of mean-field and scaling methods for the estimation 
of surface tension coefficients??????????????.. 101 
  
Table 6.2 Experimental data for ethane??????????????? 101 
   
Table 6.3 Experimental data for n-heptane??????????????103 
   
Table 6.4 Selected experimental data for a dilute aqueous n-hexane 
solution???????????????????????.. 106 
   
Table 6.5 Experimental data for dilute aqueous n-hexane solutions???? 111 
   
Table 6.6 Critical temperatures of dilute aqueous n-hexane solutions at 
various concentrations?????????????????.. 111 
  
Table 6.7 Critical temperatures of dilute n-hepane-ethane solutions at various 
concentrations????????????????????? 114 
   
Table 6.8 Experimental data for dilute n-heptane-ethane solutions????.. 114 
   
Table 6.9 Comparison of bulk phase concentration data from dilute aqueous 
n-hexane solutions at various temperatures?????????... 120 
   
Table 6.10 Comparison of bulk phase concentration data from dilute n-
heptane-ethane solutions at various temperatures???????. 121 
 
 vi 
 
List of Figures 
Figure 1.1 Simulated curved smooth interface of vapor-liquid separation in a 
droplet????????????????????????2 
   
Figure 1.2 Schematic of an asymmetric vapor-liquid coexistence 
curve????????????????????????.. 4 
   
Figure 2.1 Local Helmholtz- energy density from Landau expansion (in terms 
of the order parameter) given in Eq. (2.1.14) for 
0
9/16u =
 
and 
varying temperature distances, 
?
T?
????????????... 13 
   
Figure 2.2 Local Helmholtz- energy density from Landau expansion (in terms 
of the order parameter) given in Eq. (2.1.15) ? which includes the 
reference value for the local free energy density ? for 
0
9/16u =
 
and varying temperature distances,
 
?
T? ??????????? 13 
   
Figure 2.3 Mean-field dimensionless density profile (a) and gradient (b) as a 
function of height???????????????????.. 17 
   
Figure 2.4 Contour plot of the interface between two fluid phases as predicted 
by mean-field theory??????????????????. 18 
   
Figure 2.5 Three-dimensional universal plot of the dimensionless local 
Helmholtz-energy density 
4
0
(/ )f ?  and its dependence on height 
and density?????????????????????. 20 
  
Figure 3.1 Schematic of an asymmetric vapor-liquid coexistence curve 
comparing mean-field and scaling approximations near the critical 
point????????????????????????. 25 
   
Figure 3.2 Schematic of a T ??  diagram indicating the location of zero 
ordering field (h
1
 = 0) ?????????????????.. 29 
   
Figure 3.3 Comparison of mean-field and renormalization group theory 
density profiles as a function of height???????????. 42 
   
Figure 3.4 Contour plot of the interface between liquid and vapor phases as 
predicted by renormalization group theory?????????.. 42 
   
Figure 3.5 Simulation of a smooth interface for a droplet near the critical 
point of fluid-fluid separation using RG theory???????? 43 
   
 
 vii 
 
Figure 3.6 Representation of the interface thickness considered for the 
dimensional scaling analysis calculation of surface tension in Eqs. 
(2.4.11) and (2.4.13)-(2.4.15)??????????????? 47 
   
Figure 4.1 Simple schematic illustrating the properties of the variables used 
in the Young-Laplace relation for surface tension at the interface 
between two phases, ?  and ? ??????????????.
51 
   
Figure 4.2 
Sketch of the concept of the Tolman length 
()
?  for a droplet of 
liquid????????????????????????. 52 
   
Figure 4.3 Sketch of the concept of using the coexistence diameter to 
approximate the ratio of the Tolman length to the interface 
thickness in Eq. (4.3.1)?????????????????. 61 
   
Figure 5.1 
Phase diagram for a polymer solution (
4
10N = ) near the critical 
point????????????????????????.. 67 
   
Figure 5.2 
Plots of 
eff
a and 
2
b against inverse degree of polymerization (1/N) 
for the range N = 1 to 10
5
 ????????????????. 73 
   
Figure 5.3 Three-dimensional plot of the dimensionless near-critical Tolman 
length for a droplet of polymer in solution (from Eq. (5.2.18))?? 76 
  
Figure 5.4 Plots of the two terms from the dimensionless Tolman length for a 
droplet of polymer in solution at three temperatures?????? 77 
   
Figure 5.5 
Comparison of N-dependent amplitudes 
0
?
B  and 
0
?
?
? for a moderate 
to high degree of polymerization?????????????.. 80 
   
Figure 5.6 Universal behavior of the dimensionless Tolman length with 
respect to degree of polymerization and temperature distance to 
phase separation for a polymer-rich droplet as predicted by Eq. 
(5.4.10)??????????????????????? 88 
   
Figure 5.7 Dimensionless Tolman length for a polymer-rich droplet with 
4
10N =  exhibiting crossover between the critical and polymer 
regimes from Eq. (5.4.10)???????????????... 89 
  
Figure 6.1 Schematic of an asymmetric vapor-liquid coexistence curve??... 93 
   
Figure 6.2 Normalized symmetric mean field and Renormalization Group 
theory profiles?????????????????????96 
   
 
 viii 
 
Figure 6.3 Density profile for pure water at a temperature distance of 
2
?
10T
?
?=? ,  plotted with respect to height (z-coordinate)???.. 104 
   
Figure 6.4 Density profile for ethane at a temperature distance of 
2
?
10T
?
?=? ,  plotted with respect to height (z-coordinate)???.. 102 
   
Figure 6.5 Density profile for n-heptane at a temperature distance of 
2
?
10T
?
?=? ,  plotted with respect to height (z-coordinate)???...104 
   
Figure 6.6 Plots of the difference (a) and normalized difference (b) between 
experimental 
?
(()Tx? and theoretical 
?
(()T ??  temperature 
distances as predicted by Eq. (5.4.2) and data in Table 6.4 for a 
dilute aqueous n-hexane solution where 
c
0.015x = ??????.
107 
   
Figure 6.7 Critical locus for aqueous n-hexane solutions????????.. 112 
   
Figure 6.8 Critical locus for a mixture of ethane and n-heptane?????? 113 
   
Figure 6.9 Concentration profiles for n-hexane in water at the vapor-liquid 
interface, at a critical and average concentration of 
c
0.015xx== , and at various temperatures, plotted with respect 
to height (z-coordinate)?????????????????. 116 
   
Figure 6.10 Concentration profiles for n-hepane in ethane at the vapor-liquid 
interface, at a critical and average concentration of 
c
0.0202xx== ,  and at various temperature distances, plotted 
with respect to height (z-coordinate)????????????. 117 
   
Figure 6.11 Concentration profiles for n-hexane in water at the vapor-liquid 
interface, at a constant temperature distance 
2
?
10T
?
?=? , and at 
various critical or average concentrations, plotted with respect to 
height (z-coordinate)??????????????????. 119 
   
Figure 6.12 Concentration profiles for n-hepane in ethane at the vapor-liquid 
interface, at a constant temperature distance 
2
?
10T
?
?=? , and at 
various critical or average concentrations, plotted with respect to 
height (z-coordinate)??????????????????. 120 
   
Figure 6.13 Normalized density profile for a polystyrene-cyclohexane solution 
with degrees of polymerization 100N = , 150N =  and at a 
temperature distance of 
2
?
10T?=? ?????????????125 
   
   
 
 ix 
 
Figure 6.14 Density profiles for n-heptane in ethane at the vapor-liquid 
interface, at a critical and average concentration of 
c
0.0202x = ,  
and at various temperatures, plotted with respect to height (z-
coordinate)??????????????????????.128 
   
Figure 6.15 Density profiles for ethane and n-hepane in ethane at the vapor-
liquid interface, at a constant temperature distance 
2
?
10T
?
?=? , 
and at various critical concentrations, plotted with respect to 
height (z-
coordinate)?????????????????????. 130 
   
Figure 6.16 Density profiles for ethane and n-hepane in ethane at the vapor-
liquid interface, at a constant temperature distance 
2
??
() ( ) 10Tx T?
?
?=?? (without Fisher Renormalization), and at 
various critical concentrations, plotted with respect to height (z-
coordinate)?????????????????????? 131 
 
 
 1 
 
Chapter 1: Introduction 
 
 The manner in which two phases separate from one another can have a 
significant impact on many engineering applications, whether these phases are the 
liquid and vapor phases of a pure (or one-component) fluid, or two phases of differing 
liquids.  Directly related to interfacial phase separation is the concept of surface 
tension.  Both interfacial behavior and surface tension are important for engineering 
applications when considering droplet formation, microemulsions, capillary action, 
nucleation, wettability, and fluid flow through micropores. 
1.1   The Smooth Interface 
 In the study of interfaces, Gibbs
1
 introduced the concept of the ?dividing 
surface,? which is an important baseline for establishing the location of where the 
surface of tension acts in a droplet.
2,3
  In some fluids and on larger scales, the location 
of the ?dividing surface? can be the same as the ?thickness? or width of the interface 
between two fluids, where the interface is only on the order of the size of a molecule 
of fluid.  This case is known as the ?sharp? interface, as the transition from one fluid 
phase to the other is very abrupt.  In situations where the molecular size of a fluid is 
significantly large or the physical fluid properties are near the critical point of 
separation, the transition between two liquid phases becomes much more gradual, or 
smooth, shown in Fig. 1.1.  In this figure, the darker color represents the gradient of 
the transition from one phase to another and the light color (or absence of color) 
indicates each of the bulk phases.  This smooth interface illustrated in Fig. 1.1 is 
 
 2 
 
present in soft matter (i.e. polymer solutions and microemulsions) and in fluids near 
the critical point of separation.
4-9
  
    
1.2   Surface Tension and Curved Smooth Interfaces 
Many chemical engineering applications involve the formation or use of small 
droplets.  For example, some targeted drug delivery systems utilize micro- or 
nanodroplets that form a flexible, biodegradable and hollow sphere into which a 
pharmaceutical can be inserted.
12,13
  These tiny hollow spheres can be created 
dissolving a polymer in an immiscible solvent; this solution is then emulsified to form 
small droplets.
14
  After the droplets are formed, the solvent is then removed from the 
polymer droplet through solvent evaporation, leaving behind hollow microspheres.
14
 
Another example is the formation of aerosol micro- or nanodroplets for use in 
coatings.  Ceramic coatings, due to their insulation and thermal properties, are of 
interest in many engineering applications.  In order to create very thin ceramic films, 
Figure 1.1. Simulated 
curved smooth interface of 
vapor-liquid separation in a 
droplet.  The interface was 
simulated  using toolboxes 
in MATLAB developed by 
Adomaitis
10
 and Adomaitis 
and Chen.
11
 The darker 
color indicates the steepest 
gradient of the interface of 
separation and the lightest 
color indicates the bulk 
phases on either side of the 
interface. 
 
 
 3 
 
small droplets must be used to create an ultra-fine spray.  In one specific study, 
conducted by Balachandran, Miao and Xiao,
15
 electrostatic atomization was used to 
generate 4-5 ?m droplets of suspended ZrO
2
 and SiC to create a homogeneous thin 
ceramic film on alloy substrates.  Although there are countless other applications, 
given these engineering examples, it is important to recognize and account for the 
presence and behavior of smooth interfaces, especially in very small droplets.  
 Determining the surface tension at the separation boundary of small droplets 
that exhibit a smooth interface is not as straightforward as it might seem.  In 
particular, the effects of curvature are typically considered negligible for surface 
tension on the macroscale.  However, the curvature correction becomes increasingly 
more significant as applications and processes are scaled down, as has been the trend 
in many fields of engineering.  As a result of downscaling, different physical effects 
and properties become more pronounced.  One such effect is the curvature-correction 
to the surface tension. 
In principle, the surface tension of a curved interface will behave differently 
than that of a planar interface,
2,3,16-31
 but this curvature correction is typically ignored 
in engineering practice.  This is mainly due to differing opinions regarding the net 
effect of curvature on the surface of tension, and as a result, the magnitude of this 
correction has been the subject of controversy for several decades.
2,12,18,20,22,24-28,30,32-43
 
In addition, when the mean-field approximation is used, the correction to the surface 
tension is insignificant, even when considering a nano-sized droplet.
44
  However, 
more recent studies have shown convincing evidence to the contrary, indicating that 
this curvature correction can significantly alter surface tension calculations.
45
 
 
 
 4 
 
1.3   Asymmetry and Its Effect on Interfacial Profiles 
 Another important concept in engineering applications is the impact of fluid 
asymmetry on the behavior of an interface.   Whether or not a fluid is symmetric or 
asymmetric simply refers to the shape of its coexistence curve.  If the phase 
coexistence is a symmetric parabola, the fluid is symmetric.  Conversely, if the phase 
coexistence does not exhibit this symmetry, it is considered asymmetric.  A simple 
illustration of an asymmetric phase diagram can be seen in Fig. 1.2.  If the phase 
coexistence was completely symmetric in this figure, the coexistence would be a 
symmetric parabola about the critical point, denoted by the blue dot and critical 
density, 
c
? . 
 
 Fluid asymmetry can be inherent both in pure (or one-component) fluids as 
well as in fluid mixtures.  The addition of a small amount of solute to a one-
component fluid with a vastly different molecular volume and chemical 
characteristics can result in new fluid asymmetry which can significantly impact the 
behavior of the mixture?s interfacial profile.  
Figure 1.2.  Schematic of an 
asymmetric vapor-liquid 
coexistence curve.  The 
rectilinear (actual) diameter is 
given by the dashed green line.
 
 5 
 
Simple models, such as the lattice-gas model
46-48
 offer means to model 
symmetric fluids and fluid mixtures, but do not account for asymmetry.  Equations of 
state for a particular substance can account for fluid asymmetry, but may not 
accurately account for fluctuations near the critical point and can be particularly 
complicated if a significant amount of accuracy is desired.  Therefore, we must look 
to a method that accounts for both asymmetry and fluctuations near the critical point 
of separation. 
1.4   Methodology and Overview 
 In order to develop expressions to determine both the curvature correction to 
the surface tension and interfacial behavior of an asymmetric fluid, we must utilize a 
method that accounts for both fluctuations near the critical point of separation as well 
as fluid asymmetry.  A method that has been shown to accurately model this behavior 
is known as ?complete scaling,? developed by Fisher and Orkulas
45
 and Kim et al.
50
  
 Prior to exploring the results of complete scaling, the simple behavior of 
mean-field interfacial profiles will be addressed.  The expressions for the mean-field 
interfacial profile and surface tension will be developed in Chapter 2 through a 
mesoscopic thermodynamic approach that utilizes the Landau-Ginsburg local free 
energy functional.  As mean-field theories do not account for the effect of critical 
fluctuations, we must look to complete scaling to both account for these fluctuations 
and fluid asymmetry.  The development of expressions describing one-component 
and binary fluid behavior using complete scaling will be shown in Chapter 3.   The 
curvature correction to the surface tension, or Tolman?s length,
2
 and its implications 
 
 6 
 
will be discussed in Chapter 4.  The near-critical and general thermodynamic 
expressions for Tolman?s length will be presented as well.   
Once the complete scaling approach has been presented, the behavior of real 
fluids can be approximated.  As fluid asymmetry has been shown to be related to the 
Tolman length,
45,51
 we will look to highly asymmetric fluids, such as polymer 
solutions, to further explore the magnitude of this correction in Chapter 5.   Also in 
this chapter, the curvature correction will be addressed both near the critical point of 
separation as well as away from the critical region.  These results will be used to 
develop a crossover expression for the Tolman length in polymer solutions.   
As an alternative to the lattice-gas model, Chapter 6 provides the application 
of complete scaling to approximate the density and concentration profile behavior at 
the interface of asymmetric one-component fluids and dilute mixtures.  Based on a 
good understanding of symmetric interfaces, as presented in Chapters 2 and 3, 
asymmetry specific to a fluid or fluid mixture will be applied through scaling 
coefficients used in complete scaling.  Lastly, conclusions and potential future work 
will be presented in Chapter 7. 
 
 
 
 
 
 
 
 7 
 
Chapter 2: Mean-Field Approximation of the Smooth Interface 
 
 
The interface of phase separation provides insight as to how the surface of 
tension, a concept important to engineering calculations, will behave as it is highly 
dependent on intermolecular interactions.  As a result, not all interfaces are the same.  
A smooth interface exists when there is a gradual change in concentration at the phase 
boundary between liquid-vapor or liquid-liquid separation, and is frequently observed 
in soft matter.  Conversely, a ?sharp? interface exists when the change between two 
phases is very abrupt, much like a step function.  
Smooth interfaces are present in near-critical liquid-vapor and liquid-liquid 
interfaces, in polymer solutions and polymer blends as well as liquid membranes and 
vesicles.
4-6,9,52,53
  These interfaces can be represented by an interfacial density (or 
concentration)  profile, which will be used to determine a characteristic length scale 
or interfacial ?thickness.?  In order to determine this length scale, principles of 
mesoscopic thermodynamics will be utilized.  For initial estimations of surface 
tension, density and concentration profiles, the effects of fluctuations near the critical 
point of phase separation will be ignored and mean-field?or van der Waals?theory 
will be used to approximate a symmetric smooth interface.   
2.1  Determining the Local Helmholtz-Energy Density 
While conventional statistical mechanics has been the longstanding 
methodology and basis for mesoscopic thermodynamics, other approaches that 
 
 8 
 
emphasize universality can be successfully applied to describe critical phenomena 
and interfacial density profiles.  One such example of this approach is use of the 
Landau-Ginzburg functional.  As fluctuations are prevalent in mesoscopic 
thermodynamics, we can consider a local?or coordinate-dependent?density of an 
appropriate thermodynamic potential in order to smooth these inhomogeneities.
50-52
  
Because we are ultimately interested in calculating the surface tension, ? , we 
will start with the basic definition:  
 
(surface free energy)
(area)
d
d
? = . (2.1.1) 
We can choose an appropriate thermodynamic potential, which in this case is the 
Helmholtz energy, because it is related to the property of interest.  By using the total 
Helmholtz energy, A , the following expression can be developed from Eq. (2.1.1):  
 
[]
cB
()
local free energy density
dA
kT dz
dxdy
??
+?
??
?
==
?
, (2.1.2) 
where dxdy d=?
 
for the surface, 
c
?  is the critical density, 
B
k  is the Boltzmann 
constant, and T  is the temperature.  The remaining unknown is the expression for the 
local free-energy density, and in our case, that is the local Helmholtz-energy density.   
2.1.1  Landau Expansion and the Landau-Ginsburg Functional 
To determine the local free energy density, we will review some basic 
definitions.  The total Helmholtz-energy density, A? , can be written as the integral 
taken over the total volume of the system:
55
  
 ()AAdV??=
?
.
 (2.1.3) 
 
 9 
 
When integrated, the Helmholtz energy density can be considered as the sum of three 
terms, as indicated by the works of others:
57,58
  
 
00
() (, ) (, )
cc
AA T AT? ? ?? ?? ??=? + = + =  (2.1.4) 
where the terms 
0
??  and 
0
A  are two background terms that relate to the caloric 
properties and pressure, respectively.  The last two terms on the right hand side of this 
equation are not affected by density fluctuations and are analytic functions of 
temperature and density.  However, the first term, ( )A?? , is affected by local 
inhomogeneities, and, hence, density fluctuations and is therefore the function of 
interest.  
 Another important and related concept that addresses these inhomogeneities is 
the order parameter, ? . This position-dependent local order parameter is defined by 
an appropriate field-dependent thermodynamic potential.  In the one-component fluid, 
density is the property of interest as it exhibits strong local fluctuations.  Our 
dimensionless order parameter based on local density of the fluid is then defined as 
 
c
c
()
() ,
r
r
??
?
?
?
?
v
v
 (2.1.5) 
where ()r?
v
 
is the local density in three-dimensional space and 
c
?
 
is the critical 
density.  
The local Helmholtz-energy density can be expanded in powers of the order 
parameter and powers of gradient of the order parameter;
8
  
 
242
cBc 0 0 0
111
?
() [()] [()] () ,
2242
A k T a T r u r c dxdydz?? ? ? ?
??
?= ? + +?
??
??
?
rr
 (2.1.6) 
where the dimensionless temperature distance to the critical point, 
?
T? , is defined as 
 
 10 
 
 
c
c
?
TT
T
T
?
?? . (2.1.7) 
The remaining unknowns 
0
a
 
and 
0
u  are system-dependent coefficients, where 
0
u  
reflects the energy of interaction between fluctuations,
59
 and
 
0
c
 
is a constant related 
to the range of intermolecular interactions.
8
  These constants are also important in 
determining other thermodynamic quantities, and will be discussed in Section 2.5. 
Note that the result in Eq. (2.1.7) takes into account that at the conditions that 
define the critical point,
22
//0?? ????=? ?=, are met.  In Eq. (2.1.6), the first 
terms within the brackets are a result of Landau expansion in terms of ? , and the last 
term is the result of expansion in powers of the gradient ()/ddr?
v
.  The entire 
bracketed expression is known as the Landau-Ginzburg functional for the local 
Helmholtz-energy density.
8
   It should be noted that the order parameter symmetric, 
as is the Landau-Ginzburg functional. 
In this specific case, the interest lies with the height-dependent coordinate, z , 
which gravity selects as the direction normal to the planar surface.  Since only the 
bracketed portion of Eq. (2.1.6) can be re-written in terms of z , we can write the 
local Helmholtz-energy density, f  as 
 
2
24
000
111
?
[()] [()] [()] .
242
d
fz aTz u z c
dz
?
???
??
=? + +
??
??
 (2.1.8) 
Comparing this expression for the local Helmholtz-energy density to that in Eq. 
(2.1.6), you can see that this is merely the integrand in brackets in Eq. (2.1.6).  In the 
previous equation, the definition of the order parameter (Eq. (2.1.5)) in terms of 
z only becomes 
 
 11 
 
 
c
c
()
() .
z
z
??
?
?
?
?  (2.1.9) 
 The expression for the local Helmholtz-energy density in Eq. (2.1.8) can be 
simplified by carefully choosing values for the constants with physical meaning and 
by analyzing the derivatives, or conditions, of the order parameter and what it 
represents in terms of real physical properties.  As with other thermodynamic 
potentials, the first derivative of the local Helmholtz energy determines the conditions 
for equilibrium, or chemical potential, when equal to zero: 
Bc
//0df d k T??=? = . 
The second and third derivatives, when equal to zero, determine the spinodal (or 
region of instability) and critical point conditions, respectively.  In the case of the 
local Helmholtz-energy density, only the portion of Eq. (2.1.8) from the expansion of 
powers of the gradient (i.e. the first and second terms) will be considered:    
 
24
00 0
11
?
[ ( )] [ ( )]
24
faTz uz??=? + . (2.1.10) 
The equilibrium, spinodal and critical point conditions are then: 
 
1/2
00
0
0
?
()
   at     0
aT df
ud
?
?
??
??
==
??
??
 (2.1.11) 
 
1/2
2
00
sp
2
0
?
()
   at     0
3
aT df
ud
?
?
??
??
==
??
??
 (2.1.12) 
and 
 
3
0
CP
3
0    at     0
df
d
?
?
==, (2.1.13) 
 
 12 
 
respectively.  By applying the expression for the equilibrium condition in Eq. (2.1.11) 
to the local Helmholtz-energy density in Eq. (2.1.10), we can complete the square to 
simplify the equation so that 
 
222 4
00 0 00
11
()
44
fu u?? ?=??.  (2.1.14) 
The plot of this equation is given in Fig. 2.1, where  
0
9/16u = .   This value of 
0
u  
was chosen to match the equilibrium result in Eq. (2.1.11) along the mean-field 
coexistence curve for 
0
9/4a =  and facilitate completing the square in Eq. (2.1.14).  
This value is similar to the results of Anisimov and Sengers
60
 and Agayan,
58
 given the 
prefactors for each term in Eq. (2.1.8).  
 This ?extra? second term in Eq. (2.1.14) subtracted from the squared term 
represents the reference value of local Helmholtz-energy density.  As this term is 
independent of density, it will not affect the overall shape or behavior of a 
dimensionless density profile.  Therefore we will consider only the first term in Eq. 
(2.1.14), re-writing the expression as 
 
222
00 0
1
()
4
fu??=?. (2.1.15) 
The plot of Eq. (2.1.15) is given in Fig. 2.2.  As expected, there is overlap at 0? =  
for sets of 
?
 T??
 
values when the reference value of local Helmholtz-energy density 
is included. In comparing these figures, it is important to note that the minima on the 
axis of the order parameter and shape of in each of the plots for a given temperature 
distance remain the same.  
 
 13 
 
 
 
Now that the first term in the local Helmholtz-energy density in Eq. (2.1.8) 
has reasonably been simplified to that in Eq. (2.1.15), the new full expression for the 
local Helmholtz-energy density becomes 
?
0.01
?
0.005
?
0
?
0.005
?
0.01
T
T
T
T
T
?=
?=
?=
?=?
?=?
?
Figure 2.1.  Local Helmholtz-
energy density from Landau 
expansion (in terms of the order 
parameter) given in Eq. (2.1.14)
for 
0
9/16u =
 
and varying 
temperature distances, 
?
T? . 
?
0.01
?
0.005
?
0
?
0.005
?
0.01
T
T
T
T
T
?=
?=
?=
?=?
?=?
?
Figure 2.2.  Local Helmholtz-
energy density from Landau 
expansion (in terms of the order 
parameter) given in Eq. (2.1.15) ?
which includes the reference 
value for the local free energy 
density ? for 
0
9/16u =
 
and 
varying temperature distances,
?
T? . 
 
 14 
 
 
2
222
00 0
11
[]
42
d
fu c
dz
?
??
??
=?+
??
??
. (2.1.16) 
Then the complete expression for the surface of tension (Eq. (2.1.2)) can be re-written 
as 
 
2
222
cBc 0 0 0
() 1 1
[]
42
dA d
kT u c dz
dxdy dz
?
?? ??
+?
??
? ?
?
??
== ?+
? ?
??
??
? ?
? ?
?
, (2.1.17) 
which will assist in simplifying future calculations.  
 
2.2  Approximating the Density Profile near the Interface 
 Now that the expression for the surface of tension has been established, we 
can determine the interfacial or density profile for a symmetric fluid or fluid mixture.  
How can this interface be determined?  The surface tension can be found by dividing 
the surface?or excess Helmholtz-energy?between two bulk phases, either vapor-
liquid or liquid-liquid phases.
56
  The inhomogeneous fluid maintains stability at the 
interface because the interfacial tension compensates for the unfavorable gain in 
Helmholtz energy between the two stable densities, explained by the two terms in the 
Landau-Ginzburg functional in brackets in Eq. (2.1.17).  To maintain stability, the 
generalized local chemical potential must be constant along the interfacial profile, 
meaning 
 
()
2
22
000
2
Bc
()
0
zdf d
uc
kT d dz
??
?? ?
?
== ?+ =. (2.2.1) 
As a result, the quadratic and gradient terms must be equal to one another when 
considering the fluid characteristics below the critical point: 
 
 15 
 
 
2
222
00 0
11
[]
42
d
uc
dz
?
??
??
?=
??
??
. (2.2.2) 
This shows that to minimize the surface tension and attain a finite interface, a balance 
between a sharp interface and an infinitely diffuse interface between the two bulk-
phase densities must be achieved.
56
  
 To determine the interfacial?or density?profile, Eq. (2.2.2) is simply solved 
in terms of the order parameter, or dimensionless density, ? : 
 
1/2
0
0
00
tanh
2
u
z
c
?
?
?
? ?
??
? ?=?
??
? ?
??
? ?
. (2.2.3) 
Eq. (2.2.3) can also be re-written as  
 
1/2
0
0 0
2
tanh
?
()
c
z
aT
?
?
?
? ?
??
? ?
=?
??
??? ?
??
? ?
. (2.2.4) 
This dimensionless profile can also be re-written in terms of the characteristic 
microscopic length scale related to the thickness of the interface, or ? , which for the 
mean-field approximation is  
 
1/2
1/20
0
0
?
()
?
2( )
c
T
aT
??
??
??
==??
??
??
??
. (2.2.5) 
  
Here, 
0
?
?
 is the mean-field amplitude of the correlation length below the critical 
point, 
 
1/2
0
0
0
2
c
a
?
?
??
=
??
??
, (2.2.6) 
 
 16 
 
and the bar denotes that this is the result for the mean-field approximation.  Now the 
dimensionless interfacial profile can be written in terms of a dimensionless 
characteristic length scale, /2z ? : 
 
0
tanh
2
z?
??
? ?
=?
? ?
? ?
. (2.2.7) 
The behavior of this mean-field profile can be seen in Fig. 2.3.  Notice that the 
phase with the higher density, or larger value of 
0
/??, has the lesser ?height? on the 
/2z ?  axis.  This represents the separation of two fluids with two differing densities.  
It should be noted that for the purposes of this analysis, the effect of gravity on an 
infinite planar interface, insofar that the height of the dividing surface along its 
horizontal axis will fluctuate as a result of capillary waves,
56,61
 will be ignored. 
 
2.3  Determining the Width of the Interface 
 As discussed previously, the characteristic length, ? , was chosen as the length 
scale to determine the ?thickness? or width of the interface.  There are methods to 
approximate the interface thickness; however, we can opt to choose multiples of this 
characteristic length as a unit to represent this width to model interfacial behavior, as 
the interfacial thickness may occur at a multiple
56
 of ? . The best and most accurate 
approximation of this ?thickness? from the mean-field approximation should be 
determined from the gradient of the interfacial profile, which will allow for better 
accuracy in calculating interfacial properties.   
 
 17 
 
 
The gradient, or the probability of the location of the interface, can be seen by 
calculating at the slope of this density profile with respect to dimensionless height: 
 
20
(/ )
sec h
(/2) 2
d z
dz
??
??
??
=?
??
??
.    (2.3.1) 
Figure 2.3.  Mean-field dimensionless density profile (a) and gradient (b) as a function of height.
The dimensionless density profile is given in Eq. (2.2.7) . 
0
?
?
0
(/ )
(/2)
d
dz
??
?
Phase I Phase II
(a)
(b)
 
 18 
 
By looking at this gradient, the concept of the smooth, diffuse, or ?fuzzy? interface 
can be better visualized.  As shown in Fig. 2.4, we see a smooth, gradual transition 
between both phases at their interface.   
Also, in the case of the mean-field smooth interface in Fig. 2.4, the steepest 
gradient, and therefore the majority of the interface, lies between  ?? , yielding an 
approximated interfacial thickness of 2? .  When looking at Fig. 2.3(b), it can be seen 
that nearly 95% of the gradient?s behavior is accounted for between  2?? .  Due to 
this difference, it is important to consider the behavior of the gradient when choosing 
the interface width in order to make the most accurate calculation.   
 
 
Figure 2.4.  Contour plot of the interface between two fluid phases as predicted by mean-field
theory.  The shading indicates a larger slope of the density profile.  The dotted red lines bound ??
and the solid blue line indicates /2 0z ? = . 
 
 
 19 
 
2.4  Position and Density Dependence of the Local Helmholtz-Energy Density  
We have discussed the equation for the symmetric smooth interfacial profile; 
however, it is important to remember that this determination was based on the local 
Helmholtz-energy density.  Therefore, the behavior of the local Helmholtz-energy 
density itself should be noted.  Considering the dependence on density and height, we 
can obtain a better understanding of the contribution of these individual effects. 
 Starting with the full equation for the local Helmholtz-energy density in Eq. 
(2.1.16), using the same approximation that the quadratic and gradient terms are equal 
as before, and inserting the expression for the interfacial profile in Eq. (2.2.7), we 
obtain 
 
2
2
0
4
0
1
tanh 1
22
fz
u
??
??
??
=?
??
??
????
. (2.4.1) 
This result is a universal dimensionless expression for the Helmholtz energy density.  
This is a universal equation as it is independent of temperature due to the arrangement 
of dimensionless groups.  Its three-dimensional graphical representation, illustrating 
the individual dependences on density and height can be seen in Fig. 2.5. 
 Looking more closely at this figure, one can observe the individual affects of 
height and density on the local Helmholtz-energy density (given by the broken blue 
line), each contribute to the resultant mean-field interfacial profile, represented by the 
solid red line.  The dotted gray and solid light blue lines show the individual effects 
of density and height, respectively, on the local Helmholtz energy density.   
 
 20 
 
 
 
2.5  Mean-Field Estimation of Interfacial Tension 
As we are ultimately interested in the behavior of the surface tension, there are 
several ways in which we can approximate the surface of tension based on the 
interfacial profile itself.  For now, mean-field theory will be used to estimate this 
important engineering property.  
 Since we have a mean-field expression for the smooth interface, we can now 
develop expressions to estimate the interfacial tension based on the thermodynamic 
0
/??
/2z ?
4
0
f
?
Figure 2.5.  Three-dimensional universal plot of the dimensionless local Helmholtz-energy 
density 
4
0
(/ )f ?  and its dependence on height and density. The dimensionless local Helmholtz-
energy density is given in Eq. (2.4.1).  The solid red line shows the density profile in Eq. (2.2.7)
and the dashed blue line indicates the dependence of f on both vertical position and density. 
The gray line with circles and the solid light blue line indicate the dependence of f  on density 
and vertical position, respectively. 
 
 21 
 
properties specific to the fluid.  Referencing Eq. (2.1.17), the mean-field 
approximation for the dimensionless surface tension is  
 
2
222
00 0
cBc
11
[]
42
d
ucdz
kT dz
??
??
?
+?
??
? ?
??
=?+
? ?
??
??
? ?
? ?
?
.  (2.5.1) 
In Section 2.2, it was discussed that the energy is minimized when the two terms 
within the integral were equal to one another.  Therefore, we can safely make this 
approximation by doubling either the first or the second term in the bracketed 
expression as follows:    
 
2
222
00 0
cBc
1
[]
2
d
udzcdz
kT dz
??
??
?
+? +?
?? ??
? ?
????
=?=
? ?
??
??
????
? ?
? ?
??
.  (2.5.2) 
For the sake of simplicity, we will use the quadratic term to evaluate Eq.(2.5.1):   
 
222
00
cBc
1
[]
2
udz
kT
?
??
?
+?
??
? ?
=?
? ?
? ?
?
.   (2.5.3) 
Since the integral will be evaluated in terms of ? , the second expression in Eq. 
(2.5.2) containing the gradient will be used to perform a change of variables, which 
results in  
    
()
1/2
22
0
0
0
1
2
d
dz
u
c
?
??
=
??
?
??
??
. 
It is important to note that while performing this change of variables, the limits of 
integration change sign because /ddz?  is negative (i.e. when height increases, the 
density of the fluid ( )?  decreases).  Substituting this information into Eq. (2.5.3) and 
integrating the expression for the dimensionless surface tension we obtain: 
 
 22 
 
 
()
1/2
3
00 0
cBc
22
3
uc
kT
?
?
?
= .    (2.5.4) 
 As mentioned in Section 2.1, these mean-field constants can be used to 
describe important thermodynamic properties that can be measured experimentally.  
The mean-field correlation length amplitude, given in Eq. (2.2.6) is one of those 
values.   Another important definition is that of phase coexistence or equilibrium, as 
given by Eq. (2.1.11) for the simple mean-field approximation.  Using this relation 
shown earlier, the mean-field condition  
 
1/2
cxc 0
?
BT? =?, (2.5.5) 
where the mean-field amplitude of the phase boundary parabola, 
0
B , is 
 
1/2
0
0
0
a
B
u
??
=
??
??
. (2.5.6) 
The last important property is the isochoric heat capacity, classically defined as 
 
2
2
2
V
SA
CT T
TT
?
?
????
??
==?
????
??
??
. (2.5.7) 
At the critical point, the isochoric heat capacity exhibits a finite jump at the critical 
temperature, which can be represented in the mean-field approximation by 
 
2
0
0
2
V
a
C
u
?= , (2.5.8) 
for the parameters in this work, similar to the methodologies use by others.
57,58,60,62
 
Substituting the thermodynamic properties in Eqs. (2.2.6), (2.5.6) and (2.5.8) into Eq. 
(2.5.4), the expression for the mean-field estimation of surface tension becomes: 
 
3/2
0
cBc
8
?
()
3
V
CT
kT
?
?
?
?
=??. (2.5.9) 
 
 23 
 
This result in Eq. (2.5.9) is the result for the mean-field approximation as 
determined by both Landau and Lifshitz
63
 and Rowlinson and Widom.
56
 The density 
gradient from Landau and Lifshitz?s
63 
work corresponds term by term with the 
expression in Eq. (2.2.2).  The work of Rowlinson and Widom
56
 defines the density 
of a fluid near the critical point, and the expression for the chemical potential of this 
fluid corresponds term by term to Eq. (2.2.1). These comparisons to both Landau and 
Lifshitz and Rowlinson and Widom are given in greater detail in Appendix A.  This 
mean-field result for the approximation of the surface tension will be compared to 
other methods that take into account the effect of fluctuations near the critical point in 
Chapter 3. 
   
 
 24 
 
Chapter 3: Accounting for Fluctuations near the Critical Point 
 
3.1  Effects of Critical Fluctuations and Fluid Asymmetry 
In the traditional approach of using the mean-field approximation, inter-
particle interactions are assumed to be identical and independent of coordinate.
64
 
However, near the critical point of phase separation, the correlation length of 
fluctuations exceeds the range of intermolecular interactions, and can significantly 
modify the physical properties of the system.  The result is that the interfacial profile 
becomes very smooth and therefore infinitely thick as the critical point is 
approached.
56
  This also has implications for the surface tension of a smooth interface 
as it becomes low as a result of strong fluctuations.
7
  Mean-field approaches to the 
thermodynamics of surfaces,
12,42,65-67
 as presented in Chapter 2, ignore the effects of 
these local fluctuations.   
A schematic illustrating the effects of these fluctuations near the critical point 
can be seen in Fig. 3.1.  The actual, or singular, diameter (indicated by the broken 
green line in Fig. 3.1) is curved near the critical point, significantly departing from 
the mean-field diameter (indicated by the dotted red line).  In this figure, you can see 
that the effect of critical fluctuations causes the actual diameter to be different from 
the rectilinear diameter, or arithmetic mean between vapor and liquid densities, and 
not only changes the shape of the coexistence curve near the critical point, but lowers 
the critical point as well.
68
 As our desire is to make accurate calculations and 
approximations in the critical region, methods that account for this effect must be 
applied.  
 
 25 
 
  
Another property that must be taken into account is fluid asymmetry, as it can 
impact the behavior of a fluid both near and far away from the critical point.  This 
asymmetry is caused by the difference in interactions between phases of a single fluid 
or the difference in interactions between two or more kinds of molecules; the larger 
the differences, the larger the asymmetry.   
In a symmetric system, such as the lattice-gas model with respect to density, 
the arithmetic mean of the coexisting densities, or ?diameter? of the coexistence 
curve, is a straight vertical line, indicating a symmetric parabola.  In an asymmetric 
system, this ?diameter? is often curved near the critical point and becomes a sloped 
straight line.
68
 The slope of this line can vary dependent on the shape of the 
coexistence curve, and an illustration of this concept can also be seen in Fig. 3.1.  
Note that for a symmetric system, the line of critical density (indicated by the broken 
black line in this figure) would also represent the arithmetic mean, whereas in an 
Figure 3.1.  Schematic of an 
asymmetric vapor-liquid 
coexistence curve comparing 
mean-field (red lines) and 
scaling (blue and green lines) 
approximations near the 
critical point.  The critical 
points are each represented by 
a circle.  The dotted red line 
bisecting the curve represents 
the rectilinear (mean-field) 
diameter, the curved broken 
green line represents the actual 
(singular) diameter and the 
broken black line represents 
the actual critical density. 
 
 26 
 
asymmetric system, the arithmetic mean (indicated by the green dashed line) would 
be sloped, similar to what is shown in Fig. 3.1.   
3.2  Scaling Theory 
 One method of accounting for the effects of fluctuations as well as fluid 
asymmetry is known as scaling theory.
69
 Scaling theory connects classical 
thermodynamics to experimental observations (thermodynamic space to experimental 
space).  This theory accounts for the fact that near the critical point, all physical 
properties follow simple power laws, also referred to as scaling laws,
60
 and the 
universal exponents used within these scaling laws are known as critical exponents.   
Although various fluids and fluid mixtures display a wide range of behaviors, 
these critical exponents for analogous properties remain universal.  Scaling theory 
establishes universal relationships between power laws, and a theory known as 
renormalization group theory provides a method to calculate these universal critical 
exponents. 
69
   The basis for both renormalization group and scaling theories is the 
divergence of the correlation length of the fluctuations in the order parameter near the 
critical point.  The expression for the near-critical correlation length has been 
determined to be
56,64,70
  
 
0
?
T
?
??
?
=?  (2.2.1) 
and Fisher and Zinn
70
 and Pelisetto and Vicari.
71
 have established 0.630? ? .  In this 
section and throughout this document, ?  means approximately equal and ? means 
asymptotically equal. 
 
 27 
 
3.2.1  General Scaling Equations 
The critical behavior of real fluids and their mixtures can be asymptotically 
described by scaling theory;
72
 in other words, a system?s actual behavior can be 
related to that of a symmetric system.  This behavior is described in terms of two 
independent fields, 
1
h  and 
2
h , and one dependent field, 
3
h .  The ?ordering? or 
?strong? field?the field that exhibits strong fluctuations?is represented by 
1
h ; the 
?thermal? or weakly fluctuating field is represented by 
2
h .  These fields are all 
functions of physical fields in real space and can be combined into a form that can be 
used in a general case for fluids, using thermodynamic potentials expressed through 
conjugate order parameters.  The general form for an equation of state for any fluid in 
terms of thermodynamic potentials is: 
 
2
1
32 2
2
h
hh f
h
?
??
?
?
??
??
? ??
??
, (2.2.2) 
where f
?
 is a scaling function.
51, 73
  Within this function, the superscript ? refers to 
the sign of the independent field 
2
h , where 
2
0h >  references the region above the 
critical point and 
2
0h <  references that below the critical point.  The critical 
exponents ? and ? are universal in terms of critical point universality; these exponents 
only depend on the type of scaling or modeling used and are independent of the 
specific fluid.   The Ising values of 0.109? ?  and 0.326? ?  have been well-
established and verified experimentally.
70,71,74-77
  
 Related to the two independent fields (
1
h  and 
2
h ) are two conjugate scaling 
densities, 
1
?  and 
2
? , where 
1
?  represents the strongly fluctuating order parameter 
 
 28 
 
and 
2
?  represents the weakly fluctuating scaling density.  These scaling densities are 
defined as: 
 
2
3
1
1
h
dh
dh
?
??
=
??
??
 (2.2.3) 
and 
 
1
3
2
2
h
dh
dh
?
??
=
??
??
. (2.2.4) 
These fields and densities can be expressed in a differential form,
51,73
  
 
31122
dh dh dh??=+. (2.2.5) 
There are also three scaling susceptibilities, or second derivatives defined in terms 
of scaling densities: ?
1
, the ?strong? susceptibility associated with strong fluctuations, 
similar to compressibility; ?
2
, the ?weak? susceptibility associated with weak 
fluctuations, similar to heat capacity; and ?
12, 
the cross susceptibility.  These are 
defined by the following equations: 
 
22
2
31
1
2
11
hh
dhd
dh dh
?
?
??? ?
==
??? ?
??? ?
 (2.2.6) 
 
11
2
32
2
2
22
hh
dhd
dh dh
?
?
??? ?
==
??? ?
??? ?
 (2.2.7) 
and 
 
11
2
31
12
212
 
hh
dhd
dh dh h
?
?
??? ?
==
??? ?
??? ?
. (2.2.8) 
Since we are ultimately interested in modeling the equilibrium conditions of 
the interface, the simplification for zero ordering field (h
1
 = 0) can be applied to the 
 
 29 
 
general scaling equations.  Zero ordering field exists along the critical isochore and 
coexistence curve as shown in Fig. 3.2.   
 
 
 
The expressions for the scaling densities (Eqs. (2.2.3) and (2.2.4)) and scaling 
susceptibilities (Eqs. (2.2.6), (2.2.7), and (2.2.8)) at zero order field become 
 
2
3
1202
1
?
(0)
h
dh
fh Bh
dh
??
?
??
?=? ??
??
??
 (2.2.9) 
 
1
30
2222
2
(2 ) (0)
1
h
dh A
fhh hh
dh
??
??
?
?
??
??
=?? ?
??
?
??
 (2.2.10) 
 
2
22 22
1
1202
1
?
(0)
h
d
fh h
dh
?? ???
?
+? +?
?
??
??=? ??
??
??
 (2.2.11) 
 
1
2
2202
2
?
(1 )(2 ) (0)
h
d
fh Ah
dh
???
???
??
?
??
=??? ?
??
??
 (2.2.12) 
and 
 
1
22
1
12 0 2
222
?
(0)       (for 0)
h
hh
d
fB h
dh h h
??
?
???
??
?=? ? <
??
??
. (2.2.13) 
T 
? 
CP 
Figure 3.2.  Schematic of a T-?
diagram indicating the location of
zero ordering field (h
1
 = 0). The 
solid line represents the 
coexistence curve and the dashed 
line represents the critical 
isochore; both lines together 
represent h
1
 = 0. 
 
 
 30 
 
Here, 
0
?
B , the critical amplitude derived from the order parameter; 
0
A
?
, the heat 
capacity amplitude; and 
0
?
?
? , the susceptibility amplitude; are all system-dependent.  
As before, the ?  superscript refer to 
2
0h >  and 
2
0h < , respectively.  The ? sign in 
Eq. (2.2.9) refers to the two branches of the order parameter, 
1
0h > and 
1
0h < , 
respectively, as they approach the limit 
1
0h =  at the phase boundary.  These 
approximations in Eqs. (2.2.9)-(2.2.13) were verified against the work of Agayan.
60
  
3.2.2  Complete Scaling for a One-Component Fluid 
 Now that the general scaling equations have been established, we can use the 
concept of complete scaling, as developed by Fisher
49
 and Kim, Fisher and 
Orkoulas
50
 to model the behavior of a pure fluid.  Compared to the symmetric lattice-
gas model, complete scaling has the ability to model the asymmetric behavior in real 
fluids though scaling coefficients.  As our interest lies in the region near the vapor-
liquid critical point, the independent theoretical scaling fields 
1
h  and 
2
h can be written 
as linear combinations of physical fields. Therefore, in linear approximation, the 
scaling fields become 
 
11 2 3
??
?ha aTaP?=?+?+?, (2.2.14) 
 
2123
??
?hbTb bP?=?+?+?, (2.2.15)
 
31 2 3
??
?hcPc cT?=?+?+?, (2.2.16) 
where the sets of coefficients 
i
a , 
i
b  and 
i
c  represent the effects of asymmetry 
dependent on the type of fluid.  The dimensionless physical fields in the previous 
equations are defined as 
 
 31 
 
 
c
Bc Bc
??                
kT kT
???
??
?
??? (2.2.17) 
 
c
cc
??
                 
TTT
TT
?
??? (2.2.18) 
 
c
cBc cBc
??
                 
PPP
PP
kT kT??
?
???. (2.2.19) 
 Although they may appear to be complicated, the expressions in Eqs. (2.2.14)-
(2.2.16) can be further simplified.  Careful choice of the value of the dimensionless 
critical entropy, 
c
?
S , where
51,68
  
 
1
1c
cBc
0,cB
cxc,c
?
?
()
?
h
SP dP
Sk
kT dT
?
?
=
??
?
??
== =
??
??
?
??
??
 (2.2.20) 
allows the simplification of 
3c
?
cS=? .   Using the value of the critical entropy in Eq. 
(2.2.20), the scaling coefficients can be further reduced.  From thermodynamics and 
in terms of these dimensionless variables used in this section, 
 
???
???dP d SdT?? ?=+ . (2.2.21) 
Considering this expression in zero ordering field (
1
0h = ), we find that  
 
1
1
0,c
0,c
?
?
?
??
c
h
h
dP d
S
dT dT
?
=
=
??
??
=+
??
??
??
??
, 
therefore 
 
1
0,c
?
0
?
h
d
dT
?
=
??
=
??
??
. (2.2.22) 
Also in zero ordering field close to the critical point,  
 
1
1
11
1
123
0,c
0, 0,c 0,c
??
?
0
???
h
hc hh
dh d dT dP
aaa
dT dT dT dT
?
=
= ==
?? ????
??
== + +
?? ????
??
??
??
?? ??
. 
 
 32 
 
Therefore, based on the results in Eq. (2.2.22) 
 
1
23
0,c
?
.
?
h
dP
aa
dT
=
??
=?
??
??
 (2.2.23) 
Since only two of the physical fields are independent in each of the equations for the 
thermodynamic fields, only two of the coefficients in each equation are independent.  
Eq. (2.2.14) can be significantly simplified
51,68
 as the coefficients 
1
a  and 
1
b can be 
absorbed by the scaling amplitudes in f
?
, allowing 
11
1ab==.  Therefore, the 
simplified scaling equations for a one-component fluid become 
 
13
cxc,c
?
??
?
?
dP
haP T
dT
?
? ?
??
? ?=? + ? ? ?
??
? ?
??
? ?
 (2.2.24) 
 
223
??
?hTb bP?=? + ? + ?  (2.2.25) 
and 
 
3
cxc,c
?
??
?
?
dP
hP T
dT
?
??
=? ?? ? ?
??
??
. (2.2.26) 
In many cases, the scaling coefficients 
3
a and either 
2
b or the sum 
23
bb+  can be 
approximated or determined experimentally. 
 We can use the definition in Eq. (2.2.20) and the result in Eq. (2.2.22) to re-
write the temperature derivative of 
2
h  in Eq. (2.2.15), which becomes 
 
1
1
2
3
0,c
0,c
?
1.
??
h
h
dh dP
b
dT dT
=
=
??
??
=+
??
??
??
??
 (2.2.27) 
Therefore, by integration of Eq. (2.2.27) and incorporating the results in Eq. (2.2.23), 
the results for zero ordering field are 
 
 33 
 
 
1
2
23
3
3c
0
?
1
???
? .
h
a
hTb
a
hP ST?
=
??
=? ?
??
??
=? ?? ? ?
 (2.2.28) 
It is important to note that the expression for 
2
h  in Eq. (2.2.28) can be further 
simplified by considering basic thermodynamic definitions in terms of the 
dimensionless variables established thus far: 
 
ccB?
?
??
?
??            and            
?
?
T
dP S dP
S
dkdT
?
??
?? ?
?? ??
== = =
?? ??
?? ??
. (2.2.29) 
Considering Eqs. (2.2.5) and (2.2.20), the differential expression for 
3
h  in Eq. 
(2.2.28) can be solved in terms of the dimensionless entropy density in Eq. (2.2.29): 
 
()
()
3c23c
c23c
?
???
???? ?
? 01
?
??
1
?
dh dP d S dT b S dT
dP d dT dT
SbS
dT dT dT dT
?
???
??
?
?
=?? =+ +
?? ?? ??
??
?? =+
?? ?? ??
??
??
?? ?? ??
 
 
()
c2 3c
?
?
???
? 1
?
dP
SSbS
dT
?
??
??
==++
??
??
. (2.2.30) 
This follows that by the definition of the dimensionless entropy density,  
 
() cc c
?????
?
SSSSS?????=?=? 
the result in Eq. (2.2.30) yields: 
 
() ( )23c
??
?
1SbS???=+ (2.2.31) 
indicating that the entropy density is proportional to the weakly fluctuating scaling 
density, 
2
? .  As shown here, and as discussed by Wang and Anisimov,
68
 the 
 
 34 
 
coefficient 
3
b  does not appear to play a significant role in affecting fluid asymmetry 
and was assumed to be zero.  However, later works by Wang et al.
51
 consider and 
include the effects of 
3
b  in an effective constant, 
eff
b such that 
 
23
eff
c3
?
1
bb
b
Sb
+
=
+
. (2.2.32) 
 
3.2.3  Complete Scaling for a Dilute Binary Mixture 
 Another case that can be modeled by complete scaling is a dilute binary fluid 
mixture.  As two components are now being considered, an additional field is 
considered in the linear combination of the scaling fields near the critical point, 
namely the exchange chemical potential (
21
?? ), such that 
 
112 3 421
??
??haTaPa??=?+?+?+? , (2.2.33) 
 
2213421
??
??hTb bPb??=?+?+?+? , (2.2.34) 
 
31 213 421
??
??hcPc cTc??=?+?+?+? . (2.2.35) 
Here, 
()
111cBc
? / kT????= ?  where 
1
?  is the chemical potential of the pure solvent 
and 
()
21 21 21c B c
? / kT????= ? , where 
21
?  is the difference in chemical potential 
between the solvent and solute, dependent on mole fraction.  Not shown are 
coefficients 
1
a  and 
1
b , as only three of the scaling coefficients in each scaling field 
are independent, as discussed in the previous subsection.  It should be noted that all 
scaling coefficients and the critical parameters depend on the location of the critical 
locus, which can be specified by any of the four critical parameters:
73
  
c
T , 
c
P , 
1c
?  and 
21c
? . 
 
 35 
 
 These scaling equations for a dilute mixture can then be written in terms of 
concentration and density.  Given that from thermodynamics 
 
121
???
????dP SdT d xd?? ? ?=+ + , 
where x  represents the mole fraction of solute, the expression for concentration of 
solute in terms of scaling fields and densities can be written as
51
  
 
c4142
122
1
x ab
x
b
??
??
++
=
++
. (2.2.36) 
The density and entropy density can be written as
51
  
 
122
31 32
1
?
1
b
ab
??
?
??
++
=
??
 (2.2.37) 
and 
 
c212
31 32
?
?
?
1
Sa
S
ab
??
?
??
++
=
??
, (2.2.38) 
respectively.  It should be noted that the full expressions in both Eq. (2.2.37) and 
(2.2.38) are identical to those for one-component fluid.
51,68
 Also, similar to a one-
component fluid as discussed previously, 
1
1c = , 
2
1c =?  and 
3
?
c
cS=? , while 
4c
cx=? .   
 As given by the expression for concentration in Eq. (2.2.36), asymmetry in the 
mixture must now be accounted for in the remaining coefficients.  As we are 
considering the dilute solution, where a very small amount of solute added to a pure 
fluid or binary fluid mixture, the coefficient 
2
b  will be approximated to be the same 
as for a pure fluid and unaffected by a small addition of solute.  However, the 
coefficients 
4
a  and 
4
b  must be calculated from properties of the mixture itself. 
 
 36 
 
Using a method developed by Anisimov et al.,
78
 
4
a  and 
4
b  can be 
approximated using the derivatives of 
1
h and 
2
h with respect to the mixture term, 
21
? .  
Along the critical line, 
1
h  and 
2
h vanish, therefore 
4
a  can be determined from Eq. 
(2.2.33): 
 
1123421
??
??0haTaPa??==?+?+?+?  
 
1c c c 21
234
21 21 21 21
0
ddTdPd
aaa
dddd
? ?
????
=+ + +  
To solve for 
4
a  in terms of dimensionless groups, we obtain: 
 
1c c c
42B3
21 21 c 21
1ddTdP
aaka
?
????
? ?
?+ +
? ?
? ?
?  (2.2.39) 
Note that also along the critical line, 
21 B c c
lnkT x? = , therefore: 
 
Bc21
c
kTd
dx x
?
= . (2.2.40) 
The unknown temperature and pressure derivatives in Eq. (2.2.39) can then be 
determined as  
 
cc cc
21 21 B c
dT dT dT xdx
ddxddxkT??
== (2.2.41) 
and  
 
cc cc
21 21 B c
dP dP dP xdx
ddxddxkT??
== 
 
cc c
c
cBc
?
dP x dP
x
dx k T dx?
= . (2.2.42) 
The unknown derivative 
1c 21
/dd?? can be found from the Gibbs-Duhem relation:  
 
 37 
 
 
ii
N d SdT VdP? =? +
?
 
 
121
d SdT VdP xd??=? + ? . 
Therefore, at critical conditions  
 
1c c c c c c 21
dSdTVdPxd??=? + ?  (2.2.43) 
Taking the derivative of Eq. (2.2.43) with respect to 
21
d?  the expression becomes 
 
1c c c
ccc
21 21 21
ddT dPdx dx
SVx
xdx
?
???
=? + ? . (2.2.44) 
As the reference value of entropy of the fluid is completely arbitrary (as only 
the value of the entropy difference is of concern), the critical value of entropy can be 
chosen in such a way that is simplifies the overall expression.  In this case, the 
definition in Eq. (2.2.20) will be used, where 
()
1
cB
0,c
??
/
h
SkPT
=
=?? .  Given this 
relation and noting that 
cc
1/V ?= , Eq. (2.2.44) becomes  
 
1
1c c c c
cBc
0,c21 c B c h
d x dT dPP
kT
d k T T dx dx
?
?
??
=
? ?
?
??
=? +?
? ?
??
?
??
? ?
? ?
. (2.2.45) 
Noting that the Krichevskii parameter is defined as 
 
1
cc
0,ch
dP dTP
K
dx T dx
=
?
??
??
??
?
??
, (2.2.46) 
Eq. (2.2.45) can be re-written as 
 
()
1c c
cBc
21 c B c
dx
KkT
dkT
?
?
??
=?. (2.2.47) 
The Krichevskii parameter represents is the difference in slope between the critical 
line and saturation curve and applies the physical characteristics of the mixture and its 
dependence on concentration of solute.   
 
 38 
 
Now that the derivatives have been determined in terms of known and 
measurable values, the expression in Eq. (2.2.23) can be re-written with dimensions 
as 
 
1
23
0,ccB
1
h
P
aa
kT?
=
?
??
=?
??
?
??
. 
Given this expression, Eq. (2.2.39) can be simplified as 
 
()
4c 3
?
11ax aK
? ?
?? + ?
? ?
, (2.2.48) 
noting that the dimensionless Krichevskii parameter is defined as
cBc
?
/KK kT?? . 
Similar to the method used previously for the scaling coefficient 
4
a , a 
relationship can be established between the scaling coefficient 
4
b  with respect to 
coefficients 
1
b , 
2
b  and 
3
b  along the critical line, where 
2
0h = , using the method 
developed by Anisimov et al.,
78
 this time in terms of dimensionless variables: 
 
c1c c
4B 2 3
21 21 c 21
1dT d dP
bk b b
dd d
?
????
? ?
?? + +
? ?
? ?
. (2.2.49) 
Using the derivatives previously determined in Eqs. (2.2.40), (2.2.42) and (2.2.47), 
the expression for the scaling coefficient 
4
b  can be written as  
 
()
cc
4c 2 3
??
?
1
dT dP
bx bK b
dx dx
? ?
?? + ? +
? ?
? ?
. (2.2.50) 
Depending on how the effects of 
2
b  and 
3
b  are to be considered, there are two 
possible treatments of their approximation.  As they cannot be determined 
independently by experiment, they can be considered in two cases: one approximation 
 
 39 
 
in which 
23
bb= , and in the other approximation, where  
3
0b = .  In the case of the 
approximation were 
23
bb= , Eq. (2.2.50) becomes 
 
cc
4c 2 1 23
??
?
1    (for 1 and ).
dT dP
bx bK b bb
dx dx
??
??
?? + ? + = =
????
????
 (2.2.51) 
Alternately, in the case of the approximation where 
3
0b = , Eq. (2.2.50) becomes 
 
()
c
4c 2 1 3
?
?
1    (for 1 and 0).
dT
bx bK b b
dx
??
?? + ? = =
??
??
 (2.2.52) 
These expressions can now be used to model the behavior of real dilute solutions, 
accounting for asymmetry. 
 Similar to the effect of solute concentration on the scaling coefficients 
4
a  and 
4
b , the scaling coefficients 
3
a  and 
2
b  can also be impacted by concentration. 
Cerdeiri?a et al.
79
 and Wang and Anisimov
68
 have shown that these scaling 
coefficients show a strong dependence on molecular volume.  Therefore, when there 
is a large difference in molecular volume between solvent and solute, especially when 
a large amount of solute is present, the calculation of fluid density is affected. 
Based on the findings of Wang and Anisimov,
68
 a potential solution to this 
issue is to approximate the dependence of 
3
a  on the change in critical volume with a 
change in critical concentration of a fluid mixture as 
 
0 cc
3c 3
c
()
x dV
ax a
Vdx
?+ , (2.2.53) 
where 
0
3
a  is the scaling coefficient of the pure solvent.  Note that the correction factor 
is added to the pure component.  This is because 
3
a  should increase if the critical 
 
 40 
 
density decreases when the concentration of solute is increased (i.e. 
c
/dV dx  is 
positive).  Conversely, an increased concentration of solute with a smaller molecular 
volume and larger critical density (i.e. 
c
/dV dx  is negative) will decrease the 
asymmetry of the solution.   
In a similar manner, we can approximate that the coefficient 
2
b  is also 
dependent on concentration as,  
 
0 cc
2c 2
c
()
x dV
bx b k
Vdx
?+  (2.2.54) 
where 
0
2
b  is the scaling coefficient of the pure solvent and k  is an empirical constant 
approximated to be ~1/3k  in this work, based upon the findings of Wang and 
Anisimov.
68
  Since the values of 
2
b  tend to be much smaller in magnitude
68
 relative to 
3
a  and due to a reduction by an empirical constant k , the coefficient 
2
b  may not have 
a significant impact on fluid asymmetry. 
3.3  The Smooth Interface 
Since the modeling of real fluids, which now takes into account the affect of 
critical fluctuations, the same should also be considered for the behavior of the 
smooth interface.  The mean-field approximation of the smooth interface was 
discussed in Chapter 2; however, the effect of critical fluctuations is not considered in 
this method.  To account for these fluctuations, and their impact on the smooth 
interface, the results of renormalization group theory can be applied.  In general 
terms, renormalization group theory utilizes the Ising model, accounting for 
fluctuations near the critical point by including the effect of directional spin 
 
 41 
 
densities.
56
  This theory shows that the universal expression for the density profile can 
be written in the form, 
 
0
(/2)z
?
?
?
=? , (2.3.1) 
similar to what was expressed in Chapter 2.  Ohta and Kawasaki
80
 suggested that the 
profile for the surface can be written as follows to more accurately fit the actual 
density profile for a given substance: 
 
1/2
2
2
(/2) tanh( /2)1 sech( /2)
3
a
zz z
a
?? ?
?
? ?
?=? + ?
? ?
+
? ?
 (2.3.2) 
where  
 
3
                                            4
6
ad?? ?==?, 
and d  is the number of dimensions.  The same coordinate system as before, where 
the more dense fluid lies in the z?  direction, is applied to Ohta and Kawasaki?s 
result in Eq. (2.3.2).  Note that when 0? =  (for 4d = ), the mean-field approximation 
result is obtained as in Eq. (2.2.7).  A comparison of these results can be seen in Fig. 
3.3, where the profile gradient is smoothed further when accounting for the effects of 
critical fluctuations.     
The interface thickness can be approximated in terms of ?  by assessing the 
magnitude of the profile gradient of the renormalization group theory results shown in 
Fig. 3.4.  This figure illustrates the dimensionless density gradient based on Eq. 
(2.3.2).  The shading indicates where the magnitude of the gradient is the greatest, 
from which we can estimate the interface thickness to be ?? , or 2? .   This can be 
 
 42 
 
confirmed by a visual inspection of Fig 3.3, where the largest gradient of the density 
profile resides between  ?? . 
 
 
Figure 3.3.  Comparison of mean-field and renormalization group theory density profiles as a 
function of height.  The mean-field theory profile (based on Landau expansion of the local 
Helmholz energy density, shown in Chapter 2) in Eq. (2.2.7) is represented by the dashed red line. 
The results renormalization group (RG) theory
76
 in Eq. (2.3.2) are represented by the solid blue 
line.  
0
?
?
Figure 3.4. Contour plot of the interface between liquid and vapor phases as predicted by 
renormalization group theory.
80
   The shading indicates a larger slope of the density profile.  The 
solid blue lines bound  ??  and the two solid red lines bound 2?? . 
 
 43 
 
 The significance of the interfacial thickness is highlighted by the example of a 
sub-micron droplet of water near the vapor-liquid critical point.  A simple simulation 
of this droplet was created using toolboxes in MATLAB developed by Adomaitis and 
Chen
10,11
 and is shown in Fig. 3.5.   Based on this simulation, the interfacial thickness 
20.2R? ? , therefore, for a droplet 200 nm in radius, the interfacial width would be 
40 nm.  Taking into account Eq. (2.2.1) for the correlation length of fluctuations in 
density, approximating the correlation length amplitude as molecular size 
(
0
0.1 nm? ? ), and the critical temperature of water (
c
647 KT ? ), this simulation 
represents a temperature ~ 0.2 K  from the critical point of water.    
 
 
 
It should be noted that since the thickness of the smooth interface is 
mesoscopic in scale and can extend from nanometers to microns, the droplet size 
cannot be defined smaller than this characteristic thickness, as there would be no 
Figure 3.5.  Simulation 
(applying toolboxes in MATLAB
developed by Adomaitis and 
Chen
10,11
) of a smooth interface
for a droplet near the critical 
point of fluid-fluid separation
using RG theory.
76
 The intensity 
of shadowing represents the 
square of the
density/concentration profile in 
Eq. (2.3.2) The ?thickness? of 
the interface, given by a 
characteristic decay-length scale 
of the profile, is indicated by 
2? . 
 
 44 
 
distinction between the behaviors of the bulk phases.  This thickness is dependent on 
the properties of the fluid, but in the case of the symmetric interface given in Figures 
3.3 and 3.4, the properties of the bulk phases are not apparent until  6?? . 
 
3.4  Estimations of Surface Tension 
 Similar to the calculation of surface tension for the mean-field approximation 
given in Chapter 2, definitions of correlation length and amplitudes will be employed. 
However, in these calculations, the effects of critical fluctuations will be accounted 
for.  The methodologies discussed will include the results from renormalization group 
theory, a local approximation of the slope of the renormalization group theory profile 
and dimensional scaling analysis. 
 
3.4.1  Renormalization Group Theory Results  
Not only does renormalization group theory provide the results for an 
interfacial profile, but this theory can also assist in the estimation of surface tension.  
This simple calculation can be made by using two definitions from this theory. The 
surface energy of an area of interaction, obtained from the Ising lattice model and 
renormalization group theory has been determined by Zinn,
81
  
 
()
2
00
Bc
0.377
kT
??
+
?  (2.4.1) 
where the +  sign indicates the value above the critical point.  This equation can be 
divided by the two-scale universality factor,
48,70
  
 
()
3
0c 0
0.171A ??
++
? , (2.4.2) 
 
 45 
 
which yields the following expression for the coefficient of surface tension, based on 
the renormalization group theory, for amplitudes above the critical point:  
 
0
cBc 0 0
2.20
kTA
?
??
++
? . (2.4.3) 
Similarly, we can calculate the surface tension above the critical point by using the 
ratio of correlation lengths above and below the critical point,
82
  
 
0
0
1.96
?
?
+
?
? , (2.4.4) 
and the ratio of the critical value of the heat capacity above and below the critical 
point,
70
 
 
0
0
0.523
 
A
A
+
?
? , (2.4.5) 
where ?  indicates the value above and below the critical point, respectively.  
Applying the ratios in Eqs. (2.4.4) and (2.4.5) to Eq. (2.4.3), the coefficient of surface 
tension for amplitudes below the critical point is 
 
0
cBc 0 0
2.26
kTA
?
??
??
? . (2.4.6) 
 
 
3.4.2 Dimensional Scaling Analysis 
 
We can also develop a more general result for the surface tension using 
dimensional scaling by estimating the surface tension as the product of the density of 
the Helmholtz-energy density and the width of the vapor-liquid interface.  Because 
we are only considering the behavior below the critical point along the coexistence 
curve, we only need to consider only one variable?in this case, temperature?as 
 
 46 
 
density along the coexistence curve is a function of temperature.  The Helmholtz 
energy per unit volume scales as
8
  
 
2
0
?
(2 )(1 )
AA
T
V
?
??
?
?
?
??
??
. (2.4.7) 
Therefore taking into account this relation, the expression for the correlation length in 
Eq. (2.2.1) and assuming the interface thickness is 2?  as before, the expression for 
the dimensionless surface tension can be approximated as 
 
2
0
0
cBc
2
?
2
(2 )(1 )
AA
T
kT V
??
?
??
???
?
??
?
?
?= ?
??
. (2.4.8) 
Noting that the universal exponents 
 2(1)d?? ????= ? =. (2.4.9) 
Therefore, the relation in Eq. (2.4.8) can be written in terms of the general exponent 
in Eq. (2.4.9): 
 
0
0
cBc
2
?
(2 )(1 )
A
T
kT
?
?
?
???
?
?
??
??
. (2.4.10) 
Note that van der Waals theory,
8,54,56
 which neglects critical fluctuations, predicts an 
exponent of 3/2? = .   
The expression for the dimensionless surface tension coefficient is 
 
0
cBc 0 0
1.19
kTA
?
??
??
? , (2.4.11) 
where in the case of 3d = , 
 
2
0
?
T
?
??=?. (2.4.12) 
Above the critical point, taking into account Eqs. (2.4.4) and (2.4.5), this expression 
for the surface tension coefficient becomes  
 
 47 
 
 
0
cBc 0 0
1.16
kTA
?
??
++
?  (2.4.13) 
If the surface tension is calculated for an interface thickness of  2?? , or 4? , these 
coefficients would be  
 
0
cBc 0 0
2.37
kTA
?
??
??
?  (2.4.14) 
and  
 
0
cBc 0 0
2.32
kTA
?
??
++
?  (2.4.15) 
for values below and above the critical point, respectively. 
 
Figure 3.6.  Representation of the interface thickness considered for the dimensional scaling 
analysis calculation of surface tension in Eqs.  (2.4.11) and (2.4.13)-(2.4.15). The profile plotted is 
the result from RG theory in Eq. (2.3.2).  The red box highlights the portion of the density profile 
used to calculate the surface tension in the dimensional scaling analysis between  ?? .  The green
box highlights the portion of the density profile between 2?? .   
0
?
?
 
 48 
 
A graphical representation of the estimation of the interfacial thickness for the 
calculations in Eqs. (2.4.11) and (2.4.13)-(2.4.15) can be seen in Fig. 3.6.   Note that a 
large portion of the profile slope is neglected by the small box, whereas the large box 
is more inclusive of the behavior of the gradient.  
 
 
Table 3.1.  Comparison of mean-field and scaling methods for the estimation of 
surface tension coefficients. 
Method 
Coefficients for Amplitudes 
below the Critical Point 
Equation 
Mean-field Approximation 
0
cBc 0
8
3
V
kTC
?
??
?
=  
(2.5.9) 
Renormalization Group 
Theory 
0
cBc 0 0
2.26
kTA
?
??
??
?  
(2.4.6) 
Dimensional Scaling 
Analysis ( 2? ) 
0
cBc 0 0
1.19
kTA
?
??
??
?  
(2.4.11) 
Dimensional Scaling 
Analysis ( 4? ) 
0
cBc 0 0
2.37
kTA
?
??
??
?  
(2.4.14) 
Experimental Results
a
 
0
cBc 0 0
2.15 2.89
kTA
?
??
??
?? 
N/A 
 
a
Reference 83 
 
3.4.3  Comparison of Methodologies 
 This section and Section 2.5 both address methods to approximate the surface 
tension at an interface.  A summary of these approximations is compared to 
experimental results in Table 3.1.  Note that the result of renormalization group 
theory is well within the experimental range of values for the amplitude coefficients.  
The dimensional scaling analysis with the large interface thickness ( 4? ) also 
 
 49 
 
provides an estimate close to experimental values as compared to the mean-field 
approximation and the results of the dimensional analysis with the small interface 
thickness ( 2? ).  These results highlight both the need to take into account the effects 
of critical fluctuations as well as a careful choice of the interfacial width.  
 
 
 50 
 
Chapter 4: Curved Interfaces and Surface Tension 
 
As discussed previously, surface tension is crucial for many engineering 
applications including droplet formation, microemulsions, nucleation, flow through 
micropores and capillary action.  Usually, the effects of curvature are ignored as they 
are considered negligible on the macroscale.  However, as processes are scaled down 
to consider small droplets, such as aerosol production or modeling of biological cells, 
surface curvature can play a more significant role on the calculation of surface 
tension. 
4.1  Tolman?s Length: Curvature Correction to the Surface Tension 
4.1.1  Young-Laplace Approximation of Surface Tension 
 A well-known starting point for the calculation for the surface tension is the 
Young-Laplace equation,
84
  
 
12
11
PP
RR
??
?
?
??
?= +
??
??
. (4.1.1) 
Here, P
?
 and P
?
 represent the pressures of the phases on the concave side (or inside) 
and outside, respectively, and the radii 
1
R  and 
2
R  represent the radii of curvature.  
The surface tension, ?
?
, is considered to be that of an infinite plane without 
curvature. 
 The Young-Laplace equation can be simplified for spherical bubbles and 
droplets when all radii of curvature are equivalent (
12
RR R==), therefore 
 
 51 
 
 
2
PP
R
??
?
?
?= . (4.1.2) 
As before, the pressures P
?
 and P
?
 represent the pressures of the phases on the inside 
and outside, respectively, of the bubble or droplet.  This is illustrated in Fig. 4.1.  
 
 
4.1.2  Expression for Tolman?s Length  
It is commonly believed that the effect of curvature on surface tension is 
negligible for calculations on the macroscale, therefore the surface tension of an 
infinite plane can be used for calculations on this scale as seen in Eq. (4.1.1).  Some 
have refuted the claim that there is any effect of curvature on the surface tension;
81-83
 
however, many believe that for very small droplets and bubbles the surface tension is 
affected by strong curvature.
2,16,20-23,26,27,44,88,89
  
In 1949, Tolman
2
 considered the effect of droplet size on the calculation of the 
surface of tension as previously described by Young and Laplace and determined 
there should be a correction parameter.  This parameter, known as ?Tolman?s length? 
Figure 4.1.  Simple schematic illustrating 
the properties of the variables used in the 
Young-Laplace relation for surface tension 
at the interface between two phases, ?  and 
? . 
R 
?
? 
 
 52 
 
(? ), represents the difference in radius between the equimolar surface (
e
R ) and 
surface of tension ( R ) in a droplet of liquid,
2,16,19,56
  
 
e
RR? =?, (4.1.3) 
and can be applied to the expansion of the planar surface tension as
2
 
 
2
() 1 ..R
R
?
??
?
??
=?+
??
??
. (4.1.4) 
This is commonly known as the Laplace-Tolman equation.  Note that Eq. (4.1.3) 
establishes the sign of the Tolman length as negative for droplets and positive for 
bubbles, which has been verified by other studies.
89
 An illustration of this concept for 
a droplet of liquid can be seen in Fig. 4.2.  As it can be seen from Eq. (4.1.4), the 
smaller the radius R , the more significant Tolman?s length becomes.  
 
 
 
Figure 4.2.  Sketch of the concept of the 
Tolman length ()? for a droplet of liquid, 
where R is the radius of the surface of 
tension and 
e
R is the radius of the 
equimolar surface. 
? 
R
e 
R
 
 
 53 
 
A more general equation for the surface tension was developed later by 
Helfrich,
89
 which includes a second-order correction due to curvature in the 
expansion of surface tension,  
 
2
2
() 1 ..
k
R
RR
?
??
?
??
=?+
??
??
, (4.1.5) 
where k  is a constant that accounts for the rigidity of the interface.  However, this 
equation is not widely accepted as studies indicate that higher order terms will not 
likely provide a more accurate result.
37
  Therefore, the focus of this study will be on 
the Laplace-Tolman expression in Eq. (4.1.4). 
4.1.3  Predictions of the Behavior of Tolman?s Length 
 Studies of the Tolman length have yielded various results.  For sharp 
interfaces (as mentioned in Chapter 2), where the vapor-liquid or liquid-liquid 
interface thickness is on the order of the size of a molecule, Tolman?s length is 
expected to be very small.  This is one of the reasons that a significant curvature-
correction has only been believed to affect nano-sized droplets.  
 The sign of Tolman?s correction to the surface tension has been a subject of 
debate as well.  Some studies have confirmed that the Tolman?s length is negative for 
droplets,
16,22,26,29,31
 while molecular dynamics simulations have indicated that it may 
be positive.
23,90,91
  Other studies have found that the sign of Tolman?s length cannot 
be de determined with much certainty,
20
 while Lei et al.
40
 and Guermer et al.
34
 cite 
that the sign is only negative for droplets that are not close to molecular size and, in 
the case of very small droplets, the surface tension actually vanishes.  
 
 54 
 
 There are also varying conclusions regarding the behavior of the Tolman?s 
length for a smooth interface near the critical point.  Square gradient theories have 
provided results consistent with mean-field theory,
43,44
 as discussed in Chapter 2.  It 
should also be noted that symmetric systems, such as the lattice-gas and regular 
solution models, do not exhibit an effect on Tolman?s length, partially as a result of 
the values of the mean-field critical exponents.  Gr?n?sy
26
 and Moody and Attard
29
 
predicted that the Tolman length would become increasingly negative when 
approaching the critical temperature.  Moody and Attard
29
 also indicate that the 
Tolman length may even change sign very close to or at the critical temperature.  
These findings further support the ambiguity of the Tolman length behavior near the 
critical point due to the effect of critical fluctuations.
43
   
Studies that acknowledge the existence of Tolman?s length have given many 
different results: Tolman?s length is finite, logarithmically divergent, or algebraically 
divergent with varying exponents.  Phillips and Mohanty
92
 suggested that the Tolman 
length diverges with the same exponent as the correlation length of critical 
fluctuations
()
0.630???? , while others made different predictions.  Fisher and 
Wortis
93
 predicted two terms: one weakly diverging term with an exponent of 0.065?  
and the other, more strongly diverging term with an exponent of 0.130? . Using the 
?penetrable-sphere model? developed by Widom and Rowlinson,
94
 and Rowlinson
19,35
 
determined that the exponent for the Tolman length showed a weak divergence 
()
0.065? .  These results are commonly accepted, as they are supported by an exact 
result using scaling obtained by Fisher and Wortis.
93
   
 
 55 
 
There have been more recent developments in scaling that more properly 
account for fluid asymmetry.  Fisher and Orkoulas
49
 recognized the correction needed 
to model this behavior and developed an approach known as ?complete? scaling, 
which was elaborated by Kim et al.
50
  Anisimov and Wang verified this approach and 
further developed it for convenient use in practical applications,
68,95
 and others have 
applied these results to binary fluids.
79
  Anisimov
45
 recently indicated that a complete 
scaling approach to calculating the near-critical Tolman length yields a much stronger 
algebraic divergence than shown in previous works, with an exponent of 
0.304? ???? .  This effect is the result of critical fluctuations and its amplitude is a 
function of fluid asymmetry; therefore this behavior does not exist in the mean-field 
regime.  This result showing a strongly divergent behavior indicates that the 
magnitude of the Tolman length may result in a significant curvature correction 
calculation in highly asymmetric fluids and fluid mixtures.  
 
4.2  Tolman?s Length Near the Critical Point 
As fluid asymmetry, in addition to curvature, impacts the Tolman?s 
length
19,43,93
 the method of complete scaling can be used to account for both fluid 
asymmetry and critical fluctuations as discussed in Chapter 3.  For a general 
derivation of the Tolman length using scaling theory, Anisimov
45
 related the pressure 
difference in the Laplace-Tolman equation to the susceptibility with respect to 
fluctuations in density.  
The pressure difference between the inside of a droplet of liquid and its 
surrounding vapor is derived from two parts: a symmetric part?which is independent 
 
 56 
 
of specific fluid characteristics?and an asymmetric part, which is specific to the 
fluid,  
 
()()
sym asym
PP PP PP
?? ?? ??
?= ? + ? . (4.2.1) 
As the Laplace-Tolman equation (Eq. (4.1.5)) accounts for both curvature and 
asymmetry in the Tolman length, ? , this expression can be used to estimate the total 
pressure difference as 
 
()()
sym asym
22
1PP PP
RR
?? ??
??
?
??
?+? ? ?
??
??
. (4.2.2) 
Since the Tolman length is attributed to asymmetry,
45
 we can assume that symmetric 
contribution to this pressure difference is related to the planar surface tension 
described in Eq. (4.1.2) 
 
()
sym
2
PP
R
??
?
?
?=. (4.2.3) 
Therefore, combining this equation with Eq. (4.2.2), the ratio of Tolman?s length with 
respect to the radius of surface tension becomes
45
  
 
()
()
asym
sym
2
PP
R PP
??
??
?
?
??
?
. (4.2.4) 
 Anisimov
45
 suggested that this same pressure difference can be approximated 
using the ?critical part? of the grand-canonical potential per unit volume ( /VP?=?)  
as 
cr
P? , multiplied by the correlation length, as 
cr
P?  is related to the surface tension.  
Since R  is the length scale in Laplace?s pressure difference equation, the correlation 
length, ? , is the corresponding length scale for the pressure difference 
cr
P? .  
Following the scaling approximation, 
 
 57 
 
 
()()
cr
sym
sym
~PP R P
??
??? 
and  
 
() ()
22
cr
asym
asym
~PP R P
??
???. 
Therefore, Anisimov?s
45
 result is obtained for Eq. (4.2.4) as 
 
()
()
cr
asym
cr
sym
2
P
c
P
?
?
?
?
??
?
 (4.2.5) 
where c
?
 is a universal constant, estimated as 0.6 0.7??  from scaling theory.   
Close to the critical point, 
cr
P?  can be approximated as
45
  
 ()
2
cr
P ????? , (4.2.6) 
where the total susceptibility with respect to density fluctuations is defined as 
 
2
2
T T
P?
?
??
??????
==
????
??
????
. (4.2.7) 
Just as with the pressure differences, the total susceptibility can be written in terms of 
a symmetric and asymmetric part, analogous to the pressure in Eq. (4.2.2) and the 
critical part of the grand-canonical potential in Eq. (4.2.5) so that 
 
sym asym
?? ?=+. (4.2.8) 
Therefore Eq. (4.2.5) for the Tolman length-correlation length ratio can be re-written 
in terms of the susceptibility in Eq. (4.2.6) as 
 
asym
sym
2
c
?
?
?
??
?? . (4.2.9) 
 In order to use complete scaling as discussed in Chapter 3, Eq. (4.2.9) must be 
re-written in terms of scaling variables.  The total susceptibility in Eq. (4.2.8) is also a 
 
 58 
 
combination of the strong, weak and cross scaling susceptibilities in Eqs. (2.2.11)-
(2.2.13).  Based on the results developed by Kim
96
 and Kim et al.
50
 for complete 
scaling, keeping only the lower order terms (
1
? , 
2
?  and 
2
1
? ) and eliminating any 
product terms (e.g. 
12
?? ), the following result for the dimensionless total 
susceptibility can be obtained: 
 
22
3 31 321 23 2 32312
?
(1)(1 )5 ()2(1)().aab bb abb????? ????+ + ? ++ ++ +
??
  (4.2.10) 
 To further simplify this equation, the strength of the divergence of each term 
will be compared, and only the most dominant terms will be kept.  In the case of the 
one-component fluid, in the first ? -approximation where 2 1.26? ?=? , the 
temperature diverges as 
 
0.913
11
10.348
21
0.109
2
1 0.674
12
~~
~~
~~
~~ .
TT
TT
TT
TT
??
??
?
?
??
??
?
?
??
?? ?
??
??
??
??
??
??
 (4.2.11) 
Given these results, the total susceptibility in Eq. (4.2.10) can be simplified to 
 
2
3311 32312
? (1 ) (1 3 ) 2(1 )( )aa abb??? ??+ + + + + .  (4.2.12) 
By applying the results of scaling for a one-component fluid in zero ordering 
field (
1
0h = ) where 
 
2
23
3
??
1
a
hTb T
a
??
=? ? ??
??
??
, (4.2.13) 
 
 59 
 
the order parameter and strong and cross scaling susceptibilities given in Chapter 3 in 
Eqs. (2.2.9), (2.2.11) and (2.2.13) can be approximated as 
 
10
10
1
12 0
?
?
?
.
BT
T
BT
?
?
?
?
?
??
?
?
?
??
?? ?
??
 (4.2.14) 
Substituting these relations in Eq. (4.2.14) into the expression for the total 
susceptibility in Eq. (4.2.12) results in 
 
1
2
330 0 3230
???
? (1 ) (1 3 ) 2(1 )( )aaBTT abbBT
?? ?
??
??
?
?+ + ? ?? + + + ? .  (4.2.15) 
In terms of the approximation where 
3
0b = , Wang and Anisimov
64
 determined the 
amplitudes 
0
?
B  and 
0
?
?
?  to be 
 
()
0
0
3
0
0 2
3
?
1
?
.
1
B
B
a
a
?
?
=
+
?
?=
+
 (4.2.16) 
This provides the result for the total susceptibility in terms of variables with 
dimensions:  
 
1
3
0020
3
3
???
? 12
1
a
TBTbBT
a
???
??
??
?
??
??
??? + ? ? ?
????
+
??
??
. (4.2.17) 
where the symmetric and asymmetric parts were determined to be
45
  
 
sym 0
1
3
asym 0 0 2 0
3
?
?
3
???
? 2.
1
T
a
BT bBT
a
?
?? ?
?
??
?
?
??
?
?? ?
?? ? ? ?
+
 (4.2.18) 
 
 60 
 
 Substituting these results into the expression for the Tolman length-correlation 
length ratio in Eq. (4.2.9) yields the new expression  
 
1
30
02
3 0
?
32
?
2
?
1
aB
cBTbT
a
???
?
??
?
?+
?
? ?
?? ? ? ?
? ?
+ ?
? ?
. (4.2.19) 
Applying the expression for the correlation length in Eq. (2.2.1), the expression for 
the Tolman length becomes
45
  
 
1
0
0ef0 2
0
?
3
?
?
2
B
caBTb T
? ??? ?
?
?
??
???
?
?
? ?
????
? ?
?
? ?
m , (4.2.20) 
where the prefactor m refers to a bubble of vapor or a drop of liquid, respectively and  
 
3
eff
3
1
a
a
a
?
+
. (4.2.21)
This expression indicates that there are two contributions to the Tolman length near 
the critical point: one that weakly diverges as 
1 0.065
??
TT
?????? ?
?=? (as shown by 
others
19,35,94
 ) and a new term from complete scaling that diverges more strongly, as 
0.304
??
TT
????
?=? . 
4.3  General Thermodynamic Expression for Tolman?s Length 
Anisimov
45 
and Wang et al.
51 
also suggested a general thermodynamic 
expression for the behavior of the Tolman length both near and far beyond the critical 
region.  Based on fluid asymmetry, the Tolman length with respect to the thickness of 
the interface is suggested to be
45,51
  
 
d
?
??2
??
???
?
?
?? ??
m , (4.3.1) 
 
 61 
 
where ????  and ??? are the densities in the liquid and vapor phases, respectively.  In 
this expression, the prefactor m  refers to a droplet of liquid and bubble of vapor, 
respectively based on the definitions of coexisting phases.  The ?diameter? of phase 
coexistence is represented by 
 
d
c
?
2
??
?
?
?? ??
=  (4.3.2) 
where ???  and ?? represent the liquid and vapor branches, respectively.  A simple 
schematic of this concept can be seen in Fig 4.3. 
 
 
Equation (4.3.1) can also be verified in the critical regime using the leading 
and dominant term from Wang and Anisimov?s
68
 expression for the coexistence 
diameter 
 
2
2
dd ef0
??
??1 aB T
?
???=?? ?  (4.3.3) 
?
T?
d
??
??
?? ?
?
Figure 4.3.  Sketch of the 
concept of using the coexistence 
diameter to approximate the ratio 
of the Tolman length to the 
interface thickness in Eq. (4.3.1). 
The green line represents the 
actual diameter, the black dotted 
line is the critical density and the 
blue line is the coexistence curve.
 
 62 
 
and the definition given by Wang and Anisimov
68
 for the difference in density along 
the two sides of the phase boundary 
 
0
c
??
2
BT
?
??
?
?? ??
??. (4.3.4) 
Substituting Eqs. (4.3.3) and (4.3.4) into Eq. (4.3.1) and approximating 2 / 3c
?
?  
yields the leading term in Eq. (4.2.19).  Additionally, Eq. (4.3.1) has also been 
verified for use in the mean-field regime for a van der Waals fluid.
45
 Since these 
relations have shown to be universal, they will be applied to asymmetric solutions in 
following chapters. 
 
 
 
 
 
 
 
 
 
 63 
 
Chapter 5:  Tolman?s Length for Polymer Solutions and Other 
Dilute Binary Mixtures 
 
 
5.1  Polymer Solutions: Theory and Background 
Semidilute polymer solutions tend to become highly asymmetric as the degree 
of polymerization?or number or monomer units interconnected?increases.  This is 
because as the length of a polymer chain increases, the polymer has less preference 
for the solvent, and therefore solubility decreases. 
5.1.1  Phase Coexistence 
There are two distinct regimes in polymer solutions: the ?critical? regime and 
the ?polymer? regime.  The characteristic variable, w , first introduced by Widom,
97
 
separates these two regimes and is defined as: 
 
1/ 2
1
?
2
wNT??, (5.1.1) 
where N  is the degree of polymerization of a polymer chain.  Widom?s variable 
provides an indication of where a particular polymer solution lies on a phase diagram.   
When 1w<< , it indicates a near-critical polymer solution, when the degree of 
polymerization is small and the distance to the distance to the critical temperature is 
also very small; this is termed the ?critical? regime.  Conversely, when 1w >>  (and 
N ??), it is only dependent on the degree of polymerization and is termed the 
?polymer? regime.   
 
 64 
 
 The overall behavior of polymer solutions in general can be visualized by a 
simple phase diagram.  This behavior will be modeled in terms of known limiting 
behaviors and established relations in both the polymer and critical regimes.   
Based on Flory theory, Povodyrev et al.
98
 indicated that at lower volume 
fractions of polymer, the relation between temperature and volume fraction becomes 
linear, and the polymer regime in the limiting phase behavior approaches 
 
2
1
3
T
?
?
=? + , (5.1.2) 
where ?  is the volume fraction of polymer in solution and T  is the solution 
temperature.  The theta temperature, ? , is the critical temperature of phase separation 
for an infinitely dilute polymer solution with an infinitely long polymer chain.   This 
temperature is specific to the properties of a specific polymer and solvent 
combination.  In terms of our temperature distance variable, 
?
T? , Eq. (5.1.2) can be 
re-written for the volume fraction of the solution: 
 
cc
3
?
1
2
TT
T?
??
? ?
=? ? ? +
? ?
? ?
. (5.1.3) 
This equation can be written in a more general form, since the expression 
c
/T ?  can 
be found in terms of the degree of polymerization from Flory
99
 theory as 
 
()
c
2
1
11/
T
N
?
=
+
. (5.1.4) 
Here, it can be seen that in the limiting case as the degree of polymerization increases 
( N ??), 
c
T ?? . 
 
 
 65 
 
 In the critical regime, analogous to the difference in density along the two 
sides of the phase boundary, Eq. (4.3.4) can be re-written in terms of volume fraction 
as 
 
0
c
??
2
BT
?
??
?
?? ??
?? (5.1.5) 
where ???  is the polymer-rich phase and ?? is the solvent-rich phase.  This result is 
similar to that presented by Widom
97
 and the experimental results presented by 
Dobashi et al.
100
 The amplitude 
0
?
B  can be determined for polymer solutions of 
moderate to high degrees of polymerization,  accounting for the results of Anisimov 
et al.
101
 such that: 
 
/2
00,1
?
N
BB N
?
=
? , (5.1.6) 
where 
0, 1
?
1
N
B
=
?  is determined from the three-dimensional Ising model, which takes 
into account critical fluctuations.
50
  The development and N-dependence of the 
critical amplitude in Eq. (5.1.6) will be discussed in greater detail in Section 5.2.  
Then, in terms of the Widom variable, w , Eq. (5.1.5) becomes: 
 
()
c
2 .          (critical regime)
2
w
???
?
?? ??
?  (5.1.7) 
In the polymer regime, based on the results from Anisimov et al.,
51
 the two sides of 
the phase boundary can be approximated as: 
 
cc
3
.         (polymer regime)
222
w
?? ?
??
?? ? ???
??  (5.1.8) 
Combining what is known for both the critical and polymer regimes in Eqs. (5.1.7) 
and (5.1.8), respectively, a simple crossover expression can be developed to model 
 
 66 
 
the behavior in both regimes as well as approximate the behavior in between.  This 
simple crossover expression for the phase boundary is then 
 
()
()c
3
21
2
.          (crossover)
2
12
ww
w
?
?
??
?
?
??
+
??
???? ?
?
??
?
??
+
??
 (5.1.9) 
Lastly, based on the works of Widom
97
 and Anisimov et al.,
101
 the solvent-rich 
branches in the critical and polymer regimes are given by: 
 
()
1/ 2
c
c
6/   ( 0)
  
1/         ( )
wN w
Nw
?
?
?
?
??
?
? ?
?
?
?
?
. (5.1.10) 
Therefore, a simple crossover expression can be developed to connect the polymer 
and critical regimes for the solvent-rich phase: 
 
()
c
6
.          (crossover)
16
w
Nw
??
?
? ??
+
 (5.1.11) 
The general behavior of a polymer solution with respect to temperature 
distance to the critical point and the volume fraction of polymer (? ) from Eqs.  
(5.1.3), (5.1.9) and (5.1.11) is presented in Fig. 5.1.  In this figure, the phase 
separation curve is represented by the solid blue line and the arithmetic mean of phase 
coexistence (or diameter, 
d
? ) is represented by the dotted green line.  Similar to the 
coexistence curve for an asymmetric fluid shown in Fig. 4.4 in Chapter 4, asymmetry 
is evident from the slope of the diameter.  The red line represents the phase-limiting 
behavior at an infinite degree of polymerization ( N ??), which terminates at the 
theta temperature, ? , as predicted by Flory theory.
99
    
 
 67 
 
Also, similar to the fluids described previously the same curved diameter is 
observed near the critical point, seen in the inset in Fig. 5.1.  As before, this curvature 
is a result of fluctuations as the critical point is approached. 
 
  
5.1.2  Fluctuations 
 Semidilute polymer solutions exhibit fluctuations in two different order 
parameters that become coupled and contribute equally: one is a scalar parameter 
Figure 5.1.  Phase diagram for a polymer solution (
4
10N = ) near the critical point.  The solid
blue line represents the phase separation curve. The dotted red line represents the phase-limiting
behavior at an infinite degree of polymerization, which originates at the theta point (or theta
temperature) as predicted by Flory theory.
99
 The coexistence diameter (
d
? ) is illustrated by the
green dotted line.  ???  and ?? represent the volume fractions of polymer-rich and solvent-rich
phases, respectively. The inset shows a region very near the critical point. 
?
T?
 
 68 
 
associated with the polymer volume fraction, and the other is a vector parameter 
associated with the behavior of long polymer chains.
102
 The former is an observable 
quantity, while the latter can be modeled by de Gennes? radius of gyration
103
 for a 
polymer chain, 
g
R , 
 
g
RaN= , (5.1.12) 
where a  is the step size in a random walk.  The radius of gyration will be the 
parameter representing the correlation length of fluctuations for the polymer chain 
order parameter, which is associated with the concentration of chain ends, or 
probability that the chain ends will meet one another.
103
  
Asymptotically close to the critical point of phase separation, the correlation 
length of concentration fluctuations will dominate and become larger than the 
polymer chain?s radius of gyration, 
g
R .  In this regime, a polymer solution behaves 
similarly to an ordinary fluid near its critical point.  As the solution approaches the 
theta point, the radius of gyration will become larger than the critical fluctuations, and 
the solution will exhibit theta-point tricriticality.
101
 This behavior is very close to 
mean-field behavior, with logarithmic corrections required as a result of 
fluctuations.
97 
These scaling concepts, along with crossover between critical fluid 
behavior and tricritical theta-point behavior, will be applied to calculate the interfacial 
thickness and Tolman?s length in polymer solutions in this chapter. 
The limiting behaviors that bound this analysis are low and very high degrees 
of polymerization as well as temperatures very near and far away from the critical 
point.  For the infinite polymer chain, the polymer volume fraction tends to zero as 
the correlation length, or radius of gyration, diverges.  This condition, when present 
 
 69 
 
with a temperature above the theta temperature (? )?where the second virial 
coefficient is zero
99
 ?is analogous to a critical state.  By appropriate tuning of the 
second virial coefficient, this critical state can be modified to a first-order phase 
transition where the polymer chain collapses onto itself and phase separation occurs.  
This separation would result in the introduction of a new concentration-based order 
parameter.   
5.2  Tolman?s Length in Near-Critical Polymer Solutions 
 Complete scaling is a universal approach to modeling fluid behavior, and 
individual characteristics are taken into account by the scaling coefficients.  
Therefore, this method will be used to model the behavior of a polymer solution and 
determine the Tolman length near the critical point of separation.  In the critical 
regime, where 1w<< , the Tolman length for a polymer solution will be determined 
from Anisimov?s
41
 result given in Chapter 4 (Eq. (4.2.20)): 
 
1
0
0ef0 2
0
?
3
?
?
2
B
caBTb T
? ??? ?
?
?
??
???
?
?
? ?
????
? ?
?
? ?
m . (5.2.1) 
5.2.1  Approximation of Scaling Coefficients 
 As the degree of polymerization and volume fraction affects the asymmetry in 
a semidilute polymer solution, these are the parameters that will be considered in 
determining the scaling coefficients in complete scaling.  To approximate these 
values, Flory theory
99
 of polymer solutions will be used, where the Gibbs energy of 
mixing, 
mix
G , is 
 
mix
B
(1 ) ln(1 ) ln (1 )
G
kT N
?
?? ????=? ?+ + ?. (5.2.2) 
 
 70 
 
and the polymer-solvent interaction parameter, ? , as presented in this chapter, is 
 
RT
?
? = . (5.2.3) 
The variable ?  represents the molecular interactions between solvent molecules, 
polymer molecules and each with themselves, T  is the solution temperature and R  is 
the universal gas constant.  Assuming a high degree of polymerization where 
N ??,  the theta temperature can be expressed as
99
 = 2 / R??, therefore the 
polymer-solvent interaction parameter can be expressed as  
  =
2T
?
? . (5.2.4) 
 Since the dimensionless chemical potential of a fluid mixture is an indicator of 
its asymmetry, Wang and Anisimov
70
 found the scaling coefficients 
eff
a  (from Eq. 
(4.2.21)) and 
2
b to be 
 
34021
eff
31130
??21
??135
a
a
a
??
??
==?
+
 (5.2.5) 
and 
 
4021
2
11 11 30
??11
?? ?5
b
??
?? ?
? ?
??
? ?
? ?
. (5.2.6) 
The notation 
??
??/
ij i j
ij
T???
+
=? ? ??  in Eqs. (5.2.5) and (5.2.6) represent the 
derivatives of the chemical potential taken at the critical point, where the volume 
fraction is analogous to density.    
 The Flory theory expression in Eq. (5.2.2) can also be written in terms of the 
chemical potential by taking the first derivative of the Gibbs energy of mixing with 
respect to solvent volume fraction: 
 
 71 
 
 
mix
B
Bc c
(/)
?
GkTT
kT T
?
?
?
??
?
==
??
?
??
 
 
()
c
11
? ln ln(1 ) 1 2
2
TN
TN N T
?
??? ?
???
??
=??++?
????
??
??
. (5.2.7) 
Noting that the dimensionless volume fraction is 
 
c
?
?
?
?
? , (5.2.8) 
the derivatives needed to solve for the scaling coefficients in Eqs. (5.2.5) and (5.2.6), 
evaluated at the critical point become 
 
()
()
cc c
10 c
ccc
222
2 c
20 c 2
cc
c
333
3 c
30 c 3
cc
c
40
??
?
(/ ) 1
?? 1
?
(/ )
1
?? 1
? 2
(/ )
1
?
TT
TT
TT
T
TNT
T
TN
T
TN
??????
??
?? ? ?
???
??
?? ?
?
???
??
?? ?
?
?
?? ? ?????
===+?
?? ??
??
?
??
?? ? ?
? ?
????
??
===?? ?
????
?? ?
????
? ?
? ?
??????
===+? ?
????
?? ?
????
? ?
?
=
()
()
()
444
4
c
c 4
cc
c
22
c
11 c c
cc c
233
2
c
21 c c 222
cc
c
?? 1
6
(/ )
1
??1
?
(/ )(/ ) 1
??1
? .
(/ ) (/ )
1
TT
T
TN
T
TT T N
T
TTTN
???
?
?? ?
?
???
??
?? ? ?
???
??
?? ?
?
? ?
????
?
== ?? ?
????
??
?? ?
????
? ?
??
===+
?? ? ?
??
===?
?? ??
?
 (5.2.9) 
The derivatives in Eq. (5.2.9) can also be written in terms of known quantities 
for polymer solutions, such as the degree of polymerization, N .  According to Flory 
 
 72 
 
theory,
99
 the critical temperature and critical volume fraction of a polymer solution 
are 
 
()
c 2
11/
T
N
?
=
+
 (5.2.10) 
and 
 
c
1
1 N
? =
+
, (5.2.11) 
respectively.  Applying these relations to the derivatives in Eq. (5.2.9) yields the 
following: 
 
2
30
2
40 2
11
21
(1 1 ) 1
? 2
(1 1 ) 1
? 6
?
? 0.
TNN
NN
TN N
N
NN
NN
?
?
?
?
?
?
??
++
=
??
??
+?
??
=
??
??
+
=
=
 (5.2.12) 
Finally, substituting these relations into Eqs. (5.2.5) and (5.2.6) gives the result for 
the scaling coefficients 
eff
a  and 
2
b  in terms of the degree of polymerization:  
 
eff
31
5
N
a
NN
???
=
??
+
??
 (5.2.13) 
 
2ef
NN
ba
NN
?
+
, (5.2.14) 
which is the result of Anisimov and St. Pierre.
104
  
 
 73 
 
Plots of Eqs. (5.2.13) and (5.2.14) can be seen in Fig. 5.2.  As shown in Fig. 5.2, the 
scaling coefficient 
eff
a  approaches 3/5 as the degree of polymerization increases.  
Also, the coefficient 
2
b  follows a power law function as it approaches an infinite 
degree of polymerization. 
 
 
Figure 5.2.  Plots of 
eff
a
 
and 
2
b  against inverse degree of polymerization (1/N) for the range  
N = 1 to 10
5
.  Plots for 
eff
a and 
2
b
 
are based on Eqs. (5.2.13) and (5.2.14), respectively. 
 
 
 74 
 
5.2.2  Approximation of Critical Amplitudes 
 The necessary critical amplitudes to calculate the Tolman length can also be 
determined for polymer solutions.  The amplitudes, not surprisingly, are also 
dependent on the degree of polymerization.  Anisimov et al.
101
 and Anisimov and 
Sengers
105
 determined that the critical amplitudes (as the average molecular volume 
becomes infinitely large) become 
 
1
(1 )
2
1
000
2
(1)/2
1/2
0
0
2
31/4 1/2
0
(1 )/2
1/2
00
2
6(6 )
N
ag
c
bN
B
c
vc
N
rc
c
?
?
?
?
?
?
?
?
?
??
?=
??
??
??
=
??
??
??
?
??
??
 (5.2.15) 
where 
0
a  and 
0
u  are system-dependent constants from Landau expansion discussed 
in Chapter 2, 
0
?  is a constant related to the second virial coefficient, 
0
v  is the 
molecular volume, c  is a system-dependent constant related to the Ginsburg number, 
0
r  is the ?bare? interaction range for 1N = , and 
0
g  is a universal numerical prefactor.  
The critical exponent ?  follows the Ising relation 2 2? ??++=; therefore, 
1.239? ? . Taking into account the definition of dimensionless susceptibility for 
polymer solutions, 
 
c
?
(/ )
?
?
T
??
?
?
?? ?
?
??
?
??
, (5.2.16) 
the expressions for the critical amplitudes can be made dimensionless and simplified 
to focus only on the dependence on the degree of polymerization, such that 
 
 75 
 
 
()
()
()
(1 )/2
0
0Bc0,1
2
c
(1)/2
0
00,1
c
(1 )/2 (1 )/2
00,1 0
1
?
11/
?
1
.
N
N
N
NN
kT
N
B
BB NN
NrN
?
?
??
?
?
??
?
?
?
=
?
=
??
=
+
?
?= =?
+
== +
??
 (5.2.17) 
Based on the approximation of 2c ?  for Flory theory,
101
  we can assume that the 
coefficient is of the order of monomer or fragment size where 
0, 1 0N
r?
=
? .  
 The coefficients in Eq. (5.2.17) can also be approximated from different 
theories.  Based on the Ising model for the mean-field approximation ( 1N = ), the 
amplitudes are 
0, 1
1
N=
?? and 
0, 1
3
N
B
=
? .
50,98
  However, as discussed previously, the 
value of 
0, 1N
B
=
 decreases to unity (
0, 1
1
N
B
=
? ) due to critical fluctuations in the three-
dimensional Ising model for short-range interactions,
50
 therefore this value will be 
used. 
5.2.3  Tolman?s Length near the Critical Point 
Since the critical amplitudes and scaling coefficients can all be approximated 
in terms of the degree of polymerization as shown in Eqs. (5.2.13), (5.2.14) and 
(5.2.17), we can determine the Tolman length for a polymer solution based on the 
degree of polymerization, temperature distance to the critical point using Eq. (5.2.1).  
This equation can be made dimensionless and universal for polymer solutions by 
dividing the Tolman length by the molecular size, 
0
r , such that 
 
1
(1 )/2 0
eff 0 2
00
3
2
B
cN a B T b T
r
? ??? ?
?
?
?? ???
?
?
? ?
???
? ?
?
? ?
m . (5.2.18) 
 
 76 
 
A three-dimensional graphical representation of the dimensionless Tolman 
length for a droplet of polymer in solution, based on these approximations and the 
assumption that 2 / 3c
?
? , is given in Fig. 5.3.  Notice that in this figure there is a 
dramatic increase in Tolman?s length close to the critical point and at a relatively high 
degree of polymerization (
4
10N = ).  The magnitude of the Tolman length also 
appears to be more strongly dependent on the degree of polymerization, and thus 
asymmetry, relative to temperature distance.  
 
 
Figure 5.3.  Three-dimensional plot of the dimensionless near-critical Tolman length for a droplet 
of polymer in solution (from Eq. (5.2.18)).  It is assumed 2/3c
?
?  and 
0, 1 0N
r?
=
? . 
?
T?
1/ N
0
r
?
?
 
 77 
 
 
5.2.4  Comparison of Terms  
 Anisimov?s
45
 expression for the Tolman length given in Eq. (5.2.18) can also 
be further simplified for a more simple approximation.  By comparing the 
contribution of each term in this equation where 
4
?
10T
?
?=
3
?
10T
?
?=
2
?
10T
?
?=
0
r
?
?
Figure 5.4.  Plots of the two terms from the dimensionless Tolman length for a droplet of polymer
in solution at three temperatures.  Shown here are Eqs. (5.2.18) and (5.2.19) for the full and partial 
expressions, respectively. 
 
 78 
 
 
(1 )/2
eff 0
1
(1 )/ 2
0
2
0
3
??
first term:  
2
?
?
second term:  ,
?
cN a B T
B
cN b T
??
?
?
? ? ?
?
?
?
?
?
???
?
?
?
??
?
m
 (5.2.19) 
we can determine a way to simplify the complete expression for Tolman?s length.  In 
the first term, temperature depends as
0.304
?
T
?
? , showing a much stronger divergence 
as compared to the weak divergence in the second term, 
0.065
?
T
?
? .  A visual 
comparison of the two terms is in Eq. (5.2.19) is made in Fig. 5.4.  In this figure, we 
can see that the contribution of the first term dominates by far, due to the magnitude 
of the critical exponents.  The first term diverges strongly while the second term is 
nearly independent of temperature and only indicates a slight difference at high 
degrees of polymerization.  As a result of this comparison, a simplification can be 
made for engineering approximations which takes only the first term into 
consideration.   
 The equation for the near-critical Tolman length (Eq. (5.2.18)) can also be 
expressed in terms of Widom?s variable w  in Eq. (5.1.1).  Considering Eqs. (5.2.13), 
(5.2.14) and (5.2.17) at a moderate to high degree of polymerization ( 100N ? ), these 
values become: 
 
eff
3
( 100) 
5
aN?? (5.2.20) 
 
2
3
( 100)
5
bN N??  (5.2.21) 
 
(2 )/2
00,1
?
( 100)
N
NN
???
=
????  (5.2.22) 
 
 79 
 
 
/2
00,1
?
( 100) .
N
BN B N
?
=
??  (5.2.23) 
The comparison of the results in Eqs. (5.2.20)-(5.2.23) for a moderate to high degree 
of polymerization, and the results for the full N-dependence equation in Eq. (5.2.17) 
for the amplitudes 
0
?
?
?  and 
0
?
B can be seen in Fig. 5.5.  Note that there is little 
difference in the plots for 
0
?
?
?  and 
0
?
B , which indicate little difference for values of 
100N ? . 
By applying the values in Eqs. (5.2.21)-(5.2.23), as well as the definitions of 
w  and radius of gyration, the expression for Tolman?s length in polymer solutions 
near the critical point becomes 
 
1
gef
3
2
cRa w w
? ??? ?
?
??
???
? ?
??
? ?
? ?
m  (5.2.24) 
when 
0, 1
1
N=
?? and 
0, 1
1
N
B
=
?  using the three-dimensional Ising model, and 
1
22 1
?? ???? ???
??.  The second of the two temperature-dependent terms in this 
equation diverges weakly, as 
10.065
ww
?????? ?
? , and the leading term diverges more 
strongly as 
0.304
ww
????
? .  A comparison of the contribution of these two terms 
indicates that the only term of significance is the leading term, especially when 
approaching the critical point as shown in Fig. 5.4.  Therefore, the expression for the 
Tolman length in the critical regime simply becomes 
 
eff g
.        (critical regime)aRx
??
?
?
? m  (5.2.25) 
which assumes 2 / 3c
?
? . 
 
 80 
 
 
5.3  Tolman?s Length at High Degrees of Polymerization  
 As the correlation and Tolman?s lengths at high degrees of polymerization 
(when 1w >> ) are expected to behave differently than in the critical regime, a new 
set of equations will be developed.  In the mean-field approximation, where 
1/2? ?== , Eq. (5.2.25) for the critical regime would be independent of temperature 
(
eff g
aR? ? m ), a significant departure from the results of complete scaling.  Some 
0
?
B
0
?
?
?
Figure 5.5.  Comparison of N-dependent amplitudes 
0
?
B  and 
0
?
?
? for a moderate to high degree of 
polymerization.  Eqs. (5.2.22) and (5.2.23) represent the high N estimate and Eq. (5.2.17)
represents the equation for all values of N. 
 
 81 
 
studies indicate similar mean-field-like results are expected for the polymer 
regime.
103,105,106
  
As discussed in Section 5.1, for an infinite degree of polymerization, the limit 
of phase separation approaches an asymptotic line that extends from the theta 
temperature, shown in Fig. 5.1.  At this limit of the infinite degree of polymerization, 
the radius of gyration of a polymer chain becomes significantly greater than the 
correlation length of critical fluctuations.  The correlation length at a high degree of 
polymerization, 
N
?
??
, then becomes dependent on the radius of gyration and is given 
by deGennes?
103
 scaling theory for the semi-dilute phase of a polymer solution: 
 
11
00
~~
22
N
rr
RT
?
??
??
??
??? ?
??? ?
??? ?
. (5.3.1) 
At a high degree of polymerization, the theta temperature becomes 2/R??= , and 
Eq. (5.3.1) becomes 
 
1
0
~2
N
T
r
T
?
?
?
??
?
??
??
??
. (5.3.2) 
At a high degree of polymerization when w??, since N ?? at any given 
temperature, we can approximate the theta point temperature distance as 
 
?
T
T
T
? ?
?? . (5.3.3) 
Therefore, Eq. (5.3.2) for the polymer correlation length can be expressed as 
 
1
0
?
2
N
rT?
?
??
??, (5.3.4) 
or in terms of the Widom variable (Eq. (5.1.1)) and radius of gyration (Eq. (5.1.12)) , 
the equation becomes
104 
 
 82 
 
 
1
gN
Rw?
?
??
? . (5.3.5) 
Here, it can be assumed that the size of the monomer, 
0
r , is on the same order as the 
random walk step size in a polymer chain (
0
ra? ).  This expression in Eq. (5.3.5)
supports a similar result obtained by Szleifer and Widom
102
 for the mean-field (Flory) 
approximation for semi-dilute polymer solutions.  
It should be mentioned that there is some disagreement over the proper 
temperature variable in the polymer regime.  De Gennes
103
 and Broseta
107
 stated that 
the temperature variable away from the critical regime ( 1w >> ) should be defined as 
cc
()/()TT T??? instead of 
cc
?
()/TTTT?? ?  as shown in this work and others
97,104
 
because it accounts for the temperature at which monomer-monomer interactions 
vanish.  However, as discussed further by Widom,
97
 the temperature variable 
?
T?  and 
related variable w  provide comparable results to this suggested temperature variable 
when compared to experimental results.   
 As presented in Chapter 4, the asymmetry of a solution as indicated by the 
coexistence curve can assist in determining the Tolman length within droplets of 
solution.  Moving away from the critical regime toward the polymer regime, the 
coexistence diameter becomes linear and its slope represents the asymmetry of the 
solution.  As previously indicated in Eq. (4.3.1), this diameter can generally be used 
to determine the Tolman-correlation length ratio; in terms of volume fraction, this 
equation becomes
104 
 
d
2
??
???
?
?
?? ??
m  (5.3.6) 
 
 83 
 
where ???  is the volume fraction in the polymer-rich phase and ?? is the volume 
fraction in the solvent-rich phase.  By this definition, the prefactor ""?  refers to the 
correction for a polymer-rich droplet in solution and the ""+  refers to the correction 
for a droplet of solvent.  Away from the critical point as the solution diameter 
(
ddc
????=?) for a highly asymmetric solution approaches one half of the distance 
between the coexisting phases (???? ?? ), the Tolman-correlation length ratio for 
polymer solutions in the polymer regime becomes
104 
 1
N
?
?
??
??
=
??
??
m . (5.3.7) 
This expression for 1w >>  is quite dramatic, as it indicates that the Tolman length is 
expected to be as large as half of the interface thickness, if the interface is defined as 
2? .  This result is consistent with Rowlinson?s
31
 and Anisimov and St. Pierre?s
104 
predictions that the Tolman length is expected to be on the same order as the interface 
thickness for small droplets. 
 
5.4  Crossover Expression for Tolman?s Length in Polymer Solutions   
 Now that the behavior of Tolman?s length has been determined for the 
limiting behavior in both the critical and polymer regimes, the behavior between 
these two limits can be modeled.  In order to achieve an understanding of the Tolman 
length in this ?unknown? region, simple crossover expressions will be developed 
based on the Widom variable, w , at the known behavior at the limit N ?? and 
approaching the critical point.  As before, the defining parameters for the two regimes 
 
 84 
 
are w  and the size of fluctuations in the polymer chain relative to the radius of 
gyration: 
 
g
g
Critical regime:   1  and  
Polymer regime:  1  and  .
wR
wR
?<< <<
<< ??
 
5.4.1  Crossover Correlation Length 
Since the basis of the approximation of Tolman?s length is relative to the 
thickness of the interface, which is based on the size of the fluctuations in a polymer 
solution, the first expression developed is the crossover correlation length.  As the 
expression for the correlation length of critical fluctuations in the critical regime 
given in Chapter 3 (Eq. (2.2.1)) , 
 
0
?
T
?
??
?
=? ,     (critical regime) (5.4.1) 
is universal for any fluid, we can apply this same expression to polymer solutions in 
the critical regime, taking in the system-dependent amplitude 
0
?  in Eq. (5.2.17).  In 
writing this expression in terms of Widom?s w , it becomes
104 
 
g
,          (critical regime)Rw
?
?
?
=  (5.4.2) 
since 2  1
??
? . 
 As discussed in the previous section, the limiting behavior in the polymer 
regime is given by Eq. (5.3.5).   Combining what is known of this behavior in the 
polymer regime with that is known for the critical regime in Eq. (5.4.2),  we propose 
as a simple crossover expression:
104
  
 
()
g
1
.        (crossover)
1
Rw
w
?
?
?
?
? ?
?
+
 (5.4.3) 
 
 85 
 
A simple check of this crossover expression yields the limiting expressions: for large 
values of w , ?
?
 is the result for the polymer regime (Eq. (5.3.5)) and for small values 
of  w , ?
?
 is the result for the critical regime (Eq. (5.4.2)). 
 
5.4.2  Crossover Tolman-Correlation Length Ratio 
 The crossover concept can be taken a step further and applied to the ratio of 
the Tolman length to the correlation length to obtain a good approximation of the size 
of the curvature correction relative to the interface thickness of a droplet of polymer 
in solution.  As before, this correction for a droplet of polymer-rich phase would be 
negative, whereas the correction to a solvent droplet would be positive.  Referencing 
Eq. (4.2.19) in Chapter 4, one of the universal equations for the Tolman-correlation 
length ratio,  
 
1
30
02
3 0
?
32
?
2
?
1
aB
cBTbT
a
???
?
??
?
?+
?
? ?
????
? ?
+ ?
? ?
m , (5.4.4) 
we can determine the ratio specifically for polymer solutions.  As before, the 
characteristic variable, w  and the system-dependent amplitudes and coefficients for 
polymer solutions for moderate degrees of polymerization (Eqs. (5.2.21)-(5.2.23)) can 
be applied to this equation, therefore: 
 
1
eff
3
2
2
ca w w
???
?
?
?
?
??
? ?
??
? ?
? ?
m . (5.4.5) 
Given that the first term diverges more strongly and the results of the term analysis in 
Section 5.2 show that the first term dominates by far, we can simplify the equation to:  
 
eff
,           (critical regime)aw
?
?
?
? m  (5.4.6) 
 
 86 
 
where, as before, we can assume
41
 2/3c
?
? .  Since the coefficient 
eff
3/5a ?  at 
moderate degrees of polymerization ( 100N ? ), the scaling coefficient can be 
approximated as 
eff
1a ? , as given by Anisimov and St. Pierre.
104
  Therefore, the ratio 
of  Tolman length relative to interfacial thickness in Eq. (5.4.6) simply becomes 
 .          (critical regime)w
?
?
?
? m  (5.4.7) 
 As the Tolman-correlation length ratio for the polymer regime has been 
established to be 
()
/ 1
N
??
??
= m  in Section 5.3, the following crossover expression 
between the polymer and critical regimes can be developed: 
 
eff
1/
eff
(1 )
aw
aw
?
??
?
?
?
??
?
??
+
??
m . (5.4.8) 
As presented by Anisimov and St. Pierre
104
 for a simple approximation at moderate 
degrees of polymerization, Eq. (5.4.8) can be simplified to 
 
(1 )
w
w
?
?
?
?
?
??
?
??
+
??
m . (5.4.9) 
This equation can provide a good approximation of the Tolman?s length correction to 
the surface tension relative to the correlation length of fluctuations.  
 
5.4.3  Crossover Tolman?s Length 
 Lastly, an expression that bridges the gap between the polymer regime limit 
and the critical regime limit can be developed for the Tolman?s length.  Combining 
the relationships developed in Eqs. (5.4.3) and (5.4.8), the crossover Tolman?s length 
expression becomes: 
 
 87 
 
 
()
eff g
1
1/
eff
(1 ) 1
aRw
aw w
??
?
??
?
?
? ?
?
++
m . (5.4.10) 
For the sake of simplicity, Eq. (5.4.10) can be written as 
 
g
1
(1 )
Rw
w
??
? ?
?
?
? +?
?
+
m , (5.4.11) 
as presented by Anisimov and St. Pierre.
104
  Again, these simple crossover equations 
can be tested by considering the critical and polymer regime limits.  When w  is very 
small, Eq. (5.4.10) reduces to the expression for the Tolman length in the critical 
regime in Eq. (5.2.25) .  Equation (5.4.11) reduces to the simplified expression 
 
g
,          (critical regime)Rw
??
?
?
? m  (5.4.12) 
where 
eff
3/5 1a ??.   When w  is very large, Eqs. (5.4.10) and (5.4.11) reduce to 
the expression for the polymer regime (Eq. (5.3.5)).   
 Figure 5.6 illustrates the universal Tolman?s length behavior for a drop of 
polymer-rich phase in solution as predicted by Eq. (5.4.11).  As expected, these 
results show a divergence of the Tolman?s length with increasing degree of 
polymerization and decreasing proximity to the critical point of phase separation.  
Further analysis of this methodology revealed that this crossover expression is 
in agreement with the selected characteristic variable, w .  Fig. 5.7 illustrates the 
crossover temperature dependence of the Tolman?s length between the critical and 
polymer regimes for a given degree of polymerization.  For the case of  
4
10N = , 
careful examination of the intersection of the lines tangent to the critical and polymer 
regime behaviors ( 0.304? ????  and 1? , respectively) showed the crossover 
 
 88 
 
temperature to be 
1.67
?
10T
?
?? , yielding a value of unity ( 1w ? ), which is in very 
close agreement with the definition used in Widom?s
97
 characteristic variable. 
 
  
Two simple examples further illustrate the effects of Tolman?s length as 
shown in Fig. 5.6.  Consider a polymer solution with a degree of polymerization 
4
10N =  and a monomer size (or random walk step size) of 
0
0.2 nmr ?  and a 
temperature distance from the critical point of separation, 
4
?
10T
?
?? .   When the 
random walk step size is approximated as the monomer size (
0
ar? ), a droplet of this 
Figure 5.6.  Universal behavior of the dimensionless Tolman length with respect to degree of 
polymerization and temperature distance to phase separation for a polymer-rich droplet as predicted 
by Eq. (5.4.11) 
1/ N
?
T?
0
r
?
?
?
 
 89 
 
polymer in solution would exhibit a radius of gyration of approximately 20 nm and a 
Tolman?s length of 100 nm? ?? , given Eqs. (5.1.12) and (5.4.11), respectively.   
 
 
Another example is that of a polystyrene-cyclohexane solution 3 K from the 
critical temperature,
101
 where 
2
?
10T
?
?? .  From Eq. (5.4.11), the Tolman length 
correction would be - 20 nm with an interface thickness ( 2? ) of approximately 40 
nm.  Therefore, for a droplet of polymer with a radius of 0.4 m? , there would be a 
10% increase to the surface tension, based on the Laplace-Tolman equation (Eq. 
(4.1.4)) in Chapter 4: 
?
T?
0
r
?
?
?
Figure 5.7. Dimensionless Tolman length for a polymer-rich droplet with 
4
10N =  exhibiting crossover 
between the critical and polymer regimes from Eq. (5.4.11). The intersection of the two tangent lines 
occurs at 
1.67
?
10T
?
?? , therefore 1w ? .
 
 90 
 
 
2
() 1 ..R
R
?
??
?
??
=?+
??
??
. (5.4.13) 
5.5  Tolman?s Length in Other Polymer Applications  
 As expected by the information and models presented thus far in this work, 
Tolman?s length is expected to be present in curved surfaces of asymmetric fluids, 
including polymer solutions.  Another application of this correction to asymmetric 
solutions is polymer blends.  As published by Leermakes et al.
108,109
 and van Male 
and Blokhuis,
110
 a three-component system?such as a polymer blend consisting of a 
solvent and two different polymers?can be reduced to a two-component system of 
two polymer-rich phases.  Should these polymers not exhibit symmetry with respect 
to the Flory model and have differing degrees of polymerization, the solution will be 
asymmetric and thus require a correction to the surface tension for small droplets of 
polymer.  
 
5.6  Near-Critical Tolman?s Length in Dilute Binary Mixtures 
 Similar to the information presented in this chapter for polymer solutions, the 
Tolman length near the critical point can also be determined for a particular dilute 
binary mixture exhibiting fluid asymmetry.  Considering the full expression for the 
near-critical Tolman?s length given in Chapter 4 (Eq. (4.2.20)),  
 
1
0
0ef0 2
0
?
3
?
?
2
B
caBTb T
? ??? ?
?
?
??
???
?
?
? ?
????
? ?
?
? ?
m , (5.6.1) 
we can approximate the curvature correction to the surface tension for small droplets 
or bubbles of dilute binary mixtures given the scaling coefficients and critical 
 
 91 
 
amplitudes that are specific to the solution of interest.  In Eq. (5.6.1), it should be 
noted that the coefficient
45
 2/3c
?
? , the prefactor m  refers to a bubble of vapor or a 
drop of liquid, respectively and  
 
3
eff
3
1
a
a
a
?
+
, (5.6.2) 
as previously defined in Section 4.2.  Considering only the first and strongly 
diverging term in Eq. (5.6.1), the simplified expression for Tolman? s length 
becomes: 
 
0ef 0
?
aB T
? ?
??
?
?
??m . (5.6.3) 
This simplified expression can easily be used to approximate the curvature correction 
to the surface tension with very few inputs. 
 As discussed in Section 3.2, the scaling coefficient 
3
a  for a one-component 
fluid can be affected by a change in concentration if a small amount of solute with a 
vastly different molecular volume is added.  The general expression for the 
concentration-dependence of the scaling coefficient is directly related to the change in 
critical volume relative to a the amount of solute added, such that 
 
0 cc
3c 3
c
()
x dV
ax a
Vdx
?+ , (5.6.4) 
from Eq. (2.2.53) . 
This concept was also shown for polymer solutions, where an increasing 
degree of polymerization?which is directly related to the molecular volume of the 
polymer?initially increased the size of the scaling coefficient 
eff
a  (Eq. (5.2.13)) at 
lower degrees of polymerization.  This behavior is analogous to adding a small 
 
 92 
 
amount of solute, with a large molecular volume, to a pure fluid with a small 
molecular volume.   Equation (5.6.3) indicates that as the scaling coefficient 
eff
a  
increases, so does the Tolman length.  If an application involves the creation, 
interaction with or use of small droplets or bubbles composed of an asymmetric 
mixture, Tolman?s length can potentially be a significant correction to surface tension 
calculations. 
 It should also be noted that the expression shown in Eq. (5.6.4) must only be 
used for dilute binary mixtures as it is a linear approximation of the solution behavior.  
Therefore, the behavior of each individual dilute mixture should be analyzed to 
determine the location of the linear region and its associated concentrations that can 
be used in this approximation.  This analysis and example dilute solutions of aqueous 
n-hexane and n-heptane-ethane will be discussed more in-depth in Chapter 6.  
 
 
  
 
 93 
 
Chapter 6:  Concentration and Density Profiles for Asymmetric 
Fluids and Binary Fluid Mixtures 
 
6.1  Background and Introduction 
Fluid asymmetry?which is primarily indicated by the shape of a one or 
multi-component fluid?has been linked to many properties including heat capacity
94
 
and the curvature correction to the surface tension, known as Tolman?s length.
45,104
 
Asymmetry, either inherent in a one-component fluid or caused by the addition of a 
dissimilar component, can also affect the shape of density and concentration profiles 
of a pure fluid or mixture.  A simple asymmetric vapor-liquid coexistence curve is 
illustrated in Fig. 6.1.  
 
 
 
 Monte Carlo, molecular dynamics and Renormalization Group simulations 
can provide accurate modeling of this asymmetry, as can traditional methods of 
statistical mechanics, but they are not always expressed in simple terms or easily 
Figure 6.1.  Schematic of an 
asymmetric vapor-liquid 
coexistence curve.  The 
rectilinear (actual) diameter 
is given by the dashed green 
line. 
 
 94 
 
solved.  Instead, we will once again look to the universal method of complete scaling 
as developed in Chapter 3 to model the behavior of a dilute fluid mixture by applying 
its system-dependent amplitudes and coefficients.  Specifically, the focus will be on 
modeling the increased asymmetry due to the addition of a small amount of solute to 
a one-component fluid that has a greater preference for one phase of the fluid over the 
other.   
 
6.2  Determination of Scaling Densities for Use in Complete Scaling 
 In order to determine the density and concentration profiles using complete 
scaling, two parameters must be determined: the order parameter and thermal scaling 
density.  To choose these parameters, we must look closely at the physical fields they 
represent.   
 A first step is to look at the complete scaling expressions for density and 
entropy density for one-component fluids and binary mixtures as presented in Section 
3.2.: 
 
122
31 32
1
?
1
b
ab
??
?
??
++
=
??
 (5.2.1) 
 
c212
31 32
?
?
?
1
Sa
S
ab
??
?
??
++
=
??
. (5.2.2) 
If we expand these two equations and neglect all terms of higher order than 
?
T? , the 
density and entropy density can be expressed as  
 
2
31 3 31 2 32
??1(1 ) (1 ) ( )aaa bb?? ? ? ??=??+ + + + +  (5.2.3) 
and 
 
 95 
 
 
()2
?
?
S????,  (5.2.4) 
which is identical to the result of Wang
62
 and Wang and Anisimov,
68
 with the 
exception that 
3
b  is considered nonzero in this portion of the analysis.   As a result of 
these expansions, we can see that the order parameter, 
1
? , is closely related to the 
density of the fluid or fluid mixture, and the thermal scaling density, 
2
?  is directly 
related to the entropy density.  Therefore, we must select an order parameter and a 
thermal scaling density that incorporates these real physical densities. 
6.2.1  Symmetric Order Parameter  
 In Chapter 2, the mean-field order parameter was simply defined as the 
density difference itself, 
1
???= .  This definition, however, does not take into account 
the effect of critical fluctuations.   
One method of addressing fluid asymmetry is to apply its affects, by way of 
the coefficients from complete scaling, to a symmetric interfacial profile that accounts 
for critical fluctuations as discussed in Chapter 3.   As shown in Chapter 3 and in Fig. 
6.2, critical fluctuations cause a more gradual and smooth transition from one phase 
to another, flattening the interfacial density profile.  One such example of a profile 
that accounts for critical fluctuations is the normalized symmetric density profile 
from Renormalization Group theory obtained by Ohta and Kawasaki
80
 presented in 
Section 3.3 of this work: 
 
1/2
2
2
(/2) tanh( /2)1 sech( /2)
3
a
zz z
a
?? ?
?
? ?
?=? + ?
? ?
+
? ?
 (5.2.5) 
where  
 
 96 
 
 
3
                 and                        4
6
ad?? ?==?, 
and d represents the number of dimensions of the system.  As in Chapters 2 and 3, 
this expression is a function of height, z , and correlation length, ? , as presented in 
Section 3.2: 
 
0
?
T
?
??
?
=? . (5.2.6) 
As before, only the height-dependent coordinate, z , will be considered when 
addressing the behavior and location of the interface. 
 
    
In our case, this normalized symmetric profile in Eq. (2.3.2) can be used to 
represent the behavior of an interface both near to and far away from the phase 
boundary.  In terms of complete scaling, we can chose this profile to represent the 
order parameter at the interface of separation, 
1
? , normalized to its behavior far from 
Phase I Phase II 
Figure 6.2. Normalized 
symmetric mean field and 
Renormalization Group
76
theory profiles  
?
 
 97 
 
that interface, 
1,
?
??
, as z ???.  Therefore, our equation for the normalized order 
parameter can be re-written as: 
 
1/2
21
1,
2
tanh( / 2 ) 1 sec h ( / 2 )
3
a
zz
a
?
??
?
?
??
? ?
=? + ?
? ?
+
? ?
 (5.2.7) 
where  
 
3
6
a ?=  
when 3d = .  Far away from the interface of separation, the profile behaves as the 
phase boundary for a symmetric fluid below the critical temperature where
64
  
 
1, 0
??
     (for )BT z
?
?
??
?? ? ???  (5.2.8) 
and the positive and negative values of the expression are accounted for by  z? .  As 
before, the amplitude 
0
?
B  is specific to the fluid being examined.  Also in this 
equation, the ? prefactor accounts for the two different phases and corresponds to the 
sign of z .  Close to the interface, the model is more dependent on the shape of the 
interface as it transitions between phases, provided in the RG theory profile.   The 
expression for the symmetric order parameter, 
1
? , becomes: 
 
1/2
2
11,
2
tanh( / 2 ) 1 sec h ( / 2 )
3
a
zz
a
?? ? ?
?
??
? ?
=?+ ?
? ?
+
? ?
 (5.2.9) 
where 
1,
?
??
 is given in Eq. (5.2.8). 
6.2.2  Thermal Scaling Density  
 To apply complete scaling and model the behavior of dilute mixtures, another 
important property must be determined: the thermal, or weakly fluctuating, scaling 
 
 98 
 
density, 
2
? .   It has been shown by Wang and Anisimov
68
 that the thermal scaling 
density is the entropy per unit volume; therefore the isochoric heat capacity can be 
used to evaluate this field.  Applying Wang and Anisimov?s
68
 expression for the 
isochoric heat capacity and neglecting higher order terms, as done by Wang,
62
 the 
isochoric heat capacity affected by fluctuations can be written as: 
 
()
0cr
cr
?
V
CATB
??
?
???. (5.2.10) 
In this expression, 
cr
?
B  represents the dimensionless critical background part of the 
heat capacity, where 
cr cr B
?
/BBk? .  The critical amplitude, 
0
A
?
, is the amplitude of the 
isochoric heat capacity below the critical point, specific to the fluid of interest.  Since 
the thermal scaling density is the entropy per unit volume, along the two-phase 
region, it can be expressed as: 
 
()
cr 0
2crcr
? ?? ? ? ?
.      (for  0)
1
V
C
A
SdTTBTB
T
?
?
?
?
?
?? = ? ? ? ? ? >
?
?
 (5.2.11) 
Note that this form of Eq. (5.2.11) is of a similar structure as the general 
expression for the thermal ordering field in Eq. (2.2.10) as presented in Section 3.2. 
As we are focusing on dilute solutions, we can make the approximation that 
2
?  is 
unaffected by a very small addition of solute and maintains the bulk characteristics of 
the one-component fluid.  The weakly fluctuating field, 
2
h , can be approximated 
simply as 
2
?
hT??  in the first approximation of the incompressible lattice-gas model, 
where chemical potential and pressure are constant.
62,78
  
 It should be noted that the expression in Eq. (5.2.11) as written only accounts 
for the effect of temperature on the profile. Therefore, the thermal scaling density 
 
 99 
 
must be modified by one additional term to take into account the effect of height, or 
z-coordinate.  This adjustment can be approximated by taking into account the form 
of the mean-field approximation where the mean-field thermal scaling density is the 
entropy, or 
0
/df dT :  
 
2
201
1
2
a??=  (5.2.12) 
(from Eq. (2.1.10) in Chapter 2) and the coefficient 
0
a  is a system-dependent 
coefficient from mean-field theory.  As a good model should reduce to the mean-field 
results in the 0? =  approximation (where 4 d? =?; d  being the number of 
dimensions), we expect the thermal scaling density to be of an equivalent form of Eq. 
(5.2.12).  Therefore, for the best approximation, we should multiply the thermal 
scaling density expression in Eq. (5.2.11) by the square of the normalized order 
parameter in Eq. (5.2.7) to account for the height dependence of the entropy as well 
as fluctuations.  The resultant expression for the thermal scaling density becomes: 
 
2
01
2cr
1,
?? ? ?
1
A
TT B T
?
?
?
??
?
?
??
??
??
??? ?
??
??
?
??
??
. (5.2.13) 
6.3  Modeling Density Profiles for One-Component Fluids 
 Since complete scaling can be used to model real fluids, accounting for 
critical fluctuations, but without requiring the development of complicated equations 
of state, this method will again be used to model the density profiles of real fluids.  In 
Sections 3.2 and 6.2, the density of a fluid or fluid mixture was described in terms of 
scaling coefficients specific to the mixture and scaling densities such that 
 
 100 
 
 
122
31 32
1
?
1
b
ab
??
?
??
++
=
??
, (5.2.1) 
where 
c
? /???= .  In order to calculate the density profile of a fluid, the scaling 
densities in Eqs. (5.2.9) and (5.2.13) developed earlier in Section 6.2 (and related 
equations) can also be applied.   
6.3.1  Application to Water  
 Pure water is an example of a fairly symmetric one-component fluid.  It does 
display some asymmetry, evidenced by the size of the scaling coefficients in Table 
6.1; however, these coefficients are still relatively small and have only a small effect.  
By applying the data in Table 6.1 to the density expression (Eq.(5.2.1)), along with 
the scaling densities in Eqs. (5.2.9) and (5.2.13) to model the behavior of water, we 
obtain the density profile for 
2
?
10T
?
?=? , shown in Fig. 6.3.  In this figure, we can 
see that the profile is nearly symmetric, as the profile for water (solid blue line) 
overlaps the symmetric profile (dashed red line). 
 
2
?
10T
?
?=?
Figure 6.3.  Density profile
for pure water at a
temperature distance of
2
?
10T
?
?=? ,  plotted with
respect to height (z-
coordinate).  Eqs. (5.2.1),
(5.2.9), and (5.2.13), and
experimental data from
Table 6.1 were used to
model the behavior of the
density profile.  The
symmetric profile is
indicated by the red dashed
line. 
 
 101 
 
Table 6.1.  Experimental data for pure water 
Variable Value Author 
3
a  0.0618 Wang and Anisimov
a 
2
b  -0.0482 Wang and Anisimov
a 
0
A
?
 13.95 Wang and Anisimov
a 
0
B  2.035 Wang and Anisimov
a 
00 3
?
/(1 )BB a?+ 1.9165 (this work) 
cr cr B
?
/BBk?  5.653 
Wang and Anisimov
a 
c
?  (mol/cm
3
) 
1.79
2
10
?
?    
Levelt Sengers et al.
b
  
c
T  (K) 647 Levelt Sengers et al.
b
  
0
?
?
 (?) 0.694 Bonetti and Calmettes
c 
 
a 
Reference 68 
 
b 
Reference 111 
 
c 
Reference 112 
 
Table 6.2.  Experimental data for ethane 
Variable Value Author 
3
a  0.0014 Wang and Anisimov
a
  
2
b  -0.0603 Wang and Anisimov
a
 
0
A
?
 11.79 Wang and Anisimov
a
 
0
B  1.649 Wang and Anisimov
a
 
cr cr B
?
/BBk?  6.856 
Wang and Anisimov
a
 
c
?  (mol/cm
3
) 
6.75 
3
10
?
?    
Harmin and Thodos
b 
c
T  (K) 305.3 Ambrose and Tsonopoulos
c 
0
?
?
 (?) 0.787 Bruice
d
* 
a
 Reference 68 
b
 Reference 113 
c
 Reference 114 
d 
Reference 115 
*Correlation length (
0
1.54?
+
? ) was approximated using CH
3
-CH
3
 bond length. 
 
 102 
 
6.3.2  Application to Ethane 
 Another example of a profile that displays little asymmetry is that of pure 
ethane.  For this particular fluid, the scaling coefficients 
3
a  and  
2
b , given in Table 
6.2, are very small, which indicate there is a small amount fluid asymmetry.  The 
density profile for ethane from Eqs. (5.2.1), (5.2.9) and (5.2.13) for a temperature 
distance  
2
?
10T
?
?=?  is illustrated in Fig. 6.4.  As expected, the profile of pure ethane 
(solid blue line) mostly overlaps the symmetric profile, given by the red dashed line.  
 
 
 
6.3.3  Application to n-Heptane 
 The final example of the density profile for a one-component fluid is that of n-
heptane.   The scaling coefficients 
3
a  and  
2
b  for n-heptane (given in Table 6.7) are 
much larger than the coefficients in pure water and ethane, so the expectation is that 
this density profile would exhibit greater asymmetry.  As before, Eqs. (5.2.1), (5.2.9) 
Figure 6.4.  Density profile 
for ethane at a temperature 
distance of 
2
?
10T
?
?=? , 
plotted with respect to height 
(z-coordinate).  Eqs. (5.2.1), 
(5.2.9) and (5.2.13), and 
experimental data from Table 
6.6 were used to model the 
behavior of the density 
profile.  The symmetric 
profile is given by the dashed 
red line 
2
?
10T
?
?=?
 
 103 
 
and (5.2.13) were used to model the density profile for n-heptane, given the data in 
Table 6.3; this profile can be seen in Fig. 6.5.  
 
Table 6.3.  Experimental data for n-heptane 
Variable Value Author 
3
a  0.369 Wang and Anisimov
a
 
2
b  0.0941 Wang and Anisimov
a
 
0
A
?
 22.61 Wang and Anisimov
a
 
0
B  1.843 Wang and Anisimov
a
 
cr cr B
?
/BBk?  18.04 
Wang and Anisimov
a
 
c
?  (mol/cm
3
) 
2.35 
3
10
?
?    
Bradford and Thodos
b 
c
T  (K) 540.6 Bradford and Thodos
b
 
0
?
?
 (?) 0.787 Bruice
c
* 
a
 Reference 68 
b
 Reference 116 
c
 Reference 115 
* Correlation length (
0
1.54?
+
? ) for n-heptane was approximated using the bond length  
for CH
3
-CH
3
. 
 
 
 
As illustrated in this figure, the large scaling coefficients for n-hexane?with 
the largest contribution from 
3
a ?lead to a highly asymmetric density profile.  Here 
we can see that at the temperature distance 
2
?
10T
?
?=? , the fluid prefers the liquid 
phase ( z? ) over the vapor phase ( z+ ), as the density difference between the bulk 
liquid ( 1z << ) and critical densities is much greater than the density difference 
between the bulk vapor ( 1z >> ) and critical densities.  This can also be seen by 
 
 104 
 
comparing the n-heptane density profile (blue line) to the symmetric profile (red 
dashed line) in Fig. 6.5. 
Understanding the nature of the asymmetry and effect of the scaling 
coefficients in these one-component fluids will provide insight into the behavior of 
binary mixtures as well.  These fluids, and the approximation of their scaling 
coefficients, will be discussed in later sections. 
 
  
 
6.4  Fisher Renormalization: Temperature Correction for Concentration 
 We have approximated key expressions for the scaling densities for one-
component fluids and fluid mixtures as well as density profiles for one-component 
fluids, but it is also important to maintain accuracy by applying the appropriate 
temperature scale, as temperature can be affected by concentration.   It is well-known 
Figure 6.5.  Density 
profile for n-heptane at 
a temperature distance 
of 
2
?
10T
?
?=? , 
plotted with respect to 
height (z-coordinate). 
Eqs. (5.2.1), (5.2.9) and 
(5.2.13), and 
experimental data from 
Table 6.7 were used to 
model the behavior of 
the density profile. 
The red dashed line 
indicates the symmetric 
profile. 
2
?
10T
?
?=?
 
 105 
 
that the critical temperature of a mixture is dependent on the concentration of each of 
its components.  This difference in temperature also depends on the similarity or 
dissimilarity between the properties of each component in the mixture.  Even the 
addition of a small amount of a disparate component to a one-component fluid can 
cause a dramatic difference in its critical temperature.  
One of the points of focus within this work is to determine the concentration 
and density profiles for dilute binary solutions and to determine the profiles? 
dependence on concentration.  As theoretical scales maintain chemical potential at a 
constant value, a correction, known as Fisher Renormalization
117
 must be applied to 
account for this difference of concentration (as opposed to chemical potential) at 
certain temperature distances.  Similar to earlier notation and definitions, the 
experimental and theoretical temperature differences are defined as 
 
cc c c
cc c c
 
() ( )
??
( )        and       ( )
() ( )
TTx x TT
Tx T
Tx x T
??
?
??
?= ?=
?? ??
==
, (5.4.1) 
respectively. 
The actual?or experimental?temperature difference can be applied to 
complete scaling by converting it to scaling space, which observes a constant 
chemical potential (? ).  Anisimov et al.
118
 determined the actual temperature 
difference, 
?
()Tx? , can be converted to the theoretical temperature difference, 
?
()T ?? , by the relation: 
 
1
2
?? ?
() () ()Tx T T
?
?
? ? ?
?
??
?=? +?
??
, (5.4.2) 
where the characteristic temperature, 
 
 106 
 
 
1/
2
2 c
20ccc
(1 )
dT
Ax x T
dx
?
?
??
? ?
??
=?
? ?
??
??
? ?
? ?
. (5.4.3) 
Based on the relation in Eq. (5.4.2), it can be seen that there is a significant 
difference between the experimental and theoretical temperature distances to the 
critical point where 
2
()Tx ?? ? .  It is possible to approximate this expression by 
comparing the relative magnitude of the terms by means of the characteristic 
temperature 
2
?  (e.g. if 
2
?
()Tx ??>> or 
2
?
()Tx ??<<).  However, at temperatures very 
close to the critical point, there may not be a large difference between these terms, 
and the full expression should be used for the best accuracy.  This solution to this 
nonlinear equation was approximated using the ?fsolve? function within the MATLAB 
optimization toolbox. 
An example of this difference between experimental and theoretical 
temperatures is presented by applying the experimental data for a dilute aqueous 
mixture of n-hexane given in Table 6.4 for an n-hexane concentration of 0.015x = .  
The value of 
0
A
?
 for pure water is used to approximate the behavior of this solution, 
given in Table 6.4, as we are considering a dilute solution of water and n-hexane. 
 
Table 6.4.  Selected experimental data for a dilute aqueous n-hexane solution 
Variable Value Author 
0
A
?
 (pure water) 13.95 Wang and Anisimov
a
  
T
c 
 (
hexane
0.015
n
x
?
= ) 641.600 K De Loos et al.
b 
c
()/dT x dx  
(dilute aq. n-hexane solution) 
-352.69 K Abdulagatov
c 
a
 Reference 68 
b
 Reference 119 
c
 Reference 120 
 
 
 107 
 
 
 
For the example of a dilute aqueous n-hexane solution, where for most 
temperature distances at dilute concentrations 
2
/()1Tx? ?<<, the second term in Eq. 
(5.4.2) is the main contributing factor and ( ) ( )Tx T????.  However, both terms in 
Figure 6.6.  Plots of the difference (a) and normalized difference (b) between experimental
( ()Tx? ) and theoretical ( ()T ?? ) temperature distances as predicted by Eq. (5.4.2) and data in 
Table 6.4 for a dilute aqueous n-hexane solution where 
c
0.015x = .  
(a)
??
() ( )Tx T????
??
() ( )
?
()
Tx T
Tx
????
?
(b)
 
 108 
 
this equation must be taken into consideration very close to the critical point, where 
2
/()1Tx? ??.  At first glance, it appears that the error decreases as the critical point 
is approached for this dilute aqueous n-hexane solution, shown in Fig. 6.6 (a).  
However, further analysis shown in Fig. 6.6 (b) shows otherwise; this error relative to 
the experimental temperature distance to the critical point becomes larger near the 
critical point.  This error is nearly 15% when 
4
() 10Tx
?
?=?.  Since the characteristic 
temperature, 
2
? , (Eq. (5.4.3)) is dependent on the concentration of the solution and its 
temperature derivative, the error could be larger or smaller for other solutions.  For 
example, a solution with a steep critical temperature derivative with respect to 
concentration would exhibit a larger correction factor.  As a result, when complete 
scaling is used for calculations very close to the critical point, it is crucial to 
determine the contribution of both terms in Eq. (5.4.2) for each solution of interest for 
the highest accuracy.  For the analyses contained in this work, both terms in the 
temperature correction will be used for the best accuracy at temperatures very close to 
the critical point. 
6.5  Modeling Concentration Profiles for Dilute Binary Solutions 
 As complete scaling has been successfully utilized to determine density 
profiles for one-component fluids, this method will also be used to model 
concentration profiles for dilute binary mixtures.  In addition, since the order 
parameter and thermal density (
1
?  and 
2
? , respectively) for a dilute mixture have 
been established in terms of complete scaling in Section 6.2, we can apply the affects 
of asymmetry?either inherent in the one-component fluid or due to the mixture?to 
 
 109 
 
model real mixtures.  In Section 3.2, the concentration of solute in terms of scaling 
fields and densities can be expressed as
47
  
 
c4142
122
1
x ab
x
b
??
??
++
=
++
 (5.5.1) 
where 
c
x  is the critical composition of the mixture.  The remaining unknowns are the 
scaling coefficients, 
4
a , 
2
b  and 
4
b , which are specific to the mixture of interest. 
As discussed in Section 3.2, the expressions for the ordering and thermal 
fields of a mixture are 
 
112 3 421
??
??haTaPa??=? + ? + ? + ?  (5.5.2) 
and 
 
2213421
??
??hTb bPb??=?+?+?+?  (5.5.3) 
respectively.  Since the scaling coefficients 
4
a  and 
4
b  are coupled with the exchange 
chemical potential, 
21
??? , they represent the majority of the effect of asymmetry as a 
result of the mixture.  These scaling coefficients, as derived in Section 3.2, can be 
written in terms of the other scaling coefficients and physical fields as: 
 
()
4c 3
?
11ax aK
? ?
?? + ?
? ?
 (5.5.4) 
and 
 
cc
4c 2 23
??
?
1.   (for )
dT dP
bx bK bb
dx dx
??
??
?? + ? + =
????
????
 (5.5.5) 
In these equations, the dimensionless Krichevskii parameter, 
?
K , is defined as 
 
1
cc
0,ccBc cBc
1
?
h
dP dTKP
K
kT kT dx T dx??
=
? ?
?
??
?= ?
? ?
??
?
??
? ?
? ?
. (5.5.6) 
 
 110 
 
In Eqs. (5.5.4) and (5.5.5), the Krichevskii parameter applies the physical 
characteristics of the mixture and its associated asymmetry to adjust the one-
component fluid (solvent) coefficients, 
3
a  and 
2
b , to values that more accurately 
represent the asymmetry of the mixture. 
 The remaining scaling coefficient, 
2
b , is coupled with the chemical potential 
of the one-component substance (
1
?? ), or solvent, as shown in Eq. (5.5.3).  It can be 
safely assumed that there is little, if any, influence on the coefficient 
2
b  by the 
addition of a small amount of solute.  If it is believed that there is a very large effect 
of asymmetry on the coefficient 
2
b , Eq. (2.2.54) in Section 3.2 can be used to 
approximate this difference; however, this difference is expected to be very minor. 
Therefore, we will assume that 
2
b  for the dilute mixture is equivalent to the value of 
2
b  for the pure solvent.   
6.5.1  Application to Dilute Aqueous n-Hexane Solutions   
By applying experimental data related to the fluid mixture, these expressions 
for the scaling coefficients of the mixture can be used to approximate the behavior of 
real dilute solutions.  Mixtures of particular interest are those that exhibit a high 
degree of asymmetry with only the smallest addition of solute: where the solute has a 
strong affinity for either the liquid or vapor phase of the solvent.  For the analysis in 
this section, we will focus on a dilute aqueous n-hexane solution using the 
experimental data provided in Tables 6.1, 6.5 and 6.6.  
 
 
 
 111 
 
Table 6.5.  Experimental data for dilute aqueous n-hexane solutions 
Variable Value Author 
c
dT
dx
 -352.69 K Abdulagatov
a 
c
dP
dx
 83.573 MPa Abdulagatov
a
 
K  
178.2 MPa Abdulagatov
a
 
 
a 
Reference 120 
 
 
 
Table 6.6.  Critical temperatures of dilute aqueous n-hexane solutions at various 
concentrations 
Concentration of 
n-hexane 
(mole fraction) 
T
c
 (K) Author 
0.000 647.096 De Loos et al.
a 
0.004 645 
Yiling and 
Michelberger
b 
0.015 641.600 De Loos et al.
a
 
0.025 636.500 De Loos et al.
a
 
a
 Reference 119 
 
b
 Reference 121 
 
 
 
Before proceeding further, it is important to study the critical behavior of a 
fluid mixture of interest at varying concentrations.  Since we are applying a linear 
approximation of physical fields to this model, we should only consider those 
concentrations with critical temperatures near where the critical locus exhibits a linear 
trend.  A graphical representation of this concept can be seen in the plot of the critical 
locus in Fig. 6.7.  In the case of dilute aqueous n-hexane, we should only consider 
critical concentrations with critical temperatures that are linear in trend (indicated by 
 
 112 
 
the red dashed line in the figure inset), and therefore close to the critical point of pure 
water.  For this dilute mixture, the critical behavior of the mixture is approximately 
linear within the critical temperature range of 635 - 646.7 K, and will be our area of 
focus. 
 It should be noted that a linear approximation may not be appropriate for all 
fluid mixtures.  A linear approximation cannot be used for such mixtures exhibiting 
closed solubility loops,
122
 as they posses both an upper and lower critical solution 
temperature, with some even exhibiting double critical points. 
  
 
  
Figure 6.7. Critical locus for aqueous n-hexane solutions. CP indicates the critical point of pure 
water.  Data plotted here is from De Loos et al.,
119
 Brunner,
123
 and Rebert and Hayworth.
124
  
 
 113 
 
6.5.2  Application to Dilute n-Heptane-Ethane Solutions   
 An example of a mixture of two components with differing molecular 
volumes is an ethane-n-heptane solution. The critical locus for a binary mixture of 
ethane with a low concentration of n-heptane can be seen in Fig. 6.8.  As before, we 
are applying a linear approximation of physical fields in this model, so only those 
concentrations with critical temperatures within the range where the critical locus 
exhibits a linear trend should be used.  This linear area of the data is indicated by the 
dashed red line in Fig. 6.8, which originates at the critical point of ethane and 
includes the experimental data found by Singh et al.
125
  
 
 
Figure 6.8. Critical locus for a mixture of ethane and n-heptane. CP indicates the critical point of 
ethane.  Data plotted here is from Ambrose and Tsonopoulos,
114
 Singh, Lucien and Foster,
125
and 
Mehra and Thodos.
126
  
 
 114 
 
Table 6.7.  Critical temperatures of dilute n-hepane-ethane solutions at various 
concentrations 
Concentration of 
n-heptane in ethane 
(mole fraction) 
T
c
 (K) P
c
 (MPa) Author(s) 
0.000 305.3 4.87 
Ambrose and 
Tsonopoulos
a 
0.0087 310.4 5.28 Singh et al.
b 
0.0151 315.0 5.64 Singh et al.
b
 
0.0202 318.0 5.79 Singh et al.
b
 
0.0287 322.4 6.13 Singh et al.
b
 
0.0350 325.8 6.48 Singh et al.
b
 
a
 Reference 114 
b
 Reference 125 
 
 
Table 6.8.  Experimental data for dilute n-heptane-ethane solutions 
Variable Value Author(s) 
c
dT
dx
 588.9 K 
 Ambrose and Tsonopoulos, 
Singh et al.
a
* 
c
dP
dx
 44.8 MPa 
 Ambrose and Tsonopoulos, 
Singh et al.
a
* 
K  
-15.0 MPa Abdulagatov
b 
a
 Reference 114 
b
 Reference 120 
* Initial slopes determined by linear approximation for dilute concentrations from data in 
these references 
 
Experimental data from dilute n-heptane-ethane solutions with concentrations 
corresponding to critical temperatures within this linear region are provided in Table 
 
 115 
 
6.7.  Approximations of initial slopes based on this data along with the Krichevskii 
parameter for dilute solutions are in Table 6.8. 
 
6.5.3  Effect of Temperature on Profile Asymmetry  
 Now that the physical parameters of this model have been set for a dilute two 
dilute solutions, we can develop concentration profiles and examine the properties 
that affect asymmetry.  First, an average concentration is chosen to illustrate the 
effect of temperature on the asymmetry of a mixture, or how greatly the solute favors 
one phase over another.   
The concentration profiles for a dilute aqueous n-hexane solution with a 
critical (and average) concentration of 
c
0.015xx==  and 
c
641.6 KT =  (from 
Table 6.6) for solution temperatures near the critical point of separation can be seen 
in Fig. 6.9.  For this model and analysis, the expression for concentration in Eq. 
(5.5.1) was used.  Equations (5.2.9) and (5.2.13) were used for the order parameter 
and thermal scaling density, respectively.  The scaling coefficients were either taken 
directly from Table 6.1 for pure water or determined from Eqs. (5.5.4) and (5.5.5)  by 
applying data in Tables 6.5 and 6.6 for the mixture.  Lastly, as the effect of 
concentration is being considered, Fisher Renormalization?or theoretical 
temperature correction?in Eq. (5.4.2) was also applied to this model for increased 
accuracy.  
Identical to the process for modeling the effect of temperature on 
concentration profiles for a dilute aqueous solution of n-hexane, profiles for a dilute 
solution of n-heptane in ethane were developed.  The concentration profiles with a 
 
 116 
 
critical (and average) concentration of 
c
0.0202xx==  and 
c
318 KT =  (from Table 
6.7) for solution temperatures near the critical point of separation can be seen in Fig. 
6.10.  The same expressions (Eqs. (5.5.1), (5.2.9), (5.2.13), (5.5.4) and (5.5.5)) were 
used in conjunction with the data in Tables 6.7 and 6.8 for the dilute n-heptane-ethane 
mixture.  The value of the scaling coefficients for pure ethane were taken from Table 
6.2.  As before, Fisher renormalization in Eq. (5.4.2) was applied for increased 
accuracy. 
 
As expected, the closer the mixture is to the critical point, the phase 
coexistence becomes more symmetric and the mixture exhibits less asymmetry.  At a 
2
2
3
4
?
210
?
10
?
310
?
10
T
T
T
T
?
?
?
?
?=??
?=?
?=??
?=?
Figure 6.9.  Concentration profiles for n-hexane in water at the vapor-liquid interface, at a critical
and average concentration of 
c
0.015xx== ,  and at various temperatures, plotted with respect
to height (z-coordinate).  Eqs. (5.5.1), (5.2.9), (5.2.13), (5.5.4), (5.5.5) and (5.4.2), and experimental
data from Tables 6.1, 6.5 and 6.6 were used to model the behavior of the mixture. 
 
 117 
 
temperature distance of 
4
?
10T
?
?=? , near the critical point of the mixture, the 
concentration profile is symmetric, as shown by the red line in Figures 6.9 and 6.10.  
This indicates that near the critical point, the n-hexane distributes itself more equally 
between liquid (- z) and vapor (+ z) phases of water (Fig. 6.9).  A similar result for the 
distribution of n-heptane in the two phases of ethane is seen in the red line in Fig. 
6.10. 
 
Conversely, the further from the critical point the mixture is, the more 
asymmetric the profile becomes.  At a temperature distance of 
2
?
210T
?
?=??  (shown 
by the black line in Fig. 6.9), the aqueous n-hexane solution separates into an n-
hexane-rich vapor (+ z) phase and an n-hexane-poor liquid (- z) phase.  Since 0z =  
Figure 6.10.  Concentration profiles for n-hepane in ethane at the vapor-liquid interface, at a critical 
and average concentration of 
c
0.0202xx== ,  and at various temperature distances, plotted 
with respect to height (z-coordinate).  Eqs. (5.5.1), (5.2.9), (5.2.13), (5.5.4), (5.5.5) and (5.4.2), and 
experimental data from Tables 6.2, 6.7 and 6.8 were used to model the behavior of the mixture. 
2
2
3
4
?
210
?
10
?
310
?
10
T
T
T
T
?
?
?
?
?=??
?=?
?=??
?=?
 
 118 
 
establishes the center of the symmetric profile, it can be seen that n-hexane clearly 
has a preference for the vapor phase as the higher concentrations lie in the positive z-
direction.   
An increase in asymmetry farther from the critical point is also observed in 
the dilute n-heptane-ethane mixture.  At a temperature distance of 
2
?
210T
?
?=?? , 
shown by the black line in Fig. 6.10, n-heptane has a greater preference for the liquid 
( z? ) phase, which is expected given the larger molecular size of n-heptane relative to 
ethane.   
 Lastly, the profiles in Figures 6.9 and 6.10?even even at varying 
temperatures?all intersect at the critical (and average) concentration of the mixture 
and at 0z = , the center of the symmetric concentration profile.  This validates the 
form of the thermal scaling density used, which includes the correction of the height 
dependence of the entropy, discussed in Section 6.2.  
6.5.4  Effect of Concentration on Profile Asymmetry  
 Second, to illustrate the effect of concentration on asymmetry in a dilute 
mixture, a temperature distance to the critical point, 
?
T? , will be held constant and the 
average/critical concentration will be varied.   The equations used for this model and 
analysis are identical to those listed in the previous subsection.   
The concentration profiles for dilute aqueous n-hexane solutions of 
c
0.004xx== , 0.015 and 0.025 at a constant temperature distance 
2
?
10T
?
?=?  can 
be seen in Fig. 6.11.  The critical temperatures used and correlated to these 
concentrations of n-hexane in water are found in Table 6.6.  The concentration 
 
 119 
 
profiles for dilute n-heptane-ethane solutions of 
c
0.0087xx== , 0.0151 and 
0.0287  at the same constant temperature distance 
2
?
10T
?
?=? are illustrated in Fig. 
6.12. The critical temperatures used and correlated to these concentrations of n-
hexane in water are found in Table 6.7.  The dashed colored lines in both Figures 6.11 
and 6.12 correspond to the value of the average concentration for each profile specific 
to a given 
c
x  and are provided to assist in highlighting any asymmetry.  The 
methodologies, equations and experimental values used to model this solution?s 
behavior are the same as those used and described in Section 6.5.2. 
 
Based on these figures, it appears that there is increased fluid asymmetry with 
an increase in concentration for both the dilute aqueous n-hexane and n-hepthane-
ethane solutions.  Further analysis of both solutions showed that asymmetry did 
increase slightly with an increase in concentration in both solutions.  In the aqueous 
2
?
10T
?
?=?
Figure 6.11.  Concentration 
profiles for n-hexane in water at 
the vapor-liquid interface, at a 
constant temperature distance 
2
?
10T
?
?=? , and at various 
critical or average concentrations, 
plotted with respect to height (z-
coordinate).  The dashed lines 
indicate the average concentration 
with the color corresponding to 
each of the critical concentrations 
shown.  Eqs. (5.5.1), (5.2.9), 
(5.2.13), (5.5.4), (5.5.5) and 
(5.4.2), and experimental data 
from Tables 6.1, 6.5 and 6.6 were
used to model the behavior of the 
mixture. 
 
 
 120 
 
n-hexane solution, ratios of the bulk liquid ( 1z << ) phase relative to the average 
concentration (
c
x ) yielded relatively constant values for each concentration at a 
constant temperature, shown in Table 6.9.  However, in the bulk vapor phase 
( 1z >> ), the increased concentration of n-hexane resulted in relatively larger 
differences in fluid asymmetry. 
 
Similar results were observed in the dilute n-heptane-ethane solutions, given 
in Table 6.10.  Higher asymmetry was observed with increased concentration and 
temperature distance from the critical point in the bulk vapor phase ( 1z >> ) relative 
to the average concentration.  Concentration ratios in the bulk liquid phase relative to 
the average concentration were somewhat constant, with the exception of the 
temperature distance 
2
?
10T
?
?=? , where greater asymmetry was present with a 
change in concentration.  
2
?
10T
?
?=?
Figure 6.12.  Concentration
profiles for n-hepane in ethane at
the vapor-liquid interface, at a
constant temperature distance
2
?
10T
?
?=? , and at various
critical or average concentrations,
plotted with respect to height (z-
coordinate).  The dashed lines
indicate the average
concentration with the color
corresponding to each of the
critical concentrations shown.
Eqs. (5.5.1), (5.2.9), (5.2.13),
(5.5.4), (5.5.5) and (5.4.2), and
experimental data from Tables
6.2, 6.7 and 6.8 were used to
model the behavior of the
mixture. 
 
 
 121 
 
 
Table 6.9.  Comparison of bulk phase concentration data from dilute aqueous n-
hexane solutions at various temperatures. 
?
T?  c
x
 
liq
x
 
vap
x
 
cliq
/xx
 
vap c
/xx 
3
4.00 10
?
?  
3
2.02 10
?
?
2
1.08 10
?
?  
1.98 2.70 
2
1.50 10
?
?  
3
7.60 10
?
?
2
3.97 10
?
?  
1.97 2.64 
2
10
?
?  
2
2.50 10
?
?  
2
1.27 10
?
?
 2
6.47 10
?
?
1.97 2.59 
3
4.00 10
?
?  
3
2.75 10
?
?
3
6.06 10
?
?  
1.45 1.52 
2
1.50 10
?
?  
2
1.04 10
?
?
2
2.25 10
?
?  
1.44 1.50 
3
10
?
?  
2
2.50 10
?
?  
2
1.74 10
?
?
2
3.71 10
?
?  
1.44 1.48 
3
4.00 10
?
?  
3
3.33 10
?
?
3
4.83 10
?
?  
1.20 1.21 
2
1.50 10
?
?  
2
1.26 10
?
?
2
1.80 10
?
?  
1.19 1.20 
4
10
?
?  
2
2.50 10
?
?  
2
2.10 10
?
?
2
2.98 10
?
?  
1.19 1.19 
 
 
Based on the results presented here, a change in concentration has a larger 
impact on the asymmetry in the vapor phase.  However, the greatest impact on the 
asymmetry of concentration profiles of dilute mixtures is a temperature distance 
farther from the critical point coupled with the addition of a solute.  Since the 
analyses in this work only address dilute solutions, it is possible that higher 
concentrations of a dissimilar component?perhaps at concentrations closer to 
saturation?could yield results that indicate a change in concentration alone has a 
greater effect on the fluid asymmetry of a mixture.   
 
 
 
 122 
 
Table 6.10.  Comparison of bulk phase concentration data from dilute n-
heptane-ethane solutions at various temperatures. 
?
T?  c
x
 
liq
x
 
vap
x
 
cliq
/xx
 
vap c
/xx 
3
8.70 10
?
?  
3
9.29 10
?
?
3
3.04 10
?
?  
0.936 0.349 
2
1.51 10
?
?  
2
1.63 10
?
?
 3
6.86 10
?
?  
0.926 0.454 
2
10
?
?  
2
2.87 10
?
?  
2
3.12 10
?
?
2
1.66 10
?
?  
0.920 0.578 
3
8.70 10
?
?  
3
9.51 10
?
?
3
6.86 10
?
?  
0.915 0.789 
2
1.51 10
?
?  
2
1.65 10
?
?
2
1.20 10
?
?  
0.915 0.795 
3
10
?
?  
2
2.87 10
?
?  
2
3.13 10
?
?
2
2.29 10
?
?
0.917 0.798 
3
8.70 10
?
?  
3
9.20 10
?
?
3
8.03 10
?
?  
0.946 0.923 
2
1.51 10
?
?  
2
1.59 10
?
?
2
1.40 10
?
?  
0.950 0.927 
4
10
?
?  
2
2.87 10
?
?  
2
3.03 10
?
?
2
2.66 10
?
?
0.947 0.927 
 
 
6.6  Modeling Asymmetric Density Profiles for Dilute Binary Solutions 
Another behavior of interest is the change in the density profile of a one-
component fluid due to the addition of a small amount of solute.  It is well-known 
that if solute with a vastly different molecular volume is added to a solvent, the 
density of the mixture is significantly altered.  Since complete scaling has 
successfully been applied to model the behavior of density in one-component fluids 
and concentration in dilute mixtures, this method will also be used to model the 
density profiles of dilute binary mixtures, taking into account the effect of the 
concentration of solute or impurity added. 
 
 123 
 
6.6.1  Application to Dilute Polystyrene-Cyclohexane Solutions 
 An example of a binary solution with components with vastly different 
molecular volumes is a dilute polymer solution.  In particular, we can focus on a 
solution of polystyrene in cyclohexane.  Given the expression for density in Eq. 
(5.2.1), we can provide an analogous expression for the density profile in terms of 
volume fraction of polymer, ? : 
 
122
31 32
1
?
1
b
ab
??
?
??
++
=
??
, (5.6.1) 
where 
c
?
/???? .  Since the expressions for the order parameter (
1
? ) and thermal 
scaling density (
2
? ) in Eqs. (5.2.9) and (5.2.13) can be used for one-component fluids 
or fluid mixtures, these same expressions can also be used in this example.  It should 
be noted that since incompressible polymer solutions are modeled as a one-
component fluid where the monomers of that one-component fluid assemble to form a 
polymer chain, Fisher Renormalization is not required for this analysis. 
Recalling from Section 5.2, the scaling coefficients 
3
a  and 
2
b  in terms of 
degree of polymerization, N ,  were determined to be  
 
eff
31
5
N
a
NN
???
=
??
+
??
, (5.6.2) 
and 
 
2ef
NN
ba
NN
?
+
. (5.6.3) 
Since 
eff 3 3
/(1 )aa a?+, the expression for 
3
a  can be written in terms of degree of 
polymerization as 
 
 124 
 
 
3
31
5
31
1
5
N
NN
a
N
NN
???
??
+
??
=
???
?
??
+
??
. (5.6.4) 
However, it has also been shown in Section 5.2 that at moderate and higher degrees 
of polymerization ( 100N ? ), these expressions can be simplified to 
 
3
3
( 100)
2
aN?? (5.6.5) 
and 
 
2ef
( 100)bN a N?? . (5.6.6) 
The expressions for the critical amplitudes developed by Anisimov et al.
103
 and 
simplified in this work are: 
 
/2
00,1
?
( 100)
N
BN B N
?
=
?? , (5.6.7) 
 
(1 )/2
00,1N
AAN
??+
=
? , (5.6.8) 
 
(1 )/2
00
rN
?
?
?
? , (5.6.9) 
where 
0, 1N
B
=
 and  
0, 1N
A
=
 are both constants approximated as 
0, 1 0, 1
1
NN
BA
==
?? for the 
three dimensional Ising model and 
0
r  is the size of the monomer.  For polystyrene, 
the value
103
 
0
0.15 nmr ? .   
For this concentration profile, the intent is to be somewhat close to the 
?critical? region, where the Widom variable
93
 
1
?
1
2
wNT??< (discussed in 
Section 5.1).  A high degree of polymerization would lead to model behavior in the 
?polymer? regime, where 
1
?
1
2
wNT??>.  Therefore, in this example, for the 
 
 125 
 
temperature distance of 
2
?
10T
?
?=? , the degree of polymerization chosen is only 
moderate, where 100N =  and 150N = .  The value of the critical background (
cr
?
B ) in 
Eq. (5.2.13) was estimated to be zero for this specific case because this term 
decreases as fluctuations decrease with an increasing degree of polymerization.  The 
concentration profile for a polystyrene-cyclohexane solution at a moderate degree of 
polymerization can be seen in Fig. 6.13. 
 
 We can see in this figure that even near the ?critical? region at a moderate 
degree of polymerization, the length of the polymer chains impacts the fluid 
asymmetry of the density or volume fraction profile.  The largest volume fraction lies 
in the polymer-rich phase ( z? ) and the smallest volume lies in the solvent-rich phase  
( z+ ). 
Figure 6.13.  Normalized density 
profile for a polystyrene-
cyclohexane solution with degrees 
of polymerization 100N = , 
150N =  and at a temperature 
distance of 
2
?
10T?=? . Eqs. 
(5.6.1), (5.2.9), (5.2.13) and (5.6.5)
-(5.6.9) were used to model the
behavior of the mixture.  The red 
dashed line is the normalized 
symmetric profile. 
 
2
?
10T
?
?=?
 
 126 
 
6.6.2  Application to Dilute Ethane-n-Heptane Solutions 
Another example of a mixture of two components with differing molecular 
volumes is an n-heptane-ethane solution.  The expression in Eq. (5.2.1) can still be 
used to model the density profile.  However, this is not a polymer solution and the 
scaling coefficients 
3
a  and 
2
b  must be approximated in a different manner than in 
Section 6.6.1.  
Since the expression for the density of a mixture (Eq. (5.2.1)) does not include 
the scaling coefficients 
4
a  and 
4
b , which are coupled with the exchange chemical 
potential of the mixture in Eqs. (5.5.2) and (5.5.3), the majority of the concentration-
dependent asymmetry is introduced via the coefficients 
3
a  and 
2
b .  As indicated by 
Wang and Anisimov
64
 and discussed in Section 3.2, the coefficient 
3
a  has shown a 
dependence on molecular volume.  The coefficient 
2
b  may also be dependent on 
molecular volume as well, but not as significantly as 
3
a  (see Section 3.2).  Therefore, 
in mixtures with components of differing molecular volumes, we will approximate 
the scaling coefficients 
3
a  and  
2
b  for the mixture as 
 
0 cc
3c 3
c
()
x dV
ax a
Vdx
?+  (5.6.10) 
and 
 
0 cc
2c 2
c
()
x dV
bx b k
Vdx
?+ , (5.6.11) 
respectively, where ~1/3k  as presented in Section 3.2. 
 
 127 
 
With this approximation, the temperature correction in Eqs. (5.4.2) and (5.4.3) can be 
applied to this solution.  It should also be noted that the normalized amplitude 
0
?
B  is 
dependent on the scaling coefficient 
3
a  as presented in Section 4.2: 
 
0
0
3
?
1
B
B
a
=
+
. (5.6.12) 
Therefore, 
0
?
B  must be determined for each change in concentration, as the scaling 
coefficient 
3
a  is dependent on concentration, given in Eq. (5.6.12).    
 
6.6.2  Effect of Temperature on Asymmetry in Density Profiles  
 The first property to be analyzed is the effect of temperature on the 
asymmetry of the density profile of a dilute n-heptane-ethane mixture and its behavior 
can be seen in Fig. 6.14.  This figure represents the density profiles at a constant 
concentration of n-heptane of 0.0202x =  at various temperatures with the 
temperature correction in Eq. (5.4.2) applied.  This concentration was chosen as it 
was near the middle of the linear region of the n-heptane-ethane critical locus. 
Experimental concentration data used for this model can be found in Table 6.7.  The 
density profile was modeled using the expression for density in Eq. (5.2.1) and 
scaling densities in Eqs. (5.2.9) and (5.2.13).  The correction factor used in Eqs. 
(5.6.10) and (5.6.11) for this n-heptane-ethane mixture was 
 
c
c
1
6
dV
Vdx
?  (5.6.13) 
which was approximated by Anisimov, Gorodetskii and Shmakov
123
 by a fit to 
experimental data.  Note that this constant in Eq. (5.6.13) should have been reported 
 
 128 
 
as positive, as an increase in critical volume with respect to concentration of n-
heptane leads to an increase in fluid asymmetry.  This is because the critical density 
decreases when the concentration of solute, n-heptane, is increased, making 
c
/dV dx positive.  This constant in Eq. (5.6.13) is approximated within a small range 
of values; therefore the maximum value was selected to model the density profiles.  
Lastly, the critical concentration of the mixture was given by linear approximation, 
which is valid for these small concentrations: 
 
c c, C2 c, C7
(1 )xx????? + , (5.6.14) 
where 
c, C2
? and 
c, C7
?  are the critical densities of ethane and n-heptane, respectively 
given in Tables 6.2 and 6.3. 
 
 
  
 
Figure 6.14.  Density profiles 
for n-heptane in ethane at the 
vapor-liquid interface, at a 
critical and average 
concentration of 
c
0.0202x = ,  and at various 
temperatures, plotted with 
respect to height (z-
coordinate).  Eqs., (5.2.1), 
(5.5.2), (5.5.3), (5.6.10), 
(5.6.11), (5.6.13) and (5.4.2)
and experimental data from 
Tables 6.2 and 6.7 were used 
to model the behavior of the 
mixture.  The critical density 
of an n-heptane-ethane 
mixture of 
c
0.0202x = is 
indicated by 
c
? .   
2
3
4
?
10
?
10
?
10
T
T
T
?
?
?
?=?
?=?
?=?
 
 129 
 
As we can see in Fig. 6.14, temperature distance from the critical point does 
not appear to impact the asymmetry of the density profile for a mixture of ethane with 
a small concentration of n-heptane.  The profile closest to the critical point 
(represented by the red line in the figure) shows a very smooth transition from one 
phase to another and is nearly symmetric.  Farther from the critical point (represented 
by the black line in Fig 6.14), the very slight increase asymmetry in the direction of 
the high density or liquid phase (- z) is hardly noticeable increases in the high density 
or liquid. 
6.6.3  Effect of Concentration on Asymmetry in Density Profiles  
 The second property to be analyzed is the effect of concentration on the 
asymmetry of the density profile.  Mixtures of ethane at low concentrations of n-
heptane (x
c
 = 0.0350, 0.0202 and 0.0087) at a constant temperature distance 
2
?
10T
?
?=?  were used in this model, shown in Fig. 6.15.  The methodology and 
equations used here is the same as that described in Section 6.6.2.  The density profile 
of pure ethane (shown by the dashed red line) was also included for comparison to 
see if any asymmetry appeared due to the addition of n-heptane.  The method used to 
create this plot is the same as described earlier in this section. 
 Based on the results in Fig. 6.15, it can be seen that concentration has a large 
effect on the behavior of the density profile.   In general, as the concentration of n-
heptane increases, the density in the bulk liquid phase ( 1z << ) decreases and the 
concentration in the bulk vapor phase ( 1z >> ) decreases relative to the density profile 
of pure ethane (broken curved red line).   
 
 130 
 
The temperature correction due to concentration in Eq. (5.4.2) is a significant 
factor in this result.  Due to the steep slope of 
c
/dT dx in dilute n-heptane-ethane 
solutions (Table 6.8), Fisher Renormalization imposes a significant correction factor, 
causing a large relative reduction in the values of the temperature distance.  This is 
because the renormalization occurs along a specific path when both the total volume 
and concentration are kept constant.  Since the renormalization causes a decrease in 
temperature distance, the profiles to become smoother as these new and corrected 
temperatures are smaller and closer to the critical point.  However, if the critical 
temperature is approached at a constant chemical potential or pressure, this effect is 
absent.   
Despite the effect of Fisher Renormalization, some asymmetry can still be 
observed in Fig. 6.15.  Given the difference in behavior between the density profiles 
of pure ethane (broken curved red line) and the n-heptane-ethane mixture with a 
concentration of 
C7
0.0350x =  (curved blue line), there is increased asymmetry with 
higher concentrations of n-heptane.  This effect is highlighted when considering the 
shift in the density profiles relative to the shift in critical density.  In Fig. 6.15, the 
horizontal dashed red line represents the critical density of pure ethane and the dashed 
blue line indicates the critical density of the n-heptane-ethane mixture with a 
concentration of 
C7
0.0350x = .  By close examination, we can see in Fig. 6.15 that the 
difference in density between the critical and bulk vapor phase, relative to the 
difference between the critical and bulk liquid phase, increases with higher 
concentrations of n-heptane. 
 
 
 131 
 
 
However, since Fisher Renormalization significantly affects mixtures of n-
heptane-ethane, this correction factor will be removed in the following analysis.  In 
this specific case, the temperature distance will be held constant for all 
concentrations, at 
2
??
() ( ) 10Tx T?
?
?=?=?, in order to better examine the effect of 
concentration on the density profiles.  This result can be seen in Fig. 6.16. The same 
methodology was used to model these profiles as used in the previous two sections, 
only Fisher Renormalization in Eq. (5.4.2) was not applied in this case. 
Figure 6.15.  Density profiles for ethane and n-hepane in ethane at the vapor-liquid interface, at a 
constant temperature distance 
2
?
10T
?
?=? , and at various critical concentrations, plotted with 
respect to height (z-coordinate).  Eqs. (5.2.1),  (5.5.2), (5.5.3), (5.6.10), (5.6.11), (5.6.13) and (5.4.2)
and experimental data from Tables 6.2 and 6.7 were used to model the behavior of the mixture.  The
critical density of ethane is indicated by the horizontal dotted red line and critical density for a 
solution with
c,C7
0.0350x = is indicated by the horizontal dotted blue line.  
2
?
10T
?
?=?
 
 132 
 
 
 
When the temperature distance is held constant and considered independent of 
concentration, the density profiles shift to lower densities as greater amounts of n-
heptane is added to ethane, as expected, which is shown in Fig. 6.16.  Both bulk 
liquid and vapor phases exhibit a reduction in density with the addition of solute, as 
n-heptane has a larger molecular volume relative to ethane.  From Fig. 6.16 we can 
see that as concentration is increased, there is a much larger deviation from the 
Figure 6.16.  Density profiles for ethane and n-hepane in ethane at the vapor-liquid interface, at a 
constant temperature distance 
2
??
() ( ) 10Tx T?
?
?=?? (without Fisher Renormalization), and at 
various critical concentrations, plotted with respect to height (z-coordinate).  Eqs. (5.2.1),  (5.5.2), 
(5.5.3), (5.6.10), (5.6.11), and (5.6.13),  and experimental data from Tables 6.2 and 6.7 were used to
model the behavior of the mixture.  The critical density of ethane is indicated by the horizontal 
dotted red line and critical density for a solution with
c,C7
0.0350x = is indicated by the horizontal 
dotted blue line. 
2
??
() ( )
10
Tx T?
?
?=?
=?
 
 133 
 
critical density in both bulk liquid and vapor phases as concentration of n-heptane is 
increased.    
6.7  Conclusions 
 Compete scaling has been successfully employed to describe the physical 
behaviors of real one-component and dilute solutions by using experimental data to 
apply the effects of fluid asymmetry through scaling coefficients.  Both dilute binary 
mixtures of aqueous n-hexane and n-heptane in ethane showed a change in 
concentration and density profile behavior, respectively, with increased 
concentration.  The addition of a small amount of solute coupled with an increased 
temperature distance from the critical point resulted in the greatest increase in 
asymmetry in concentration profiles, as the solute favored one phase over the other.  
The addition of a small amount of n-heptane to ethane resulted in a shift in the density 
profiles to lower densities, as expected.  An increase in n-heptane concentration also 
increased the asymmetry of the density profiles.  However, an increase in temperature 
distance appeared to have little to no effect on density profiles.   
 
 134 
 
Chapter 7:  Summary and Outlook 
 
In this work, complete scaling has been successfully applied to model the 
interfacial behavior of real asymmetric fluids and fluid mixtures.  Selection of the 
order parameter and thermal scaling density to model asymmetric interfaces was 
based on a good knowledge of symmetric profiles, both mean-field and those that 
account for fluctuations near the critical point.   By applying mean-field critical 
exponents, it was possible to determine the proper form of the thermal scaling 
density, as the mean-field behavior is well-known.  Use of the symmetric 
renormalization group theory profile accounted for fluctuations near the critical point 
of separation.    
 In complete scaling, fluid asymmetry can be applied to symmetric profiles 
through the scaling coefficients of real one-component fluids.  This method was used 
since it is impossible to directly minimize the free energy of an inhomogeneous 
asymmetric fluid and take into account critical fluctuations using another method.  
This is because scaling equations of state are not continuous through the unstable 
region, and therefore the corresponding free energy cannot be minimized in a simple 
manner.    
When considering the behavior of the density profile of a one-component 
fluid or fluid mixture, the scaling coefficients 
3
a  and 
2
b  for dilute mixtures should be 
adjusted if there is a large difference in molecular volume between the solvent and 
solute.  For polymer solutions, this difference in molecular volume is expressed by 
degree of polymerization, N .  This study showed that a larger difference in 
 
 135 
 
molecular volume led to a higher degree of fluid asymmetry in polymer solutions, 
where the addition of larger molecules showed a preference for one phase over the 
other.  The concentration of an added solute, specifically in dilute n-heptane-ethane 
solutions, caused shift in the profiles toward the density of the solute.  This increased 
concentration of solute also contributed to asymmetry in density profiles.  
When considering dilute mixtures, the scaling coefficients 
4
a  and 
4
b  can be 
approximated using experimental data in the linear portion of the critical locus of the 
mixture, which in most cases is only at dilute concentrations.   Asymmetric 
concentration profiles for dilute mixtures of aqueous n-hexane and n-heptane-ethane 
were modeled in this work based on the approximation of 
4
a  and 
4
b  from 
experimental critical data of these dilute mixtures.  This study showed that a very 
slight increase in concentration, coupled with an increasing temperature distance from 
the critical point of separation, yielded the greatest increase in fluid asymmetry in 
concentration profiles. 
Directly related to the behavior of the interfacial profile is the curvature 
correction to the surface tension, or Tolman?s length.  This correction becomes very 
significant when the interface thickness is very large near the critical point as well as 
when there is a very high degree of fluid asymmetry.  By utilizing complete scaling, 
this study showed that the Tolman?s length becomes mesoscopic and diverges when 
approaching the critical point of separation (
?
0T??) and with increasing degree of 
polymerization ( N ??).   The near-critical Tolman length will strongly diverge as 
0.875
0.348
?
~ NT? .  Farther from the critical point of separation in polymer solutions, 
 
 136 
 
when the phase diagram becomes more asymmetric, the Tolman length becomes a 
significant correction as it approaches the same magnitude as the width of the 
interface and diverges more strongly as 
1
?
~ T
?
? , but is independent of N .  Given 
these expressions for the general thermodynamic expression for near-critical Tolman 
length, crossover expressions to approximate the curvature correction based on 
Widom?s characteristic variable, 
1
?
2
wNT??, and radius of gyration, 
g0
RrN= .  
In addition to the effects shown in polymer solutions, it is expected that 
Tolman?s length will be a significant curvature correction to the surface tension of 
small droplets in highly asymmetric systems.  There are many engineering 
applications that utilize asymmetric fluids and dilute fluid mixtures where this impact 
can be significant, such as the creation of nanodroplets, adsorption of small droplets 
on surfaces and microemulsions.     
 Two-dimensional phase-separated systems, such as Langmuir-Blodgett films, 
and separation on the surfaces of vesicles or small droplets, have many chemical and 
bioengineering applications.  Since the results in this work for three-dimensional 
asymmetric systems have yielded a divergence of the Tolman length near the critical 
point for small curved surfaces, it is expected that an even greater effect will occur in 
two dimensions.  Considering the expression for Tolman?s length near the critical 
point in Eq. (4.2.20), it is apparent that the values of the critical exponents have a 
significant effect on the curvature correction to the line tension.  Applying the critical 
exponents from the two dimensional Ising model,
56
 the exponent for the first term is 
7/8? ??=?  and the exponent for the second term is 1 1/8? ? ????=? .  This 
 
 137 
 
result highlights that as the number of dimensions decreases, fluctuations near the 
critical point increase.  These exponents suggest that the divergence of the Tolman 
length when approaching the critical point will be even greater than what has been 
presented in this research for three-dimensional systems, where 0.304? ????  and 
1 0.065? ? ?????? .   
 
 138 
 
Appendix A:  Comparison of Mean-Field Surface Tension 
Calculations 
 
 
 
As discussed in Chapter 2, the mean-field approximation for the surface 
tension was compared with both the work of Landau and Lifshitz
63
 as well as the 
work of Rowlinson and Widom.
56
 This section will describe the direct comparison 
between these results and those shown in this work. 
A.1  Comparison with Landau and Lifshitz 
 Landau and Lifshitz
63
 have provided a similar result for the local Helmholtz-
energy density as shown in this work.  We can make a comparison, beginning with 
their results
63
 for the gradient of the order parameter in terms of specific volume, 
v (vice density, in our case),  
 
()
22
0
dv
vv
dx
?
?
=? (A.1.1) 
where x  is the coordinate on which the order parameter is evaluated, ?  and ?  are 
system-dependent constants analogous to our constants 
0
u  and 
0
c , and 
0
v  is the 
reference order parameter with respect to volume.  This is Landau and Lifshitz?s 
equation for the gradient of the order parameter that satisfies the stability 
requirements, similar to Eq. (2.2.2) in this work: 
 
2
222
00 0
11
[]
42
d
uc
dz
?
??
??
?=
??
??
    (2.2.2) 
which can be re-written as 
 
 139 
 
 
()
1/2
220
0
0
2
ud
dz c
?
??
??
=?
??
??
. (A.1.2) 
 By evaluating the derivative expression in Eq. (A.1.1) , we obtain the profile 
for Landau and Lifshitz?s order parameter,  
 
00
tanhvv v x
?
?
??
=
??
??
. (A.1.3) 
This result is similar to Eq. (2.2.3) in this work: 
1/2
0
0
00
tanh
2
u
z
c
?
?
?
? ?
??
? ?=?
??
? ?
??
? ?
.    (2.2.3) 
Note that the density-based order parameter profile in Eq. (2.2.3) is affected by 
height, where lower height, is in the z?  direction where the more dense of the two 
phases exists.  In the case of Landau and Lifshitz
63
 as shown above in Eq. (A.1.3), 
their profile does not assign the direction of the coordinate x . 
Comparing Landau and Lifshitz?s results in Eq. (A.1.3) and the result in Eq. 
(2.2.3), we can make a term-by-term comparison and obtain   
 
1/2
0
0
2
u
c
?
?
??
=
??
??
. (A.1.4) 
Similar to the estimation of surface tension given in this work in Eq. (2.5.9) , Landau 
and Lifshitz?s expression for the surface tension at the interface, a , is given as
63
  
 
3
0
k
4
3
av
V
??
= , (A.1.5) 
where 
k
V  is the critical volume.  Given the relation provided in Eq. (A.1.4), assuming 
0
/2u? =  and 
0
c? = , and assuming our density-based order parameter is similar to 
 
 140 
 
that of Landau and Lifshitz?s specific volume-based order parameter, we can 
substitute our variables into Landau and Lifshitz?s expression for surface tension Eq. 
(A.1.5) and make a comparison to Eq. (2.5.9) in this work, where 
()
3/2
0
cBc
8
?
3
v
CT
kT
?
?
?
?
=??.                (2.5.9) 
Noting also that 
1/ 2
0
0
0
?
()
,
aT
u
?
??
??
=
??
??
                                      (2.1.11) 
1/2
0
0
0
2
c
a
?
?
??
=
??
??
,                                          (2.2.6) 
and  
2
0
0
2
V
a
C
u
= ,     (2.58) 
we can substitute these values into Eq. (A.1.5) to compare to our expression for the 
dimensionless surface tension in Eq. (2.5.9).  Therefore the expression for Landau 
and Lifshitz?s surface tension for the mean-field approximation becomes 
 
()
0
3/2
k
Bc
8
?
3
V
aV
CT
kT
?
?
=??, (A.1.6) 
when made dimensionless, which compares directly to the expression in Eq. (2.5.9). 
 
A.2  Comparison with Rowlinson and Widom 
Rowlinson and Widom
56
 also discuss the behavior of interfaces near the 
critical point, defining the density near the critical point as: 
 
[]
1/2
,
~()/
lg
cc
AT T B???? . (A.2.1) 
 
 141 
 
Noting the definitions given in Eqs. (2.1.7) and (2.1.9) 
c
c
?
TT
T
T
?
??
    
 (2.1.7) 
 
c
c
()
()
z
z
??
?
?
?
? ,  (2.1.9) 
and the density difference along the liquid-gas coexistence curve can be defined as  
 
0c
2
lg
?? ???= , (A.2.2) 
we can determine the relationship between Rowlinson and Widom?s
56
 variables A  
and B  in Eq. (A.2.1) and the variables in this work.  Given these equations, this 
relation becomes: 
 
1/2
1/2
0c
cc c
0c
1/2
1/2
0
0
()
()
c
c
aT T A
TT
uT B
aA
uT B
?? ?
?
???
??
?= = ?
????
??
??
??
??
=
????
??
??
 
 
2
c0
0c
aA
BuT
?
= . (A.2.3) 
 If we use the expansion of the chemical potential near the critical point 
developed by Rowlinson and Widom,
56
 
 
3
cc c
(,) ( )( ) ( ) ..TATT B?? ? ? ? ? ??= ? ? + ? +, (A.2.4) 
we can separate the coefficients A  and B  and determine their values.  In terms of the 
dimensionless variables in this work, we can duplicate this expansion of the chemical 
potential as 
 
3
00
Bc
(,)
?
T
aT u
kT
?? ?
??
?
=? + , (A.2.5) 
 
 142 
 
which can also be written with dimensions  
 
()() ()
3
B0 Bc0
cc c
3
cc
(,)
ka kTu
TTT?? ? ? ? ? ?
??
?= ? ? + ? . (A.2.6) 
A comparison of each term in Rowlinson and Widom?s expansion of the chemical 
potential
56
 in Eq. (A.2.4) to the substitution of the variables in this work, shown in 
Eq. (A.2.6), gives the solution 
 
0B
c
0Bc
3
c
.
ak
A
ukT
B
?
?
=
=
 (A.2.7) 
This result can be verified with Eq. (A.2.3). 
 Following this same method of comparing terms, we can also look to 
Rowlinson and Widom?s expression for the chemical potential difference at the 
interface of two fluids, related to the density gradient,
56
  
 
2
2
(,)
d
mT
dz
?
?? ?=?, (A.2.8) 
where ( , )T??  is the chemical potential in the homogeneous fluid of density ?  from 
the mean-field approximation and ?  is the chemical potential in the two-phase 
region.  We can compare this equation the expression for the chemical potential in 
Eq. (2.2.1) of this work, 
()
2
22
000
2
Bc
()z d
uc
kT dz
??
?? ?=?+,   (2.2.1) 
re-writing Eq. (2.2.1) in the same form as Eq. (A.2.8) to yield the expression 
 
()
2
22
0Bc 0Bc 0
2
()
d
ckT z ukT
dz
?
????=? ?. (A.2.9) 
 
 143 
 
Noting that  
 
22
22
1
c
dd
dz dz
??
?
= , 
Eq. (A.2.9) becomes 
 
()
2
220Bc
0Bc 0
2
c
()
ckT d
zukT
dz
?
????
?
=? ?. (A.2.10) 
By a term-by-term comparison, we find  
 
0Bc
c
ckT
m
?
= . (A.2.11) 
 Since we have determined the values of the unknown constants in Eqs. (A.2.7)
and (A.2.11), and the difference in density along the coexistence curve in Eq. (A.2.2), 
we can substitute these constants and relation into Rowlinson and Widom?s 
expression for the surface tension near the critical point,
56
  
 
()
1/2
3
1
62
lg
mB
? ??
??
=?
??
??
. (A.2.12) 
Eq. (A.2.12) can be re-written in terms of mean-field variables in this work, which 
provides the same result as Eq. (2.5.9): 
3/2
0
cBc
8
?
()
3
v
CT
kT
?
?
?
?
=??.                                      (2.5.9) 
 
 
 144 
 
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