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CHARACTERIZATION OF AMORPHOUS POROUS CARBONS
BASED UPON STRUCTURAL ENHANCEMENT OF SORPTION
POTENTIALS

By

Robert J. Dombrowski

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Chemical Engineering

200]

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Abstract

CHARACTERIZATION OF AMORPHOUS POROUS CARBONS BASED
UPON STRUCTURAL ENHANCEMENT OF SORPTION POTENTIALS

By
Robert J. Dombrowski

Analysis of gas sorption isotherms may yield information about structural
heterogeneity that can be used to control selectivity of separations processes, catalytic
reactions, and gas storage. Disordered porous materials may be analyzed via
simulation of sorption isotherms using methods that rely on simple geometric
structures with a single source of heterogeneity to render the problem more tractable.
Molecular simulations (MS) and density functional theory (DFT) are considered exact
for a given simple pore structure and chemical makeup. DFT model isotherms have
been used to analyze experimental isotherms for several amorphous porous carbons at
77K for both argon and nitrogen. The pore size distributions (PSD) of the two gases
generally agreed within 8% for both the total volume and mode of the distribution.
Although both probe gases yielded similar results the argon is recommended for use
as the probe gas of choice, as it is smaller than nitrogen, is a monatomic molecule,
and has no permanent multipole moment. Elemental analysis of the carbons indicates
that the two probe gases are in better agreement when the carbons do not contain
functional groups.

Sorption analysis is very dependant upon the model parameters used to describe the
potential functions. The potential parameters may be determined from bulk physical
characteristics of a molecule such as the polarizability and the magnetic

susceptibility, or may be fitted to isotherm data. The monolayer filling pressure of

carter: D“?-
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single-v a! l ~.
determine

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D” or MS
MS and DP}
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carbon pores larger than 10 molecular diameters approaches the maximum therefore
choosing the parameters to match the monolayer filling pressure of a large slit pore
with that of a nonporous carbon is an excellent way to fit the potential parameters.
An alternate means to accomplish the same goal is to fit an isotherm to a known
distribution of pore sizes. Transmission electron microscope pictures may be taken of
single-walled carbon nanotubes to determine their PSD, which can be used to
determine the potential parameters. The technique has be verified through
comparison PSD of isotherms generated with the use of nonporous carbon fitted
parameters with the TEM data resulting in an error of less than 3% .

Similar to the DPT and MS methods the Horvath-Kawazoe (HK) method relies on the
superposition of the adsorbent—adsorbate potential to enhance adsorption in
micropores, and may be used to determine isotherms for a pore of a given size and
structure. The HK method with a generalized potential function requires less than
one hundredth of the time required to generate a series of isotherms using either the
DPT or MS methods, while remaining remarkably accurate for its simplicity. Both
MS and DPT predict pore wall wetting prior to condensation in the pore, whereas the
original HK method predicted that the pores are either completely empty or
completely full. The HK method was modified to approximate pore wall wetting at
pressures lower than the filling pressure, yielding closer agreement of predicted pore
size as compared to DPT. The addition of the monolayer filling in the HK method
has the advantage of rendering it possible to fit the potential parameters of the system

to a nonporous reference, in addition to improved accuracy for the mesopore range.

 

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Acknowledgements

I thank my advisor Dr. Christian Lastoskie. Our conversations discussing research
were invaluable to my progress and growth as a researcher. Through our discussions
I learned several techniques and approaches to problems that I foresee using
throughout my career in matters concerning research and daily life. I am also grateful
to my fellow graduate students each of whom touched my life in a unique way. I
must also thank the many undergraduate researchers whom I have had the pleasure of
working with, and hope that their experience was as beneficial as it was to me.
Special thanks are due to Dan Hyduke who intelligently aided my research and
became a friend. Additionally, I would like to thank my committee members Dr.
Miller, Dr. Boyd, and Dr. Teppen, for the knowledge that they imparted to me
through classes and personal discussions.

My utmost praise and thanks are extended to my fellow graduate student Jamaica
Prince, who aided my progress in innumerable ways and became my closest friend.
Through Jamaica I learned several lessons about professionalism and life in general
and I will always be grateful. I also thank my parents Richard and Judith

Dombrowski who always encouraged me to excel and gave me my core beliefs.

iv

 

List of Ta‘
List of Fit
1. L'lt'oc't.

1.2 L":

 

MGR,
Malian)

Table of Contents

List of Tables
List of Figures

1. Introduction to Adsorption
1.1 Rationale
1.2 Introduction to Sorption Theories
1.2.1 Sorbent Structural Modeling
1.2.2 Classical methods
1.2.3 Semi-empirical Methods
1.2.4 Statistical Thermodynamic Models
1.3 Choice of Method
1.4 References
2. Pore Size Analysis of Activated Carbons from Argon
and Nitrogen Porosimetry Using Density Functional Theory
2.1 Introduction
2.2. Theoretical Section: Density Functional Theory
2.3. Experimental Section: Argon and Nitrogen Porosimetry
2.4. Fitting of the DPT Model Parameters
2.5. DPT Model Isotherms for Carbon Slit Pores
2.6. PSD Analysis of Porous Carbons
2.7. Discussion of Argon and Nitrogen Porosimetry Methods
2.8. Conclusion
2.10 References
3. Carbon Nanotubes as an Alternate Reference Adsorbent
3. 1. Introduction
3.2. Thermodynamic Model
3.3. Method for Fitting the Potential Parameters
3.3.1 Potential Function
3.3.2 Electric properties
3.3.3 Nonporous carbon method
3.3.4 Nanotubes as a Reference Adsorbent
3.4. Results
3.4.1 General information
3.4.2Verifying validity
3.4.3 Carbon dioxide adsorption
3.4.4 Test of the Carbon dioxide Parameters
3.5. Conclusions
3.6. References
4. The Horvath-Kawazoe Method Revisited
4.1 Introduction

viii

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4.3 C:
4: D;
4.4 R.
4.5 D
4.5 C. '
4.‘ R.’
5. A Tw-
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5.1 Ir:
5.: D.

5.3 RL"
5.4 o'
6. ConclllxF
6.1 SUI
6.2m“

 

4.2 Critique of Original HK Method

4.3 Description of Generalized HK Method

4.4 Results

4.5 Discussion

4.6 Conclusion

4.7 References

5. A Two-Stage Horvath-Kawazoe Adsorption Model

for Pore Size Distribution Analysis

5.1 Introduction

5.2 Description of Two-Stage Horvath-Kawazoe Method
5.2.1 Review of the Original HK Model
5.2.2 Description of the Two-Stage HK Pore Filling Model

5.3 Results

5.4 Conclusions

5.5 References

6. Conclusion and Future Directions

6.1 Summary of Results

6.2 Topics for Future Investigation
6.2.1 Further Improvements to the HK model
6.2.2 Investigation of Adsorption at Alternate Temperatures
6.2.3 Study of Alternate Probe Gases
6.2.4 Study of Irregular Adsorbents

vi

98

105
117
126
132
135

137
137
139
139
142
146
163
166
168
168
170
171
172
172
173

Chapter I
Table 2.1 '

lb]: 2.2 E

 

Chapter 3
Table 3.1 E

Chapter 4

Iah’e 4.1: ii
and wet-51' '
Table 4.5. :
ltmxtalur.

Chapter 5
Table 5.1 l
altrage por

 

List of Tables

Chapter 2

Table 2.1 Lennard-J ones Potential Parameters 25
Table 2.2 PSD Maxima 45
Chapter 3

Table 3.1 Potential Parameters 78
Chapter 4

Table 4.1: Fitted parameters for unweighted

and weighted local density models 113
Table 4.2: Dimensionless solid-fluid potential and

temperature for probe gases on graphite. 114
Chapter 5

Table 5.1 The maximum pore size (Hmax),
average pore size (H) and the pore volume Vp. 162

vii

 

 

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Chapter 2
Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 2.8
Figure 2.9
Figure 2.10
Figure 2.11
Figure 2.12
Figure 2.13
Figure 2.14

Chapter 3
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6
Figure 3.7
Figure 3.8
Figure 3.9
Figure 3.10
Figure 3.11
Figure 3.12
Figure 3.13
Figure 3.14

Chapter 4
Figure 4.1:
Figure 4.2:
Figure 4.3:
Figure 4.4:
Figure 4.5:

Figure 4.6:
Figure 4.7:
Figure 4.8:

List of Figures

Schematic of a Slit Pore

Evaluation of Surface Area

Nonporous Argon Adsorption

Argon Specific Excess Adsorption

Nitrogen Specific Excess Adsorption

Pore-filling Correlation

Saran char Adsorption Isotherms

Coconut char Adsorption Isotherms

Granular Activated Carbon Adsorption Isotherms
Powdered Activated Carbon Adsorption Isotherms
Saran char Pore Size Distribution

Coconut char Pore Size Distribution

Granular Activated Carbon Pore Size Distribution
Powdered Activated Carbon Pore Size Distribution

Nonporous Argon Adsorption

TEM Photograph of Single-Walled Carbon Nanotubes
Relevant Region of Isotherm Fitting

TEM Pore Size Distribution

Nitrogen Specific Excess Adsorption in SWNT

Argon Specific Excess Adsorption in SWNT
Comparison of the TEM, Argon and Nitrogen PSD

A Carbon Dioxide Adsorption Isotherm

Excess carbon dioxide adsorption isotherms in SWNT
Carbon dioxide excess adsorption isotherms in Slit Pores
Comparison of Surfaces for Fitting Parameters
Adsorption Isotherms on Granular Activated Carbon
PSD on Granular Activated Carbon

Adsorption isotherms with SWNT Potential Parameters

Pore-filling Correlations

Comparison of Pore-filling Correlations

Local Density Profile

Local Densities

Contact Layer and Interior Layer Peak Heights and
Variances

Comparison of Local Density Profiles

19
26
28
30
31
33
34
35
36
37
38
39
4O
42

66
67
7o
72
73
74
76
77
79
8o
81
83
84
87

97

104
115
116

118
120

Pore filling Correlation for Unweighted Horvath-Kawazoe 122

Pore filling Correlation for Density-weighted Horvath-

viii

 

 

harsh

Haze 5.1

fifiril

Figure 4.9:

Figure 4.10:
Figure 4.1 1:
Figure 4.12:

Chapter 5
Figure 5.1:

Figure 5.2:
Figure 5.3

Figure 5.4:
Figure 5.5
Figure 5.6

Figure 5.7 a-d.
Figure 5.8 a-d.
Figure 5.9 a-d

Figure 5.10 a-d
Figure 5.11 a-d

Figure 5.12 a-b.

Kawazoe

Comparison of Pore Filling Correlations
Comparison of Model Isotherms
Nitrogen Adsorption Isotherm
Comparison of Pore Size Distributions

Schematic of Two-stage Horvath-Kawazoe Pore
Filling Method

Fitted Correlation of the Slope of 206 Pore

A Comparison of DFT Model Isotherms and
Two—stage HK

Fitted Correlation of the Slope

Comparison of a Fitted Two-stage HK Model Isotherm
Spatially Varying Potential in a 2 Molecular Diameter
Slit Pore

DPT Model Isotherms

Two-stage HK Model Isotherms

Filling Pressure Correlations of DPT and Two-stage
HK Methods

Fitted Isotherms to Experimental Data

Pore Size Distributions Calculated from Fitted
Isotherms

Fictitious Isotherm Fit and Comparison

of the PSD Predictions

 

123
125
128
129
131

143
147

148
149
152
154
156
157

158
160

161

164

 

1.1 Ratim

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Chapter 1.

Introduction to Adsorption

1.1 Rationale
Characterization methods for porous materials are required to effectively utilize their

properties such as pore size and adsorptive capacities for a variety of purposes. Size
selective adsorbents may be used for a variety of catalytic reactions as well as separations
based solely on the physical size and potential interactions of the bulk fluid molecules.
Several techniques based upon the size variations are employed, such as restriction of the
mass transfer of the larger compounds. In a reactive system it may be desirable to
selectively control the species that have access to the catalyst. Larger species can be
excluded from the catalyst, allowing the smaller species to selectively react. [1] Or
conversely smaller species may be allowed to preferentially diffuse out of the reactive
area, leaving the larger species to react further and possibly increase the yield of the
smaller species, as in the production of xylene. [2] The pore size may also be used to
separate gases based upon there diffusivity. Uranium isotopes may be enriched through
the minor decrease in the Knudsen diffusion of the heavier isotopes in pores that are
narrower than the mean free path of the gaseous compound. In addition to diffusion
properties, it has been demonstrated that altering the size of a pore by less than one
angstrom may increase the selectivity of one compound over another by several orders of
magnitude. [3] Altemately, the storage of hydrogen in carbon nanotubes is a strong
function of their size and tube separation. [4] Therefore a means to predict the relative
affinities is required to optimally utilize any sorbent used in temperature or pressure

SWing adsorption.

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1.2 Introduction to Sorption Theories
Several techniques exist to determine information about the surface characteristics of a

solid; all require a probe, an interaction, and a response. [5] Probes may be either waves
or matter, each with various limitations and strengths. Wave probes such as
electromagnetic radiation or electrons suffer from the lack of depth which they allow one
to “see” into the surface therefore can only be used with materials that contain a regular
pore size throughout. X-ray diffraction [6,7] and electron microscopy [7] may be used to
determine the pore size of regular materials such as carbon nanotubes or zeolites. Matter
probes such as an adsorbing gas have the ability to penetrate into the material beyond the
layers where wave probes are absorbed, however they may experience size exclusion
preventing them from entering narrow or bottlenecked pores. Gas adsorption is still the
principle technique [8] used to determine the structure of amorphous porous media.

Sorption measurements are based upon the heterogeneity of the sorbent, the cause of the
heterogeneity may come from the surface chemistry, surface charges, or structural
variations. The heterogeneity displayed in the filling pressures of an isotherm may be
regarded to be derived from the range of pore widths in chemically homogeneous
materials. An integral equation can be written which relates the measured isotherm HP)

to the source of the heterogeneity, which is expressed in equation (1.1) solely as the size

ofapore H.
”max

F(P)= jr(P,H)F(H)dH (1.1)
Hmin

Where HRH) is the thermodynamic model isotherm, at pressure P, and size H, and F (H )
is the pore size distribution. Although the adsorption integral is a convenient way to

characterize a solid it is based upon assumptions to render the problem tractable while

 

      
 

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sacrificing some degree of realism. Some typical assumptions chosen to simplify the
model are, independent noninteracting pores, a particular geometric structure is assumed
to represent the pore width such as a slit or cylinder, and the surfaces are assumed to be

chemically homogeneous.

1.2.1 Sorbent Structural Modeling
While the previously mentioned assumptions do simplify inversion of the adsorption

integral to determine the pore size distribution (PSD), they are less accurate than a
molecular simulation of an exact representation of the solid. However, since the exact
pore matrix within an amorphous carbon is unknown the nearest representations are
based upon either physical data gathered from the system studied or an amorphous
representation. The representations of the adsorbent require statistical data of the systems
they are used to study, for instance pore glasses may be used [9] at various quench times
to control the pore size and connectivity. These models have the advantage of realistic
irregularly shaped and interconnected pores, thus attaining some degree of realism for
micropores, however the model fails to yield accurate results for the high pressure range
of an isotherm which is usually attributed to filling of meso and macropores. An
alternative to the controlled pore glass is the reverse Monte Carlo method where the
radial distribution function (RDF) of carbon plates are chosen to match the RDF
determined by X-rays or neutron diffraction which are incapable of measuring the effect
over mesopores. Like the pore glass method the experimentally measured isotherm and
that of the disordered solid determined via simulation agree only in the low-pressure

range. The meso and macropores that are generally considered to be responsible for the

  
  

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filling at higher pressures cannot be determined from the diffraction patterns due to their

negligible influence on the opposing pore wall.

1.2.2 Classical Methods
The simple geometric models have the advantage of requiring less computational time

and the ability to account for pores of any size range. There are a variety of established
methods based upon simple geometric models that predict the thermodynamic model
isotherm used to invert the adsorption integral (equation 1.1), reducing the problem to
selecting the model which most appropriately describes the system of interest. The
classical methods are based upon measured bulk fluid properties and fail as pores
approach molecular size, where continuum approximations no longer apply. The Kelvin
equation or modified versions may be used for pores that are larger than 75 A [10]

otherwise the pore size will tend to be underestimated.

P -27
In i = ’ (1.2)
[Po] RTpr(rk—t)

In the modified Kelvin Equation )7 is the surface tension, p, is the liquid density of the

 

adsorbate, t is the adsorbate film thickness, rk is the radius of the meniscus, R is the gas
constant, T is the temperature, PC and P0 are the filling pressure and saturation pressures
respectively. Equation 1.2 is a form of the modified Kelvin equation for a perfectly
wetting fluid with constant molar volume, which relates the change in mechanical
pressure to the change in molar free energy. The addition of a term accounting for the
thickness of the film that occurs due to wetting prior to condensation increases the
accuracy and range of the model. Micropores are known to have coarse structure in the

adsorbed liquid phase, as well as enhanced adsorption due to fluid wall interactions. The

Keilln 59.35
trustee is I:
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predict hp:
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Kelvin equation is derived to describe the equilibrium mechanical pressure where the
interface is a continuum, therefore cannot be applied to pores with structured fluids which
strongly interact with the pore walls. The simplicity of the model and its ability to
predict hysteresis based upon changes in the radius of curvature for advancing and
retreating interfaces, has lead many to attempt modifications that render it applicable to
smaller pores. Recent studies of MCM—4l adsorbents [11,12,13] it was found that the
modified Kelvin equation could be brought to closer agreement to measured capillary
condensation pressures by subtracting a constant from the (rk-t) term of the right hand
side of equation 1.2. The value of this constant was found to vary for different gases and
is based solely upon empirical observations, therefore there is no means presented to
calculate such a parameter for different gas solid systems.

The Kelvin equation has been further modified to account for the effects of surface
layering and adsorbent—adsorbate interactions for a cylindrical pores. [14,15,16]
Assuming that the vapor-liquid interface is represented by a cylindrical meniscus during
adsorption and by a hemispherical meniscus during desorption, and invoking the Defay-
Prigogene expression for a curvature—dependent surface tension [17], the equilibrium

condition for capillary coexistence in a cylindrical pore is obtained as

d(AG)
dN

V1700 (R - t)
(R — t — 11/2)2

_ Pov ~ " 2v,ym(R_t)2
—lpg gdP+¢(R’R) [(R—t)(R—t-4)+AGfl/4](R—t_l)

 

 

P ~
=0=jp:vgdp+¢(t,R)-—

(1.3)

 

where d(AG)/dN is the Gibbs free energy change per mole adsorbed; vg and v, are the

respective gas and liquid molar volumes at the saturation pressure P0; y is the surface

 

tension of 1'
pore radiu~
inter ‘tion

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adsorbent .1
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tension of the vapor-condensate interface; 03y is the adsorbate molecular diameter; R is the
pore radius; and A = (2/3)“20'ff . [14] The potential field $(r,R) represents the net

interaction of an adsorbate molecule at radial coordinate r with the surrounding
condensate and solid. This field is given as the difference between the adsorbate-
adsorbent and adsorbate-adsorbate potentials, and it is represented using the 9-3 potential
for an adsorbate confined within a cylindrical pore of a structureless Lennard-Jones solid.
The adsorbed layer thickness 1 can be determined for a given pore radius and potential
field by solving the equality given by equation (1.3). The capillary coexistence curve can
then be constructed by calculating the condensation pressure Pg from equation (1.3) using
the known values of film thickness and pore radius. This method was shown to give
good agreement with the capillary coexistence curve and pore criticality predicted by
DPT for adsorption of nitrogen [14,15,16] and other probe gases [15] on MCM-41
materials. The method also qualitatively predicts some wetting features of the isotherm
below the capillary condensation pressure. [14,15] This model is therefore capable of
predicting continuous filling of micropores with a transition to condensation of pore
vapor to pore liquid.

1.2.3 Semi-empirical Methods

Semi-empirical methods have also been developed to account for the enhanced
adsorption in micropores due to the superposition of the potential of the pore walls. The
Horvath-Kawazoe method HK relates the change in the free energy of the system to the
enthalpy and entropy change that occurs due to adsorption. [18] With the assumption
that the entropy change of the system is small the change in free energy of the system

approaches the change in enthalpy of the system. The original HK was based upon the

 

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Lennard-Jones 12-6 potential for a plane of graphite, and was integrated to yield the
familiar 10-4 potential. The potential well depth was calculated using the Frenkel Halsey
Hill theory and utilized the Kirkwood-Muller expression for the adsorption potential.
Unfortunately, these were erroneously applied, as will be discussed in Chapters 4 and 5.
However, the HK method was a milestone in the advancement of adsorptive studies, as it
was specifically designed for use with micropores and the enhancement of adsorption due
to the potential between the adsorbent and adsorbate. The HK method led to the
development of similar models with different geometries such as the Saito-Foley method
for cylindrical pores [19] or the Yang method for spherical pores [20,21]. Further
attempts to improve the HK method have involved improvements to the potential
function [20,22] and modifications to make the density profile within the pore more
realistic. [20,23,24] In addition to the work done to improve the HK method as a stand
alone technique it has been modified to the better correlate with the results obtained via
density functional theory DPT, such alterations include modification of the average
density profile and addition of a step to account for pore wall wetting prior to pore
condensation. A more detailed discussion of the modifications to the HK method will be
presented in Chapters 4 and 5.

1.2.4 Statistical Thermodynamic Models

Statistical thermodynamic models such as the density functional theory DPT
[25,26,27,28] produce model isotherms and predict the density profile within a pore that
rivals the predictions of molecular simulation MS for simple geometries. [10] The DPT
method is only applicable to pores that are of a simple geometry such as a sphere, slit, or

cylinder, and do not contain much chemical heterogeneity of the surface atoms, however

 

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for simple systems it provides accurate results in less than one hundredth of the time
required for MS. When utilizing the DPT method all pores in a particular carbon are
assumed to have a common shape with size being the only source of heterogeneity. The
details of DPT will be discussed in Chapters 2 and 3, therefore only a brief description
will be provided here. The grand potential of the pore is written as a function of the

density profile within a pore
£2[p(r)1=Flp(r)l—Jp(r)lu—V...(r)ldr (1.4)

where .Q is the grand potential, F is the Helmholtz free energy functional, p( r) is the local
fluid density, r is the position in the pore, u is the chemical potential and Vex, is the is a
spatially varying potential. The grand potential is then minimized for a specific
temperature, chemical potential, and pore size with respect to the smoothed density
profile. Once the density profile is obtained the volume average of the density is taken
across the entire pore volume and used to represent the density at a particular bulk
chemical potential and pore size. Although local minima may exist for several density
profiles, the lowest grand potential indicates the equilibrium profile. A series of
isotherms are generated over the pressure range of interest and used in conjunction with
equation (1.1) to determine the pore size distribution.

1.3 Choice of Method

The choice of method used to determine the PSD of a particular material is dependant
upon the known physical characteristics of the system of interest. The most accurate
method is a MS based upon assumptions about the materials that are being studied.
When the simplistic pore models are chosen the DPT method predicts results that are

comparable to the MS while requiring one hundredth of the time, additionally the density

 

 

,.

proxies l.

 

macropire
range it is

and mtxlzt‘.

 

 

 

Hinge, DO“

whim; ft)

OI MS 1“)!

(hitter 5 3

profiles in the pores are also predicted. The advantage of using either the DPT or MS
methods is that they may are capable of predicting the behavior of pores of any size.
Methods such as the HK and Kelvin equation are limited in the ranges where they can be
used appropriately. The Kelvin equation is capable of producing accurate results for
macropores, while modified versions appear to extend the applicability to the mesoporous
range it is doubtful however that this may be applied to the micropore range. The HK
and modified versions are capable of predicting the PSD in the micro and mesopore
range, however is less accurate than either the DPT or MS methods, however is
advantageous in the efficiency of determining the model isotherms. Therefore when DPT
or MS isotherms are not in existence the modified two-stage HK method described in

Chapter 5 is recommended.

   
    

1.4 Rtitli

——

1. (her:
3. LEM
let the
Will.
3. Mali.
Bucit
i- “it;
5- his:
Neil

6 inou

1.4 References

 

1. Chemical engineering progress Jan 2001 p 18.

2. Lastoskie, C.M. A Statistical Mechanics Interpretation of the Adsorption Isotherm
for the Characterization of porous Sorbents PhD. Dissertation Cornell university
1994.

3. Maddox M.W, and Gubbins KE. Molecular Simulation of Fluid Adsorption in
Buckytubes. Langmuir 1995; 11 3988.

4. Wang, Q.; Johnson, K. Journal of Chemical Physics 1999, 110, 577.

5. Adamson A.Physical Chemistry of Surfaces 5th ed. John Wiley and Sons Inc,
New York 1990.

6. Inoue S, Ichikuni N, Suzuki T, Kaneko K. Capillary Condensation of N2 on
Multiwall Carbon Nanotubes The Journal of Physical Chemistry B 1998; 102 (24)
4689.

7. Rinzler, A.G.; Liu, J .; Dai, H.; Nikolaev, P.; Huffman, C. B.; Rodriguez-Macias,
F.J.; Boul ,P.J.; Lu, A.H.; Heymann, D.; Colbert, D. T.; Lee, R. S., Fischer, J. E.,
Rao, A.M. Eklund, P.C. Smalley R. E. Large Scale purification of single-wall
carbon nanotubes: process, product, and characterization. Appl. Phys. A.. 1998,
67, 29.

8. Gregg SJ, Sing KSW. Adsorption, Surface Area and Porosity. Academic Press,
London 1982.

9. Gelb, L.D. and KB. Gubbins, Langmuir 15 (1999) 305.

10

f

10. Lastoskie. 1
Molecular
Engineenr.

11.51.10“ .1:

12.81. Krill, _\

13.11.1(nilt. .\

14. CG. Sonu

15. CG. Sonxx

16' S-K. Bilatz.‘

17. R, Dem) :1
“id C 0.. B

18‘ H0“ ath G
Dismbutio
Japan 16 i l

19.3211]O A. an
Adgwpmm

20. Rege‘ S an.
Distributioi

21. Cheng‘ L a
Sphen'm P
Enigineen.n

7?

ys- DOmbIO“ \i

Model for P

 

10. Lastoskie, C. and Gubbins, K. Characterization of Porous Materials Using
Molecular Theory and Simulation. Submitted to Advances in Chemical
Engineering 2001.

1 l. M. Kruk and M. Jaroniec, Chem. Mater. 12 (2000) 222.

12. M. Kruk, M. Jaroniec, J.M. Kim and A. R. Ryoo, Langmuir 15 (1999) 5279.

13. M. Kruk, M. Jaroniec and A. Sayari, Langmuir 13 (1997) 6267.

14. CG. Sonwane and SK. Bhatia, Chem. Eng. Sci. 53 (1998) 3143.

15. CG. Sonwane, S.K. Bhatia and N. Calos, Ind. Eng. Chem. Res. 37 (1998) 2271.

16. SK. Bhatia and CG. Sonwane, Langmuir 14 (1998) 1521.

17. R. Defay and I. Prigogine, Surface Tension and Adsorption: Longmans, Green
and Co., Bristol ( 1966).

18. Horvath G. and Kawazoe, K. Method for the Calculation of Effective Pore Size
Distribution in Molecular Sieve Carbon. Journal of Chemical Engineering of
Japan 16 (1983), 470.

19. Saito A. and Foley, H. Curvature and Parametric Sensitivity in Models for
Adsorption in Micropores. AIChE Journal 37 (1991) 3, 429.

20. Rege, S and Yang, R. Corrected Horvath-Kawazoe Equations for Pore-Size
Distribution. AICHE Journal 46 (2000) 4, 734.

21. Cheng, L. and Yang, R. Improved Horvath-Kawazoe Equations Icluding
Spherical Pore Models for Calculating Micropore Size Distributions Chemical
Enigineering Science 49 ( 1994) 16, 2599.

22. Dombrowski, R. and Lastoskie, C. A Two-Stage Horvath-Kawazoe Pore Filling

Model for Pore Size Distribution Analysis. Submitted to the Journal of AIChE.

11

 

 

———

23. Llilushlé. C.

Studies in Surf.

 

14. Dombrov ski. I~
Relisited. C 0".
35.1.:«8105kie. C; (
Slit Pore: A De
36. Seaton. .\'.; We}

IlRaxikovitch. P;

 

 

28. Nilson. R and C

 

23. Lastoskie, C. A modified Horvath—Kawazoe Method for Micropore Size Analysis.
Studies in Surface Science and Catalysis 128 (2000) 475.

24. Dombrowski, R.; Lastoskie, C.; and Hyduke, D. The Horvath-Kawazoe Method
Revisited. Colloids and Surfaces A in press 2001

25. Lastoskie, C.; Gubbins, K.; Quirke, N. Pore size Heterogeneity and the Carbon
Slit Pore: A Density Functional Theory Model.Lamgmuir 9 (1993) 2693.

26. Seaton, N.; Walton, J.; Quirke, N. Carbon 27 (1989) 853.

27. Ravikovitch, P.; Haller, G.; Neimark, A. Adv. Coll. Int. Sci. 76 (1998) 203

28. Nilson, R and Griffith, S. J Chem Phys. 1 11 (1999) 4281.

12

 

Dialect 5111 R1. 11)
from argon and mung.
16111151111511.5101

 

Chapter 2.

Dombrowski RJ, Hyduke DR, Lastoskie CM .Pore size analysis of activated carbons

from argon and nitrogen porosimetry using density functional theory. LANGMUIR
16(11): 5041-5050 MAY 30 2000.

13

Pore =

and Kim

2.1. Introduction
Actuated cart
and a selectixe '8
mixtures. The t1
actuated carbon
Characterization
im0hmguwth
Comme
Various feedstt
The Carbonize
dIOXIdQ 10 dcx
0f manUfactu

Althou

8h suc

111319311" QCI]

Chapter 2.

Pore Size Analysis of Activated Carbons from Argon

and Nitrogen Porosimetry Using Density Functional Theory

2.1. Introduction

Activated carbons are used in the chemical process industries as gas storage media
and as selective adsorbents for the separation and purification of vapor- and liquid-phase
mixtures. The thermodynamic behavior of a fluid, or a fluid mixture, adsorbed on an
activated carbon depends principally on the pore size distribution (PSD) of the adsorbent.
Characterization of the PSD is therefore an important problem in industrial applications
involving the use of activated carbons.

Commercial activated carbons are manufactured by therrnolytic decomposition of
various feedstock materials, including cellulosic matter and synthetic polymer resins.
The carbonized thermolysis products are “activated” via exposure to steam or carbon
dioxide to develop the accessible internal pore volume of the adsorbent [1]. This method
of manufacture yields activated carbons that have semiamorphous microstructures.
Although such materials defy a simple topological description, we might envision a
“typical” activated carbon as a contiguous network of graphitic microcrystals, joined
along their edges by surface functional groups, with each crystal containing on the order
of two hundred atoms, stacked into two or three graphite layers [2]. The size, shape and

connectivity of the pores in an activated carbon depend in a complex manner on the

14

leelstorl miteri.

 

oi resin or CO;

 

conditions and the
little on $1411111t'
110141 mtsuplf
it hits diiirctct.

determined indtre

 

Extracting PSD in‘
make several asst
solution of the P
Carbon is com
homogeneous. g
activated curlior
presented :8 est

ill order to r

feedstock material used, the temperature and duration of the carbonization, and the extent
of steam- or COz-activation. At the present time, the relationship between the processing
conditions and the resultant carbon structure is poorly understood.

Unlike crystalline adsorbents such as zeolites, or templated porous materials such as
MCM-41 mesoporous molecular sieves, activated carbons are not readily characterized
by X-ray diffraction or by neutron scattering. Rather, the PSD of a carbon is generally
determined indirectly through interpretation of gas adsorption measurements [3].
Extracting PSD information from the experimental adsorption isotherm requires that we
make several assumptions regarding the structure of the adsorbent, thus rendering the
solution of the PSD numerically tractable. Usually, it is assumed that the activated
carbon is composed of noninterconnected, slit-shaped pores with chemically
homogeneous, graphitic surfaces. By invoking these assumptions, the PSD of an
activated carbon, f(H), can be determined by solving the adsorption integral equation
presented as equation 1.1.

In order to obtain accurate results for the adsorbent PSD, a realistic pore filling model
must be employed to construct the model isotherms HRH) that appear on the right-hand
side of equation (1.1). Lastoskie reported a thermodynamic model for nitrogen
adsorption on graphitic slit pores at 77K based upon density functional theory (DPT) [4].
Comparisons of DPT results to Monte Carlo molecular simulations have shown that the
DPT model accurately describes the density profile of the inhomogeneous adsorbed fluid
with simple pore structures such as carbon slit pores [5]. DPT thus provides correct
model adsorption isotherms for use in equation (1.1), at a small fraction of the

computational time needed to obtain the same isotherms from molecular dynamics or

15

Monte Carlo moi
lielxin equation
frequently Used :
that these C1;i.\.\lt'
in micropores.
18.9]. Sloreoxe
the Kelvin equ.
pore filling mo
Nitrogen i
inerpensne. r
e‘ll‘et‘lmentsl
Dermal iiquit
temperailllc 5
Choice‘ Sitter
maintained
SituatiOnS it
Carbons 1h
interdtlign.
its; Carbc
titles “01 2
for anhf e

Monte Carlo molecular simulation. Other, simpler methods that predate DPT, such as the
Kelvin equation [6] or the Horvath-Kawazoe (HK) equation [7], have also been
frequently used as pore filling models. However, it has been convincingly demonstrated
that these classical pore filling models do not provide a realistic description of adsorption
in micropores, and therefore they are not suitable for micropore size characterization
[8,9]. Moreover, the classical models are limited in the scope of their applicability, e.g.
the Kelvin equation for mesopores and the HK method for micropores, whereas the DPT
pore filling model describes adsorption over the entire range of carbon pore sizes.
Nitrogen is the probe gas most commonly used for PSD analysis because it is
inexpensive, readily obtained, inert, and well-studied in the adsorption literature. The
experimental isotherm on the left-hand side of equation (1.1) is usually measured at the
normal liquid nitrogen boiling point of 77 K, which is sufficiently below the critical
temperature so that a large uptake of the adsorbate is realized. This is again a convenient
choice, since liquid nitrogen is inexpensive and isothermal conditions can easily be
maintained using liquid nitrogen as a cryostat. Despite these merits, there may arise
situations in which a probe gas species other than nitrogen is desirable. On activated
carbons that have substantial chemical heterogeneity, there may be significant
interactions between the quadrupolar nitrogen molecule and various functional groups
(e.g. carboxylic acids) on the carbon surface. The DPT model for nitrogen adsorption
does not account for the nonspherical shape of the nitrogen molecule, nor does it allow
for any electrostatic interactions other than dipole and induced dipole interactions, of
nitrogen molecules with the carbon surface. The neglect of these features may result in

errors in the calculation of the carbon PSD.

16

  
  
  
  

Argon is
heterogeneous :tc'.
moments. and it
extensitely for PI
oxides [3.10]. D
normal boiling po'
Egon as the tetr
more expensite
0111 1116 argon aj
Therefore. in t‘
Mes at 77
rePorted in S:-
ODiained {QT

Illifil'prg‘ exp

22' Theo"

A d
1llfifamre [.
the local d.

funcr‘mnu]

Q

Argon is nearly an ideal probe gas for characterizing the pore structure of
heterogeneous adsorbents, since it is a monatomic, spherical molecule with no multipolar
moments, and it is similar in size to nitrogen. Argon porosimetry has been used
extensively for PSD analysis of activated carbons, silicas, aluminas and mesoporous
oxides [3,10]. DPT model isotherms have been computed for argon adsorption at its
normal boiling point of 87 K [11]. This temperature is readily maintained by using liquid
argon as the temperature bath for the sorption measurement. Because liquid argon is
more expensive than liquid nitrogen, however, it is convenient and economical to carry
out the argon adsorption measurement at the normal liquid nitrogen boiling point of 77 K.
Therefore, in this work we present a DPT model for argon adsorption in graphitic slit
pores at 77 K. This model is described in Section 2.2 and the model isotherms are
reported in Section 2.3. We then compare and analyze, in Section 2.4, the PSD results
obtained for several activated carbons, using the argon and nitrogen DPT models to

interpret experimental argon and nitrogen isotherms measured on the carbon adsorbents.

2.2. Theoretical Section: Density Functional Theory

A detailed description of the DPT method has been presented extensively in the
literature [4], and so we will provide a short summary of the theory. In the DPT method,
the local density profile p(r) of the inhomogeneous adsorbed fluid at phase equilibrium is
determined by solving for the density profile that minimizes the grand potential

functional [2 of the confined fluid. The grand potential functional is given as

Q[p(r)1= Hptrit — ldrprritu — rem] (2.1)

17

there F is the 11.

 

the tthorhate-soly
laden 01611116 en'.
H.1ocrtled heme:1
directions thigurt
coordinate. the div.

Valzl = 0.1:
hequation 12.21. t

and H is the sepur.

 

 

011116011?“ng pit;
Seeing A = 3.35

potential [11]

Where on W E" a

it" .
hit a €41an arm

White 13368 ha

lassuming a Sport '-

51 ~_

db. Tills approt
a .
oneiltmtgnmnL1

The, 11311111101!

turn of the hard \

“30113316 interact

where F is the Helmholtz free energy functional; p is the chemical potential; and Vex, is
the adsorbate-solid potential at spatial coordinate r. The integration in equation (2.1) is
taken over the entire pore volume of the adsorbent. In a slit pore of physical pore width
H, located between two homogeneous graphitic slabs that are unbounded in the lateral
directions (Figure 2.1), the external potential Vex, depends only upon one spatial

coordinate, the distance z of the adsorbate center-of-mass from the adsorbent surface:

V... (2) = ¢., (z) + 4)., (H - z) (2.2)
In equation (2.2), «12,, is the adsorbate interaction potential with one of the bounding slabs,
and H is the separation distance between the carbon nuclei in the surface graphite layers
of the opposing pore walls. For a solid density p, = 0.114 A'3 and a graphite lattice plane
spacing A = 3.35 A, the adsorbate-solid potential is well-represented by the 10-4-3
potential [12]

o *2178 6281-2-(file—[flj4 03’ 1 (23)
"" ‘fp‘ " [5 z z 3A(z+o.6iA)-‘j '

where 03:” and 83f are the Lennard-Jones parameters for the adsorbate molecule interaction

 

with a carbon atom in the porous solid. In equation (2.3), the lateral structure of the
graphite layers has been replaced with an integration over the lattice plane surface area
(assuming a spatially uniform carbon atom density) and a summation of the layers in each
slab. This approximation conveniently reduces the DPT grand potential minimization to
a one-dimensional solution of the adsorbate density profile p(z).

The Helmholtz free energy functional is computed from perturbation theory as the
sum of the hard sphere Helmholtz free energy functional F i, (for the repulsive adsorbate-

adsorbate interactions), and a mean field contribution for the dispersion interactions:

18

F‘JI‘IrI
r .
c A

“I ‘19

ALL 1

.1.
'i'

Q Q
Q Q
Q Q
Q Q
Q Q
Q Q
Q Q
Q Q

QQQQQQQQ
QQQQQQQQ

   

l
.1
"1'

Figure 2.1: Schematic of graphitic slit pore model showing definition of physical pore

width H.

liplrl1=h1ilf

 

The first term if
tttractire fluid-:7
Chandler-Anders.

octr—r'

there r, : ‘1 " 0,

whose centers of :1

 

 

op‘lrlzl

12.6 i
The Parameters 0.
m3 adsorbate-ad“.
The hard-5pm.

111631 and an “ch

1
Flp(r)l = lilp(r);d] +-2- Hdrde p(r)p(l’ >¢...dr — r1) (2.4)

The first term in equation (2.4) is calculated for hard spheres of diameter d, and the
attractive fluid-fluid potential (than in the second term is obtained from the Weeks-
Chandler-Anderson representation [13]

r—r'l) = (1)” (Ir—r'l), Ir—r'l > rm

 

¢‘"’( (2.5)

— E I, , Ir — r | < rm
where rm = 2”6 of, and or, is the Lennard-Jones potential for a pair of adsorbate molecules

whose centers of mass are separated by a distance r:

.,..-..,,r[:._iaio_tri

(2.6)
The parameters 0}, and 817 are the Lennard-Jones molecular diameter and well depth for

the adsorbate—adsorbate pair potential.

The hard-sphere Helmholtz free energy functional in equation (2.4) is divided into an

ideal and an excess component
fl[p(r);dl = ledrp(r)[ln(A3p(r)) — 11+ kTIdrp(r)f...lE(r);d]
(2.7)
In equation (2.7), the first term on the right-hand side represents the ideal gas
contribution to the hard-sphere free energy; A=h/(271mk7')”2 is the thermal

wavelength, T is the temperature, m is the adsorbate molecular mass, and h and k are the
Planck and Boltzmann constants, respectively. The excess molar Helmholtz free energy,

fa, that appears in the second term on the right-hand side of equation (2.7), is calculated

20

using a northern
density profile p
adsorbed within r

etlects [l-l]. Tr.Q

 

oscillatory dct‘olli
profiles hate alxt
micropores [5].
minimization of
calculation of th:
is obtained in t1
and molecular 8
To pro;
13.71, the dens
account for the
density PTOlile
Elf) :
\ymOUS presc
“011:, We 300
this nonlocai
thereby). 18.” 0b[

“71‘“1th an

“31.

using a nonlocal or “smoothed” density profile fir) instead of the local adsorbed fluid

density profile p(r). It is known from surface force microscopy measurements that fluids
adsorbed within micropores have highly structured density profiles due to confinement
effects [14]. The high degree of fluid ordering at the pore wall surface results in an
oscillatory density profile that dampens toward the center of the pore. Similar density
profiles have also been observed in molecular simulation studies of fluids confined in
micropores [5]. The DPT model does not yield an oscillatory density profile from the
minimization of the grand potential functional, if the local density profile is used in the
calculation of the excess Helmholtz free energy. A relatively unstructured density profile
is obtained in the local version of DPT; such a profile is inconsistent with experimental
and molecular simulation results.

To properly represent the excess free energy of the adsorbed fluid in equation
(2.7), the density profile that is input to the f“ functional must be weighted so as to
account for the sharp spatial gradients in the confined fluid density. A smoothed input

density profile is therefore calculated as
Ed) = lde p(r')W11r - from] (2.8)

Various prescriptions for the weighting function w have been proposed [15]. In this
work, we adopt the weighting function suggested by Tarazona; a detailed description of
this nonlocal version of DPT can be found elsewhere [16]. The excess Helmholtz free
energy is obtained from the Camahan-Starling equation of state for a hard sphere fluid
[17], with an equivalent hard sphere diameter d given by the Barker-Henderson diameter

[18].

21

 

 

 

Adsorption ts
adsorbate densrt,
115011th fluid (‘
adsorption PRH

111:11.11 Of 1116 pure

rlP.H)=-'_
H.

 

“here p13. Pei an
POWIIULU “ The 1

#195) :P

13(95):,

Adsorption isotherms for pores of different width H are constructed by solving for the
adsorbate density profiles p(z) that minimize the grand potential functional of the
adsorbed fluid over a range of chemical potential values p. The specific excess
adsorption ITP,H) is then obtained from the line average of the density profile across the

width of the pore

F(P.H) = £1: [p(z) — pb ldz (2.9)

where p(z), p1, and P are the density profile, bulk gas density and pressure at chemical
potential u. The bulk fluid equations of state that relate p1,, P and ,u are given by the set

of mean-field equations

p(pb) = Nh(Pb,d)+an (2.10)

1
P(p,,) = Ph(pb,d)+§apb2 (2.11)
where 11,, and P}, are the Carnahan-Starling hard sphere chemical potential and pressure

[17], respectively, and a is the van der Waals attractive parameter

,0 2
a :41: 10 r ¢an(r)dr (2.12)

In the argon and nitrogen adsorption experiments (reported in the next section), helium
adsorption is used to calibrate the pore volume. This helium-calibrated pore volume is
then used to convert the absolute adsorption to an excess adsorption, by subtracting the
product of the bulk gas density and the calibrated pore volume from the absolute
adsorption. Neimark and Ravikovitch have noted that the same convention should be
used in calculating excess adsorption isotherms from DPT for comparison with

experimental results [19]. They used DPT to compute a helium-calibrated pore volume,

22

and found thi-
carton slit per
in the case 0
cdibrated por-
oidth smaller
thus: iii-access

use equation

13 EXperi
At;
1116 select

actitated c
All urge“
OIdEr 10 ,
graphitic
hours; in
“aisle”
EOIhcm
and hit
“10mg,
the but
Obtain

Using

and found that the excess isotherms calculated for supercritical methane adsorption in
carbon slit pores decrease by 4 to 5% when the correction for the pore volume is applied.
In the case of argon or nitrogen adsorption at 77 K, the correction for the helium-
calibrated pore volume is negligibly small (less than 2%), except in slit pores of physical
width smaller than 6 A. Pores that are this narrow exclude argon and nitrogen, and are
thus inaccessible to PSD analysis using argon or nitrogen porosimetry. We may therefore
use equation (10), rather than the helium-calibrated DPT pore volume, in calculating the
specific excess adsorption with minimal error.
2.3. Experimental Section: Argon and Nitrogen Porosimetry

Argon and nitrogen isotherms were measured on four porous carbons at 77 K.
The selected carbons were Aldrich powdered activated carbon 16155-1, granular
activated carbon 4021-8, a 100-200 mesh Saran char, and a 50—200 mesh coconut char.
An argon isotherm was also measured on Sterling PT, a nonporous graphitized carbon, in
order to obtain the solid-fluid potential parameters for argon adsorption in the model
graphitic slit pores. All samples were outgassed at 350 °C for a period of at least 20
hours; in all cases, ultimate pressures below 10’5 torr were achieved. Samples were then
transferred to an insulated Dewar containing a temperature bath of liquid nitrogen.
Isothermal conditions at 77.3 i 0.1 K were maintained during all experiments. Argon
and nitrogen adsorption isotherms were measured using a Coulter Omnisorp 100
automated system, over a pressure range from approximately 10’5P0 to 0.8P0, where P0 is
the bulk saturation pressure of the adsorbate. The specific excess adsorption isotherm is
obtained from the absolute amount of gas adsorbed by correcting for the bulk gas density

using helium calibration of the pore volume. All isotherms are reported in terms of

23

elitist pressure P. P
the experimental \a:

pressure of solid in;

It fitting 01 the 1
The Lennar
titted5 to match th
toiling point ol 7';
temperature of bu
lennard-lones p:
supercooled liqui
hi this work. use
Supercooled liqt
12.101 and (2.11
solid phase toe
Slli‘eicoolecl her
353168 well “1‘.
Saturated liqmd
that DH is in-
dfinsiry profih
superCogled 11
theOTEIiCa] m

tom-1mm in t

relative pressure P/Po. For liquid nitrogen at 77 K, P0 = 762 i 2 torr; for argon at 77 K,
the experimental value of P0 was found to be 204 :4 torr, which is the saturation

pressure of solid argon.

2.4. Fitting of the DFT Model Parameters

The Lennard-Jones parameters Off and 8,7 for bulk nitrogen have been previously
fitted5 to match the saturated liquid nitrogen density of 0.02887 mol/cm3 at its normal
boiling point of 77.347 K. The fitted parameters are listed in Table 2.1. The triple point
temperature of bulk argon is 83.8 K; hence, at 77 K, one may elect to fit the bulk argon
Lennard-Jones parameters to either the solid argon (0.0411 mol/cm3; 205 torr) or the
supercooled liquid argon (0.0365 mol/cm3; 229 torr) density and saturation pressure [20].
In this work, we have chosen to fit the model parameters using the physical properties of
supercooled liquid argon; this yields the fitted values listed in Table 2.1. Equations
(2.10) and (2.11) describe only vapor-liquid coexistence, and cannot predict the vapor-
solid phase coexistence of bulk argon at 77 K. However, the saturation pressure of
supercooled liquid argon calculated from equations (2.10) and (2.11) has been found to
agree well with the experimental saturation pressure obtained by extrapolating the
saturated liquid coexistence line below the argon triple point [21]. It should also be noted
that DPT is incapable of predicting solid-phase transitions; hence, the condensed-phase
density profiles computed from DPT for argon adsorption at 77 K correspond to
supercooled liquid argon, not solid argon. In principle, this mismatch between the
theoretical model and the experimental isotherms (which indicate that solid argon

condenses in the carbon pores) may cause errors in the PSD analysis. As we shall later

24

  
   

anonstrttehoscset
titieeondensed at; '
urinal.
idle 21'. lennard-l

in el.

 

 

Parameter L42";
1 ' Sut‘crct
otii . 3.3'

-‘
‘

The Lennarc
llllh carbon atoms
adsorption isothen
Values listed in Tc
the argon-carbon
dameter is set e
molecular diamet
is then determine
hel‘ieen the expo
isotherm for argr
mesopore is sulfr
1185111 pore 1111K-
DFl isotherm in

1 DOIlPOTOUS ) 9mm

demonstrate, however, the computed DPT isotherms are largely unaffected by the choice
of the condensed argon phase, and so the effect of this choice on the PSD analysis will be
minimal.

Table 2.1: Lennard-Jones potential parameters for nitrogen [5] and argon used in DPT

 

 

 

 

 

 

 

model.
Parameter Argon Nitrogen
Supercooled liquid Solid
6.0.) 3.375 3.27 3.572
em’k (K) 1 10.2 1 13. 93.98
O}{(A) 3.362 3.31 3.494
8,,Jk (K) 58.02 61.5 53.22

 

 

 

 

 

The Lennard-Jones parameters 03f and 85f for the interaction of adsorbed nitrogen
with carbon atoms in a graphite surface have been previously fitted [5] to the nitrogen
adsorption isotherm on the nonporous graphitized carbon Vulcan at 77 K (parameter
values listed in Table 2.1). We follow a similar procedure for fitting the parameters of
the argon-carbon interaction potential. The argon-carbon effective intermolecular
diameter is set equal to the arithmetic mean of the argon-argon and carbon-carbon
molecular diameters. The Lennard-Jones well depth 851‘ for the argon-carbon interaction
is then determined by selecting the value that provides the best quantitative agreement
between the experimental argon isotherm of the nonporous reference carbon and the DPT
isotherm for argon adsorption in a model graphitic slit pore of width H = 67.5 A. This
mesopore is sufficiently large so that the adsorbed films on the two opposing surfaces of
the slit pore have no significant potential interactions; i.e. at low relative pressures, the
DPT isotherm in this slit pore can be regarded as the adsorption isotherm on two isolated

(nonporous) graphite surfaces. To directly compare the experimental and DPT model

25

 

o
is:

Amount Adsorbed (m mollg)
p
—h

0

1:1:me 2.2: E
COil‘tl‘itl’ison [.
lSO-therm; the

 

P
or

      

p p p

N w uh
1T11L11

1

Amount Adsorbed (mmollg)
O

llllllllllllLllli

 

 

O 2 4 6 8 10 12
Theoretical Film Thickness (A)

Figure 2.2: Evaluation of the specific surface area of Sterling nonporous carbon from
comparison to the universal film thickness curve. The circles are the experimental
isotherm; the solid line shows the linear fit of the data through the origin. The slope of

this fitted line yields the specific surface area.

26

rodents. we must i
shlch in the case 01“
deBoerer al. [231- 1"

the argon uptake \67 ~
set of 11.3 milg \s.
contert the DFlcal.
apore 101111116 basis

he nelldepth parar‘,
the experimental an;
experimental isother'
same of £1,me con
the DH isotherm, ,
closely matched rh.
isotherm,

1‘ his been 1'.
this inflection Plum
[Mi-11501116111] 15 1);;
whereas the Sterling
degree of SUrlace h;
graphlllkd. Suilacti
dismAbullon of the

shallovter slope that

Should also be not

ea

 

 

isotherms, we must first determine the specific surface area of the reference carbon,
which in the case of argon is a Sterling PT graphitized carbon. Following the method of
de Boer et al. [22], the specific surface area is obtained by fitting a line through a plot of
the argon uptake versus the theoretical film thickness (Figure 2.2). A specific surface
area of 11.3 m2/g was measured for the Sterling carbon; this value was then used to
convert the DPT-calculated specific excess argon adsorption in the 67.5 A-slit pore from
a pore volume basis to an adsorbent mass basis, as described in our previous work [5].
The well-depth parameter 83f that yielded the best agreement between the DPT model and
the experimental argon isotherm is reported in Table 2.1, and a comparison of the
experimental isotherm and the fitted DPT isotherm is shown in Figure 2.3. The best-fit
value of 8,, was considered to be the one for which the low-pressure inflection point in
the DPT isotherm, corresponding to the formation of an adsorbed monolayer, most
closely matched the inflection point in the monolayer region of the experimental
isotherm.

It has been found that the DPT isotherm always exhibits a greater slope around
this inflection point than the experimental isotherm. This is to be expected, since the
DPT isotherm is based upon adsorption on a homogeneous model graphitic surface,
whereas the Sterling reference carbon, and any other nonporous carbon, will retain some
degree of surface heterogeneity, no matter what the extent to which the adsorbent was
graphitized. Surface physical and/or chemical heterogeneities broaden the site energy
distribution of the adsorbent surface, and cause the measured isotherm to have a
shallower slope than it would if the surface were energetically homogeneous [23]. It

should also be noted that we have deliberately chosen to fit the solid-fluid potential

27

o
—A
a,
\

c:

2..

N
%l

 

 

 

 

 

Amount Adoorbod (mmollg)
o

s:
2
L I

 

1‘

0.0001

figure 2.3 Argo

Dfi'fillfid isoth

1118 humid-Jpn

endl‘ollllsl and s

0.16

 

P
d
N
a
r

0.08 a-

0.04 .2

Amount Adsorbed (mmollg)

 

 

 

 

l l l
o r rrrrrrrr r rrrrrrrr r rrrrrrrr r rrrrrrr

0.0001 0.001 0.01 0.1 1
PIPO

Figure 2.3: Argon adsorption on nonporous carbon surface at 77 K. The circles are the
experimental isotherm measured on Sterling PT graphitized carbon. The lines are the
DPT -fitted isotherms in a model graphitic slit pore of width H = 67.5 A, obtained using
the Lennard-Jones potential parameters calculated from the supercooled liquid (square

endpoints) and solid (triangle endpoints) argon bulk physical properties.

28

 

pgflllflfll's to the rr‘
art-emptied to obmi
experimental isothe'
1051. the principal ;
cation atoms: ssh:
substantial as the

multilayer/condens.
rhe DPT model, be.
RI it relative Pressu:
inconsequential to 1
can be aCCOmphg]

reference isotherm

35. DPT Model I
In Figure

.mpliitic slir pores

Table 2.1. For cor

\
0 er the same size
l'tllIOg

5“ Isotherm

111 the
m -
ebOporeS. (

1 '0 - -
a‘eer 5111 mte‘
\r

Il_rt~~- .
cable diffemncb

ads '
OrpIIOI'] mOdL‘I [

parameters to the monolayer region of the nonporous reference isotherm, and have not
attempted to obtain quantitative agreement between the DFT isotherm and the
experimental isotherm at higher relative pressures. At low relative pressures (i.e. P/Po <
103), the principal potential interactions are between the argon molecules and the surface
carbon atoms; whereas at higher pressures, argon-argon interactions become more
substantial as the adsorbed films thicken. It is generally not possible to fit the
multilayer/condensation region of the nonporous (BET-type) isotherm at P/Po > 0.4 using
the DFT model, because the model isotherm will exhibit a condensation phase transition
at a relative pressure specific to the pore slit width. This lack of agreement, however, is
inconsequential to the fitting of the Lennard-Jones solid-fluid potential parameters, which
can be accomplished very satisfactorily using only the low-pressure portion of the

reference isotherm.

2.5. DPT Model Isotherms for Carbon Slit Pores

In Figure 2.4, the model DFT isotherms are shown for argon adsorption in
graphitic slit pores at 77 K, calculated using the potential parameters summarized in
Table 2.1. For comparison, the DFI‘ isotherms for nitrogen adsorption at 77 K in pores
over the same size range are presented in Figure 2.5. The overall shape of the argon and
nitrogen isotherms are similar: in the micropores, the pores fill in a single step; whereas
in the mesopores, condensation is preceded by the formation of a monolayer, and, in the
larger slit pores, wetting of one or more additional adsorbed layers. There are some
notable differences between the argon and nitrogen DFI‘ isotherms. In the nitrogen

adsorption model, the isotherms of the micropores smaller than 10 A are continuous;

29

1 moi/om.)
L m 8 DR 8 2::
m

._b
O

1E-10

Figure 2": Argr
calculated {mm l

milling from left

 

ALIAAAAIAAAI
I T

\

Hmmollcm’)
3

 

 

 

 

 

0|
1 L
1'

 

 

 

o 3 A
1E-10 1E-8 1E—6 154 15-2 1E+O
PIP,

Figure 2.4: Argon specific excess adsorption in model graphitic slit pores at 77 K,
calculated from DFI‘. Isotherms are shown for slits having a physical pore width H

(reading from left to right) of 6.8, 8.4, 10.5, 11.8, 13.5, 16.9, 20.2, 33.8, and 67.5 A.

30

 

 

AREU\_CEEV.—

 

 

3 20 ~-

E i /
E :

E 15 “' /

 

 

 

 

 

10 €r
5 ir
o: 4 . .
1E-10 1E-8 1E-6 1E-4 15-2 1500

PIP.

Figure 2.5: Nitrogen specific excess adsorption in model graphitic slit pores at 77 K,
calculated from DFI‘S. Isotherms are shown for slits having a physical pore width H

(reading from left to right) of 7.1, 8.0, 8.9, 10.7, 13.4, 17.9, 28.6, and 42.9 A.

31

 

.l-gv

[fill-v-

V")

4‘-
s

whereas for the argon model, a phase transition is predicted in these pores. In the pores
larger than 14 A, the nitrogen model predicts continuous pore filling during growth of the
adsorbed film, followed by a single phase transition at the capillary condensation
pressure. The argon DPT isotherms, in contrast, exhibit multiple phase transitions at the
monolayer and multilayer filling pressures, in addition to the capillary condensation
pressure. The more step like structure of the argon DPT isotherms is not unexpected,
since these isotherms are calculated at a lower reduced temperature T/TC than the nitrogen
isotherms. (The critical temperature TC of argon, obtained from mean-field theory using
equations (2.11) and (2.12), is 164.7 K; for nitrogen, it is 140.5 K. Both of these values,
it should be noted, are approximately 14 K higher than the actual critical temperatures of
the gases, since mean-field theory does not correctly predict the curvature of the liquid-
vapor coexistence curve near the critical point.)

The pore filling pressures are shown as a function of slit pore width in Figure 2.6
for argon and nitrogen adsorption at 77 K. It can be seen that for a given pore width,
argon and nitrogen condense at nearly the same pressure. The adsorbed argon density,
however, is greater than the adsorbed nitrogen density when complete pore filling has
occurred. (This can be seen by comparing Figures 2.4 and 2.5.) A higher adsorbed-fluid
density is achieved for argon because it has a smaller molecular diameter than nitrogen,

and it is adsorbed at a lower reduced temperature than nitrogen.

2.6. PSD Analysis of Porous Carbons
Experimental argon and nitrogen adsorption isotherms at 77 K are reported in

Figures 2.7-2.10 for four porous carbons. The total excess adsorption of argon exceeds

32

1E

E
1

I
I

E

1

DC....nu
float-n-
elson. 0»:

1E

 

hgunil

mquen

 

151-0 E

 

 

 

, D
i o D '3'”
In

. 15-2 if

98 i «P

‘L i

9.’ 115-4 4; 'is:

3 a

3 4 .9

e a”

g, 156: IE

.5 .

= 3

tr. 2

1E-8 g7 D
.
2 D
1E-10 Trititt’tiriiriiririirri
0 1o 20 30 40 50
Pore Width (A)

Figure 2.6: Pore filling correlation to slit pore width for DPT models of argon and

nitrogen adsorption at 77 K. Filled squares: argon; open squares: nitrogen.

33

r
llfl

§
1:
1

 

4

6 l4.

4
8

0
A 351.9..va- n: v IVQA-LavZ-le a: =3:-<

N... a

1E

re 2.7:

Fitu

b

*1 nstuns

130“ and
l‘nLLlerm 0f

 

14

 

 

 

 

A 12 7—
é”
'2 10 I ’ d
O "“7 - ‘:"'
a I _ ,. -
e "- '
pa 8 +- - I
O I
Q I
s '
'8 6 q” 1'
<1 1.
§ 4* ,7
E .
< H
2 ~~ .
0 - l i i l
lE-06 1E-05 0.0001 0.001 0.01 0.1 1
lP/Po

Figure 2.7: Adsorption isotherms on Saran char at 77 K. Filled squares: argon uptake;
Open squares: nitrogen uptake. The lines show the respective DFT isotherms obtained for
argon and nitrogen using the PSD that best fits the DFT model to the experimental

isotherm of each probe gas. The PSDs for argon and nitrogen are fitted separately.

34

Aflxv—CEE v 593...}..le a: 3::- <

‘
.

Figure

5:506 {LS 1

 

i—s i—t l—I i—t
N b O\ 00
1 1 1

Amount Adsorbed (mmole/g)
S

 

 

 

 

O ‘ I l i
1.E-06 1 .E-04 1 .E-02 1.E+00

P/Po

Figure 2.8: Argon and nitrogen isotherms on coconut char at 77 K. The notation is the

same as in Figure 7.

35

4
l1 . - »\L [hk 4

All\-nvll-lllv IU.IDI-InvI-v< idol-II-Av>

0—

1
l

)C.

Gilli

here 2.9

llOL‘tliun it

 

14

12~

10*

Volume Adsorbed (mmollg)
OO

 

 

 

o
6 ~ 9 , .
o .
4 q .
2 _
0 d I i r r
0.000001 0.00001 0.0001 0.001 0.01 0.1
Relative Pressure (P/Po)

Figure 2.9: Argon and nitrogen isotherms on granular activated carbon at 77 K. The

notation is the same as in Figure 7.

36

 

NAHilllltll

n0\h0.0€£v UQQuOmU<

.CDOExx

Hm 3. l1

“0019.011 15 '

 

_—l-_l—il—l
[95000
1111
T1 ll

 

 

 

 

0 — l l
10015-06 1001204 1005-02 1.005+00

Amount Adsorbed (mmoles/g)
5‘

P/Po

Figure 2.10: Argon and nitrogen isotherms on powdered activated carbon at 77 K. The

notation is the same as in Figure 7.

37

 

 

\

A“

that of nitrogen in all four cases; this is consistent with the predictions of the DFT models
shown in Figures 2.4 and 2.5.

The adsorption isotherm of each carbon is fitted using the appropriate DFT pore
filling model in equation (1.1), to obtain the PSD of the adsorbent. For convenience, we

have elected to fit the PSD of each carbon to a trimodal gamma distribution (m = 3)

)il3
aiO’iH
_ «H 2.1
f(H )= g— F031); €XP(Y, ) ( 3)

which has nine adjustable parameters: the amplitude 01,, mean [3,, and variance 7, of each
mode 1' of the distribution. In equation (14), F refers to the gamma function. It is
important to note that this choice of a fitting function is completely arbitrary, and that
there is no physical basis for assuming that the PSD of an adsorbent is uniquely described
by the gamma distribution, or any other sum of mathematical distributions. There is in
fact no need whatsoever to use a distribution function to fit the PSD; an equally valid
approach is to subdivide the PSD into several dozen or more pore width increments, and
assign an independently adjustable amplitude coefficent a,- to each pore width increment
j. The only mathematical constraint on the PSD function is that it must be nonnegative
for all pore widths H. If the PSD function contains too few adjustable parameters (e. g. a
unimodal distribution), the choice of the fitting function may unduly constrain the PSD
results. It has been our experience, however, that a trimodal distribution provides ample
flexibility for determining the PSDs of activated carbons. We have previously
demonstrated, for example, that the nitrogen adsorption isotherm of a porous carbon can

be fit very accurately with the DFT model using either a trimodal gamma distribution or a

38

r‘ l

0
A<oflxnp50u flaw—oflfiahomnnu 08.7» 0......-

C

iilzur

b

931051.11

l

3? 911

V

0.4

 

P
(a
I I 1

Pore Size Distribution (cm3/g'A)
p o
- N

 

 

 

 

Pore Width (A)

Figure 2.11: PSDs obtained from argon (filled squares) and nitrogen (open squares)
porosimetry for Saran char using DFT method. The PSD is reported as cm3 pore volume

per gram of adsorbent per A of pore width interval.

39

1!. IE.
. I 0
. 0

0 0
.Jxou>h:3 :O_u5n._.3.r._: “Surf. 0......-

‘
.

guts

DFT metho

h

 

 

 

 

 

02 _

in 0.2 :r

E -«

3 -

'5' 0.15 -r

z; -

:1 -

.n

'5

m

a 001 :—

0

.5 H

m —l

0 -l

E 0.05 :“

0 q . .

0

 

Pore Width (A)

Figure 2.12: PSDs obtained from argon and nitrogen porosimetry for coconut char using

DFT method. The notation is the same is in Figure 11.

40

0 fl 1" I!
A<OI\I.F:QV :cadzh-nhdaufl- ”Sue," “Vt-AVI-

q
'C
.
' a

F9
A...
L

£1th 1

0.05

 

 

 

 

.2 j :1.
é” 0.04 f . '
E i .
a q .
g 0.03 + _ '
*5 i '
.D u
'5 Z ' .
m
a 0.02 f 1
VJ ‘ El
2 001 :3 - u .
O ' I
D-u : I .
0 ~ r T T I i I l 1 1 1 1 I I I I 4'13==:=l_I..'-."_.
0 10 20 30 40 50 60

Pore Width (A)

Figure 2.13: PSDs obtained from argon and nitrogen porosimetry for granular activated

carbon using DFT method. The notation is the same is in Figure 11.

41

boa
.N.” 0
230.520
0 as Envv :0
.4. a

Flare 2

v

“41110110

 

0.2

Pore Size Distribution (ems/90A)
o

O L.

- on

P
8
1 l L

 

 

 

o _ I - r. ——4 - LighL!‘L-‘-

Pore Width (A)

Figure 2.14: PSDs obtained from argon and nitrogen porosimetry for powdered activated

carbon using DFT method. The notation is the same is in Figure 1 l.

42

 

I
'. J
nu

.1

titans:
as“. :1
the PS]
nitrous
uteri.“
£15557);
miut
13112113 '
tritdel.
lif‘ihsr
P103011

DH 11

Cail‘tei

trimodal lognormal distribution to represent f(H) [4]. The PSDs obtained using these two
distributions are essentially indistinguishable from one another, underscoring that it is the
numerical values of the PSD itself that are unique, not the function that is arbitrarily
chosen to represent the PSD.

The adsorption integral, equation (1.1), is evaluated for each probe gas/adsorbent pair
using numerical integration. A least-squares error minimization criterion is used to
determine the best-fit PSD parameters for each carbon. The best-fit model isotherm for
each experimental isotherm is shown as a solid line in Figures 2.7-2.10. A comparison of
the PSDs that yield the best-fit argon and nitrogen DFT isotherms is shown for each
porous carbon in Figures 2.11-2.14. We note that a good fit of the DFT model to the
experimental data is obtained for all eight measured isotherms. The cases where there are
observable differences between the DFT model and the experimental isotherm can be
attributed to the model isotherm being more structured than the measured isotherm. This
is due to the assumption of physical and chemical homogeneity in the graphitic slit pore
model, and to the effects of heterogeneity in smoothing the structure of the experimental
isotherms. The differences between the DFT and experimental results are slightly more
pronounced for the set of argon isotherms, because of the phase transitions predicted by
DFT for this adsorbate at 77 K, as noted earlier. Overall, however, an acceptable fit of
each experimental isotherm is obtained using a trimodal gamma distribution for the PSD.

The PSD for each adsorbate/adsorbent pair (Figures 2.11-2.14) is truncated at a lower
bound corresponding to the width of the slit pore that condenses at the lowest pressure
measured in the experimental isotherm. This lower pore size bound is determined from

interpolation of the DFT pore filling correlations shown in Figure 2.6. Our justification

43

FL

1'0»;
~.J
V

F
1d:

311m;

\

11b: 41' .

l

)
”‘3'
r2;
:1

for truncating the PSD is that all the pores that are narrower than the lower bound pore
width are already filled with condensate at the start of the experimental isotherm
measurement. Consequently, no phase transition will be observed in these pores over the
course of the experiment, and no information regarding the PSD of these micropores may
reliably be obtained. (A slight increase in the condensate density does occur in pores of
all size as the pressure increases, due to compression of the adsorbed fluid; however, this
compression effect alone is insufficient to unambiguously determine the PSD of
micropores smaller than the lower bound pore width.

All four porous carbons have a large micropore volume, as can be seen from the
PSD peak exhibited by each adsorbent in the 11 to 13 A range. Some mesopore volume
also appears to be present in the granular and powdered activated carbons, as evidenced
by the broad secondary peak in the PSDs of these samples in the 40 to 60 A range. In
each of the four samples, the PSD interpreted from the argon porosimetry data has a
somewhat broader distribution of pore sizes than the PSD obtained from the nitrogen
data. A comparison of the pore volume and the maximum in the PSD of each adsorbent
is presented in Table 2.2. The percentage difference in the values of the PSD maxima for
argon and nitrogen, calculated as

100% x (absolute difference in Ar and N2 PSD maxima)/(Ar PSD maximum)
ranges between 3 and 16%, with an average differential of 8.7% for the four samples
tested. A larger difference was found in the pore volumes calculated from the argon and
nitrogen PSDs, than for maxima in the PSD. The pore volumes reported in Table 2.2 are
obtained by integrating the PSD over all pore widths. The pore volume predicted from

argon porosimetry consistently exceeds the value derived from nitrogen porosimetry; the

fifium.g.
ninlwng:
lhfilihl
Dfl inten‘ret.

__—-——-

cm

Sunnu

 

 

COIOl'lUl C

 

. 1W Jere

2“l- Discu

W:
3f '3 W100
PSDmn
DH C0111
Pftvbg £115
10 51701 11
Wham
3011mm

10 Obiuln

differential, again computed relative to the argon results, ranges between 16 and 31%,
with an average difference of 25%.
Table 2.2: PSD maxima and pore volumes of the four carbon samples as determined from

DFT interpretation of argon and nitrogen porosimetry data.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Carbon Adsorbate PSD Maximum (A) Pore Volume (ems/g)

Saran char Ar 1 1.8 6.54
N2 1 1.4 4.99

% difference 3.4 23.7

coconut char Ar 13.5 5.77
N2 11.4 4.85

% djfierence 15.6 15.9

granular AC Ar 12.5 4.66
N2 12.9 3.20

% difierence 3.2 31.3

powdered AC Ar 11.1 9.15
N2 12.5 6.28

% difference 12.6 31.4

 

 

 

 

 

2.7. Discussion of Argon and Nitrogen Porosimetry Methods

With regard to fluid separation applications, the most important physical property
of a porous carbon is the maximum in its PSD. The average differential of 8.7% in the
PSD maxima calculated from the argon and nitrogen porosimetry measurements using
DFT constitutes may be regarded as reasonably good agreement between the two separate
probe gas analyses. This is particularly so given that the calculation of the PSD is subject
to error resulting from any of three separate inputs: (1) the precision of the experimental
isotherm measurement; (2) the accuracy of the DFT pore filling model; and (3) the
soundness of the numerical method used to invert the Fredholm integral of equation (1.1)

to obtain the PSD. Of these three contributing factors, the second is most likely to be

45

.. '1 I
961118105 M
. ‘chm:
.‘l-‘f‘n‘en ¢\ 1 I

. _ 1‘ “"11
> 1.11“
:- lii >1

111110 the 1

'9 “15.10
331.11. sh.

1"”0’71‘11‘1‘11
up. e.

1:5 to it ex
11: rims:

111 15512111"

mixer-item.

liters is re
1311013: the
3011 5.18 ”C i

iOlUme Ofl

responsible for the observed differences in the PSDs computed from the argon and
nitrogen adsorption measurements. The construction of the DFT isotherms HRH) used
in the solution of f(H) in equation (1.1) requires that we invoke major simplifications in
regard to the structure of the model carbon adsorbent. The assumptions of a slit pore
shape, surface homogeneity, neglect of connectivity, and so forth are at best a crude
approximation of the actual structures of the porous carbons studied in this work. Hence,
it is to be expected that some inconsistencies will arise in the PSDs calculated from argon
and nitrogen porosimetry as a result of the DFT model assumptions (or, for that matter,
the assumptions made in any other pore filling model). In the case of a more ordered
adsorbent, such as a mesoporous molecular sieve, we might anticipate better agreement
between the argon- and nitrogen-derived PSDs, since our pore model construct for a
highly-ordered porous material will more closely match the actual pore structure of the
adsorbent. For example, excellent agreement was found in PSD results obtained by using
DFT to interpret argon and nitrogen adsorption isotherms at 77 K and 87 K on MCM-4l
type adsorbents, a class of mesoporous molecular sieves that have one-dimensional,
cylindrical oxide channels with pore diameters between 32 and 45 A [21].

One might then ask the question: which of the PSDs obtained, from argon and
from nitrogen porosimetry, is the more reliable representation of the actual carbon PSD?
There is reason to favor the PSD interpreted from argon porosimetry, on the basis of two
factors: the interaction of the nitrogen quadrupole with carbon surface functional groups,
and steric exclusion of the larger nitrogen molecule from portions of the internal pore

volume of the adsorbent. As we shall now discuss, these two effects collectively provide

46

 

the 11101103:
are. In

drainer
1101:1111:
of quadni
5111:1111
tie user
111 :hemi
Emular;
L0 dfllt‘u‘

[1371110)

surface.

@111 mtro

llr site 1
8115011111,
511131011

111111111

a plausible explanation of the differences in the PSDs and pore volumes obtained from
the argon and nitrogen isotherm measurements.

A comparison of Figures 2.4 and 2.5 indicates that the argon and nitrogen DFT
models both predict that an adsorbed monolayer will form at FIFO 5 10“4 in pores larger
than 13 A. In the experimental isotherms of the activated carbons (Figures 2.9 and 2.10),
the monolayer filling pressures for argon and nitrogen were indeed observed to be the
same. In the case of the char samples (Figures 2.7 and 2.8), however, the nitrogen
monolayer formed at a pressure nearly a full order-of-magnitude lower than the argon
monolayer. We speculate that the lower nitrogen monolayer filling pressure is the result
of quadrupolar interactions of the nitrogen adsorbate with functional groups on the
surface of the chars. Argon, which has no permanent multipole, is largely unaffected by
the presence of these functional groups, whereas nitrogen adsorbs at a lower pressure on
the chemically heterogeneous carbon surface on account of its quadrupole. Unlike the
granular and powdered activated carbons, the Saran and coconut chars were not subjected
to activation during their synthesis. We therefore hypothesize that activation of the
thermolyzed carbon source material eliminates functional groups from the adsorbent
surface. This would explain the consistency in the monolayer filling pressures for argon
and nitrogen on the activated carbon samples.

The quadrupole moment of nitrogen has been invoked to explain differences in
the site energy distributions of controlled pore glasses calculated from argon and nitrogen
adsorption at 77 K [24], and the PSDs of microporous carbons determined from
supercritical methane adsorption and subcritical nitrogen adsorption [25]. Electrostatic

interactions are not explicitly accounted for in the DFT pore filling model presented in

47

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this work, which assumes that the adsorbate molecules are spherically symmetric and
interact only via van der Waals (i.e. dispersion/repulsion) forces. If the neglected
quadrupolar interactions are substantial, then DFT analysis of the nitrogen isotherm will
yield PSD estimates that are skewed toward pore widths that are too small. This might
explain some of the observed differences in the PSDs obtained for the chars. It is
therefore our recommendation that argon porosimetry be used to interpret the PSDs of
porous carbons if there is reason to believe that the adsorbent is chemically
heterogeneous. While this procedure will not eliminate the effects of other
heterogeneities (e.g. surface defect sites, pore junctions) that may be present, and that
equally impact argon and nitrogen adsorption, it will at a minimum avoid the potentially
confounding effects of the nitrogen quadrupole on the analysis of the PSD.

For all of the carbon samples, the total pore volume, per unit mass of adsorbent,
calculated from argon porosimetry is substantially larger than the pore volume obtained
from nitrogen porosimetry (Table 2.2). One possible explanation, that would reconcile
the average pore volume differential of 25%, is that the internal pore volume of each
adsorbent is partitioned into an array of void spaces, connected to one another by way of
narrow pore linkages or bottlenecks. The nitrogen molecular diameter is approximately
0.2 A larger than the argon diameter (Table 2.1). Therefore, if some of the bottlenecks
have a pore width smaller than the nitrogen diameter, but larger than the argon diameter,
nitrogen diffusion will be kinetically limited, or perhaps excluded altogether, in these
linkages, whereas argon may be able to diffuse unhindered through the bottlenecks. If
this were the case, nitrogen as an adsorbate would be unable to probe the void volume

accessible only through the narrow pore linkages, whereas the argon adsorbate would

48

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penetrate a greater extent of the internal pore volume of the adsorbent. (The additional
volume sampled by argon may also account for some of the observed differences in the
argon and nitrogen PSDs.)

It is conceivable that polar surface functional groups, spanning the edge planes of
adjacent graphitic microcrystals, might act as pore bottlenecks in the microstructures of
the porous carbons. If this were so, then the aforementioned quadrupolar interactions of
the nitrogen molecules with the polar functional groups would “corral” the nitrogen
adsorbate at these sites of heterogeneity and inhibit further penetration of nitrogen into
the carbon internal pore volume. The concept of adsorbate size exclusion in micrOpores
has been recently proposed as a means of probing the connectivity of a porous adsorbent,
by using percolation theory to interpret the adsorption isotherms of different-sized
adsorbate molecules [26]. In the DFT pore filling model, the connectivity of the pore
volume is altogether neglected, and is substituted with a conceptual model of isolated slit
pores of infinite aspect ratio. If there is evidence that the network of voids in the
adsorbent contains many narrow pore junctions, a more accurate sampling of the internal
pore volume will probably be obtained using the smaller argon, rather than nitrogen,
molecule as the probe gas.

It has been noted that the Lennard-Jones parameters for the argon-argon pair
interaction potential can be fit to physical properties of either solid argon or supercooled
liquid argon at 77 K. The argon DFT isotherms and PSD results reported in this work
were obtained using the supercooled liquid argon properties to fit the model parameters.
The saturation pressure measured in the argon adsorption experiments, on the other hand,

indicates that solid argon condenses in the porous carbons. In order to assess the possible

49

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errors caused by the mismatch of the condensed argon phase in the theory and
experiments, the potential parameters for the argon-argon and argon-carbon interactions
were recalculated using the solid argon physical properties, following the same procedure
outlined in Section 2.4. The best-fit Lennard-Jones parameters using the solid argon
properties as the basis for fitting are summarized in Table 2.2, and a comparison of the
DFT model isotherms, fitted to the Sterling reference isotherm, is shown in Figure 2.3 for
the two different sets of argon potential parameters. It is found that using the solid argon
physical properties increases the argon-argon potential well-depth by 3% and decreases
the argon molecular diameter by the same amount. Using the arithmetic mean for the
argon-carbon effective diameter, and a recalculated specific surface area of 10.1 mZ/g
from the thickness curve plot, it is found that the Sterling isotherm is best fit with an
argon-carbon well depth that is 6% larger than the well depth fitted from the supercooled
liquid data. As can be seen from inspection of Figure 2.3, however, either set of potential
parameters will yield a set of DFT isotherms that reproduces the experimental monolayer
filling pressure of ~104P0. The only observable difference in the fitted isotherms is in
the adsorbed phase density. Thus, the two sets of model parameters will yield different
predictions of pore volume, but the location of the PSD maxima (as determined from the
pore filling pressure correlation) will be nearly the same for the two argon models. Based
upon the rather small value of 037 obtained by fitting the parameters to the solid data, and
other considerations discussed in Section 2.4, it is our recommendation that the
supercooled liquid properties be used to model argon adsorption at 77 K. An alternative,

to avoid uncertainty in the parameter fitting, is to conduct the argon experiments and

50

23. (

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lite m.

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modeling at the normal liquid argon boiling point of 87 K, using liquid argon instead of
liquid nitrogen as the temperature bath.

Finally, it should be noted that the PSD results reported in Figures 2.11-2.14 are
obtained without the use of regularization methods. Regularization refers to a numerical
technique whereby additional constraints are specified in order to convert a
mathematically ill-posed problem - the solution of the PSD function f(H) in equation (1)
- into a well-posed problem [27]. The effect of regularization is to “smooth” the PSD by
suppressing perturbations caused by experimental error in the isotherm data [28]. Given
that we have not used any regularization in solving the adsorption integral equation for
the best-fit PSD, the general agreement between the argon and nitrogen results for the
PSD maxima is quite good. If regularization were to be utilized, we would expect to find
even closer agreement between the argon- and nitrogen-derived PSDs. For example, a
large value of the regularization “smoothing” parameter would convert the bimodal PSD
fits obtained for some of the carbons into unimodal PSDs. The relatively good agreement
between the argon and nitrogen porosimetry results, even in the absence of regularization,

may be taken as evidence of the fundamental soundness of the DFT pore filling mode].

2.8. Conclusion

In this work, we have shown that a DFT model of argon adsorption at 77 K in slit-
shaped carbon pores yields PSD results from argon porosimetry that are comparable
those obtained by interpreting nitrogen adsorption at 77 K using a similar DFT model.
The maximum in the PSD of an adsorbent can be bracketed to within nine percent, on

average, by comparing the PSDs determined from the two separate probe gas

51

2‘ .
[11.1.13
11,1}...
1 “0.”:
LIMA

l
4" ,1 1
\l\5l:

‘
l V)”
51.5-0

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11,...
lLirht

111 :0

11131

measurements. Larger differences are observed in the calculation of the internal pore
volume of the adsorbent using argon and nitrogen porosimetry. The DFT model does not
incorporate pore connectivity effects or nitrogen quadrupolar interactions. Thus,
divergent PSD results can be expected if the argon and nitrogen DFT models are used to
interpret the isotherms of porous carbons with narrow bottleneck pores or polar surface
functional groups. In such instances, it is recommended that argon porosimetry be used
in conjunction with the argon DFT model to determine the PSD and the pore volume of

the adsorbent.

52

10.

ll.

12.

13.

14.

15.

2.10 References

Yang, R.T. Gas Separation by Adsorption Processes; Butterworths: Boston, 1985.

Bonsal, R.; Donnet, 1.; Stoeckli, F. Active Carbon; Marcel Dekker: New York, 1988.
Gregg, S.J.; Sing, K.S.W. Adsorption, Surface Area and Porosity; Academic Press: New
York, 1982.

Lastoskie, C.; Gubbins, K.E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786.

Lastoskie, C.; Gubbins, K.E.; Quirke, N. Langmuir 1993, 9, 2693.

Barrett, E.P.; Joyner, L.G.; Hallenda, P.P. J. Am. Chem. Soc. 1951, 73, 373.

Horvath, G.; Kawazoe, K. J. Chem. Eng. Japan 1983, I6, 474.

Lastoskie, C.; Quirke, N.; Gubbins, K.E. In Equilibria and Dynamics of Gas Adsorption
on Heterogeneous Solid Surfaces; Rudzinski, W.; Steele, W.A.; Zgrablich, G., Eds.;
Elsevier: Amsterdam, 1997; Vol. 104, p. 745.

Ravikovitch, P.I.; Haller, G.L.; Neimark, A.V. Adv. Coll. Int. Sci. 1998, 76, 203.
Ravikovitch, P.I.; Haller, G.L.; Neimark, A.V. In Mesoporous Molecular Sieves;
Rudzinski, Bonneviot, L.; Beland, F.; Danumah, C.; Giasson, S.; Kaliaguine, S., Eds.;
Elsevier: Amsterdam, 1998; Vol. 117, p. 77.

Olivier, J.P. Carbon 1998, 36, 1469.

Steele, W.A. Surf. Sci. 1973, 36, 317.

Weeks, J.D.; Chandler, D.; Andersen, H.C. J. Chem. Phys. 1971, 54, 5237.
Israelachvili, J.N. Intermolecular and Surface F orces; Academic Press: London, 1992.

Evans, R. In Inhomogeneous Fluids; Henderson, D., Ed.; Marcel Dekker: New York,

1992; Chapter 5.

53

l6. Tarazona, P. Phys. Rev. A 1985, 31, 2672; ibid, 1985, 32, 3148. Tarazona, P.; Marini
Bettolo Marconi, U.; Evans, R. Mol. Phys. 1987, 60, 5743.

17. Carnahan, N.F.; Starling, K.E. J. Chem. Phys. 1969, 51, 635.

18. Barker, J .A.; Henderson, D. J. Chem. Phys. 1967, 47, 4714.

19. Neimark, A.V.; Ravikovitch, P.I. Langmuir 1997, 13, 5148.

20. CRC Handbook of Chemistry and Physics, 61“ ed.; Weast, R., Ed.; CRC Press: West

Palm Beach, FL, 1981.

21. Neimark, A.V.; Ravikovitch, P.I.; Grun, M.; Schuth, F.; Unger, K.K. J. Coll. Int. Sci.
1998, 207, 159.

22. de Boer, J.H.; Linsen, B.G.; Osinga, T]. J. Catal. 1965, 4, 643.

23. Olivier, J.P. In Proceedings of the Fifth International Conference on Fundamentals of
Adsorption; LeVan, M.D., Ed.; Kluwer Academic Publishers: Boston, 1996; p. 699.

24. Olivier, J.P. In Proceedings of the Sixth International Conference on Fundamentals of
Adsorption; Elsevier: Amsterdam, 1998; p. 207.

25. Quirke, N.; Tennison, S.R.R. Carbon 1996, 34, 1281.

26. Lopez-Ramon, M.V.; Jagiello, 1.; Bandosz, T.J.; Seaton, N.A. Langmuir 1997, I3, 4435.

27. V. Szombathely, M.; Brauer, P.; Jaroniec, M. J. Comp. Chem. 1992, I3, 17.

28. Davies, G.M.; Seaton, N.A. Carbon 1998, 36, 1473.

54

C.

Chapter 3.

Carbon Nanotubes as an Alternate Reference Adsorbent

3.1. Introduction

Carbon nanotubes were first discovered by Iijima[l] in 1991, and have been
found to have excellent physical and electrical properties. Nanotubes are fullerene
molecules that have repeat hexagonal units that extend their length, essentially becoming
sheets of graphite that are rolled upon themselves [2]. Nanotubes may be either single or
multi-layered dependent upon the means used to produce them. The size of a typical
nanotube diameter varies between 1 and 20 nm for multi-layered tubes [3], with lengths
that are typically greater than 100 times the diameter or nearly infinite on a molecular
scale[4]. The nanotube length can be dramatically shortened by sonication in acid to a
length that is more desirable [5]. The mean tube diameter may also be tuned during the
production process by altering the production conditions [4]. Consequently nanotubes
can be modified to suit a particular application via altering their thickness resulting in
stronger surface potentials, their diameters resulting in higher selectivity, and their
lengths to maximize transport properties.

Carbon nanotubes have attracted much attention due to their electrical properties
and their mechanical strength; these same attributes make them appealing lattice material
[3] or size selective catalysts. Carbon nanotubes have potential applications as a means to
separate mixtures of molecules based upon their relative size [6]. Another practical
application is safer cheaper gas storage through adsorption [7,8]. In either application the

diameter of the tube is the controlling factor in determining the relative affinity for the

55

adsorbates of different sizes [6]. Varying the diameter of the buckytube by even a
nanometer can change the selectivity for one molecule by several orders of magnitude
[6].
The determination of the size of a nanotube can be performed via several methods, each
with benefits and associated problems. Transmission electron microscopy (TEM) can be
used to generate cross sectional images of tubes, which can then be directly measured to
find the tube diameter [4]. Although, the measurement of the tube image can give precise
measurements for particular nanotube, this is a time consuming process when applied to
many nanotubes and does not measure tube length. While the information gained from
TEM may be exact for determining the PSD for tubes interior diameters it gives no
indication of the exterior properties of the tubes. The exterior surface properties, such as
the effect of the interstitial spaces between the tubes, are extractable using isotherms [9].
Although evidence has recently come to light that the interstitial channels within tube
bundles do not adsorb gases [10] there is however a great deal of evidence of sorption to
the exteriors of nanotubes [7,8,9]. The importance of the exterior properties are shown in
their application, for storage purposes optimizing the interstitial spaces between tubes is
just as imperative as optimizing the actual tube size [7,8] for gases.

The isotherm may yield data on the size heterogeneity through inversion of the

adsorption integral for cylindrical pores which is equivalent to the slit pore Equation (1.1)
Rmax

r(P)= jr(P,R)F(R)dR (3.1)

Rmin

56

Where R is the physical pore radius and the subscripts refer to the maximum and
minimum values, P is the pressure, F (P) is the excess adsorption, F(P,R) is
thermodynamic model isotherm, and F (R) is the pore size distribution. The
thermodynamic model must be based upon a model that matches the physical properties
of the system, such as pore shape, potential function, and solid fluid interaction energies.
In the case of a single walled nanotubes (SWNT) a model that represents the system is a
cylinder of infinite length, with a single shell of carbon atoms. For a given
thermodynamic model, p(P,R), a PSD, F(R), may be chosen that reproduces the excess
adsorption isotherm RP) when combined in Equation (3.1). Although, the uniqueness of
any particular solution may be questioned, applying the following rules minimizes the
likelihood of obtaining multiple solutions:

1. There can be no negative pore volumes.

ii. The pore width must be large enough to avoid steric repulsion, usually two
adsorbate molecular diameters.

iii. The filling pressures of the thermodynamic models must be within the
pressure range measured experimentally.

Potential parameters are fitted based upon the adsorption integral found in
equation (3.1). One typically measures an isotherm of a reference material under the
same conditions as the unknown sample isotherm. Of the three variables in the
adsorption equation two must be known to obtain an estimate of the third. The easiest of
the variables to know is the experimental excess isotherm HP), followed by the PSD,
F(R). In this chapter the thermodynamic model ITP,R) is chosen to be the density

functional theory model, which is based upon the Lennard-J ones interaction potential.

57

Carbon nanotubes have regular structure, high surface area [9] and are chemically
homogeneous, which makes them an excellent standard for determining reference
isotherms. Additionally the length of a nanotube is typically over 100 nanometers, while
the end effects of the tube determined by molecular simulation extend 2 nanometers into
the tube [11]. The micropore size distribution of a SWNTs can be determined using
TEM for a relatively small but representative sample. Once the PSD is known a solid-
fluid energy well depth can be chosen to determine the thermodynamic models used in
equation (3.1) that result in reproducing the initial filling pressure of the reference

isotherm.

3.2. Thermodynamic Model
The density functional theory (DFT) is explained for the cylindrical geometry in
several other sources, therefore a brief summary is offered [12,13,14,15]. The grand

potential function .(2 for a cylindrical pore is given by equation (3.2).

91pm] = F[p(r)l — jdrprritu - V...(r>1 (3.2)
Where F is the Helmholtz energy, 11 is the chemical potential, and Vex. is the potential
energy field between the fluid and the solid. The local density p(r) is chosen to minimize
the overall grand potential. The potential energy Vex, is a function of the pore radius and

spatially varies with the distance from the centerline of the pore and represents an

integrated form of the 12-6 Lennard-Jones potential [16,17].

58

 

63' '"m 2 l
_ L[2_L) F[__9_,_3;1{1_L]]
V (AR) 1 2 2 32-04 R 2 2 R

4£———=—erro
T

117‘ " ‘f ’ 1'4
r r
_3_2__

Gal R1

I. - d ‘

(3.3)

 

 

 

 

Where F[01,B;y,x] is the hypergeometric series, est is an energetic parameter, 0'51 is the
solid fluid intermolecular diameter, r is the distance from the pore centerline, R is the
tube radius, K is the Boltzman constant, and T is the temperature. The values of es; and
0'31 are determined by the methods found in section 3.3. It is important to note that the
potential function is solely for the interior of the pore, and approaches the limit of a 10-4
potential of a flat slab as the radius R approaches infinity. The potential of an isolated
nanotube exterior is less than that of an infinite plane [18], however the difference in the
potential minimum is small. The occurrence of an isolated nanotube is doubtful, as the
tubes are known to grow in bundles. The exterior environment of a nanotube, or the
interstitial space between tubes can be modeled as a nanotube with a larger radius to
match the appropriate potential. For the limits of this calculation a flat plane is used to
model the potential of a very large tube.

The DFT method for a cylindrical pore is very similar to the method described in
Section 2.2, with the exception that the externally varying potential function is that
outlined by Equation 3.3. The Helmholtz free energy F is a function of the smoothed

local density p(r). The smoothed local density is weighted based upon the Tarazona

method, although it is modified to match the cylindrical geometry[19]. The method that
Tarazona [19] describes uses a power series expansion, of which the first three terms are

used to give very good agreement for the density profiles of Lennard-Jones fluids near

59

attractive walls, which is given in Equation 2.8. The grand potential is minimized with
respect to the local density profile, which may result in multiple minimums, representing
a condensed and expanded phase. The phase change occurs when the grand potential is
equal for both the liquid and vapor minimum. The stable or equilibrium minimum is the

one that has the lowest grand potential.

3.3. Method for Fitting the Potential Parameters

3.3.1 Potential Function

The potential parameters used in this paper refer to the Lennard—Jones 12-6 parameters,
which are related to the size and potential well depth associated with any two molecules
or atoms. The Lennard-Jones potential function consists of an attractive term raised to

the sixth power and a repulsive term raised to the twelfth power.

4).. 248.. E}— 12— EE- 6 (3.4)
" " d d

Where 80' is the potential well depth between i and j type atoms, 0;,- is the collision
diameter of i and j type atoms, and d is the distance that separates atoms i and j.
Although isolating two atoms and measuring their interactions is a very difficult task, the
individual terms can be resolved from bulk data. The fluid-fluid Lennard-Jones
parameters may be calculated from the saturation data of the adsorbate at the
experimental temperature. Relating the liquid molar density p, the pressure P, and the
temperature T to their corresponding non-dimensional units one can determine the values

of 0' and 8.

60

 

 

 

P _ P03 i 1 _ P* (35)
pRT e kT po3NW p*T* '

At the saturation there is a single degree of freedom, therefore specifying a particular
temperature would constrain the remaining parameters for both the condensed and gas
phases. The correct values of T*, P* and p" and therefore 8 and 0' are found by equating
the values for both phases at saturation. The carbon molecular diameter is well known to
be 3.35 A for a graphitic carbon [20]. The arithmetic mean is used to determine the
interaction distance between the carbon molecule and the adsorbate. Matching the solid
to fluid potential well depth is slightly more difficult. The adsorption integral in equation
(3.1) is the key to this dilemma, one must use the reference isotherm F (P) and the PSD
F(R) to ascertain the parameters which influence the thermodynamic model. The
thermodynamic model may be chosen at the discretion of the user. However, for a given
pore size and chemical composition Molecular Simulations MS and DFT are generally
considered to be exact. There are however semi-empirical methods such as the Horvath-
Kawazoe HK model, which produce results that are similar to the DFT with a reasonable
tradeoff in accuracy [21,29]. The fore mentioned techniques all depend upon the
potential interaction between the adsorbate and adsorbent. Other classical models such as
the Kelvin equation, may also be used, however they do not accurately predict the results
in pores that are smaller than 75 A and are not dependent upon the adsorbate-adsorbent
interaction, therefore will not be considered in this paper. Although the MS method is
the most exact means to determine a thermodynamic model its results are rivaled by
DFT, which requires 1% of the computational time. The HK model although recently

shown to have surprising accuracy [21,29] has not been improved sufficiently for the

61

pores with a single filling step and cylindrical geometry. The DFT thermodynamic
model will therefore be utilized to determine the Lennard-Jones interaction parameters
between the solid and fluid. The DFT model as previously mentioned is based upon the

Lennard-Jones parameters all of which are determined from the saturation data except the

solid fluid potential well depth 85f.

3.3.2 Electric properties

Although, the most accurate data is obtained through fitting the Lennard-Jones

parameters at the experimental conditions, they may also be determined via several

schemes based upon the electric properties of the atoms. Due to the range of the forces it
is easily shown that ion to ion and ion to dipole interactions account for the bulk of the
electrostatic forces, followed by dipole interactions then greater multi-pole interactions.
In the absence of ions the dipole-to-dipole interaction accounts for over 85% of the
attractive force [22]. It is well known that dipoles maybe induced in an electric field,
61/611 a field generated by an adjacent dipole. It is easily shown that dipoles with their
Orientations weighted upon the energy, dipole to induced dipole, and induced dipole to
indneed dipole interactions all are dependant on the inverse r6 dependence [22,23]. The

Value of the potential due to dipole attraction is determined in the following ways [22]:

 

 

3.6
—C -C -C
U: 1+ 2 + 3
r6 r6 r6
2
c. =_2_ fl;
3kT 47w,
z #301}-
2 47:80

. . 1.1.
‘ 2 ’ 1,.+Ij.

62

where U represents the attractive potential, C 1 is the dipole to dipole interaction,C 2 is the
dipole to induced dipole, and C3 is the induced dipole to induced dipole interaction 11 is
the dipole moment, k is Boltzman’s constant, 80 is the permittivity of free space, a is the
polarizability volume, I is the ionization energy, and the subscripts i and j refer to atoms i
and j. The natural radial dependence of a dipole can be shown to vary as the inverse of
r6. The radial dependence of quadrupoles varies with the inverse of r8, and r10 for dipole
and quadrupole interactions of freely rotating molecules based upon and energy weighted
orientation. Therefore the methods that fit the Lennard-Jones parameters to the
properties of the dipole neglect all other multipole interactions, and may yield potential
well depths that are to shallow.

The Lennard-Jones potential may be integrated over the dimensions of a flat
surface to yield the 10-4 potential, or over an infinite cylinder to yield Equation 3.3.
Alternate theories exist to determine the potential well depth. The concept, first put forth
in the Frenkel Hasley Hills FHH adsorption theory, is that the excess free energy change
of adsorption is equal to the difference in the interaction energy of the adsorbate molecule
with the slab of adsorbent, less the adsorbate interaction energy with a similar slab of
pure adsorbate [24]. Horvath and Kawazoe used FHH theory to determine the potential (1)
between two infinite sheets of graphite separated by a distance H, as demonstrated in the

following equation (corrected from its form shown in the reference [25]):

114:1, (zi-i. (z): ”-‘A-12;.’.V “‘1 [(9] [9) .1 H1: z j” [ H1: 2 H 3.7

where z is the distance from the surface; 0' is the solid fluid molecular diameter; N is the

 

 

 

molecular density (the number of molecules per unit surface area); A is the dispersion

63

coefficient; the subscripts s and f refer to the adsorbent and adsorbate, respectively, and

the dispersion coefficients are calculated using the Kirkwood-Muller equations [26]:

6mc2a5af

sf :1:—a— 3.8
___5_+_L
Z: If

A, =3mc2a,2, 3.9

In the above equations, m is the mass of an electron; c is the velocity of light; and a and x
are the polarizability and magnetic susceptibility, respectively, of the adsorbent atom or
adsorbate molecule. In both of the methods presented the potential well depth is
determined by the polarizability of the molecule. This method however predicts a

potential well depth that is only 40% of the values obtained via fitting methods [21,29].

3.3.3 Nonporous carbon method

One method to fit the Lennard-Jones parameters is to find an experimental
isotherm of a sample with a known PSD, then to determine the L] that allows the
thermodynamic model to minimize the error in the adsorption integral Equation 3.1.
Utilizing the slit shaped pore DFT thermodynamic model with the adsorbate-adsorbate
data fit to the saturation properties of the system the remaining variable is solid fluid
potential well depth. Large pores have been shown to fill in layers [27,28,29] with the
monolayer relative filling pressure approaching a maximum value as the pore exceeds 20
molecular diameters in width. The filling pressure is controlled predominately by
relative temperature, molecular size and solid-fluid potential well depth. The temperature
may be controlled using a temperature bath or a cryostat. However, the potential well

depth and size are fixed for a given adsorbate-adsorbent system. The energetic

64

heterogeneity is assumed to be due to physical separation of opposing pore walls, hence
for sufficiently large pores the superposition of potential is negligible and the monolayer
filling pressure is determined by the potential exerted by only one wall. The limiting
value of the monolayer filling pressure is fortuitous because it allows one to treat the pore
size distribution of a nonporous solid as a mono-disperse PSD despite the ambiguity of its
actual size distribution as shown in figure 3.1. The thermodynamic model can be related

to the experimental isotherm though the following relationship for a Dirac distribution for

 

slit pores.
1R*SF P,R*
Vads(P):—2‘ 2( ) 3.10
oflNav

Where Vad,( P) is the volume adsorbed, R* is the mono-disperse pore size, S is the surface
area of the sample, and F(P,R*) is the thermodynamic model. The thermodynamic model
is often sharper than the reference isotherm, in which case the inflection points of the
monolayer filling should be matched.

The shortcomings of fitting the parameters to a nonporous surface are the
ambiguity of the shape and size of a pore in a “nonporous” carbon, measuring the high
relative pressure of the monolayer for some gases, matching the distribution of the pore
sizes and the reduced sensitivity to potential well changes. The ambiguous size of an
appropriate isotherm makes an exact fit less likely, although the monolayer filling
approaches a maximum it still is a function of the pore size and increases with pore size.
Additionally nonporous references generally contain a distribution of meso and

macropores which results in a reference isotherm which displays heterogeneity resulting

65

 

0.16

0.12 —~

0.08 —

I

Amount Adsorbed (mmollg)

 

 

 

 

0.0001 0.001 0.01 0.1 ‘l
PIPO

Figure 3.1 Argon adsorption on a nonporous graphitic carbon surface at 77K. The
circles are the experimental isotherm and the solid line represents the fitted slit pore

isotherm with a pore width of 67.5 A.

66

 

Figure 3.2 TEM photograph of single-walled carbon nanotubes. The dotted

circles represent the least squares fit circle to the SWNT wall.

67

in filling over a broader range of relative pressures. The exact point of inflection may not
lie on a measured data point reducing the ability to correctly fit the isotherms. The
nonporous standard has the shortcoming of a relative filling pressure that is much higher
than that of a micropore. Some gases such as carbon dioxide require pressure vessels to
reach the monolayer filling pressures at room temperature, which may increase cost. The
approximation of a nonporous solid indicates that the adsorbate is influenced by only one
wall. The superposition of potentials results in greater sensitivity to potential well depth
in micropores compared to meso and macropores.
3.3.4 Nanotubes as a Reference Adsorbent

The use of a microporous reference can obviate many of the difficulties associated
with the nonporous reference. Nanotubes have properties that make them an excellent
alternative reference material, such as their simple geometric structure, chemical
homogeneity, and multiple means of measuring their size. Using a micropore will have
the result of increased sensitivity to the potential well depth due to the super position of
potential wells. Additionally the model of a cylindrical pore represents nanotubes well,
whereas a nonporous carbon may have structural heterogeneity due to the varying shape
and size of the surface. As stated earlier it is necessary to determine the PSD of the
reference adsorbent. The PSD of a nanotube sample may be determined via measuring
the cross section of a nanotube photographed using transmission electron microscopy
(TEM). Nanotubes diameters may be found using a least squares approximation to fit a
circle to points that lay midway in the pore wall as shown in Figure 3.2. Utilizing a
method similar to that of the nonporous surface a solid fluid potential chosen and used to

generate a series of thermodynamic model isotherms, which are used in the inversion of

68

the adsorption integral Equation (3.1). The PSD which is generated for a particular solid-
fluid potential well depth may be compared to a simple histogram of the pore sizes
measured by the TEM to reveal the validity of the chosen potential

Both the nonporous and nanotube reference isotherms contain extraneous information at
relative pressures greater than the monolayer filling pressure. In the case of the
nonporous reference the isotherm the data obtained at pressures greater than the
monolayer “knee” may be misleading. Whereas for a nanotube the relative pressure
range is bounded by the monolayer filling pressure of the smallest and largest pore in the
distribution determined by TEM. As shown in Figure 3.3 the monolayer region as
defined by the thermodynamic model, is the most relevant region for fitting the potential
parameters. At relative pressures larger than the monolayer filling pressure the fluid-
fluid interactions play a larger relative role in pore condensation, therefore are less
reliable for determining the solid-fluid potentials. For nonporous surfaces the
heterogeneity of the meso and macropore sizes is also evident in the second layer
transition, reducing the reliability of data fitted to that region of the reference isotherm.
Data obtained from microporous references at relative pressures higher than the
monolayer filling pressures are suspect for the same reason as the nonporous reference
with the addition of ambiguity associated with the sorption to the tube exteriors and
interstitial spaces.

3.4. Results

3.4.1 General information

A sample of purified single walled carbon nanotubes, which was sonicated in strong acid

for 48 hours to shorten the nanotubes, remove the end caps and impurities, was dried and

69

 

45 18

Volume Adsorbed (mmollg)
F(mmol/cm3)

 

 

 

 

I I

0.0000001 0.00001 0.001 0.1
Relative Pressure (P/Po)

Figure 3.3 The relevant region of the isotherm to fitting is determined by the isotherm
generated predicted by the largest pore in the TEM photo(solid line). The circles show

the relevant region of the isotherm while the squares represent data from the nanotube

exterior.

70

photographed using TEM. The TEM photographs were then analyzed to determine the
histogram of pore sizes, based upon the least squares fit of a circle to the cross-section of
a nanotube, as shown in Figure 3.2. The mode pore size was determined to be 12.5 A as
specified by the manufacturer using TEM (see Figure 3.4). The preparation of the SWNT
for the TEM photograph may alter their relative positions to one another, therefore no
information about the interstitial spaces between tubes or tube bundles may be
ascertained from the analysis of the TEM data. The PSD obtained via TEM therefore
contains data based upon the tube diameter alone and is independent of the external tube
conditions.

3.4.2Verifying validity

The validity of the nanotube as a standard reference may be verified by comparing the
TEM PSD with the PSD obtained by porosimetry measurements based upon nonporous
references. Nitrogen and Argon Lennard-Jones parameters obtained for graphitic
nonporous carbons in the method described in section 3.3 [20,28], were used to generate
a series of cylindrical pore isotherms shown in Figures 3.5 and 3.6 respectively. For
convenience the pores were assumed to be in a trimodial gamma distribution (M=3),
although the PSD could have been determined over discrete intervals or with another

distribution function [20].

maiW
— .H 3.11
H=> :;—— 1mm ——:'exp< y. >

It has been demonstrated that during the manufacture of SWNT, bundles of tubes are
formed rather than individual tubes. The external surface of a tube is available for

sorption, however it has been shown that the interstitial spaces between tubes in bundles

71

 

 

 

Number of Pores
Ur

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 5 10 15 20 25 30
Pore Size (A)

Figure 3.4 TEM Pore size distribution. The number of pores with a diameter that lies
between half-angstrom intervals is displayed; the total number of pores measured is 72.

The mode in the distribution is the same as that predicted by the manufacturer. [4]

72

 

25

N
O

h—d
£11
1

 

F(mmol/cm3)
S

 

 

 

0 1
1.00E—08 1.00E-06 1.00E-04 1.00E-02 1.00E+00

Relative Pressure (P/Po)

Figure 3.5 Nitrogen specific excess adsorption in model single walled carbon nanotubes
at 77K. The isotherms shown from left to right have physical pore diameters of 10.7,

12.1, 13.2, 14.3, 17.9, 21.4, 28.6, and 35.7 A.

73

 

 

 

 

F(mmol/cm’)

 

 

 

0 ‘ i i 1 1
113-09 0.0000001 0.00001 0.001 0.1

Relative Pressure (P/Po)

Figure 3.6 Argon specific excess adsorption in model single walled carbon nanotubes at
87K. The isotherms shown from left to right have physical pore diameters of 11.3, 12.4,

13.1, 13.9, 15.8, 18.8, 22.5, 30, 37.5, and 75 A.

74

are too small for sorption to occur, [30,31] however modified tubes will adsorb on the
exterior [8,32,33]. The external surface of a SWNT has a weaker potential than a flat
surface [18], therefore tends to adsorb gases at a higher pressure. However, bundles of
tubes will have interstitial spaces that may be approximated as the interior of a tube with
a radius that is larger than the separation of tube bundles. These interstitial spaces of tube
bundles are expected to be larger than the size of an actual nanotube. Therefore the first
mode of the gamma distribution is anticipated to measure the PSD of the SWNT
interiors, while the remaining modes are due to the external surface area of the tube
bundles. As shown in Figure (3.7) the first mode of both the Ar and N2 overlap the PSD
determined via TEM and each falls within the binning of the histogram and within 2% of
the center of the peak. The two probe gases also predict a second mode that is within 4%
of one another, which was not be detected via TEM, presumably due to interstitial spaces.
3.4.3 Carbon dioxide adsorption

Carbon dioxide is a linear three-centered molecule with a permanent quadrupole moment,
with properties that make it a desirable alternate probe gas. Carbon dioxide is subcritical
at room temperature, making it a desirable probe gas where equilibrium cannot be
achieved in reasonable time scales at cryogenic temperatures. Unfortunately, the high
saturation pressures require special apparatus to measure isotherms above the monolayer
filling pressure at room temperature. The filling pressures of micropores however are at
low enough pressures to be measured with common machinery. The saturation properties
of carbon dioxide were used to determine the Lennard-Jones potential for fluid-fluid

interactions and may be found in Table 3.1. An isotherm was measured for carbon

75

 

0.07 16
_ Z
0.069 1413:
5'
A “129‘
$0.05~ 2,
"E ~10;
o -1
V0.04—l 8
Q +—
E 8 g
:1
30.03— g.
> *6 g
‘13 :1.
a 0.02“ F4 3
v-i
111
0.017 C2 2
0‘ t0

 

 

 

Pore Size (A)

Figure 3.7 Comparison of the TEM, argon and nitrogen PSD, note that the first peak of
each of the distributions are within 3% of one another. Also note that the TEM is

incapable of generating the secondary mode predicted by both probe gases.

76

 

 

 

 

3
A 2.5 - .
“l o
E o
e 2‘ °
'6 O
3 9
1g 15 1
w:
a
15’ 1 -
2
o
>
0.5 1
O 7 r r 1 1
0.00001 0.0001 0.001 0.01 0.1 1
Relative Pressure (PIPO)

Figure 3.8 A carbon dioxide adsorption isotherm measured on single walled carbon
nanotubes at 273K the squares represent the isotherm data and the line represents the best

fit DFT model isotherm for carbon dioxide.

77

dioxide on SWNT at 273Kas shown in figure (3.8) A series of cylindrical pore isotherms
were generated using a guess for the solid-fluid potential well depth as shown in figure
3.9. The isotherms were used in conjunction with the isotherm measured for SWNT to
determine a PSD, which would likely reproduce the reference isotherm. The value of the
solid-fluid potential well depth was adjusted to cause the first mode of the PSD to align
with the PSD obtained by TEM. The solid-fluid potential well depth which best
reproduced the mode in the PSD was 110K*kB and was then used to generate slit pore
isotherms for carbon dioxide between graphitic slabs. It was found that increasing the
magnitude of the potential well depth increased the size of the mode of the distribution.
The carbon dioxide slit pore isotherms are shown in figure 3.10 using the potential
parameters found in Table 3.1. The carbon dioxide slit pore isotherms are at a

Table 3.1 Potential Parameters

Parameter Ni Carbon dioxide

k

 

k

considerably higher relative temperature and therefore contain considerably less structure
than either the argon or nitrogen isotherms, which were generated for 87K and 77K
respectively. Like the argon and nitrogen slit pore isotherms [28] (Figures 2.4 and 2.5), a
transition is noted near 3.70' marking the transition to multilayering isotherms [29].
Carbon dioxide however has the monolayer filling pressure of large pores over a broader
range for a particular pore size, which would result in difficulty fitting the Lennard-Jones

potential parameters with a nonporous carbon. As shown in figure 3.11 the carbon

78

 

16

144

12-

10*

F(mmol/cm3)
oo

 

 

 

O _ ‘ I I I
113-06 113-05 0.0001 0.001 0.01 0.1 1

Relative Pressure (P/Po)

Figure 3.9 Excess carbon dioxide adsorption isotherms in model single walled carbon
nanotubes at 273K for a solid fluid potential well depth of 110K*k3, From left to right

the pores are 10.7, 11.4, 12.1, 12.8, 13.5, 14.3, and 17.8 A.

79

 

25

20-

151

 

10*

F (mmol/cm3)

51

 

 

 

 

0a ’ .
0.00000001 0.000001 0.0001 0.01 1

Relative Pressure (P/Po)

Figure 3.10 Carbon dioxide excess adsorption isotherms in model graphitic slit pores at
273 K Calculated using DFT. The isotherms shown left to right for pores having physical

widths of 7.1, 8.0, 8.9, 11.6, 14.2, 17.8, 21.4, 35.6 and 71.2 A.

80

 

50

 

 

 

 

l
45 7 .1.
g 40 - '
U) I
© 35 “ I.
s” x
3 30 ~ ,'
E 25 ~ 2"
In l
g 20 T 0"
<2 ' A
E 15 T I, ‘
E 10 ~ ,' ‘
> I A
5 2 I
0 1 1 1 r
1E-06 lE-05 0.0001 0.001 0.01 0.1 1
Relative Pressure (P/Po)

Figure 3.11 Comparison of surfaces for fitting. The nonporous carbon dioxide isotherm
measured by Bottani [34]is represented by triangles, the open diamonds are experimental
data for carbon dioxide adsorption on single walled carbon nanotubes, the solid line is the
fit of the DFT potential parameters and the dotted line is the DFT slit pore isotherm with

the potential determined via fitting to nanotube data.

81

dioxide isotherm obtained on a nonporous carbon does not display the characteristic
“knee” required to utilize the standard nonporous fitting technique. However, as seen in
the SWNT isotherm(figure 3.8) and the cylindrical pore model isotherms (figure 3.9 ) one
is capable of measuring the PSD.

3.4.4 Test of the Carbon dioxide Parameters

To test the reliability of the potential parameters obtained via the nanotube reference, a
comparison was made between carbon dioxide parameters fit to SWNT and argon and
nitrogen parameters, which were fit, based upon a nonporous surface. A granular
activated carbon (BPL 4x6) was examined with argon, nitrogen and carbon dioxide probe
gases to determine their respective isotherms as shown in figure 3.12. The argon and
nitrogen isotherms were determined at 77K and the carbon dioxide measurements were
preformed at 273K. It is not unexpected that the carbon dioxide isotherm should fill at
higher relative pressures at this higher relative temperature. The PSD for each gas was
determined based upon the inversion of the adsorption integral and is displayed in figure
3.13. The mode of each distribution was found to reside in a narrow range between 11.86
and 14.42 A. The argon and carbon dioxide modes were within —10% and +11% of the

nitrogen mode. As determined by the following equation:

|M0de0fGasX — NitrogenMode
NitrogenMode

x100% 3.12

 

Percentdiffemce =

The PSD predicted by the carbon dioxide isotherm is within a reasonable degree of error
when compared to the either nitrogen or argon probe gases. The relative proximity of the
carbon dioxide and nitrogen modes indicated a reasonable fit of the potential parameters.
There are several reasons why one would not anticipate the same result for all three probe

gases: (1) the precision of the experimental isotherm measurement; (2) the accuracy of

82

 

 

 

 

 

14

12 -
é” o
E 10 ‘ O ’ 0
E x
a 8 — . .
.n .
3 . '
M
'5 v
< 6 1 9 . ’
GE) ' a
2 4*
o
>

2 fl , o.

0 d l l T

1E-06 1E-05 0.0001 0.001 0.01 0.1 1

Relative Pressure (P/Po)

Figure 3.12 Adsorption isotherms on Granular activated carbon at 77K for argon
and nitrogen represented by filled and open squares and carbon dioxide at 273K
represented by filled circles. The solid lines represent the respective best fit DFT model
to the experimental isotherm, each of which was fitted separately under the same

conditions as the experiment.

83

P
o
a:

 

'2

111111111111111L1111111

P
c
c»

P
o
N

0.01

Pore Size Distribution (ems/90A)

 

 

 

 

0111111111111111
0 10 20 30 4O 50 60
Pore Width (A)

Figure 3.13 PSD obtained from the best fit DFT model isotherms on granular
activated carbon. The argon is represented by the filled squares the nitrogen by the open

squares and the carbon dioxide by the solid line.

84

the DFI‘ pore filling model; and (3) the soundness of the numerical method used to invert
the Fredholm integral of Equation 3.1 to obtain the PSD. The assumptions made in
determining the DFT Model isotherms are a simplification of the actual systems and
therefore will necessarily involve some degree of error. Neither nitrogen nor carbon
dioxide are actually single centered molecules as used in the model, therefore their
anisotropy is neglected. The model also simplifies the structure of the sorbent, the
assumptions of slit shape, no connectivity, and perfectly homogeneous surfaces are
simplifications to make the problem more tractable [28]. Results indicate that either a
nonporous surface or a SWNT will yield comparable potential parameters, however the
equilibrium isotherm may be questioned for the validity. Further investigations are
required to verify the isotherms. Due to the protracted times required for diffusion in the
nanotubes it is questionable whether the nanotubes are in true equilibrium with the
surroundings. One test to determine if an isotherm is in equilibrium is to increase the
temperature at which the isotherm is measured, DFT and MS predict that increasing the
temperature will increase the filling pressure and decrease the total volume adsorbed.
Changing the cryostatic temperature bath from liquid nitrogen to liquid argon shifts the
temperature from 77K to 87K, however the saturation pressure of the adsorbate is not
concurrently measured. As seen in Figure 3.14 sets of nitrogen and carbon dioxide
isotherms taken at different temperatures yield results that overlap one another. The
nitrogen isotherms are nearly overlapping and do not display significant differences
despite the prediction that at 87K the pores should fill at higher relative pressures. The
carbon dioxide isotherms display significant deviations from the anticipated results. The

reason for these deviations remain unclear, both temperature measurements are taken at

85

high relative temperatures and should display the opposite results. Possible reasons for
the unanticipated results include stretching of the bonds in the nanotubes at higher
temperatures, resulting in larger tubes with greater storage capacity. An alternate view is
that the rate of diffusion through the nanotubes is very slow, giving rise to non-
equilibrium isotherms.
3.5. Conclusions

The nanotubes are capable of serving as alternate reference adsorbents to
nonporous carbons, as illustrated by the overlap in the predicted PSD of the argon and
nitrogen on the SWNT and carbon dioxide on activated carbon, despite the apparent non-
equilibrium state. The data taken with a reference fluid at saturation indicates that the
nanotubes fill at the anticipated pressures, however deviations are noted for fluids that do
not have a saturated reference fluid. The apparent success in fitting the carbon dioxide
potential parameters cannot be trusted entirely despite its ability to match the PSD
predicted by both argon and nitrogen with a high degree of accuracy. When the
temperature is altered the values for these constants tend to change significantly therefore
their validity is questioned. Further analysis over a wider array of temperatures is
required to determine the appropriate equilibrium isotherms. While the mechanism
responsible for the non-equilibrium status is poorly understood it is known that increasing
the temperature has the result of faster diffusion times, it is therefore more likely that the

higher temperature isotherms are closer to the equilibrium isotherms.

86

 

 

 

 

7
In
I

6 ~ .13
”:71 ID
% 5 _ In
E .1:
E .0
z 4 - In
Q
"*3 c. '"
V)
'6 3 . 1:!
5 2 ‘ DID If
'5 I} IA‘ AA
>

1 “ 0 BBQ “‘ A&

in. A‘ AA
a E E A‘ A AAA
0 . .
0.0000001 0.00001 0.001 0.1 10
Relative Pressure (P/Po)

Figure 3.14 Adsorption isotherms of on single walled carbon nanotubes of nitrogen at
77K (open squares) and 87K (filled squares), and carbon dioxide at 273k (open triangles)
and 293K (filled triangles).

87

3.6. References:

 

1. Iijima, S. Naturel99l; 354: 56.

2. Ebbesen, T. W. “Cones and Tubes: Geometry in the chemistry of Carbon. ”
Accounts of Chemical Research. 1998;31 558.

3. Dujaurdin, E,; Ebbesen, T.; Krishnan, A. and Treacy M. Wetting of Single Shell
Carbon Nanotubes. Advances Materials 1998;10, 1472.

4. Rinzler, A.G.; Liu, J .; Dai, H.; Nikolaev, P.; Huffman, C. B.; Rodriguez-Macias,
F.J.; Boul ,P.J.; Lu, A.H.; Heymann, D.; Colbert, D. T.; Lee, R. S., Fischer, J. E.,
Rao, A.M. Eklund, P.C. Smalley R. E. Large Scale purification of single-wall
carbon nanotubes: process, product, and characterization. Appl. Phys. A.. 1998,
67, 29.

5. Lui, J.; Rinzler, A.G.; Dai, H.; Hafner, J.H.; Bradley R. K.; Boul ,P.J.; Lu, A. H.;
.Iverson, T.; Shelimov, K.; Huffman, C. B; Rodriguez-Macias, F.J.; Shon, Y.;
Lee, R. S.; Colbert, D. T.; Smalley R. E. Fullerene Pipes. Science 1998;
280,1253.

6. Maddox, M.W.; Sowers S.L.; Gubbins, K.E. Molecular Simulation of Binary
Mixture Adsorption in Buckytubes and MCM-41. Adsorption 1996; 2, 23.

7. Darkim, F. and Levensque, D. Monte Carlo Simulations of Hydrogen Adsorption
in Single-Walled Carbon Nanotubes. Journal of Chemical Physics 1998; 109
4981.

8. Wang, Q.; Johnson, K. Journal of Chemical Physics 1999, 110, 577.

88

 

10.

ll.

12.

13.

14.

15.

16.

17.

18.

Eswaramoorthy, M.; Sen, R.; Rao, C.N.R. Chemical Physics Letters 1999; 304,
207.

Talapatra, s.; Zambano, A.E.; Weber, SE. and Migone, A. D. Gases Do Not
Adsorb on the Interstitial Channels of Closed—Ended Single-Walled Carbon
Nanotube Bundles Physical review letters 2000, 85 138.

Maddox M.W, and Gubbins KE. Molecular Simulation of Fluid Adsorption in
Buckytubes. Langmuir 1995; 11 3988.

Lastoskie, C.M.; Quirke, N.; Gubbins, K.E. Stud Surf Sci and Cat 1997, 104, 745.
Schumacher, K., Ravikovitch, A. Du Chesne, A. , Neimark, A., Unger , K.
Characterization of MCM-48 Materials Langmuir, 2000; 16, 4648.

Ravikovitch, P.,Haller, G., Neimark, A. Density Functional Theory Model for
Calculating Size Distributions: Pore Structure of Nanoporous Catalysis. Advances
in Colloid and Interface Science 1998; 76 203.

Ravikovitch, P., Domhanaill, S. C., Neimark, A. Schuth, F. and Unger K.K.
Caplary Hysteresis in Nanopores: Theoretical and Experimental Studies of
Nitrogen Adsorption on MCM-41. Langmuir 1995 ; 11, 4765.

Balbuena PB, Gubbins KE.The Effect of Pore Geometry on Adsorption Behavior
COPS III International Symposium on the Characterization of Porous Solids
Marseille May 1993.

Tjatjopoulos GJ, Feke DL, Mann JA. Molecule—Micropore Interaction Potentials
J .Phys. Chem 1988; 92:4006—4007.

Stan, G.; Cole, M.W. Low Coverage Adsorption in Cylindrical Pores. Surface

Science 1998; 395, 280.

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19 Tarzona, P. Phys. Rev. A 1987, 60, 573

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

Lastoskie, C.; Gubbins, K.E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786.
Dombrowski RJ, Hyduke DR, Lasktoskie CM. The Horvath-Kawazoe Method
Revisited Colloid and Surfaces, A: Physiochem. and Eng. ASAP

Atkins PW. Physical Chemistry Fourth Edition. New York NY: W. H. Freeman
and Company. 1990:645-662.

Gregg, S.J.; Sing, K.S.W. Adsorption, Surface Area and Porosity; Academic
Press: New York, 1982.

J. Frenkel, "Kinetic Theory of Liquids," Clarendon Press, Oxford (1946).
Horvath, G.; Kawazoe, K. J. Chem. Eng. Japan 1983, 16, 474.

A. Muller, Proc. Roy. Soc. London A154 624 (1936).

Lastoskie, C.; Gubbins, K.E.; Quirke, N. Langmuir 1993, 9, 2693.

Dombrowski RJ, Hyduke DR, and Lastoskie CM. Langmuir 2000, 5040?
Dombrowski IR, and Lastoskie C Submitted to AICHE Journal.

Talapatra S, Zambano AZ, Weber SE, Migone AD. Gases Do Not Adsorb on the
Interstitial Channels of Closed-Ended Single-Walled CarbonNanotube Bundles.
Physical Review Letters 2000; 85(1): 1 38- 141 .

Darkrim F, Levesque D. High Adsorptive property of Opened Carbon Nanotubes
at 77K. J. Phys Chem B. 2000; 104 6773-76.

Mao Z, Sinnott SB. A Computational Study of Molecular Diffusion and Dynamic
Flow through Carbon Nanotubes. J. Phys Chem B. 2000; 104 4618-24.

Yin YF, Mays T, McEnaney B. Adsorption of nitrogen in Carbon Nanotube

Arrays. Langmuir 1999; 15 8715-8718.

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34. deTorre LEC, Bottani EJ, Steele WA Amorphous carbons: Surface structure and

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91

Chapter 4.

Dombrowski RJ, Hyduke DR, Lastoskie CM . The Horvath-Kawazoe method
revisited. Colloid and Surfaces, A: Physiochem. and Eng. Vol 187-188, pp 23-39
ASAP.

92

Chapter 4.

The Horvath-Kawazoe Method Revisited

4.1 Introduction

The determination of the pore size distribution (PSD) of microporous adsorbents
and catalysts continues to be an important research problem. Gas adsorption porosimetry
remains a convenient and popular technique for probing the PSD of a microporous solid.
Recently, novel methods have been proposed for characterizing the structure and
adsorbent properties of disordered porous solids. However, these promising new
approaches are not yet developed to a point where their widespread implementation is
possible. The PSD characterization problem, therefore, becomes one of selecting the best
available thermodynamic adsorption model F(P,H) to describe gas uptake in pores of
different width over the experimentally-sampled range of bulk gas pressures and
temperatures. An assortment of pore filling models have been put forth for this purpose;
among the first models suggested were the BET equation [1], the Dubinin-Radushkevich
equation [2], and the Kelvin equation [3]. It has been recognized for some time that
these "classic" models of adsorption do not accurately represent the filling process in
micropores; therefore, more recent models have been proposed that, in some manner,
incorporate gas-solid energies of interaction into the calculation of the model isotherm.

These include methods based upon statistical thermodynamics, such as density functional

93

theory [4] and molecular simulation [5]; and semi-empirical models such as the Horvath-
Kawazoe (HK) method [6].
The HK method is the first pore filling model specifically developed to describe

adsorption in microporous solids. In the original HK method, an analytic pore filling
correlation is obtained by calculating the mean free energy change of adsorption 47 that

occurs when an adsorbate molecule is transferred from the bulk gaseous phase to the
condensed phase in a slit pore of width H:
1n(PC/P0) = $(H)/kT 4.1

In equation (4.1), P0 is the bulk gas saturation pressure; PC is the pore filling pressure; k is
the Boltzmann constant; and T is the temperature. The assumed exponential dependence
of filling pressure on the free energy change of adsorption is also invoked in Polanyi
potential theory [7] and in the Frenkel-Halsey-Hill or FHH theory [8—10]. In the original
HK method, the mean free energy change upon adsorption is calculated from an
unweighted average of the adsorbate interaction potential 4), measured over the volume of

the slit pore located between z = 0,, and z = H - asf:

IH—o’f ¢(Z)dz J.;—o“ ¢(z)dz

05f sf
05f

 

 

5 = 4.2
where z is the perpendicular distance of the adsorbate molecule center-of—mass from the
plane of atomic nuclei in the solid surface layer, and as; is the arithmetic mean of the
adsorbate molecule and adsorbent atom diameters. The interaction energy between the

adsorbate molecule and a slit pore wall is represented by the 10-4 potential

94

«Io-4<z>=.’:—’1[£%T°-(%T]

which is obtained by integrating the Lennard—Jones 12-6 pair potential over an unbounded
planar surface [11]. In equation (4.3), N is the molecular density (the number of
molecules per unit surface area); A is the dispersion coefficient; and d = (2/5)”60fi,f =
0.8580}; is the distance from the solid surface at which the 10-4 potential has a zero
interaction energy.

The adsorbate potential within the slit pore, (b in equation (4.2), is modeled in the
original HK method as the sum of the gas-solid potential 0),; and a second term

representing the adsorbate-adsorbate interaction, 0,7:

[Ii—1°13)“+I..<:.I‘°-(,;:.fl

where the subscripts s and f refer to the adsorbent and adsorbate, respectively, and the

 

 

 

dispersion coefficients are calculated using the Kirkwood-Muller equations (as reported

in reference [6]):

6mc2aa
Asf=———‘——f— 4.5
91,9:
ls Xf
3 2
Aff ZEmC (1fo 4.6

In the above equations, m is the mass of an electron; c is the velocity of light; and a and z
are the polarizability and magnetic susceptibility, respectively, of the adsorbent atom or

adsorbate molecule. Substitution of equation (4.4) into equation (4.2) and integration

95

yields, when combined with equation (4.1), the original analytic HK correlation between

the pore filling pressure and the slit pore width:

NSAS +N A 9 3 9 3
W 9 ”1(2) 4(3) -1(—“-) AILI
kTo(H—2d) 9 d 3 d 9 H—d 3 H—d

Correlations similar to equation (4.7) have also been derived for cylindrical [12] and

 

spherical [13] pore geometries. The pore filling correlation predicted by the original HK
equation for nitrogen adsorption in carbon slit pores at 77 K is shown in Figure 4.1. It
can be seen that the original HK method correctly predicts lower micropore filling
pressures than are obtained from Kelvin-equation based pore-filling models such as the
Barrett-Joyner-Halenda (BJH) method [3]. The introduction of the HK method thus
brought about a major improvement to the PSD analysis of microporous solids, since
unlike the Kelvin equation, the HK model yields physically realistic PSDs for adsorbents
that uptake gases at very low relative pressures (i.e. at P/PO < 0.01).

In the last decade, a dramatic improvement in computer capabilities has led to the
application of numerically intensive statistical thermodynamics methods to model
adsorption in microporous solids. The advantage of using molecular simulation to
calculate the theoretical adsorption isotherms F(P,H) is that the isotherms so obtained
may be considered "exact" for the assumed pore geometry, surface composition and
intermolecular potential functions. For adsorbents with pore structures that are well
approximated by simple geometries such as slits or cylinders, theoretical adsorption
isotherms can be efficiently calculated using density functional theory (DFT) [4, 14-15].

Adsorption isotherms and pore filling correlations obtained from DFT rival molecular

96

 

1E+00
15-01 + 8%
115-02 HK
21:15-03
0- 1 E-04
2
a 1 E-05 +
2 1E-06
D.
g 1 E-O7
E 1E-08
1 E-09
115-1o

 

DFI'

 

 

o 5 1o 15 20 2530354045 50
Pore Width (A)

Figure 4.1: Pore filling correlations predicted by the Barrett-Joyner-Halenda (BJH)
method, the original Horvath-Kawazoe method (HK), density functional theory (DFT),
and Gibbs ensemble Monte Carlo molecular simulation (points) for nitrogen adsorption in

carbon slit pores at 77 K [19].

97

simulation results in accuracy (Figure 4.1) and can be computed in a small fraction of the
time required for molecular simulations. Molecular simulation, however, remains the
principal method of modeling adsorption in materials that have irregular pore geometries

(e.g. controlled pore glasses) or interconnected pore structures (e.g. zeolites) [16-18].

4.2 Critique of Original HK Method

The original HK model yields an analytic equation that is easily applied to
adsorbent PSD analysis by using a simple spreadsheet, whereas a Fortran or C++
computer algorithm must typically be written and executed to obtain theoretical isotherms
from DFT or Monte Carlo simulation. A comparison of the pore filling correlations
obtained for the original HK, DFT, and Gibbs ensemble Monte Carlo (GEMC) adsorption
models is shown in Figure 4.1 for nitrogen adsorption in graphitic slit pores at 77 K [19].
It can be seen that, relative to the DFT and GEMC results, the HK model, as originally
posed, overestimates the nitrogen condensation pressure in carbon micropores. There are
several reasons for the difference in the predicted HK and DFT/GEMC pore filling
correlations:

(1) The gas-solid interaction potentials that were used in the HK model and in the
DFT/molecular simulation approach are not the same. The original HK method uses the
10-4 potential for adsorbate interaction in a slit pore bounded by a single graphitic carbon
layer. In the reported comparisons with DFT and Monte Carlo simulation, meanwhile,
the 10-4-3 potential [20] for adsorbate interaction in a slit pore bounded by layered, semi-
infinite graphitic slabs has been used. The 10-4-3 potential for an adsorbate molecule at

distance 2 from a single graphite slab is

98

 

 

 

10 4 4
2 0’ 0’ 0'
¢10—4—3 : ZMSfPSO'Z'fA _ Sf _ Sf - Sf 4'8
5 z z 3A(z + 0.61A)3

where p, = 0.114 A'3 and A = 3.35 A are the graphite density and graphite layer spacing,
respectively; and 83f and 65f are the Lennard-Jones well depth and effective molecular

diameter, respectively, for the gas-solid interaction potential. The "3" term of the 10-4-3
potential represents the dispersion energy between the adsorbate molecule and the
subsurface layers of the graphite slab. If equation (4.3) is written in terms of the

Lennard-Jones gas-solid potential parameters, one obtains the result

 

 

2 0f 10 0f 4
_ 2 _ s _ s
¢10-4-3-27T€sfpsdsf4 5[ z J { 2 J 4.9

with NA = 1011'.t;‘,,p,Ad6 from equation (4.3). Comparing equations (4.8) and (4.9), it can
be seen that, for the same potential parameters, the 10—4-3 potential has a greater net
attraction for the adsorbate molecule at all distances 2 than does the 10—4 potential.
Therefore, the DFT adsorption model with the 10-4-3 potential might be expected to
predict lower pore filling pressures than the HK model with the 10-4 potential.

(2) The gas-solid potential parameters selected for the HK model are not the same
as the parameters fitted in the DF T model. In the original HK method, the dispersion
coefficients are fitted using the polarizability and magnetic susceptibility of the bulk
adsorbate (nitrogen) and adsorbent (carbon), as noted in equations (4.5) and (4.6). Values
of these and other needed physical properties, such as the molecular diameter and the
monolayer density, are culled from various references [6]. In the DFT method, the

Lennard-Jones parameters for the adsorbate-adsorbate pair potential are fitted to the

99

physical properties of bulk saturated liquid nitrogen, and the parameters for the
adsorbate-adsorbent potential are fitted to the monolayer uptake region of the nitrogen

adsorption isotherm on a nonporous graphitized carbon black. The gas-solid potential

parameters obtained in the DFT model for nitrogen adsorption at 77 K were 05f: 3.49 A
and Es/k = 53.2 K. By converting the parameter values NA and d in equation (4.3) to
their Lennard-Jones potential equivalents as written in equation (4.9), one recovers 03f:

3.20 A and ES/k = 21.3 K as the values used in the original HK model. Thus, there is

substantial disagreement in both the form of the potential and the values of the potential

parameters used in previous comparisons of the HK and DFT pore filling correlations.

The HK solid-fluid diameter is smaller than the DFT 05f value because a nitrogen

molecular diameter of 3.0 A was selected for the HK calculation [6]. This is an unusually
small value for the nitrogen molecular diameter; it is generally accepted that diatomic
nitrogen has an effective diameter that is larger than a carbon atom, with a value typically

in the range 3.5 to 3.8 A. The gas-solid well depth in the MK model is also smaller than

that used in the DFT model; in this case, the 85f value for the HK method is only 40% of

the fitted DFT well depth. This amplifies into a large difference between the HK and
DFT micropore filling pressures because the filling pressure varies exponentially with the
free energy change on adsorption in equation (4. 1).

(3) The adsorbate-adsorbate interaction energy is not calculated correctly in the
original HK model. Four conceptual flaws have been identified in the derivation of the
original HK pore filling correlation shown in equation (8). First, there is a factor of two

error in the Kirkwood-Muller formula reported in the original HK paper for the adsorbate

100

dispersion coefficient (equation (4.6)). The correct formula for A If is noted by Muller [see
the appendix of reference 21] and should read as
Afl =3mc20tfxf 4.10

A second error related to the adsorbate—adsorbate energy contribution is that the
adsorbate pair interaction should be subtracted from the adsorbate-adsorbent interaction,
rather than added to it as is done in the original HK method. The concept, first put forth
in the FHH adsorption theory, is that the excess free energy change of adsorption is equal
to the difference in the interaction energy of the adsorbate molecule with the slab of
adsorbent, less the adsorbate interaction energy with a similar slab of pure adsorbate [7].
In effect, one must remove a slab of adsorbate and replace it with a similarly-shaped slab
of the solid adsorbent to obtain the correct free energy change due to adsorption. In the
original HK treatment, however, the two free energies are not subtracted from one
another but instead are added together. No justification is given for summing the
interaction energies rather than taking the difference. Using the values reported in the

original HK paper for the monolayer densities and dispersion coefficients, the HK

potential parameters are given by equations (4.5) and (4.6) as NSAsf = 3.50X10'18 J A4 for

the adsorbate-adsorbent potential and NjAff = 2.40x10'19 J A4 for the adsorbate-adsorbate

potential [6]. Since the adsorbate-adsorbate parameter is about 7% of the magnitude of
the adsorbate-adsorbent parameter, the error in the size and sign of the adsorbate pair

potential term in equation (4.7) slightly changes the HK pore filling correlation. Using

the FHH form of the free energy difference NA = NsAsf - NfAff, with Alf given by equation

(4.10), the corrected value of NA is 3.02x10’18 J A4, 19% less than the originally reported

101

HK value of 3.74x10'18 J A4 for nitrogen adsorption on a carbon surface. This correction

actually shifts the HK pore filling correlation to higher condensation pressures, in greater
disparity with the DFT correlation of Figure 4.1.

A third defect of the original HK method is the ad hoc manner in which the
adsorbate-adsorbate potential is modeled. Equation (4.7) implies that the adsorbate pair
interaction energy can be calculated by placing the adsorbate molecules in the same
region of space occupied by the solid adsorbent atoms. This is physically unrealistic,
since pairwise adsorbate energetic contributions must clearly arise from adsorbate
molecules situated within the accessible volume of the slit pore, and not within the slab
wall. The original HK paper gives no explanation for superimposing the adsorbate-
adsorbent and adsorbate-adsorbate interactions onto exactly the same 10—4 potential,
aside from a nonspecific reference to the latter as being "an implicit function of the
adsorbate-adsorbate-adsorbent interactions" [6].

A possible fourth shortcoming of the adsorbate-adsorbate energy calculation in
the original HK method is that the mean free energy change due to adsorption is
calculated from an unweighted average over the accessible volume of the micropore. It is
known from molecular simulation studies that the local adsorbate density in the pore
varies strongly with position due to fluid layering near the pore walls. In particular, the
contact layer or monolayer is highly structured because it adsorbs within the large energy
well that is present in both the 10-4 and the 10-4-3 potential at a distance of one
molecular diameter from the pore wall. Other fluid layers similarly are not uniform, but
occupy discrete positions within the pore, giving rise to a structured local density profile

that is oscillatory. By neglecting the high degree of ordering in the adsorbed fluid, and in

102

particular by undercounting the adsorbate density in the monolayer region, the HK
method, as originally posed, underestimates the mean heat of adsorption computed by
equation (4.2), and thus overestimates the micropore filling pressures calculated from
equation (4.1). Consequently, the PSD results obtained for the original HK model in
equation (1.1) are less accurate than those given by DFT or molecular simulation.
Although several limitations of the HK method have been recounted here, the
original HK concept should not be dismissed. The basic concept of the model is
meritorious in that it attempts to relate the micropore filling pressure to the solid-fluid
interaction potential. One method of improving the HK correlation that is suggested by
the previous discussion is to represent the local density in the slit pore as a weighted
profile rather than as an unweighted profile. If the mean free energy change upon
adsorption is then calculated from an integration of the potential that is property weighted

with respect to the local adsorbate density p(z)

If” p(z)¢(z>dz
1p“: ”Hm 4.11
l ‘f p(2)dz

03f

 

the HK pore filling correlation given by equation (4.1) should be in better agreement with
the DFT and simulation results. In a recent paper, an improved HK pore filling
correlation for argon adsorption in carbon slit pores at 77 K was reported in which the
local density profile was represented by a sum of Gaussian distributions [22]. The
Gaussian distribution modes were fitted with a uniform amplitude and variance; the shape
parameter values were chosen so as to give the best agreement with local density profiles

calculated from DFT over a range of micropore widths. It was found, using equation

103

1.E+OO

 

1.5-01 _
O 1.E-02 ~

323

1.5-06 ‘

 

1.5-07 1 -CI— Modified HK
1 E-08 - . DFT
‘ -A- Original HK

Filling Pressure PJP

1.E-09 ~ .

 

 

1.5-10 . . . . 1 . . . . 1 . . . .
5 10 15 20

 

Pore Diameter (Al

Figure 4.2: Comparison of pore filling correlations for argon adsorption in carbon slit
pores at 77 K using DFT, the original HK method, and a modified HK method in which

the local adsorbate density profile is represented as a sum of Gaussian distributions [22].

104

(4.4) for the interaction potential as in the original HK paper, that substituting the
weighted local density profile in place of the unweighted density profile improved the
agreement between the HK and DFT pore filling correlations (Figure 4.2). However, a
general method of calculating the Gaussian shape parameters of the local density profile
from the adsorbate-adsorbent physical properties was not reported in this work.

In this paper, a generalized HK method is described for the PSD analysis of
adsorbents that have slit-shaped pores. Methods of calculating the micropore filling
pressure correlation for the unweighted and weighted local density assumption are
presented. A comparison of the HK and DFT predictions is also reported for the first
time using the same adsorbate-adsorbent potential (the 10-4-3 potential) and the same
potential parameter values in the comparison. By representing the gas-solid interactions

equivalently in both adsorption models, a truer comparison of the models is achieved.

4.3 Description of Generalized HK Method

In the generalized HK model, the filling pressure of a micropore is determined
exclusively from the characteristic temperature T" = kT/efl and solid-fluid potential
strength E* = esflefl of the adsorbate-adsorbent system. A description of the generalized
adsorption model follows.

Solid-fluid interaction potential: The adsorbate-adsorbent interactions are
modeled using a 10-4-3 potential in which there is a single characteristic molecular
diameter 0', that of the adsorbate molecule. With this approximation, 03f: A = 0' and the

10-4-3 potential of equation (4.8) reduces to

105

¢*(z *)=—¢—= 27rE*p, *[3(z*)“0 —(z 1)4 —l(z*+0.61)‘3] 4.12
Efl' 5 3

where z* = Z/O' and p,* = p303 = 5.2 for graphitic carbon [20]. The simplification in the
10-4-3 potential per equation (4.12) is justified on the grounds that, for most probe gases
used to characterize the PSDs of activated carbons, the adsorbate molecular diameter is
nearly equal in size to the carbon atom spacing in the porous solid. For example, for
nitrogen as the probe gas and graphitic carbon as the adsorbent, 0': 3.57 A and 03f: 3.49
A = 0.980' E 0' [23]. Helium is a notable exception to this rule; however, this is not a
concern in the development of our adsorption model because helium is usually employed
to characterize the pore volume of the adsorbent, rather than the pore size distribution.

The adsorbate-adsorbate potential is not included in the calculation of the HK pore
filling correlation presented in this dissertation. As noted in the introductory section, the
adsorbate-adsorbate interaction is erroneously interpreted in the original HK model. In
the absence of a suitable method of correcting this defect, the adsorbate pair interaction
energy is simply excluded from the free energy calculation. This decision is justified on
the grounds that the adsorbate—adsorbate potential is relatively small compared with the
adsorbate-adsorbent potential, and thus it is the gas-solid interaction energy that primarily
determines the filling pressure of a micropore.

With the imposed uniformity of the adsorbent layer spacing to match the adsorbate
molecular diameter, the conditions of the adsorbate-adsorbent system are described by
two only independently variable parameters, the dimensionless solid-fluid potential

strength E* and the dimensionless temperature 7“.

106

Unweighted density representation: In the unweighted HK method, the local
adsorbate density in the slit pore is assumed to be uniform everywhere in the accessible
volume of the pore. The accessible pore volume is defined as the region of the pore
where the solid-fluid potential has a negative value; i.e. where there is a net attractive
interaction between the adsorbate molecule and the adsorbent surface. The local density
profile in the unweighted density approach is therefore represented as p(z) = pm. for d < z
< H - d; and p(z) = 0 for z < d and z > H - d, where d 5 0.8510 is the distance from the
surface at which the 10-4-3 potential has a zero energy. The unweighted HK model thus

yields an analytic equation for the mean free energy change of adsorption:

 

H*-d*
_* 1d,, pav¢*(2*)dZ*
¢ = Had.
J'd* pavdZ*
27EE* * 4 _ _
—11 91

+%[—(d *)-3 + (H *—d *)‘3]+%L (d *+0.61)‘2 +(H *—d *+0.61)'2]}

The adsorption isotherm HRH) is then calculated as
HRH) = 0 for P < PC

HRH) = p0,.(H - 2d)/H for P > PC 4.14
—* * . . .
where PC = P0 exphp / T ) 18 the filling pressure for a pore of Width H calculated from

equations (4.1) and (4.13). The unweighted density pa,.(7"‘, E*) is obtained from a
correlation to DFT results as described later in this section. Note that in both the

unweighted and the weighted HK model, the specific excess adsorption is assumed to be

107

equal to the absolute amount adsorbed, because for temperatures that are well below the
bulk adsorbate critical temperature, the gas-phase adsorbate density is very small.
Weighted density representation: In the density-weighted HK method, the local

adsorbate density profile in the slit pore is modeled as an n-modal Gaussian distribution

2v2

p(=z) 2A exp[————— ( ”——'——)2] 4.15

where A., V,, and p,- are the amplitude, variance, and mean position, respectively, of mode
i of the distribution. Two types of Gaussian curves are used in the model. Peaks in the
local density profile that correspond to an adsorbed layer that is in contact with a wall are
represented by Gaussian curves with amplitude Am and variance Vm. Local density
maxima that are not adjacent to a pore wall are represented by Gaussian curves with
amplitude Ac and variance V... The selection of a Gaussian distribution with two distinct
types of distribution curves to describe the local density profile is based upon two
considerations.

First, it is known, from molecular simulations and from DFT calculations, that an
adsorbed fluid is strongly structured by the presence of the adsorbent surface. A confined
adsorbed fluid thus exhibits a local density profile with an oscillatory structure that
decays toward a bulk-like fluid density (if the pore is sufficiently large) at the pore
centerline. The effect of this ordering is represented in the weighted HK model by the
spacing of the Gaussian distribution modes. Second, by virtue of the short-ranged
dispersion forces near the adsorbent surface, the contact layer of adsorbed fluid is much
more highly ordered than interior layers of adsorbate that are not in direct contact with the

pore wall. This difference in the ordering of the contact and interior adsorbate layers is

108

acknowledged by assigning different parameters to describe the shape of the contact and
interior peaks of the local density profile. The quartet of shape parameters (A,,,, V,,,; A,
V,) are fitted to a two-parameter correlation of the dimensionless temperature 7“ and
dimensionless solid-fluid potential well depth E* using local density profiles obtained
from DFT for adsorption on a "nonporous" surface (see next section for a description of
the fitting method).

The number of modes it used to represent the local density profile is determined
from the physical width of the slit pore according to the pair of formulas

n = int(H/O') - 1 + (-)(f) 4.16

f: H/o- int(H/O') 4.17
where int(x) is the truncated integer value of x, and O is the Heaviside step function:
9(x) = 0 for x S 0, and (-)(x) = 1 for x > 0. The above equations mathematically express
the simple rule that, for pore widths that are non-integer ratios of the molecular diameter
0', an additional mode is added to represent the local density profile. For example, for H
= 2.00; a unimodal distribution is used; for H = 2.50', a bimodal distribution; and for H =
3.10”, a trimodal distribution.

The positions a,- of the distribution modes are assigned according to the following
rules:

For n = 1 (H S 20'), the Gaussian curve is centered in the middle of the slit pore,
so that the density profile attains its maximum at position 111 = H/2.

For n = 2 (20'< H _<_ 30'), the two distribution modes are centered at 111 = 0' and [12

= H - 0', respectively.

109

For n = 3 (30'< H S 40'), the contact peaks of the distribution are centered at [.11 =
0'and p2 = H - 0', and the interior peak is centered at 113 = H/2.

For n = 4 (40'< H S 50'), the interior peaks are spaced a distance of one adsorbate
diameter from the adjacent contact layer peaks; i.e. the positions of the maxima are at [.11
=0,u2=H-0,u3=20;and,u4=H-20'.

For an odd number of peaks n > 4, the nth peak is placed on the centerline of the
pore (i.e. 11,. = H/2). The other peaks are spaced at intervals of one adsorbate diameter
from the pore walls at z = 0 and z = H; i.e. peaks 1 through n-l placed in pairs at positions
(,uk, [11,“) = 00', H -j0'), wherej = (k+l)/2.

For an even number of peaks n > 4, the peaks are placed in pairs at coordinates (0',
H - 0), (20', H - 20'), (1'0", H - j0'), in successive spacings one adsorbate diameter
removed from the pore walls at z = 0 and z = H.

This simple placement algorithm yields density profiles that have peak positions
that match those obtained from DFT calculations in pores that contain an odd or even
number of density profile maxima. The amplitudes A, of the Gaussian distribution modes
are determined according to the following formula:

For n = 1, the amplitude of the peak is given as A1 = f Am, where f is the fractional
dimensionless pore width given in equation (18). Thus, for H = 2.00', A. = Am, whereas
for H = 1.80',A1= 0.8Am.

For n = 2, the peak amplitudes are calculated as A, = A2 = (l+f)Am/2. For

example, for H = 3.00', A. = A2 = Am, whereas for H = 2.60', A1 = A2 = 0.8Am.

110

For n = 3, the amplitude of the contact peaks are A. = A2 = A.,, and the amplitude
of the interior peak A3 = f AC.

For n = 4, the contact peaks have amplitudes AI = A2 = Am, and the interior peaks
have amplitudes A3 = A4 = (l+f)AC12.

For an odd number of peaks n > 4, the contact layer peaks have amplitudes A1 =
A2 = Am; and the n-th interlayer peak (the one positioned on the pore centerline) has an
amplitude A,, = f AC. The other interlayer peaks have amplitudes equal to AC.

For an even number of peaks n > 4, the contact peaks have amplitudes A1 = A2 =
A.,; the two innermost interior peaks (i.e. those closest to the pore centerline) have
amplitudes given as A,” = An = (1+f)AO/2; and the other interior peaks each have an
amplitude of Ac.

This algorithm for assigning the peak amplitudes ensures that, as additional
distribution modes are added for larger pore widths, the shape of the weighted local
density profile changes smoothly for increasing pore size. In a previous version of the
density-weighted HK method, an additional full-size peak was added each time the pore
width ratio exceeded a half-fractional molecular diameter (i.e. at H = 2.50', 3.50', 4.50',
and so on) [22]. This previous weighting scheme, however, produced discontinuous
jumps in the HK pore filling correlation, whereas the current density-weighting scheme
avoids such discontinuities.

Finally, the variances V; of the Gaussian distribution modes are assigned as
follows. For n S 2, the peak variances are set equal to V,,,. For n > 2, the contact layer

peak variances are V, = V2 = Vm, and all interior layer peaks have a variance equal to VC.

111

The adsorption isotherm HRH) for the weighted HK method is calculated as

HRH) = 0 for P < PC
H-d
HRH) = Id p(z)dz/H for P > P. 4.18

where R. is the filling pressure for pore width H calculated from equations (4.1) and
(4.11).

Fitting of Local Density Model Parameters: The unweighted local density model

parameter pm, was fitted to a bivariate logarithmic correlation of the form
p“(7*, E*) = ( a0 + al ln(7"‘) )( bO + bl ln(E*)) 4.19
where (do, a], b0, ’71) are the set of fitted coefficients (Table 4.2). The Gaussian shape

parameters (Am, Vm; Ac, VC) used for the density-weighted representation of the local

density profile were each fitted to a correlation of the same form as given above. The
correlation coefficients were chosen so as to yield the best possible agreement with

condensed-phase local density profiles calculated from DFT for adsorption in a slit pore
of width H = 200' at the saturation pressure P0. This pore width is sufficiently large so
that adsorbate molecules that are adsorbed at or near one pore wall are far removed from,
and essentially do not interact with, the other wall of the slit pore. Thus, adsorption in the
H = 200' pore is effectively equivalent to adsorption on a nonporous surface. The local

density profiles calculated from DPT for adsorption in the H = 200' pore can therefore be

used to develop correlations of the local adsorbate density on isolated planar surfaces.

112

Table 4.1: Fitted parameters for unweighted and weighted local density models.

 

 

Parameter a0 aI bn bI
P... -1.2 2.02 0.12 3.45
A -05: 1.08 3.48 7.66
Vm 0.41 0.98 -0.02 0.03
Am -05 0.37 3.00 7.26
v: 0.01 0.02 -471 4.64

Nineteen density profiles were compiled from DFT over a range of temperatures
0.60 S 7* S 1.0 and solid-fluid potential well ratios 0.32 S E* S 0.96. As shown in Table
4.2, this range encompasses the (E*, T") values of most of the probe gases used for PSD
analysis of adsorption isotherms measured at 77 K or 273 K. A detailed description of
the DFT adsorption model used in this work has been reported in section 2.2 [24]. All of
the DFT density profiles were calculated using a grid spacing interval of 0.010‘ in the z
coordinate for the numerical solution of p(z). Indistinguishable results were obtained for
grid spacings of 0.0050', 0.010’ and 0.020'.

The DFT local density profile in the H = 200' pore at 7“ = 0.70 and E* = 0.80 is

shown in Figure 4.3. To fit the correlation of the unweighted local density as a function

of T* and E*, the value of pm, that represents the mean density in the slit pore is

determined for each (7“, E*) pair. Graphically, this is the height that gives the
rectangular region shown in Figure 4.1 the same area as the area under the DFT local
density profile. The P... values obtained from the DFT results are reported in Figure 4.4.

A least-squares optimization was performed to find the set of (a a1, bu, b.) coefficients

(1’

113

 

 

 

 

 

 

 

 

 

 

 

 

 

 

graphite.
Adsorbate E * T *
Ar 0.526 0.702
C6H6 0.252 0.621
COZ 0.235 0.541
CF4 0.429 0.954
He 1 .441 0.406
H2 0.801 0.482
Kr 0.389 0.708
CH4 0.435 0.754
N 2 0.566 0.823
Xe 0.360 0.747

 

 

Table 4.2: Dimensionless solid-fluid potential and temperature for probe gases on

114

10—

 

 

      

 

 

82_
Mb .4
e
3' an
m .4
5 l
E 4~~
(U
8 « 1
.l
2__
.i w. I *
o_*'1'lb' . .

0 2 4 6 8 10
Distance from Pore Wall (zlo)

Figure 4.3: Local density profile at the saturation pressure calculated from DFT (points)
for a slit pore of width H*=20 filled with condensate at 7"‘=0.70 and E*=0.80. For
clarity, only the region from the left-hand wall to the pore centerline is shown; the density
profile is symmetric across the centerline of the pore axis. The lines show the fitted
profiles at this set of conditions using the unweighted (horizontal line) and weighted

density model.

115

 

1.1

Unweighted Density pm, 03
9
co
RA

 

 

0.8 —— M
0.7 ~-
0 6 L L 1 1 1

- I r I f I I I I T 17 I

 

0 0.2 0.4 0.6 0.8 1 1.2
Solid Fluid Potential E*

Figure 4.4: Local densities (points) obtained from an unweighted fit of DFT density
profiles at T*=0.60 (circles), 0.70 (triangles), 0.80 (squares), and 1.00 (diamonds) over a
range of E* values. The lines show the fitted bivariate logarithmic correlation of the

unweighted local density for the corresponding set of temperatures.

116

that best reproduces the pawl": E*) correlation. The fitted coefficients are reported in

Table 4.1 and the best-fit correlation results are shown as the lines in Figure 4.4.

To fit the Gaussian shape parameters for the density-weighted HK model, the

amplitude and variance of the contact layer Gaussian peak (Am and Vm) were first fitted to

match as closely as possible the height and weight of the contact peak obtained in each
DFT calculation. The variance Vc of the interior layer (condensate) peak was next chosen
so as to reproduce the width of the second layer DFI‘ peak (i.e. the peak in the local
density profile adjacent to the contact layer). Finally, the amplitude AC of the interior
layer peak was assigned so that the integrated areas under the DFT and weighted

Gaussian density profiles (see Figure 4.3) were the same. The Gaussian parameters (Am,

Vm; AC, VC) obtained by fitting the DFT results are shown in Figures 4.5(a—d), and the

logarithmic correlation coefficients and correlation results are reported in Table 4.2 and
shown in Figure 4.5, respectively. The unweighted and weighted model shape
parameters are all well fit using the bivariate logarithmic correlation which has four

adjustable coefficients.

4.4 Results

Micropore Local Density Profiles: Using the set of correlation coefficients reported in
Table 4.2, local density profiles were computed using the density-weighted sum-of-
Gaussians model for a set of micropores ranging in size between H = 20' and 60'. A
comparison of the local density profiles predicted by the weighted density model, and

those calculated from DFT at the saturation pressure, is shown in Figure 4.6 for six

117

.5
h

 

0.08

 

0.07 ~-

0
0.06
0.05

0.03 -

d
N
l
I

d
O
I
U

I

Contact Peak Amplitude A ,,.
Q
Contact Peak Variance V,"

 

 

 

 

2 1 JV 1 = I LI fi = T 7 0.02 ' l T l I I T l I I

0 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2
Solid Fluid Potential 5* Solid Fluid Potential E‘

 

 

6 0.25

. O
. I
0.15 -~
4 A
‘ O

0.05 -L

 

 

O
N

O
.5
I

 

Interior Peak Amplitude A c
(at
Interior Peak Variance V,

 

 

 

 

 

 

04:-:.:.:-¢. 01.1.1.4...
0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2
Solid Fluid Potential E‘ Solid Fluid Potential E‘
Figure 4.5: Contact layer and interior layer peak heights and variances (points) obtained
from a density-weighted fit of DFT density profiles, and fitted two-parameter correlations
(lines) of the peak heights and variances. The symbols denote the same temperatures as

in Figure 4. (a). Contact peak height. (b). Contact peak variance. (c). Interior peak height.

((1). Interior peak variance.

118

Local density (poi)

Local density (po’)

 

A l A A
V

N
A l A A
V

 

0 1 2
Position 2"

 

 

 

 

 

Position 2*

Figure 6c

 

119

 

Local density (po’)
‘0’

 

 

Figure 6b

 

U :5 0|

ALLALLAAJIA
Y

M
A 1 A A
Y

Local density M

 

 

 

Position 2‘

Figure 6d

 

 

 

 

 

 

 

 

 

. 5
1 .
- ‘1 4 «-
fl ‘ A ‘
g 1 ”b i
z. 3 1* 3 3 ..
a i E .
c 1 g ‘
§ 2 1 § 21
8 ‘ " I
-' 3 § 1
1 “ 1 .1
0 o
0 1 2 3 4 0 1 2 3 4 5
Position 2‘ Position 2'
Figure 6e Figure 6f

Figure 4.6: Comparison of local density profiles calculated from DFT (points) and from
the density-weighted model (lines) for adsorption in microporous slits at T"=0.80 and

E*=0.32. (a). H*=2.5. (b). H*=3.25. (c). H*=3.75. (d). H*=4. (e). H*=4.5. (f). H*=5.25.

120

different pore sizes at 7"‘=0.80 and E*=0.32. It can be seen that the DFT local density
profiles are reasonably well approximated by the sum-of-Gaussians model over the full
range of micropore widths. The best agreement is generally obtained for slit widths that
are an integral multiple of the molecular diameter 0'. However, the weighted-density
model also reproduces the local density profile in pore sizes that are not integral multiples
of 0' surprisingly well, considering that the density-weighted model algorithm contains
only four Gaussian shape parameters.

Comparison of Pore Filling Correlations: Using the unweighted and weighted
model representations of the local adsorbate density profile, pore filling correlations were
calculated using the HK method to estimate the mean heat of adsorption. The variation in
the filling pressures for slit pores between 20 and 60' in width is shown for the
unweighted HK model for adsorption at different temperatures T" at fixed E*=0.80 in
Figure 4.7a, and for different wall potential strengths E* at fixed 7"‘=0.8 in Figure 4.7b.
The same set of results is reported for the weighted HK model in Figures 4.8a and b. As
expected, the filling pressure for a given pore size is strongly dependent on the strength of
the solid-fluid interaction potential, with a large decrease in filling pressure occurring as
the value of E* increases. The filling pressure is also sensitive, to a lesser extent, to the

temperature at which adsorption occurs. As the value of 7* increases, both the filling

pressure P and the saturation pressure P0 increase, but the saturation pressure increases

more rapidly with temperature, so the net effect is that the relative filling pressure P/Po

decreases as 7“ increases.
A comparison of the unweighted HK and weighted HK pore filling correlations

for identical 7“ and E* values in Figures 4.7 and 4.8 reveals that the density-weighted

121

1E+00

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

g 115-03
5
2 1E-06
E 1E-09
O
~2-
'u'.’ 15-12

115-15 .

o
Porerdth (11')
Figure7a

154-00
3 113-033
g :
2 16-06 a
3 E
Q _
G _
5 115-09 5
a E
.E 2
E 115-12 _

1E-15

0 1 2 3 4 5 6
PoreWidth (H‘)
Figure7b

Figure 4.7: Pore filling correlation for unweighted Horvath-Kawazoe method. (a).
Correlation for E*=0.32 (circles), 0.48 (squares), 0.64 (triangles), and 0.80 (diamonds) at
fixed temperature 7"‘=0.80. (b). Correlation for 7"‘=0.60 (squares), 0.70 (diamonds), 0.80
(triangles), and 1.00 (circles) at fixed solid-fluid potential E*=0.80. The lines serve as a

guide to the eye.

122

1E+00

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A 1E-03
§
2» 1E-06
2
§ 115-09
2
a.
a 1E-12
.F.
‘L 115-15
1E-18 _
0 1 2 3 4 5 6
Pore Width (H')
Figure8a
1E+00
A 115-03
8
g 1E-06
g
g 1E-09
2
a
m 1E-12
.E
“ 15-15
1E-18 4L
0 1 2 3 4 5 6
Pore Width (11')
Figure8b

Figure 4.8: Pore filling correlation for density-weighted Horvath-Kawazoe method. The

notations and (7", E*) values in parts (a) and (b) are the same as reported in Figure 4.7.

123

HK model predicts a lower filling pressure for a given pore width than does the
unweighted HK model. This result is to be expected since the density-weighted model
was specifically designed to account for the large local adsorbate density in the
monolayer region adjacent to the pore wall. A comparison of the unweighted HK,
weighted HK, and DFT pore filling correlations is presented for nitrogen adsorption at
77K in Figure 4.9a and for argon adsorption at 77 K in Figure 4%. The DFT pore filling
correlations at 77 K for nitrogen [23] and argon [24] have been previously reported.
Interestingly, it is found that the unweighted HK adsorption model yields a pore
filling correlation that is in relatively good agreement with the DFT correlation for both
nitrogen and argon adsorption at 77 K. This finding contradicts previous comparisons of
the DFT and HK pore filling models, in which the unweighted HK model overestimated
micropore filling pressures relative to DFT. The previous comparisons, however, were
carried out using different potentials, and different potential parameters, in the two
models to represent the interactions between the adsorbate and the surface atoms.
Specifically, the original HK method used the 10-4 potential, whereas DFT results were
reported for the 10-4-3 potential. As noted earlier, the 10—4 potential does not include the
dispersion interactions of the adsorbate with subsurface solid atoms in the pore walls, and
hence predicts a smaller mean heat of adsorption than if the 10-4-3 potential is used.
Furthermore, the solid-fluid potential well parameter 85f calculated using the Kirkwood-
Muller formula in the original HK paper is considerably smaller than the 85f value that has
been employed in recent DEF adsorption models. In the present analysis, both the DFT

and the HK model results are calculated using the 10-4-3 potential, with the same 817 and

85f values, to represent the solid-fluid interaction. The only difference in the solid-fluid

124

Filling Pressure PcIPO

Figure 9a

Filling Pressure Pcfl’o

Figure 9b

1E+00

1E-02

1E-04

1E-06

1E-06

1E-1O

1E-12

1E+00

1E-02

1E-04

1E-06

1E-06

1E-1O

1E-12

 

 

 

 

 

 

 

 

 

 

J

 

 

5 10 15 20
Pore Width (A)

 

 

 

 

 

 

 

 

 

 

Pore Width (A)

Figure 4.9: Comparison of pore filling correlations calculated from DFT

(squares), the unweighted HK method (triangles), and the density-weighted HK method

(circles) for adsorption of (a). nitrogen and (b). argon in carbon slit pores at 77 K. The

temperature and solid-fluid potential values used to calculate the pore filling correlations

are 7"‘=0.823, E*=0.567 for nitrogen and 7"‘=0.702, E*=0.526 for argon.

125

potentials used for DFT and for the HK method in Figure 4.9 is that for DFT, the 10-4—3
potential of equation (9) is used, with 6,): (6,, + ofl)/2 = (3.40 A + 0')/2, and A = 3.35 A;
whereas in the unweighted and weighted HK models, the simplified potential of equation
(13), with 0',; = A = 0', is employed. This results in a small difference in the solid-fluid
potentials for the DFT and HK calculations. Nonetheless, the comparison between the
DFT and HK pore filling models in this work is a purer comparison of the adsorption
models themselves, rather than of the potentials used in the models

Another somewhat surprising result observed in Figures 4.9a and b is that the
density-weighted HK pore filling model is in poorer agreement with the DFI‘ model than
the unweighted HK model. Because of the good representation of the local adsorbate
density profile obtained using a weighted sum of Gaussians ( Figure 4.6), one would
expect the density-weighted HK pore filling correlation to agree more closely with the
DFT predictions than the unweighted HK correlation. The density-weighted HK model,
however, consistently underestimates the pore filling pressure relative to DFT for all
micropore widths for both nitrogen and argon adsorption. The unweighted HK
correlation, by contrast, overlaps the DFT correlation reasonably well for both of the

probe gases.

4.5 Discussion

It should be noted that, although the filling pressure correlation of the unweighted HK
model closely matches that of the DFT model, the adsorption isotherms constructed from
DFT and from the HK method for a specific pore size are dissimilar, particularly for

mesoporous slits. In the HK method, a steplike adsorption mechanism is assumed in

126

which a pore is completely empty at pressures below its filling pressure, and completely
filled with adsorbate at pressures above its filling pressure. Figure 4.10 shows a
comparison of the nitrogen adsorption isotherms predicted by the unweighted HK,
weighted HK, and DFT models in slit pores of width H = 30' and H = 50' at 77 K. For the
case of the weighted HK model isotherm, the phase transition always occurs at a lower
pressure than in the counterpart DFT isotherm; whereas for the unweighted HK model,
the condensation pressure may lie slightly above or below the DFT filling pressure, as
indicated by Figure 4.9. Note that the HK-based model isotherms in Figure 4.10 do not
exhibit an increase in adsorbate density above the filling pressure due to compression of
the condensed fluid as P/P0 increases. The more accurate DFT adsorption model
accounts for the adsorbate compressibility, and also correctly predicts the formation of a
precondensation film or monolayer prior to capillary condensation in slits that are larger
than 3.50' in width. The increased realism of the DFT adsorption model comes, however,
at the expense of an increased computational effort needed to assemble the requisite set of
isotherms to carry out adsorbent PSD analysis.

Pore Size Distribution Comparison: To illustrate that different results are obtained
for the adsorbent PSD based on the choice of adsorption model, the unweighted HK,
weighted HK, and DFT pore filling models were each used to interpret the PSD of an
Aldrich powdered activated carbon. The adsorption isotherm of the activated carbon for
nitrogen adsorption at 77 K is shown in Figure 4.11. Also shown in Figure 4.11 are the
best-fit model sorption isotherms obtained for the three pore filling models, using an
optimal pore size distribution function F (H) determined numerically for each adsorption

model by a sum-of—squares error minimization analysis of the adsorption integral

127

 

P
on

 

 

 

0.7 ‘ r - I
”o
0- 0.6 ~ I "
> - - -
'17: 0.5 6' ' I
5 u u
D 0.4 - l— l
2 I I
g 0.3 ~ I
i .2 '
2 ' ' I ' I

0.1 -| I

1E-6 1E-4 1E-2 1E+0

Relative Pressure (PIPO)

Figure 4.10: Comparison of model isotherms calculated from DFT (solid line), the
unweighted HK method (dashed line), and the density-weighted HK method (dotted line)

for nitrogen adsorption at 77 K in a slit pore of width 10.7 A (left) and 17.9 A (right).

128

.s
N

 

—L
O
1
1

Volume Adsorbed (cc/g)
Oi

 

 

 

 

1
I I IIIIIII #1 IIIIIII

   

I I IIIIII] I I IIIIIII

1E-6 1 E-4 1 E-2 1 E+0

I IIIIIII

Relative Pressure (PIPO)

Figure 4.11: Nitrogen adsorption isotherm at 77 K measured on Aldrich powdered
activated carbon (points), and fitted isotherms from DFT (solid line), the unweighted HK
method (dashed line), and the density-weighted HK method (dotted line). The lines are
difficult to distinguish because all three adsorption models give a plausible fit to the

experimental adsorption isotherm.

129

equation (1.1). It can be seen in Figure 4.11 that all three pore filling models give a
satisfactory fit to the experimental adsorption isotherm. The best-fit PSDs that yield the
fitted model isotherms of Figure 4.11 are shown for each adsorption model in Figure
4.12. The weighted HK model predicts a larger mean pore size for the adsorbent than is
obtained from the presumably more accurate DFT model. This is to be expected since the
weighted HK model consistently underestimates the filling pressure over all pore size
ranges. For the unweighted HK model, the PSD maximum and the mean pore size may
coincide reasonably well with the DFT results, or they may lie either above or below the
DFT-calculated values, depending on the characteristics of the adsorbent isotherm being

interpreted. For the case of the powdered activated carbon of Figure 4.11, a large amount

of gas uptake occurs over the relative pressure range P/P0 between 10'5 and 10.4. Over

this range of filling pressures, the DFT model predicts that slit pores between 11 and 13
A are being filled (see Figure 4.9a), while the unweighted HK model predicts
condensation in pores between 10 and 12 A. It is thus not surprising to find that, for this
particular isotherm, the unweighted HK model yields a PSD that is shifted to a smaller

pore size range relative to the PSD obtained from DFT. For a different experimental

isotherm, a different outcome might be observed; for example, for an adsorbent isotherm

with a large nitrogen uptake volume between P/P0 = 10'3 and 10-2, the unweighted HK

model would predict a larger mean pore size than would the DFT model (Figure 4.9a).

It is somewhat unexpected that the unweighted HK model more closely
reproduces the DFT pore filling correlation than the weighted HK model. Given that the
local adsorbate density profile is more realistically represented in the weighted HK

model, one would expect this model to predict pore filling pressures more accurately than

130

 

P
01

P
a

lllllllllllltll

 

 

Pore Size Distribution (cc/90A)

 

llllIILll

 

 

 

1o 12 14 16 18 20
Pore Width (A)

Figure 4.12: Comparison of pore size distributions obtained for Aldrich powdered
activated carbon using DFT (solid line), the unweighted HK method (dashed line), and

the density-weighted HK method (dotted line).

131

the unweighted model. However, the weighted HK model consistently underestimates
the condensation pressures, even though the adsorbate-adsorbate interactions are
altogether neglected in the generalized method. It is apparent that equation (4.11)
overestimates the free energy change due to adsorption. One possible explanation of the
difference between the DFT and weighted HK model, at least for pores that are larger
than 3.70' in width, is that the mechanisms of capillary condensation are different in the
two models. In the DFT model, condensation is preceded by monolayer wetting on each
pore wall, whereas in the HK method, the free energy change of adsorption into the
contact adsorbate layer adjacent to the pore wall is included in the free energy calculation
of equation (4.11). The weighted HK model may therefore underestimate the pore filling
pressure because, unlike DFT, it does not recognize monolayer formation and capillary
condensation as separate events that occur in mesopores. For pore widths smaller than
3.70', pore filling occurs in a single step in both the DFT and HK models, and the
difference between the weighted HK and DFT models is more difficult to explain.
Equally surprising is the finding that, when the unweighted HK model and the
DFT model are compared using the same gas-solid potential and potential parameter
values, the two models yield similar pore filling correlations. Thus, contrary to previous
findings, it appears that the original HK method gives a plausible representation of the
DFT pore filling correlation, provided that the correct 10-4-3 potential parameters are
used in the HK model. Furthermore, the agreement between the DFT and unweighted
HK pore filling correlations for both nitrogen and argon adsorption at 77 K suggests that
similar results may be obtained for other probe gases. Given the assumption of an

unstructured local density profile made in the unweighted HK model, the agreement

132

between this model and DFI‘ is to some degree fortuitous. It does appear, however, that
the simple unweighted integration of the gas-solid potential in equation (4.2) avoids
overcounting the energy in the region of the potential minimum where the monolayer
forms, and thus gives a more accurate estimate of the pore condensation pressure than

does the density-weighted integration.

4.6 Conclusion

The Horvath-Kawazoe method is the simplest adsorption model that is able to
predict pore filling pressures with some quantitative accuracy. Previous comparisons of
the original HK method with the DFT adsorption model yielded poor agreement between
the methods, principally because different gas-solid potentials and different potential
parameters were used in the models for calculating the pore filling correlation. In this
work, it has been shown that good agreement between the unweighted HK and DFT
adsorption models is obtained if the same potential and parameter values are used in both
models. A generalized HK adsorption model that involves the use of a weighted density
profile in the free energy integration has also been developed. However, the density-
weighted HK method predicts pore filling pressures that are too low relative to DFT
results. The unweighted HK method does not incorporate important features such as pore
wall wetting into the model adsorption isotherm, and therefore it should not be used if an
adsorption model from DFT or molecular simulation is available to interpret experimental
isotherms for a particular adsorbate-adsorbent system. If DFT model isotherms are

unavailable for the probe gas/adsorbent system of interest, however, the generalized HK

133

method can be conveniently implemented using a spreadsheet to obtain reasonably

accurate PSDs for microporous adsorbents with a minimum of computational effort.

134

4.7 References

1.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

S. Brunauer, L.S. Deming, W.S. Deming and E. Teller, J. Amer. Chem. Soc. 62 1723
(1940).

M.M. Dubinin and L.V. Radushkevich, Doklady Akad. Nauk. S.S.S.R. 55 327 (1947).
BR Barrett, L.G. Joyner and PP. Halenda, J. Amer. Chem. Soc. 73 373 (1951).
N.A. Seaton, J .R.P.B. Walton and N. Quirke, Carbon 27 853 (1989).

V.Y. Gusev, J.A. O’Brien and N.A. Seaton, Langmuir, 13 2815 (1997).

G. Horvath and K. Kawazoe, J. Chem. Eng. Japan 16 474 (1983).

RA. Pierotti and HE. Thomas, Surface and Colloid Science 4 241-245 (1971).

J. Frenkel, "Kinetic Theory of Liquids," Clarendon Press, Oxford (1946).

GD. Halsey, J. Amer. Chem. Soc. 74 1082 (1952).

T.L. Hill, J. Chem. Phys. 17 668 (1949).

D.H. Everett and J .C. Fowl, J. Chem. Soc. Faraday Trans. 1 72 619 (1976).

A. Saito and HG Foley, AIChE J. 37 429 (1991).

LS. Cheng and RT. Yang, Chem. Eng. Sci. 49 2599 (1994).

CM. Lastoskie, K.E. Gubbins and N. Quirke, Langmuir 9 2693 (1993).

A.V. Neimark and P1. Ravikovitch, Studies Surf. Sci. Catalysis 128 51 (2000).

L. Sarkisov and RA. Morison, Studies Surf. Sci. Catalysis 128 21 (2000).

L.D. Gelb and K.E. Gubbins, Studies Surf. Sci. Catalysis 128 61 (2000).

M. Heuchel, E. Buss and R.Q. Snurr, Langmuir 13 6795 (1997).

CM. Lastoskie, N. Quirke and K.E. Gubbins, Studies Surf Sci. Catalysis 104 745

(1997).

135

20. W.A. Steele, Surf. Sci. 36 317 (1973).

21. A. Muller, Proc. Roy. Soc. London A154 624 (1936).

22. CM. Lastoskie, Studies Surf. Sci. Catalysis 128 475 (2000).

23. CM. Lastoskie, K.E. Gubbins and N. Quirke, J. Phys. Chem. 97 4786 (1993).

24. RJ. Dombrowski, D.R. Hyduke and CM. Lastoskie, Langmuir 16 5041 (2000).

136

Chapter 5.

A Two-Stage Horvath-Kawazoe Adsorption Model
for Pore Size Distribution Analysis

5.1 Introduction
Porosity and pore structure are properties of catalysts and adsorbents that may control

transport phenomena, selectivity of reactions, and selective physisorption in separation
processes [1,2]. Several experimental methods exist for the determination of the pore size
distribution (PSD) of porous solids as discussed in Chapter 1. Gas sorption porosimetry is
one such method that may be used over the range of mesoporous and microporous materials.
The pore size distribution F (H) is obtained from the specific excess adsorption isotherm by
solving the adsorption integral equation

The PSD obtained by solving equation 1.1 will depend on the choice of thermodynamic
adsorption model HRH) that is used. Pore filling models based upon the Kelvin equation
are valid only in pores larger than 7.5 nm [1,3,4]; in smaller pores, continuum
thermodynamic models do not accurately represent the properties of the confined fluid. In
micropores, the filling pressure decreases sharply on account of the enhanced gas-solid
potential in such pores [5]. Adsorption models based upon molecular simulation (MS),
density functional theory (DFT), or the Horvath-Kawazoe (HK) method all attempt to
account for the effect of the gas-solid interaction potential in predicting model isotherms
HRH), for pores of specified size and shape. The HK, DFT and MS thermodynamic
models are valid for micropores that have widths approaching molecular dimensions [4].
The reliability of the PSD obtained depends not only on the choice of adsorption model, but
also on the form of the gas-solid potential and potential parameters used to represent the
molecular interactions. For example, the gas-solid potential in a mesoporous or macroporous

activated carbon might best be represented by the 10-4-3 potential [6], whereas a

137

microporous carbon with thin pore walls (e.g. a ribbon-like microstructure) may have a gas-
solid potential that is better described using the 10-4 potential. The PSD results will also
depend on the assumed pore geometry. HK pore filling correlations, for example, have been
derived for slit-shaped, cylindrical, and spherical pore geometries [7,8,9]. Similarly, MS
[2,10,11] and DFT [12] calculations have been carried out on various regular and irregularly
shaped pore geometries.

Although MS methods are considered to be the most accurate method of obtaining model
isotherms for PSD analysis, a large amount of MS computational time is required to
assemble a set of isotherms HRH) in sufficient detail so as to solve the adsorption integral
equation. DFT requires less CPU time than MS to generate an equivalent number of
isotherms, and it predicts nearly the same filling pressures and adsorbed-fluid densities as
MS [4]. Both statistical thermodynamics models (MS and DFT) are robust in the sense that
they can be confidently applied to either microporous or mesoporous solids for PSD analysis.
For adsorption in micropores at cryogenic temperatures, the DFT and MS adsorption models
predict IUPAC Type I adsorption isotherms with continuous filling to maximum capacity at
very low relative pressures. For mesopores, the two models predict IUPAC Type IV
isotherms, with pore wall wetting followed by a discontinuous phase transition at the
condensation pressure. For very large mesopores, the DFT and MS models predict capillary
condensation preceded by multilayer film growth suggestive of the IUPAC Type VI isotherm
observed in low temperature isotherm measurements [14].

The DFT method of PSD analysis is computationally more efficient than the MS approach,
but numerical solution using a computer algorithm is still required in order to obtain the DFT
model isotherms. An alternative model of adsorption in microporous solids, one that can be
solved using a simple spreadsheet, is the HK pore-filling model. The HK method yields an

analytic pore filling correlation that is calculated semi-empirically from an estimated mean

138

free energy change of adsorption. In the original HK approach, the pores of the adsorbent are
assumed to be completely filled with adsorbate or completely empty, depending on whether a
given pore is above or below its respective filling pressure. The HK model further assumes
that filling occurs in a single step and that no pore wall wetting or compression of the
condensed phase occurs as the bulk gas pressure increases. Because of these assumptions,
model isotherms constructed per the original HK method do not exhibit the wetting and film
growth features that are obtained for DFT- and MS-derived isotherms. In spite of such
approximations, it has been recently shown that the HK method reproduces the DFT pore
filling pressure correlation with surprising accuracy when the two models are compared
using the same gas-solid potential and dispersion parameters[15].

Various modifications of the original HK method have been proposed. These involve
corrections to either the shape of the pore [8,9] or the method of calculating the mean
adsorption free energy change [3,14,15]. A few other analytic adsorption models have been
suggested that derive a relationship between the pore filling pressure and the Gibbs energy of
adsorption [16,17]. As with the variations of the HK method, however, these adsorption
models do not account for the multiple filling events that occur in mesopores. To improve
the realism and the range of applicability of the HK method, it would be useful to modify the
original HK model so that the significant effect of pore wall wetting is accounted for in some
manner. A new adsorption model based upon this concept, referred to as the two-stage HK
pore-filling model, is presented in this paper. The two-stage HK model is described in
Section 5.2, and a comparison of the PSD predictions of this model with results obtained for
the predecessor "single-stage" HK method are presented in Section 5.4. Original DFT
calculations are reported in Section 5.4 for comparison with HK results. A summary of
findings and recommendations for PSD analysis is reported in Section 5.5.

5.2 Description of Two-Stage Horvath-Kawazoe Method
5.2.1 Review of the Original HK Model:

139

The premise of the HK method is that the pressure at which pore filling occurs may be
obtained from the mean free energy change of adsorption from the bulk gas phase to the
adsorbed phase through the equation 4.]. Due to the fact that equation 4.1 and consequently
equation 4.7 both have analytical results, the generation of model isotherms requires far less
time to generate than the iterative solution found using DFT. The pore filling correlation
described by equation 4.7 is applicable to pore widths H > 2031-, and it predicts that the pore
filling pressure decreases monotonically with decreasing pore width. In the original HK
formulation, the density of fluid adsorbed in a pore is set equal to the liquid adsorbate density

p, if the bulk gas pressure is greater than the filling pressure given by equation (4.7).

Otherwise, the pore is assumed to be empty. The model isotherms HRH) generated in the

original HK method are thus a set of Heaviside step functions
HRH) = P1911” - PHK(H)] 5.1

where O[x] = 0 for x < 0, and O[x] = 1 for x 2 0. Consequently, in the HK method, the gas
uptake that occurs at bulk gas pressure P in an experimental isotherm measurement can be
uniquely assigned to condensation in pores of only a single width.

As indicated by equation (4.7), the HK model in its original form assigns a contribution to
the potential from interactions between adsorbate molecules. However, there are some
conceptual difficulties with this treatment. The mathematical form of equation (4.7) implies
that adsorbate molecules occupy the same space as the adsorbent atoms [3,15]. Clearly this
is physically unrealistic. There are also errors in the original HK method associated with the
magnitude and sign of the term representing the adsorbate-adsorbate interactions in equation
(7). As noted in a recent paper [15], the Kirkwood-Muller equation for the A17 dispersion
coefficient, given as equation (4.6) in the original HK paper, is too small by a factor of two;
Muller [18] reports the correct form of this equation. Also, for the HK method to be

consistent with its conceptual underpinnings in the Frenkel-Halsey-Hill theory, the term in

140

equation (4.7) that represents the adsorbate-adsorbate interaction should be subtracted from,
rather than added to, the term for the adsorbate-adsorbent interaction.

In practical applications of equation (4.7), the adsorbate-adsorbent interaction energy is often
an order of magnitude larger than the adsorbate-adsorbate interaction. In such cases, the
aforementioned errors regarding the magnitude and sign of the Art term are not significant.
The original HK paper, for example, presents a model for nitrogen adsorption in carbon slit
pores at 77 K; in this model, 93% of the total potential energy is attributed to nitrogen-carbon
interactions, and only 7% to nitrogen-nitrogen interactions. Given the uncertainties
regarding the correct form of the adsorbate pair interaction in equation (4.7), it may therefore
be sensible to disregard these interactions altogether, and calculate the pore filling pressure
solely from the gas-solid potential which accounts for the bulk of the free energy change.
One of the principal shortcomings of the original HK method is that it is unable to accurately
portray the full sequence of filling events that occur within mesopores. The shape of the HK
model isotherm given by equation (5.1) is similar to that of the Type I isotherm, which is
characteristic of adsorption in a microporous solid. The original HK method does not
generate realistic model isotherms for mesoporous adsorbents, which have Type IV
isotherms in which pore wall wetting is followed by capillary condensation. To develop a
robust HK method, suitable for PSD analysis of either microporous or mesoporous materials,
one might consider extending the HK model to incorporate film wetting and growth in
mesopores. In this extended HK method, the mesopore isotherm features two or more filling
steps, rather than just a single step as in the original HK method. The first step corresponds
to formation of the monolayer on the pore walls; the final step represents capillary
condensation in the center of the pore.

In large mesopores, stepwise (Type VI) film growth is known to occur at low temperatures.

In principle, each growth step could be modeled as a separate filling event in the new HK

141

formulation. From a practical standpoint, however, the development of the multilayer is a
secondary structural feature of the mesopore isotherm. Of greater importance, in the context
of the PSD analysis, is that the initial pore wall wetting and the final capillary condensation
events be modeled as accurately as possible. Therefore, a two-stage HK pore-filling model,
with monolayer formation followed by condensation, has been developed for the PSD
interpretation of mesoporous solids. The new HK model integrates seamlessly with the
original HK method for microporous adsorbents, and so the improved model can be used to
interpret the PSDs of materials that have a combination of microporosity and mesoporosity.
A description of the two-stage HK model follows.

5.2.2 Description of the Two-Stage HK Pore Filling Model

The same methodology of the original HK model is applied to the two-stage HK model, with
the addition of new features that describe the wetting and growth of a monolayer film. The
two-stage HK model is generalized so that it may be applied for any adsorbate/adsorbent pair
at any subcritical temperature. The adsorbate-adsorbate and adsorbate-adsorbent interactions

in the new HK model are modeled using the Lennard-Jones potential with respective well-
depth parameters eff and 85f. The adsorbate and adsorbent molecular diameters are assumed
to be identical, i.e. off: 0',, = 05f = 0'.

The two-stage HK model isotherms are of the general form illustrated in Figure 5.1. The
filling transition pressures and the adsorbed fluid densities are calculated according to the
following procedure.

(1). The pressure PC at which capillary condensation occurs in the two-stage HK model is

calculated in the same manner as in the original HK method:

142

 

Excess Adsorption

 

 

 

 

Pressure

Figure 5.1: Schematic of two-stage Horvath-Kawazoe pore filling method showing the
principal features of the model mesopore isotherm: (1) capillary condensation pressure; (2)
condensed fluid interval; (3) film wetting pressure; (4) empty pore interval; and (5) film

growth interval.

143

 

 

[5“ 2rpm. z)dz iii/26901. z)dz
le P P =_.H= 0 = 0 5.2
do ”
QHJ) = ¢sj(z) + ¢sj<H " Z) 53

where the gas-solid potential 0,, is symmetric about the pore centerline at H/2. Equation
(5.2) is the same as that given by equations (4.2) and (4.3), except that the lower integration
endpoint has been changed to do, the distance from the surface at which the gas-solid
potential has a zero value. The implicit assumption is that the accessible pore volume is the
region in which the gas-solid potential has a negative value. In this paper, results are

reported for two gas-solid potentials, the 10-4-3 potential and the 10-4 potential

 

10 4 3
:1: 2 O’ O'\ 0'
¢lO-4—3(Z):27r£sfps [§[—] -[— — 3] 5.4
Z Z ) 3(z+0.610’)
10 41
t 2 O' O’
¢10-4(Z):27r£sfps |:§[—] ’(_J 5.5
2 z

 

For both potentials do 5 0.850'. The dimensionless adsorbent density p,* = 0,03 is set equal
to the density of graphite, for which p,* = 5.2.

(2). For bulk gas pressure P > PC, the adsorbed fluid density HRH) is assumed to be
constant, with a value p, calculated from the dimensionless temperature T* = kT/Efl and the
dimensionless gas-solid potential well depth E * = £,/£,,- as

p f (T*,E*) = p0(1-c01n(T*))*(l+c1ln(E*)) 5.6
where p0 = 0.834, c0 = 0.600 and c1 = 0.0375. This correlation was fitted to DFT results
obtained over a range of (T*, E *) values for the mean density at the bulk fluid saturation
pressure of a Lennard-Jones fluid adsorbed in slit pores of width H = 200' [15]. The

condensate density is a strong function of temperature and a weak function of the gas-solid

well depth.

144

(3). The pressure R., at which a monolayer film adsorbs is given as

d + d +
j 0 O¢(H,z)dz j 0 C,d’(1'1l.z)dz
d0 = d0

len(P,,,/PO)=6,,,(H) = 5.7

 

 

do +0 0'
d
d0 z

Equation (5.7) is applicable to slit pores of width greater than H,,, = 2( do + 0'). For the 10-4
or 10-4-3 potential, this corresponds to a cutoff pore width H,,, = 3.700'. Thus, for pore
widths H > Hm, the model isotherm has two filling transitions, one for monolayer wetting and
one for capillary condensation; whereas for pore widths H S Hm, a single transition occurs at
the filling pressure given by equation (5.2). Consequently, for pore widths H S Hm in the
two—stage HK model, the model isotherm does not have the shape shown in Figure 5.1, but
rather is a Heaviside step function similar to that described in equation (5.1) for the single-

step HK pore filling model.
Equation (5.7) implicitly assumes that the mean free energy change (3", for monolayer

adsorption may be calculated from a weighted average of the gas-solid potential, in the
region of the pore volume extending one molecular diameter from the zero-point coordinate
of the gas-solid potential. The definition of equation (5.7) is such that as the pore width
approaches the limiting value Hm, the wetting pressure given by equation (5.7) maps
seamlessly into the filling pressure correlation given by equation (5.2). Also, in the large
pore limit H —> +00, the monolayer wetting pressure Pm asymptotically approaches a limiting
value that corresponds to the wetting pressure on a planar nonporous surface. Equation (5.7)
is therefore both mathematically well defined and physically consistent with the wetting
behavior observed in DFT model calculations.

(4). For bulk gas pressure P S Pm (or P S PC in the case of pores smaller than Hm), the
pore is assumed to be completely empty: HRH) = 0.
(5). For bulk gas pressures Pm < P S R, the adsorbed fluid density is assumed to have a

logarithmic dependence on the pressure as given by the following equation:

145

 

20' P
FHH = T*,E*-—+ST*,E*,H1 — 5.8
( > M )H ( rum

C

where the coefficient S = (8178 ln(P/PC )] H is given by the correlation

 

T*,E*,
a a b 0—2(d +0) [)1
S(T*.E*,H)=SO “1 1-—'— 0 0 ' 5.9
T* E* H—2(d0+o)

where SD = 0.0222, a0 = 0.536, a1 = 1.32, b0 = 20 and bl = 0.6. The first term on the right
hand side of equation (5.8) is the adsorbed fluid density at the saturation pressure, multiplied
by the fraction of the pore volume occupied by the monolayer film. It is assumed that at the
onset of capillary condensation, the adsorbed fluid on each pore wall has a thickness equal to
0'. Also, the wetted film is assumed to have the same molar volume as the condensed fluid
phase.

The second term on the right hand side of equation (5.8) is a correction that accounts for film
growth over the interval between the onset of wetting and the condensation transition to the
completely filled state. The (T*,E*) dependence of equation (5.9) was obtained by fitting
DFT results at twenty different (T*, E*) values, as shown in Figure 5.2, so as to reproduce
the average slope of the isotherm for a pore of width H = 200' between the first and second
phase transitions (see Figure 5.3 for an illustration). The pore width dependence of equation
(5.9) was then correlated to the average slope of the DFT isotherm between the first and
second filling events for the range of pore widths 3.700’< H S 200', as shown in Figure 5.4
for fixed 7* = 0.70 and E* = 0.80. The form of equation (5.9) is chosen so that as the pore
size decreases toward Hm, the slope S goes to infinity. This is physically consistent with a
transition from two-step to single-step pore filling at the critical pore width Hm, an important
feature of the two-stage HK adsorption model.

5.3 Results
Selection of the Potential Parameters: As noted in the introduction, one of the principal

reasons why the original HK method and the DFT method predict different pore

146

 

0.04

0.035 -

0.03 -

0.025 ~

0.02 -

Slope of Monolayer
o
'2
01

0.01 ~

0.005 1

 

 

 

 

0 0.2 0.4 0.6 0.8 1
Solid Fluid Potential E* (6.18")

Figure 5.2: Fitted correlation of the slope of the isotherm of a slit pore of width H = 200 over
the interval corresponding to growth of the wetted film. Symbols denote results from DFI‘
for 7“ = 1.0 (triangles), 0.8 (circles), 0.7 (squares), and 0.6 (diamonds). Lines show the

correlation fit to the DFT data.

147

 

0.9

0.8 ~ Hr

0.7 7 /
0.6 1

0.5 1

Pi

 

 

 

 

 

 

 

 

 

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00
Relative Pressure (PIPo)

Figure 5.3 A comparison of DFT model isotherms and two-stage HK model isotherms the

slope of the monolayer region is given by the dotted lines

148

0.25

 

0.2 1

0.15 1

5*

 

0.1 1

0.05 1

 

 

 

0 . r .
0 20 40 60 80

Pore Size (A)

 

Figure 5.4: Fitted correlation of the slope of the isotherm for slit pores of different width over
the interval corresponding to growth of the wetted film. The symbols denote DFT results for

7* = 0.7 and E* = 0.8; the line shows the correlation fit at the same conditions.

149

filling correlations is that a different procedure is used to fit the potential parameters of each
model. In this work the DFT parameters of the adsorbate-adsorbate interaction potential are
fitted to the physical properties of the saturated liquid adsorbate at its normal boiling point
temperature. The parameters of the gas-solid potential, meanwhile, are selected so that the
DFT model isotherm of a large pore (e.g. H = 200') is in good agreement with the low-
pressure (monolayer) portion of the experimental isotherm of a nonporous reference
adsorbent. Ideally, the reference material that is chosen has a chemical composition and
surface structure similar to the porous adsorbents for which PSDs are to be determined.

In the original HK approach, the potential parameters for the adsorbate-adsorbate and
adsorbate-adsorbent interactions are obtained from the polarizability and magnetic
susceptibility of the adsorbate and adsorbent using the Kirkwood-Muller formulas. Because
different methods are used to fit the potential parameters of the DFT and HK adsorption
models, divergent potential parameter values are obtained for the two methods. For example,
the DFT fitting procedure gives es/k = 53.2 K for nitrogen adsorption at 77 K on a
graphitized nonporous carbon [4]. By contrast, for the HK method a nitrogen-carbon well
depth of varies with the size of the pore as shown in equation (4.7) using physical data from
the original HK paper [7]. Because the HK well depth determined by the Muller equations is
only 40% of the DFT-fitted value, the original HK model predicts nitrogen condensation
pressures in small carbon mesopores that are 1 to 3 orders of magnitude too large.

It has been noted elsewhere that the Kirkwood-Muller parameter fitting approach may
introduce large errors into the adsorption model [19]. Also, for gas-solid systems that have
not previously been studied, the physical property data needed to apply the Kirkwood-Muller
formulas may not be readily available from the published literature. Given these

considerations, a new, experimentally facile procedure is proposed for fitting the potential

150

 

parameters of the two-stage HK model. As in the original HK method, the adsorbate
molecular diameter 0' is specified as an unfitted parameter. However, the potential well

depths 8,] and 8,; are fitted using a procedure akin to that used to fit the gas-solid well depth in
the DFT model, as shown schematically in Figure 5.5 and explained further in section 2.2.

In order to fit the potential well depth, one must have a reference material that is chemically
similar to the material being analyzed and may be regarded as nonporous. The potential
function that best describes a nonporous sample the 10-4-3 Steele potential, as it is capable of
accounting for the influence of multiple layers of sorbent, which is physically analogous to
such a system. A nonporous surface is expected to have a potential that is similar to that of a
slit pore whose walls are separated by a distance large enough that the superposition of the
potentials is comparable to that of a single wall. An experimental isotherm contains two
forms of information, the relative pressure (P/Po) and the amount adsorbed (mmollg). A
nonporous adsorbent will typically have an isotherm that exhibits a step like behavior, these
steps can be attributed to layer formation. The region of an isotherm before the second layer
transition is the most relevant section for fitting the solid fluid parameters. The monolayer
density pm within a pore can be described by the following using the multiple filling model as
given by equation 5.8.

For pores that are significantly large (267 A) the monolayer density is a weak function of the
pore size H. The density in a pore may be related to the volume adsorbed V though the

following equation.

_ HSpm
03A,,
Where H is the pore size, S is the surface area, A, is Avagodro’s number. The monolayer

V 5.10

filling pressure is also a function of the pore size, dimensionless temperature and energy.

The value of the surface areas, S, may be the BET surface area, however this has been shown

to under predict the actual value, therefore the method of using the film thickness as

151

 

 

 

p
4:.

O
.9 or
(D 0'!

 

.0

to

01
I

Amount Adsorbed (mmollg)
O
N

 

 

 

0.15 1
0.1 1
I
0.05
I
o J v I r
1 .00E-05 1 .00E-03 1.00E-01

Relative Pressure (PIPM)

Figure 5.5 Comparison of a fitted two-stage HK model isotherm (solid line) and a reference

isotherm Argon adsorption on sterling activated carbon square symbols.

152

 

 

described by [13]Lastoskie should be utilized. Following the method of de Boer et al.[18],
the specific surface area is obtained by fitting a line through a plot of the argon uptake versus
the theoretical film thickness figure 2.2. A specific surface area of 11.3 mz/g was measured
for the Sterling carbon; this value was then used to convert the multiple filling model-
calculated specific excess argon adsorption in the 67 A-slit pore from a pore volume basis to
an adsorbent mass basis, as described in previous work [21].

Utilizing the 10-4-3 Steele potential and substituting into equations 5.7 and 5.8, one finds
that the monolayer filling pressure is a function of the ratio of E * / T*. After the proper
selection of the E*/T* ratio, the monolayer filling pressure has been determined, and the
values of E* or T* can be fitted, while maintaining the ratio determined earlier, to match the
volume adsorbed, at the relative pressure just below the formation of a second layer, as seen
in Figure 5.5.

The selection of the potential is determined by the physical characteristics of the material
being studied. If pores are assumed to have only one layer of atoms as a forming a wall, then
the 10-4 potential should be used, however if two or more layers of adsorbent or adsorbent
and adsorbate are expected the 10-4-3 potential should be used. As illustrated in Figure 5.6
the potential for the two or more layers is approximately the same as for a semi-infinite
plane. The 10-4 potential and the 10-4-3 potentials should bracket all the possible potentials
due to the structural inhomogeneities for a slit pore.

Utilizing the 10-4 and 10-4-3 potentials of the slit pore for both argon and nitrogen at
77K four series of fourteen isotherms, whose size ranged between two to twenty molecular
diameters, were generated using DFT. The potential parameters for argon were chosen to
match those found in the literature [5] with the exception of the fluid-fluid molecular

diameters being chosen to equal the molecular diameter of the adsorbent (3.35A). The error

153

 

Dimensionless Potential
r'o
o

 

 

 

-4o . .
0.7 0.9 1.1 1.3

Distance from Pore Wall (z/o)

 

Figure 5.6 Spatially varying potential in a 2 molecular diameter slit pore at 77K for single
planes of graphite (triangles), for a double layer of graphite planes(crosses), and for a semi-

infinite slab of graphite (solid line)

154

associated with the change in the fluid molecular diameter was found to be insignificant for
the argon. (isotherm not shown) The potential parameters for the nitrogen were refit using the
method described by Lastoskie et al.[21] for the generalized 10-4—3 potential. The net effect
was an increase in the potential well depth to 58.46/k, when the nitrogen molecule was
assumed to have a diameter equal to that of a carbon atom. The 10-4 potential isotherms
displayed similar trends to the isotherms generated using the 10-4-3 potentials [5] for the
same conditions, with the notable exception that the filling occurred at slightly higher relative
pressures as seen by comparing Figures 5.7a-d. The higher filling pressures are expected as
the 10-4 potential has a higher net energy well depth. In all cases the DFT isotherms show a
single layer or single filling for slit pores smaller than 3.70‘ and multiple filling steps for
larger pores. All DFT isotherms also display adsorption before and after the relative filling
pressure, representing the compression of the adsorbed layer or layers. The two-stage HK
model mimics the filling behavior of the DFT isotherms for both the single and semi-infinite
layers of adsorbent for both monolayer and multilayer condensation. The two-stage HK
model correctly displays the filling of a monolayer and its density relative to the final density
at the saturation pressure, while correctly displaying the limiting value between single and
multiple filling isotherms. Additionally, the model isotherms as seen in figures 5.8a-d
correctly predict that the monolayer filling pressures approach an asymptotic limit similar to
the behavior of the DFT isotherms.

The filling pressures can be broken into two categories the monolayer filling pressure and the
condensation filling pressure for the two-stage model. In most cases the two-stage HK
model, like the original HK model with an improved potential function, under predicts the
multilayer filling pressure compared to DFT.(see figure 5.9a-d ) With the same potential
function and potential parameters the multiple filling model generally over predicts the

monolayer filling pressure, this apparent disparity indicates that the potential is

155

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

40 40
35 1 35
30 1 30
"E 25 / A
" 25 1
5 i / 5
E 20 ‘5’ 20
.5. 15 ~ 5
1.. I: 15 .
10 1
10 1
5 4
5 .
0 .
1.E-09 1.E-06 1.E-03 1.E+00 0'
1E-12 1E-09 1E-06 0.001 1
Relative Pressure (PIPo)
Relative Pressure (PlPo)
4o . 40
35 _ 35
30 1 3o .
6" :
325 E" ME 25 (
o20 5- g
E g 20
1:153 15. 15.
10 5- ‘—
5 i 10 ‘
o 3 5 1
1E-10 1E-6 1E—6 1E-4 1E-2 154-0 0 , /

 

1E-12 1E-09 1E-06 0.001 1

Relative PressureP/Po Relative Pressure (PIPo)

Figure 5.7 a-d. DFT model isotherms for a) argon 10-4 potential pore sizes 6.7, 7.5, 8.3, 10.1,
13.4, 33.5, and 67A b) nitrogen 10—4 potential pore sizes 6.7, 7.5, 8.3, 10.1, 13.4, 26.8, and
67A c) argon 10—4-3 potential pore sizes 6.7, 7.5, 8.3, 10.1, 13.4, 33.5, and 67A d) nitrogen

10-4-3 potential pore sizes 6.7, 7.5, 8.3, 10.1, 13.4, 26.8, and 67A

156

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

40 40
35 1 35 ,
301 F 30
'2 25 "E 25,
0 o
= =
g 20 g 201
E —' g . ..
L. 15 ‘ — — [_ 15 I.
10 - 10 1
5‘ 5i 1 i
o l— 0 ~ . ~ ~
"5.09 0.000001 0.001 1 1.E-11 1.E-08 1.E-05 1.E-02
“9'3“" 9'93““ (9’90) Relative Pressure (PIPo)
40 4o
35 1 35 _
30‘ 1 30, _
6‘ P A
5 25‘ ”E 25
=o g
E 20« _ g 204
.5. 15- E 154 _ -
" I:
10 ‘ F 10 fl
5
5 _
0

 

 

1.E-10 1.E-07 1.E-04 1.E-01 '
1.E-11 1.E-08 1.E-05 1.E-02

Relative Pressure (PIPo)
Relative Pressure (P/Po)

Figure 5.8 a-d. Two-stage HK model isotherms for poresizes 6.7, 7.5, 8.4, 9.2, 10.1, 10.9,
11.7, 12.6, 13.4, 15.1, 16.8, 20.1, 26.8, 33.5, and 67A for a) argon with a 10-4 potential b)

nitrogen 10-4 potential c) argon with a 10-4-3 potential (1) nitrogen with a 10-4-3 potential

157

Relative Filling Pressures(Pc/Po)

Relative Filling Pressures(Pc/Po)

1.E+OO

 

1.E-01 <
1.5-02 -
1.5-03 J
1.E-04 ‘
1.5-05 «
1.E-06 «
1.E-O7 «
1.E-08 ~
1.E-09 .
1.E-10 «

 

1.E-11

 

 

1.E+OO

1O 15 20

Pore Width A

25 30

 

1.E-01 <
1.E-02 .
1.E-03
1.E-04 ‘
1.E-05 ~
1.E-06 4
1.E-07 -
1.E-08 4
1.E-09 ~
1.E-10 ‘

L

f'

 

 

1.E-11

T

10 15 2O

Pore Width A

 

25 30

Filling Pressure (PclPo)

 

1.E+00
1.E-01 4
1.E-02 4
1.E-03 .
1.E-04 4
1.E-05 J
1.E-06 <
1.E-O7 ‘
1.E-08 .
1.E-09 1

Filling Pressure (Pa/Po)

1.E-10

1.E+00

 

 

 

15 25
Pore Width (A)

 

1.E-01 ~
1.E-02 <
1.E-O3 ~
1.E-O4 «
1.E-05 ~
1.E-06 .
1.E-07 ~
1.E-08 1
1.E-09 ~
1.E-1O 4

 

 

 

1 5 25
Pore Width (A)

Figure 5.9 a-d Filling pressure correlations of DFT solid symbol and two stage HK methods

open symbols for a) argon 10-4 potential b) nitrogen 10-4 potential c) argon 10-4-3 potential

d) nitrogen 10-4-3 potential all at 77K

158

simultaneously too strong and too weak. While it is tempting to attribute this behavior to the
density profile within a pore, Dombrowski et al [15] and Lastoskie [21] have shown that the
density weighted average of the potential would increase the average potential, improving the
monolayer fit at the expense of the multilayer fit.

The solutions to the inversion the adsorption integral equation 1.1 have been found for a
powdered activated carbon for each potential and adsorbate system. The isotherm fits to the
experimental data are reported in figures 5.10a—d. A simple inspection reveals that all the
isotherm fits are plausible. The multiple filling model closest fit isotherms display a
consistent irregularity near the limit of the monolayer filling pressure, which are not
displayed by the DFT model isotherm fit. The characteristic malformation of the isotherms
is due to the monotonic increase in the final density and zero slope after the single filling
transition of the isotherms smaller than 3.70’, where as the DFT model isotherms do not
display a consistent increase in the density following pore filling. The simplistic
representation of a single filling of the generalized version of the HK method also does not
display any characteristic abnormalities, however none are expected with such a model. The
resulting PSDs obtained from each of the three methods are displayed in figure 5.11a-d
clearly indicate that the multiple filling model provides better agreement to the DFT
calculations than the generalized HK method. For pores that are smaller than 12.4 A the
multiple filling model and the generalized HK methods are identical, therefore the true
comparison lies in the prediction of larger pores. The generalized and multiple filling HK
model isotherms for specific pore sizes become less accurate for smaller pores than for large
pores, upon which the model is based. As demonstrated by Dombrowski et al [13] the local
density of the adsorbed phase is higher than the interior local density, and a model which is
based upon a large pore, would tend to weigh the interior density higher than the monolayer

density. Although there is a systematic error produced by using this one size fits all pores

159

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20 14
18 .. A _,
a g: 12
= 16‘ g
3 e 10*
g 14 1 V
E 12 E 8
1: 3.
< E 4 ,
2 5‘ g
'3 41 < 2 ~
>
2 * ’ o . T
o . . , 1.E-06 1.E-04 1.E-02 1.E+00
0.000001 0.0001 0.01 1
Relative Pressure (PIPo) Relative Pressure (PIPO)
20 14
18 4 A 12 J °
A m
g 16 « 3
E .
g 144 g 10
g 12 ‘ 3 8 q
10 < O
“a '3 6 ‘
< 8* <
~ o
5 6‘ E 4 ‘
o 2
E 4 ‘ o 2 4
< >
2 4
o ‘ ' ' 0 I I

Relative Pressure(PlPo)
Relatlve Pressure (PlPo)
Figure 5.10 a—d Isotherm fits to experimental data(diamonds) on powdered activated carbon
at 77K, DPT (dotted line), generalized HK (crosses), two stage HK(solid line). The data
overlap indicates that each of the models fit well. For a) argon 10-4 potential b) nitrogen 10-4

potential c) argon 10-4-3 potential d) nitrogen 10-4-3 potential.

160

0.6

 

Pore Volume (cc/9A)

 

 

 

Pore size

0.16

 

0.14 ~
0.12 ~

0.1 4
0.08 .
0.06 4

Pore Volume (cc/9A)

0.04 <

0.02 4

 

 

 

 

Pore size

Pore Volume(cm3IgA)
C
N

0.4

 

0.35 .

S3
on

0.25 i

Pore Volume (cm’lgA)
O
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Figure 5.11 a-d Pore size distributions calculated from isotherm fits in figure 5.10. The DFT

fit(squares) is presumed most accurate, the two stage HK method (triangles) is closer to the

DFT than the generalized l-lK method(diamonds).

161

Table 5.1 The maximum pore size (Hmax), average pore size (H) and the pore volume Vp.

Model Hmax H Vp
10.4.3 Ar DFT 11.055 21.8588 0.5512
10.4.3 Ar 2HK 10.438 17.5402 0.7298
10.4.3 Ar HK 11.725 17.4206 0.6106
10.4.3 N2 DFT 11.2895 20.3730 0.536

10.4.3 N2 2HK 11.225 18.8422 0.6287
10.4.3 N2 HK 12.7635 21.3935 0.5292
10.4 Ar DFT 9.38 15.9592 0.7397
10.4 Ar 2HK 10.552 17.4906 0.7251
10.4 Ar HK 11.5575 15.9529 0.709

10.4 N2 DFT 9.74 18.4298 0.5927
10.4 N2 2HK 10.0835 16.8587 0.7706
10.4 N2 HK 9.715 15.1630 0.877

local density profile it contains a significant improvement, through its ability to relate the
local density to the temperature and solid fluid interaction energy. The result of the low
density predicted in the smaller pores is an increase in the volume of such pores predicted by
the inversion of the adsorption integral (equation 1.1). The increased volume of the smaller
pores results in the average pore sizes being shifted to smaller values than those predicted by
DFT. Table 5.1 illustrates the pore sizes associated with either method compared DFT. As
stated earlier it is anticipated that the average pore size will be skewed towards smaller
values for both methods, while the pore volumes are anticipated to be higher, than that
predicted by DFT for small pores. In cases where DFT predicts micropores the generalized
and multiple filling models should be interchangeable, however, for systems that contain
some of the smaller mesopores the multiple filling model should behave more similarly to the
DFT than the generalized version. Pores that are larger than about 40 A begin to see a
diminishing return from the addition of the multiple filling steps. As the pore size increases
to larger mesoporous sizes the relative contribution from the monolayer rapidly decreases,
and the second filling transition becomes the dominant trait of the isotherm. In the case

where DFT predicts PSD whose peak is larger than 1 l A the multiple filling HK model is in

162

closer agreement than the generalized HK model. While the generalized model is closer to
the DFT when the pore size is anticipated to be small. For comparison a fictitious DFT
isotherm was generated for a mesoporous size distribution as shown in Figures 5.12a-b, the
two-stage HK method predicts a mesopore maximum in the PSD that is within 2.1% of the
maximum predicted by DFT, whereas the original HK predicts a maximum with an error of
10.7%. The over prediction of the monolayer filling pressure in the two-stage HK model
results in the erroneous prediction of micropores and an under prediction of small mesopores
to balance the volume of the falsely predicted micropores. The over estimation of the filling
pressure in the original HK method results in the under prediction of the pore size
culminating in an incorrect PSD in the micropore regime.

The tendency of the original HK method to over predict the size of small pores coupled with
the under prediction of the size of mesopores reduces the applicability of the model. Simply
altering the potential function overcomes these difficulties and displays the true robustness of
the HK method. Through the incorporation of multiple filling steps and altering the potential
function, the two-stage HK model can yield results that are comparable to DFT, while
offering analytical results that may be processed using a simple spreadsheet program.

5.4 Conclusions
The generalized and two-stage HK models are relatively efficient tools to measure PSD and

are recommended for use when DFT isotherms are not available. DFT does provide a more
realistic version of the isotherms for individual pore sizes, however it is time consuming to fit
the Lennard-Jones parameters and to generate a series of isotherms required consuming to fit
the Lennard-Jones parameters and to generate a series of isotherms required to invert the
adsorption integral. The two-stage HK model is advantageous in the simplicity that the

Lennard-Jones potential parameters and their respective isotherms may be generated

163

Volume Adsorbed (mmollg)

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Relative Pressure (PIPo)

Figure 5.12 a and b. Fictitious isotherm fit and comparison of the PSD predictions. Figure 3
the DFT (squares) isotherm corresponds to the selected DFT PSD in figure b. The two-stage
HK method(triangles) is in close agreement in the mesoporous regime. The original HK
method (diamonds) is in closer agreement with the isotherm but worse agreement with the

PSD.

 

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0.01

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164

Pore Volume (cm3/gA)

 

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Pore Size (A)

in minutes with a spreadsheet program as opposed to the days required for the DFT
counterparts. The two-stage HK method is superior to the generalized HK version for pores
whose size is between 12.4 and 50 A. It is important to note that there has not been an
efficient method suggested for fitting the Lennard-Jones potential parameters to any
experimental results for the single-stage HK method, whereas a simple method is presented
for the two-stage model. The effects of the wrong potential parameters can propagate and
result in inaccurate PSDs, therefore the two-stage model is suggested despite the

characteristic flaw that occurs near the maximum attainable monolayer filling pressure.

165

5.5 References

l. Storck, S., H. Bretinger, and W. Maier, “Characterization of micro— and mesoporous
solids by physisorption methods and pore-size analysis.” Applied Catalysis A., 174, 137
(1998).

N

. Maddox M. W., S. L. Sowers, and K.E. Gubbins. “Molecular Simulation of Binary
Mixture Adsorption in Buckytubes and MCM-41.” Adsorption, 2, 23, (1996).

Lo)

. Rege, S. U. and R. T. Yang. “Corrected Horvath Kawazoe Equations for Pore-size
Distribution.” AICHE J. 46, 734 (2000).

A

. Lastoskie C., K. E. Gubbins, and N. Quirke. “Poresize Heterogeneity and the Carbon Slit
Pore: A Density Functional Theory Model.” Langmuir. 9, 2693 (1993).

Lil

. Dombrowski R. J ., D. R. Hyduke, and C. M. Lastoskie. “Pore size Analysis of Activated
Carbons from Argon and Nitrogen Porosimetry Using Density Functional Theory.”
Langmuir, 16, 5041, (2000).

ON

. Steele, W. A. “The Physical Interaction of Gases With Crystalline Solids.” Surface
Science. 36, 317, (1973)

7. Horvath, G., K. Kawazoe. “Method For The Calculation Of Effective Pore Size
Distribution In Molecular Sieve Carbon.” Journal of Chemical Engineering Japan. 16,
470, (1983).

00

. Satio A., H. C. Foley. “ Curvature and Parametric Sensitivity in Models for Adsorption in
Micropores.” AICHE Journal. 37, 429, (1991).

\D

. Cheng, L. S., and R. T. Yang. “Improved Horvath-Kawazoe Equations Including Spherical
Pore Models For Calculating Micropore Size Distribution.” Chemical Engineering
Science. 49, 2599, (1994).

10. Gelb, L. D. and K. E. Gubbins. “Characterization of Controlled Pore Glasses: Molecular
Simulations of Adsorption.” Studies in Surface Science and Catalysis. 128, 61, (2000).

11. Maddox, M. W. and K. E. Gubbins. “Molecular Simulation of Fluid Adsorption in
Buckytubes." Langmuir. 11, 3988, (1995).

12. Lastoskie, C. M., N. Quirke, and K. E. Gubbins. “Structure of Porous Adsorbents:
Analysis Using Density Functional Theory and Molecular Simulation.” Studies in
Surface Science and Catalysis, 104, 745, (1997).

1.3. Lastoskie, C., K.E. Gubbins, and N. Quirke. “Pore Size Distribution Analysis of
Microporous Carbons: A Density Functional Theory Approach.” The Journal of Physical
Chemistry. 97, 4786, ( 1993).

14. Gregg, S. J ., S. W. Sing. Adsorption, Surface Area and Porosity. New York : Academic
Press,(ch 2. )1982.

166

15. Dombrowski R. J ., D. R. Hyduke, and C. M. Lastoskie. “Horvath-Kawazoe Method
Revisited.” Journal of Colloid and Surface Chemistry A. Submitted 2000.

16. Nilson, R. H., Giffiths, S. K. “condensation pressures in small pores: An analytical model
based on density functional theory.” Journal of Chemical Physics, 111, 4281, (1999).

17. M. J aroniec, K. P. Gadkaree, J. Choma, Relation between adsorption potential
distribution and pore volume distribution for microporous carbons, Colloids and Surfaces
A: 118 (1996) 203-210.

18. Muller A. Proc. Roy. Soc. London A154 624 (1936).

19. Sadus R. J. Molecular simulation of fluids : theory, algorithms, and object-orientation
New York : Elsevier, (Ch. 4) 1999.

20. de Boer, J .H.; Linsen, B.G.; Osinga, T.J. J. Catal. 1965, 4, 643.

21. Lastoskie C. M. “A Modified Horvath-Kawazoe Method For Micropore Size Analysis.”
Studies in Surface Science and Catalysis, 128, 475, (2000).

167

Chapter 6.

Conclusions and Future Directions

6.1 Summary of Results
Several amorphous porous carbons were analyzed using argon and nitrogen as probe
gases at 77K, the pore size distributions (PSD) were found to agree well when analyzed
with DFT model slit pores. The DFT model is designed to predict the equilibrium
between a bulk gas and a liquid like adsorbed phase. The triple point of argon is 83.8K
therefore the dense adsorbed phase must be either super-cooled liquid or a solid. In this
work it was found that excellent agreement between the PSD predicted by the two probes
was attained when the potential parameters of argon were fit as a super-cooled liquid in a
slit pore at 77K. The specific pore volume, mean pore width and the mode pore size on
average were within 8% of one another for the two probes. The DFT method presumes
that the probe gas is a monoatomic gas with no permanent quadrupole moment, and the
surface is defect free and chemically homogeneous. These assumptions are known to be
simplifications of the actual system. However, the argon probe more closely fits these
criteria than nitrogen, which has a quadrupole moment, is diatomic and known to interact
more strongly with surface defects and functional groups. Therefore it is recommended
that argon be used as a probe gas in lieu of nitrogen when surface functional groups are
known to be present.

Characterization of microporous materials requires that the method of generating the
model isotherms accounts for the enhanced adsorption due to the superposition of the
potential of opposing pore walls. Therefore the correct potential between the adsorbent

and adsorbate is essential for determining the proper pore size distribution. MS, DFT and

168

the HK methods all use the Lennard-Jones potential, whose parameters may be
determined by either the physical properties of the system or via fitting to the
experimental conditions. Potential parameters found by fitting have been proven to be
more accurate than those obtained from the bulk properties. The current state of the art
fitting methods require the use of a nonporous reference adsorbent that is chemically
similar to the system being probed. Through the use of the adsorption integral equation
1.1, a nonporous surface is considered to be a Dirac delta function PSD, and the
monolayer filling pressure can be matched to a an isotherm of a nonporous material. The
potential parameter may also be determined by fitting a known distribution of micropores
to an experimental isotherm. This method has the advantage of increased sensitivity to
potential due to the superposition of the wall potential. Additionally the micropores tend
to adsorb at pressures lower than flat surfaces, which allows one to determine the
potential when the monolayer filling pressure is to high to measure without the use of
special high pressure equipment. In addition to the fore mentioned advantages SWNTs
have a high surface area therefore tend to yield accurate results with a small sample size.
One of the limitations faced when studying nanotubes with small diameters is the rate of
diffusion into the tubes, due to their small size attaining an equilibrium isotherm may
require protracted times to obtain and isotherm. Therefore it becomes necessary to either
fracture the tubes or shorten their lengths to decrease the time required to diffuse through
the tubes. An alternate method to shorten the diffusion time would be to raise the
temperature at which the experiment is conducted, unfortunately this requires the

construction of a cryostat to maintain the temperature within a strict tolerance. As shown

169

in Chapter 3 despite the diffusion limitations the SWNTs show promise as an alternate
reference adsorbent.
Determining the proper potential parameters is essential for determining the PSD of
micro and mesoporous carbons. Although the DFT and MS are exact for idealized
systems, they are iterative techniques that are computationally very expensive. The HK
method may be modified to yield PSD results that are comparable to DFT while requiring
less than one hundredth of the time. When the potential function of the HK method is
altered to the 10-4-3 potential or the 10-4 potential with fitted parameters the filling-
pressures of the DFT method and HK method coincide with one another for the
unweighted HK method. It may be demonstrated that the PSD predicted by the HK
method with improved potential function yields PSD modes that are within 15% of the
values predicted by DFT for microporous carbons.
A novel modification to the HK method was also proposed to account for the multiple
filling events that are shown to occur in DFT and MS isotherms. Through the addition of
a monolayer filling pressure the two-stage HK isotherm more closely represents the
mesopores in a carbon when compared to the DFT predicted PSD. It has been
demonstrated that the two-stage HK method can predict the mesopore mode with an error
of 2% compared to the DFT mode while the original HK method predicts a mode that is
in 12% error. The addition of a monolayer filling pressure enables one to fit the Lennard-
Jones potential parameters when a nonporous reference isotherm is provided.
6.2 Topics for Future Investigation

The techniques used in this dissertation were limited by several constraints that may

be investigated by future researchers. The models used to determine the PSD were

170

limited to simplistic representations of adsorbent adsorbate systems, many common
adsorbents contain some degree of chemical as well as structural heterogeneity. Also
further development of the factors that influence the equilibrium sorption isotherms
require investigation. Some relevant extensions of this work include:

6.2.1 Further Improvements to the HK model

The HK model simplistically relates the free energy of adsorption to the heat of
adsorption, while neglecting the effects of entropy. It is known that the entropy of a
system typically decreases with the order and density of a system. DFT and MS have
shown that the adsorbed fluid is typically ordered and on average 1000 times as dense as
the bulk gas. Therefore although the magnitude of the entropic effects are less than those
of the enthalpic effects they are none the less important. The ordering of the adsorbed
phase would indicate that the proper average heat of adsorption in a pore should be
weighted with respect to density, however when entropic effects and adsorbate-adsorbate
interactions are neglected, the density-weighted HK performs poorly. As demonstrated
in Chapter 4 the density profile within a pore can be represented by a series of Gaussian
distributions. Integration of the area of these distributions would correctly identify the
density for a pore of any size, which would be an improvement to the one size fits all
density suggested, which would reduce the pore volume that is attributable to micropores.
In addition to the improper saturation density selection in micropores the density has
been shown to increase after the filling event. The addition of increased adsorption after
filling would ameliorate and in some cases remove the discontinuous fits of the two-stage
HK model near the monolayer filling pressure. The current work investigated the HK

method and improved models for systems that could be modeled with slit pores, however

17]

there may be adsorbents that are best modeled with cylindrical or spherical pores. It is
recommended that the two-stage method be expanded to alternate pore geometries.
Therefore, further investigation topics include: the study of entropic effects, the effect of
adsorbate-adsorbate interactions, seeking the correct micropore density, study of
compression of adsorbed layers, and study of other pore geometries.

6.2.2 Investigation of Adsorption at Alternate Temperatures

Sorption isotherms must be measured at a single temperature. Current methods to keep
the temperature static involve the use of a temperature bath where a material is at its
normal boiling point or sublimation point. The use of a phase change cryostat constrains
the temperatures that may be studied to the few readily available cryogenic fluids.
However, it has been shown that for very small pores diffusion may require protracted
times to reach equilibrium. To determine if an isotherm is in apparent or true equilibrium
the temperature may be increased to determine the effect. When adsorption is exothermic
the amount adsorbed should decrease and the filling pressure should increase with
increasing temperature. The limitations of studying SWNTs at liquid nitrogen and liquid
argon temperatures did not allow for sufficient temperature change to determine if the
isotherms had reached equilibrium. The construction and use of a cryostat that controls
the temperature with a tolerance of i001 K is recommended for studies pertaining to the
equilibrium sorption in small pores as well as alternate adsorbents.

6.2.3 Study of Alternate Probe Gases

The porosimetry of amorphous materials can be conducted with a variety of commonly
used probes. The current work investigated the use of argon, nitrogen and to some extent

carbon dioxide. While these gases are representative of most probes they do not meet all

172

the criteria for studying all materials. The study of nearly spherical molecules best
matches the assumption of a single centered Lennard-Jones molecule used in the DFT
and HK calculations. A molecule that is nearly described by these criteria is methane,
which does not have a permanent multipole moment. With the construction of a cryostat
the study of methane is recommended at relative temperatures comparable to either
nitrogen or argon at 77K. Previous studies have shown that methane adsorption on
graphite at 77K yields results that conflict with argon and nitrogen porosimetry.

6.2.4 Study of Irregular Adsorbents

As discussed in the introductory chapter molecular simulations have progressed to the
point where it is now becoming possible to conduct simulations of irregular porous
materials. It has been demonstrated that soil organic matter is composed of
macromolecules that have characteristics that are similar to polymers. Although,
simulations of pure polymeric systems of larger than forty carbon units have been unable
to reach equilibrium with current methods, recent advances in computing technology and
combinations of displacement moves have vastly improved the state of the art. Sorption
to soil organic matter continues to be one of the principle mechanisms of contaminant
retention by soils, therefore it makes an interesting and relevant topic of study. Quench
time models have shown that quasi-equilibrium states are suitable for adsorptive studies
and are capable of producing realistic irregular micropores. Current attempts to model
the behavior of an adsorbate such as carbon tetrachloride in a dense polymer model of
soil have met with limited success. It is recommended that hybrid Monte Carlo

simulations be attempted where the polymer adsorbent is not treated as a fixed lattice.

173

The study of such a system could incorporate several forms of heterogeneity such as the

structural variances due to irregular pores as well as chemical heterogeneity.

174

 

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