ADSORPTION OF SUPERCRITICAL CARBON DIOXIDE ON MICROPOROUS ADSORBENTS: EXPERIMENT AND SIMULATION
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
by Weihong Gao, M.S.
The Ohio State University 2005
Dissertation Committee Approved by Dr. David L. Tomasko, Adviser Dr. Isamu Kusaka ------Dr. James F. Rathman Adviser Graduate Program in Chemical Engineering
ABSTRACT
Supercritical carbon dioxide is an efficient solvent for adsorption separations because it can potentially be used as both the carrier solvent for adsorption and the desorbent for regeneration. Recent results have demonstrated an anomalous peak or
“hump” in the adsorption isotherm near the bulk critical point when adsorption isotherm is plotted as a function of bulk density. This work presents new data for adsorption and desorption of carbon dioxide on NaY zeolite over a wide range of pressures (vaccum-
2800psia) at temperatures near the critical point of carbon dioxide (32.0 to 50.0°C). The results indicate a strong affinity for CO2 as well as a significant “hump” near the critical
point. The lattice model previously developed by Aranovich and Donohue is applied to
correlate adsorption isotherms. The model successfully predicted the adsorption isotherms at the whole pressure range but failed to predict the adsorption “hump” near the
critical point with physically meaningful parameters. To investigate this behavior in more
detail, molecular simulation is executed to explore adsorption of CO2 on activated carbon
and Na Y zeolite at 32.0°C. We checked the effect of pore width on the adsorption, and
compared simulation with experiment data. The excess adsorption by simulation is larger
than experiment data, and simulation did not catch adsorption “hump” near the critical
point.
ii
This dissertation is dedicated to my parents.
iii
ACKNOWLEDGMENTS
I wish to thank my adviser, Dr. Tomasko, for his patience and guidance throughout the past four years. His enthusiasm, encouragement and good humor made earning this degree an enjoyable experience.
I would like to thank Dr. Kusaka for his instruction of molecular simulation.
I am grateful to Derrick Butler for his contribution to modeling part of this work.
I would like to thank Dr. Xueqing Wang for measuring of adsorbent surface area,
Paul Matter for pretreatment of zeolite, Alissa Park for measuring of particle size distribution.
I also wish to thank all SCF group members, Hongbo Li, Xiangmin Han, Yong
Yang, Dehua Liu, Max Wingert, Kemi Ayodeji, and Zhihua Guo, for the happy life in this group.
I am indebted to my parents, my sister, my brother, and my wife. Their support and sacrifices make this achievement possible.
iv
VITA
September 1974……… Born - Zhejiang, China
June 1995 …………… B.S., Chemical Engineering, East China University of Science and Technology, Shanghai, China.
April 1998…………… M.S., Chemical Engineering, East China University of Science and Technology, Shanghai, China.
1998-2000…………… Process Engineer, Shanghai Chemical Institute, Shanghai, China
2000-2005…………… Research Assistant, The Ohio State University, Columbus, Ohio.
PUBLICATIONS
1. Gao, W.; Butler, D.; Tomasko, D. L.; “High Pressure Adsorption of CO2 on Na-Y zeolite and Model Prediction of Adsorption Isotherms.” Langmuir. (2004), 20(19), 8083-8089.
2. Zhang, X.; Zhang, Ch.; Xu, G.; Gao, W.; Wu, Y.; “An Experimental Apparatus to Mimic CO2 Removal and Optimum Concentration of MDEA Aqueous Solution.” Ind. & Eng. Chem. Res (2001), 40(3), 898-901
FIELDS OF STUDY
Major Field: Chemical Engineering
v
TABLE OF CONTENTS Page
Abstract ………………………………………………………………………………….ii
Dedication………………………………………………………………………..………iii
Acknowledgments……………..…………………………………..…………………….iv
Vita………………………………………………………...…………………….………..v
List of Tables………………………………………………..…………….…….…..….viii
List of Figures…………………………………………………………..……….…..…..ix
Chapters:
1. Introduction………………………….…………………..…….…..……………..1
1.1 Motivation……………………………………………………………………..1
1.2 Research outline……………………………………………………………….8
1.3 Thesis outline………………………………………………………………….9
2. High pressure adsorption of CO2 on NaY zeolite and model prediction of
adsorption isotherms……………………..…………………………………….10
2.1 Introduction……………………….……………………………….…...…….10 vi 2.2 Experimental………………………..…………………...…………….……..14
2.2.1 Materials……………..……………………..……….…..…….……..14
2.2.2 Pretreatment and loading……………..………...………..…………..15
2.2.3 Gravimetric adsorption apparatus……………………….…………..16
2.3 Results and discussion…………………………...………………………..…17
2.3.1 Buoyancy correction………………….………………………...……17
2.3.2 Adsorption isotherms………………………….…….…….…………19
2.4 Thermodynamic analysis and modeling……………………………………..22
2.4.1 Heat of adsorption……………………………………...…………….23
2.4.2 Ono-Kondo lattice model…………………..……………….………..27
2.5 Conclusion…….……………………………………...……………….……..39
3. Molecular simulation of supercritical CO2 on activated carbon and NaY
zeolite……………………………………………..………...………..…….…….40
3.1 Introduction………………………………………….…...…..……..………..40
3.2 Algorithm and modeling……………………………..……………....………43
3.3 Results and discussion…………………………………..………..….………47
3.4 Conclusion………………………………………………..………….………76
vii 4. Summary and recommendations……………………………………..………..77
4.1 Summary……………………………………………….…………..…….…..77
4.2 Recommendations……………………………………….………..….………78
Appendix A……………………………………………………………..……………….81
Appendix B…………………………...…………………………...………….…………98
References……………………………………………………………..………..…….107
viii
LIST OF TABLES
Table Page
2.1 NaY Zeolite Properties………………..………..….…….………………...…….14
2.2 Adsorbed Phase Properties Calculated from High Pressure Adsorption Data.….27
2.3 Result of fitting all 4 Parameters in the Adsorption Model……..…….…………30
2.4 Results of fitting only εA and am in the Adsorption Model (2.5 parameters)….…32
3.1 Potential parameters for NaY framework…………………..………………..…..47
ix
LIST OF FIGURES
Figure Page
2.1 Zeolite Pore Size Distribution……………………………………………………15
2.2. Calculated Volume Difference from Helium Adsorption as a Function of
Pressure…………………………………………………………………………..18
2.3. Adsorption-Desorption Isotherms of CO2 on NaY Zeolite, CO2 Density (line) is
Plotted on the Right Axis………………………………………………………...20
2.4 Excess Adsorption as a Function of CO2 Density showing Anomalous “Hump”.
(Open symbols represent data from Hocker et al)…………………………...…..21
2.5 Isosteric Heat of Adsorption Calculated directly from Adsorption Data……..…25
2.6 Total adsorption as a Function of CO2 Density (Open symbols represent data from
Hocker et al)…………………………………………………………...…...……26
2.7 Bimodal micropore model of adsorbent…………………………………...…….31
2.8 Comparison of the lattice model (2 and 3 layers in S1 and S2 respectively) with
experimental data. (Globally fitted)…………………………….….…………….34
x 2.9 Comparison of the lattice model (2 and7 layers in S1 and S2 respectively) with
experimental data. (Globally fitted)…………………………………….……….35
2.10 Comparison of the lattice model (2% external surface) with experimental data.
(Globally fitted)……………………………………………………....………….37
2.11 Comparison of the lattice model (10% external surface) with experimental data.
(Globally fitted)………………………………………………………….………38
3.1 Framework of NaY zeolite (Jaramillo, Grey et al. 2001)…………….………….46
3.2 Gas-liquid coexistence curve…………………………………………………….49
3.3 Density of CO2 fluid at 32.0ºC ……………………………………………...…..50
3.4 Compressibility of CO2 fluid at 32.0ºC ………………………...……………….51
3.5 Pore size distribution of activated carbon……………………………..…………53
3.6 Adsorption simulation on carbon wall with different size (P=933.6 psia,
H=14.73σ, t=32.0 ºC)………………………………………………..…….…….54
3.7 Adsorbate density inside carbon slit……………………………………………..55
3.8 Excess density inside carbon slit..………………………………………………..57
3.9 Excess adsorption inside carbon slit (T*=1.0028)……………………………….59
3.10 Excess adsorption inside carbon slit (T*=1.029)…………………...……………60
3.11 Excess adsorption inside carbon slit (T*=1.093)..……………………………….61
3.12 Density probability inside carbon slit……………………………………………63
3.13 Simulation system of pore fluid……………....…………………………….……64
3.14 Compressibility of pore fluid (T*=1.0028)………………………………………65
3.15 Density of pore fluid (T*=1.0028)……………………………………………….66
xi 3.16 Compressibility of pore fluid (T*=1.093)………………………………………..68
3.17 Density of pore fluid (T*=1.093)………………………………………………...69
3.18 Excess adsorption of CO2 in carbon slit pore with width of 60σ………………...70
3.19 Excess adsorption of CO2 on activated carbon with 2% mesopores (32.0ºC)…...71
3.20 Excess adsorption of CO2 on activated carbon with 5% mesopores (32.0ºC)…...73
3.21 Excess adsorption of CO2 on activated carbon (40.0ºC)………………………...74
3.22 Excess adsorption of CO2 on NaY zeolite (32.0ºC)……………………………..75
xii
CHAPTER 1
INTRODUCTION
1.1 Motivation
When the temperature and pressure of a fluid is above its critical temperature and critical pressure, the fluid is at supercritical state. Supercritical fluids have both liquid- like densities and gas-like transport properties, and minor adjustments of pressure and temperature may bring big changes in those properties. The viscosity of supercritical fluid is as low as that of gas and the solute diffusion coefficient is usually an order of magnitude higher than that in liquid. Also, liquid-like density makes the supercritical fluid a strong solvent for some compounds. Since the density near critical point is adjustable, the solubility can be changed significantly. With those properties, supercritical fluids are widely used in research and industry (McHugh and Krukonis
1993; Sihvonen 1999).
Supercritical carbon dioxide is the most widely used supercritical fluid because it is low-cost, nontoxic, nonflammable and environmentally benign. The critical temperature of carbon dioxide is about 31.25°C; it is near ambient temperature, so supercritical carbon dioxide can be used to process thermal sensitive compound. With its special properties, supercritical carbon dioxide has received much attention as a substitute
1 for organic solvent in environment protection (Sako 1997), extraction (McHugh and
Krukonis 1993), chromatographic (Pól, 2004), separation (Iwai et al. 1994), reaction
(Jessop 1995), and polymer process (Sihvonen 1999).
Supercritical carbon dioxide is extensively used in extraction of organic
compound. Since a plant-scale set-up of extraction of caffeine from coffee bean was built
about two decades ago (McHugh and Krukonis 1993), there have been more than 100
supercritical extraction vessels (Sihvonen 1999). Most of them are used to extract natural
products such as coffee, hops, tea and tobacco. In recent years, researchers also use
supercritical carbon dioxide to extract some high-value compounds from food and fruit.
For example, supercritical-liquid extraction is applied to extracting lycopene (Pól, 2004)
from solid matrices, flavor from orange oil (Shen et al. 2002), and β-carotene from carrot
(Rosenthal 1995). Besides those agriculture productions, supercritical carbon dioxide is
also used to extract alcohol from aqueous azeotropic mixtures (Gros et al. 1998) and to
extract metal from aqueous solution (Erkey 2000). A favorable property of supercritical
extraction is that supercritical carbon dioxide can be connected on-line with
chromatographic techniques (Pól 2004).
In supercritical extraction, usually pure solute is obtained by decompressing
carbon dioxide. Decompressing and again compressing carbon dioxide increase the cost of operation, though it is still acceptable for high-value extract. For some extraction processes, such as the extraction of caffeine from coffee bean, however, decompressing carbon dioxide is not cost-benefit effective since the extract is not the objective product
In this type of processes, recycle supercritical carbon dioxide is highly recommended to decrease cost. Recently, researchers tried to separate solute from supercritical carbon 2 dioxide fluid with membrane (Afrane and Chimowitz 1996; Chiu and Tan 2001; Tan,
Lien et al. 2003). With selected membrane, Chiu et al (2001) got 100% caffeine rejected
from supercritical carbon dioxide fluid at near critical point with high flux permeation of
CO2. But they found the highest rejection rate and the highest permeation rate occur at near critical point only (Chiu and Tan 2001).
Recently, more and more research focuses on separation of enantiomer with supercritical fluid chromatography (SFC). A lot of drugs in use are enantiomers; usually only one of the enantiomers has pharmacological effect while the other one sometimes has side or poisonous effects (Ward 2000). Regulating authorities require higher purity of those pharmaceutical productions. Advanced analysis technique, mainly chromatography technique, is required to follow the new regulation. In the face of this requirement, researchers have paid attention to the separation of enantiomers with SFC. Because of the low viscosity, high density and high solubility of supercritical fluid, SFC has shorter equilibrium time and faster separation. The mobile phase fluid in SFC is generally supercritical carbon dioxide. The disadvantage of carbon dioxide is its weak polarity, so usually some additives are added to separate polar molecule (Gübitz 2001). In SFC, a packed column is used as stationary phase (Ward 2000; Yaku and Morishita 2000); several kinds of stationary phases are reported for different chiral separation (Gübitz
2001). Nozal et al (2002). separated albendazole sulfoxide enantiomers with supercritical carbon dioxide chromatography using alcohol-type modifiers. Yaku et al (2000) compared separation of dilatiazem enantiomers with SFC and HPLC (high pressure liquid chromatography). They found SFC has higher selectivity, resolution and efficiency
3 (Yaku and Morishita 2000). In addition, capillary SFC technique is used in some separations (Phinney 2000).
Another separation application is the adsorptive separation of isomers. Because
the diffusion coefficient of solutes in the SC-CO2 is higher than that in liquids, SC-CO2 may be a better solvent for adsorption separation than liquid solvents, and is especially attractive for difficult-to-separate mixtures such as isomers. The similar boiling points and size of isomers render them difficult to separate by conventional methods such as distillation, but slight differences in their steric properties and electronic structure can have large effects on their adsorption characteristics. With the proper choice of adsorbent and operating conditions, one component of the mixture is adsorbed more strongly or faster than are the other components, resulting in a concentration of the other ones in the fluid phase. Iwai et al (1994) successfully separated 2,6- and 2,7-dimethylnapthalene
(DMN) with supercritical carbon dioxide by using NaY zeolite, and found that the separation coefficient is much higher with supercritical CO2 on the properly selected
zeolite and that the highest separation efficiency is obtained at high pressure. In further
research, they found the saturation adsorption amount of 2,7-DMN is about twice as large as that of 2,6-DMN due to the difference of affinities between isomers and the adsorbent surface(Uchida et al. 1997). Their findings indicate that, by using supercritical CO2 as
solvent, we can efficiently separate isomers with a proper adsorbent that possesses
suitable surface characteristics.
In adsorption separation for nonvolatile compounds, a suitable solvent is required
to desorb the product from the adsorbent. If a separate desorption solvent is avoided, the
separation cost and the environmental hazard can be reduced. Recently, researchers have 4 demonstrated the potential of SC-CO2 as a carrier for both adsorption and desorption(Tan
and Tsay 1990; Lin and Tan 1991; Sakanishi et al. 1996; Uchida et al. 1997).
In the application of either adsorption separation or SFC, a solid phase is used.
Even in supercritical extraction, a solid phase is involved in recycle of supercritical fluid.
It is important to explore the adsorption behavior of supercritical carbon dioxide on or
inside a solid matrix. There is a vast literature on supercritical separation, but few
researchers discuss the adsorption of supercritical carbon dioxide, the carrier fluid itself.
It is important to account for the behavior of CO2 in separations research because the
competitive adsorption of CO2 may affect the selection of operation conditions(Uchida et
al. 1997). Some researchers have studied supercritical CO2 adsorption on microporous
adsorbents (Cerofolini et al. 2000; Grajek 2000; Yong et al. 2002). Most of the data are
for relatively low pressure while high-pressure isotherms typically consist of few data
points(Strubinger and Parcher 1989; Giovanni et al. 2001; Krooss et al. 2002; Hocker et
al. 2003). The lack of resolution in typical discrete measurements is a challenge for
understanding adsorption of supercritical fluids because of the existence of an anomalous
“hump” near the critical point (Findenegg 1983; Humayun and Tomasko 2000; Hocker et al. 2003).
In this project, Near-continuous carbon dioxide adsorption/desorption isotherms
on NaY zeolite are measured with a gravimetric microbalance system (Humayun and
Tomasko 2000), from vacuum to 2800psi at 32.0°C, 35.0°C, 40°C and 50°C. At 32.0°C
there is a significant “hump’ if the excess adsorption is plotted as a function of bulk
density. Other researchers also found similar phenomena in supercritical adsorption on
5 activated carbon (Humayun and Tomasko 2000), on 13x zeolite embedded on clay
(Hocker et al. 2003) and on flat surface (Findenegg 1983).
Adsorption models are widely used to predict adsorption isotherms(Aranovich and Donohue 1997); however, since most of these models contain saturation vapor pressure as a parameter, they cannot be used to describe adsorption above the critical point. A 2-D Peng-Robison EOS was developed to describe pure gas adsorptions(Zhou et al. 1994) over a wide pressure range and accurately correlates data at temperatures well above the critical temperature but only qualitatively describes isotherms near the critical point(Humayun and Tomasko 2000). Aronovich and Donohue successfully adapted the
Ono-Kondo lattice model to describe adsorption of supercritical fluids (Aranovich and
Donohue 1996; Aranovich and Donohue 1997; Donohue and Aranovich 1998). With this model, Hocker et al(2003) qualitatively predicted adsorption isotherms on zeolite/clay pellets over a wide pressure range but also the adsorption “hump” near critical point. An
Ono-Kondo lattice model(Aranovich and Donohue 1997) is used to correlate the adsorption isotherms of CO2 on zeolite crystals, and the correlation is compared with our experimental data. The model fitted the experiment data perfectly at low and high pressure but did not accurately fit the “hump” near the critical point (Gao 2004).
The third part of this research is molecular simulation. We know that experiments yield macroscopic data such as the heat of adsorption or the adsorption isotherms.
However, the behavior of molecules adsorbed in the pores of a microporous material is very difficult to study directly by experiment. So simulation is chosen to get this microscopic information. Computer simulation provides a direct path from the statistical mechanics of a system to the macroscopic properties of experimental interest. One of the 6 strengths of molecular simulation is its ability to provide guidance in optimizing the
properties of materials by taking snapshots of dynamic and equilibrium behavior.
Molecular simulation is widely used in supercritical adsorption research (Takaba et al.
1996; Nicholson 1998; Du et al. 1999; Mart and Gordillo 2002; Do et al. 2003; Turner
and Gubbins 2003; Cao et al. 2004). Some papers about molecular simulation of
adsorption of carbon dioxide on solid adsorbents (Bakaev et al. 1999; Rutherford and Do
2000; Samios et al. 2000; Zhou and Wang 2000; Zhou 2001; Qinglin et al. 2003; Cao et al. 2004; Jia and Murad 2004) are published. These papers either focus only on low-
pressure adsorption, or generally talk about high-pressure adsorption above critical
pressure. This research focuses on the simulation of adsorption of CO2 on micropore
adsorbent near the critical point. In this part, Grand Canonical Monte Carlo method is
used to simulate adsorption of supercritical carbon dioxide on activated carbon and NaY
zeolite. In order to compare the simulation with experimental data, pure CO2 fluid is also
simulated and the simulation is compared with calculation with EOS Equation of state.
One difficulty in this approach is how to get the critical point in simulation. From
experiment data, the significant adsorption “hump” occurs at 32.0ºC, which is only
0.28% above the critical point (31.25ºC). An accurate critical temperature is critical for
this project. Usually, GCMC simulation is executed to get liquid-vapor coexistence line
and then get the critical temperature, but this traditional method is somewhat time-
consuming. Here, the transition matrix Monte Carlo (TMMC)(Errington 2003) is used
simulation. TMMC simulation can be used to directly determine the liquid-vapor
coexistent point at any temperature with model system. With the coexistence data at
different temperature, the critical temperature is calculated by fitting coexistence data 7 with scaling law. Adsorptions on activated carbon pore at different size are simulated.
The size effect on adsorption is checked. Also, adsorption on activated carbon and NaY
zeolite at 32.0°C are simulated, and then the simulation is compared with experimental
data.
1.2 Research Outline
The main objective of this research can be described as
• Study the adsorption/desorption of carbon dioxide on NaY zeolite. The effect of
temperature and pressure on adsorption/desorption is checked.
• Correlate adsorption isotherms with Ono-Kondo lattice model under the whole
experiment pressure range.
• Simulate pure carbon dioxide fluid and adsorption of SFC on activated carbon and
NaY zeolite at near critical temperature. Investigate molecular behavior of CO2
inside micropore adsorbent.
In the adsorptive separation, the adsorption of CO2 plays an important role in the
separation. It is important to investigate the adsorption behavior of CO2 on solid
adsorbent, especially at near critical point. In this research, the adsorption of CO2 on NaY zeolite at near the critical point shows some special characters. The adsorption behavior near critical point is still unknown. Understanding the molecular behavior of CO2 molecules confined in micropores, especially at near critical point is helpful to predict the
8 possible competitive adsorption between CO2 and solute in future separation research.
This will lead to greater understanding for separation of difficult-to-separate mixtures.
1.3 Thesis Outline
In Chapter 1, previous related research is introduced and the motivation of this research is presented.
Chapter 2 describes experimental procedure of CO2 adsorption/desorption on
NaY zeolite. The set-up and the experimental protocol are introduced in this chapter.
Adsorption/desoprtion isotherms at 32.0°C, 35.0°C, 40.0°C and 50.0°C are presented.
Experimental pressure is from vacuum to 2800 PSI. An Ono-Kondo lattice model is used
to correlate adsorption isotherms on NaY zeolite at different temperature. Details of the
model and the selection of parameters are described. The correlation result is compared
with experiment data.
Molecular simulation of adsorption on activated carbon and NaY zeolite at
32.0°C is presented in Chapter 3. Algorithms and parameters of Transition Matrix Monte
Carlo method and Grand Canonical Monte Carlo method are described. Effects of slit
pore size and temperature on adsorption are checked. Simulation results are compared
with experiment data.
Chapter 4 describes the conclusion obtained from this research and some
recommendations for future work.
9
CHAPTER 2
HIGH PRESSURE ADSORPTION OF CO2 ON NaY ZEOLITE AND
MODEL PREDICTION OF ADSORPTION ISOTHERMS
2.1 Introduction
Supercritical fluids (SCFs), specifically supercritical carbon dioxide (SC-CO2), have received much attention as environmentally benign solvents. SCFs have liquid-like densities and gas-like transport properties and minor adjustments of pressure and temperature can bring large changes in those properties. Because the diffusion coefficient of solutes in the SC-CO2 is higher than that in liquids, SCFs may be better solvents for adsorption separation than liquid solvents, and are especially attractive for commercial adsorptive separation of complex, difficult to separate mixtures such as isomers. The similar boiling points and size of isomers render them difficult to separate by conventional methods such as distillation, but small differences in their steric properties and electronic structure can have large effects on their adsorption characteristics. With the proper choice of adsorbent and operating conditions, one component of the mixture adsorbs more strongly or faster resulting in a concentration of the other in the fluid phase.
Iwai et al(1994) separated 2,6- and 2,7-dimethylnapthalene (DMN) with supercritical 10 carbon dioxide by using NaY zeolite. They successfully separated mixtures and
compared the separation coefficient with that in liquid octane separation. They found the
separation coefficient is much higher with supercritical CO2 on properly selected zeolite and that the highest separation efficiency is obtained at high pressure. In further research, they found the saturation adsorption amount of 2,7-DMN is about twice as large as that of
2,6-DMN owing to the difference of affinities between isomers and the adsorbent surface(Uchida et al. 1997). This indicates that we can efficiently separate isomers with a proper adsorbent having suitable surface characteristics using supercritical CO2 as solvent.
In adsorption separation for nonvolatile compounds, a suitable solvent is required to desorb the product from the adsorbent. If a separate desorption solvent can be avoided, separation cost and environmental hazards can be reduced. Recently, researchers have demonstrated the potential of SC-CO2 as a carrier for both adsorption and desorption(Tan and Tsay 1990; Lin and Tan 1991; Sakanishi et al. 1996; Uchida et al.
1997). For example, Tan and Tsay found that CO2 desorbed all the xylene from silicalite
pellets (silicalite in a 20% aluminum oxide binder)(Tan and Tsay 1990).
There is a vast literature on supercritical separation, but very few papers discuss
the adsorption of supercritical carbon dioxide, the carrier fluid itself. It is important to
account for the behavior of CO2 in separations research because the competitive
adsorption of CO2 may affect the selection of operation conditions(Uchida et al. 1997).
Researchers have studied CO2 adsorption on microporous adsorbents(Cerofolini
et al. 2000; Grajek 2000; Yong et al. 2002) but most of the data is for relatively low
pressure while the few high-pressure isotherms typically consist of widely spaced data 11 points(Strubinger and Parcher 1989; Chen et al. 1997; Giovanni et al. 2001; Krooss et al.
2002; Hocker et al. 2003). The lack of resolution in such discrete measurements is a
challenge for understanding adsorption of supercritical fluids because of the existence of
an anomalous “hump” near the critical point(Thommes et al. 1994; Humayun and
Tomasko 2000; Hocker et al. 2003). Humayun et al. have reported near-continuous
carbon dioxide adsorption /desorption isotherms on activated carbon with a gravimetric
microbalance system(Humayun and Tomasko 2000), and this data was recently verified
by another group using a different technique (Robinson, 2003).
This chapter presents new data for the adsorption of CO2 on NaY zeolite from
vacuum to 20 MPa at temperatures near the critical temperature of CO2. Zeolites are a
class of microporous materials, which possess pores and channels of regular dimensions
within the crystalline lattice. These materials have periodic cavities and channels
throughout the structure with sizes on the order of a few nanometers. Because the pore
sizes are of the same magnitude as many industrially relevant molecules, zeolites possess unique sorption, catalytic and transport properties. Further, their well-defined structure enables the application of precise characterization and modeling techniques. Zeolites are therefore the subject of a vast body of adsorption experiment and molecular simulation research(Fitzgerald et al. 2003).
Adsorption models are widely used to predict adsorption isotherms(Aranovich and Donohue 1997) but since most of theses models contain saturation vapor pressure as a parameter, they cannot be used to describe adsorption above the critical point.
Supercritical adsorption isotherms do not fit easily into the IUPAC classification(Donohue and Aranovich 1998), and represent a challenge for quantitative 12 modeling efforts. A 2-D Peng-Robison EOS was developed to describe pure gas adsorption(Zhou et al. 1994) over a wide pressure range and accurately correlates data at temperatures well above the critical temperature but only qualitatively describes isotherms near the critical point(Humayun and Tomasko 2000). Chen et al(Chen et al.
1997) applied the Simplified Local Density model developed by Lira(Subramanian et al.
1995) to describe their adsorption data on activated carbon with moderate success.
Aronovich and Donohue successfully adapted the Ono-Kondo lattice model to describe adsorption of supercritical fluids(Aranovich and Donohue 1996; Aranovich and Donohue
1997; Donohue and Aranovich 1998). With this model, Hocker et al(2003) qualitatively predicted adsorption on zeolite/clay pellets over a wide pressure range and the adsorption
“hump” near the critical point. In this chapter, the Ono-Kondo lattice model(Aranovich and Donohue 1997) is used to describe the adsorption isotherms of CO2 on zeolite crystals, and compare them with experimental data. While the model, with four parameters, can quantitatively describe the adsorption behavior over the entire pressure range, it only captures the enhanced adsorption near the critical point with physically unrealistic values of some parameters. It is concluded that the lattice model does not accurately correlate near critical adsorption on microporous adsorbents and details will be discussed in the modeling section.
13 2.2 Experimental
2.2.1 Materials
The NaY zeolite was obtained from Zeolyst International (CBV 100). The zeolite
properties as provided by the manufacturer are listed in Table 2.1, and the mean particle
size was 1.8 µm. The pore size distribution was determined via nitrogen adsorption with
the MP method(Brunauer 1967) and is presented in Figure 2.1. The powder was
pretreated as described in the next section. Grade 4 carbon dioxide (4.0 Instrument,
PRAXAIR) was used for adsorption; Helium (Grade 4.5, PRAXAIR) was used for
buoyancy measurements.
SiO2/Al203 Nominal Na2O Weight Unit Cell Size Surface Area
Mole Ratio Cation Form % A m2/g
5.1 Sodium 13 24.65 900
Table 2.1 NaY Zeolite Properties
14 1200
1000
/g) 800 2
600
400 Surface area
Surface Area (m Area Surface Cumulative surface area
200
0 0 5 10 15 20 25 30
Pore Diameter (A)
Figure 2.1. Zeolite Pore Size Distribution
2.2.2 Pretreatment and Loading
To prevent contamination of the zeolite, a careful pretreatment and loading protocol was followed. The zeolite was first soaked in a concentrated NaCl aqueous solution for 24 hours to exchange out all ionic impurities (e.g. magnesium). It was then washed in demineralized water and dried in a furnace. To eliminate recontamination during measurement and loading of the sample onto the balance, a known weight of the 15 dried Zeolite was packed into the sample holder in between 2 layers of quartz wool with a
wire mesh on top to keep the sample in the holder. The sample holder was then weighed
and placed in a furnace and heated to 500ºC under oxygen flow for 5 hours. The sample
holder was then quickly transferred to a pre-weighed glass vial and sealed. The vial was then weighed to determine the weight of the sample holder and the amount of “clean” zeolite.
2.2.3 Gravimetric Adsorption Apparatus
The adsorption apparatus, experimental procedure and data rectification and analysis were as described previously(Jwayyed et al. 1997; Humayun and Tomasko
2000), with slight revisions. An OXY-Trap (AllTech) was added to further reduce oxygen impurities in the CO2 gas. Also a low-pressure digital gauge (Sensotec, ±0.05%)
was added for accurate measurements of low-pressure adsorption (vacuum~20 psi).
Temperature is measured with a thermistor (DP25-TH, OMEGA).
The balance was zeroed beforehand by placing weights equivalent to the sample
holder on the sample side and adding/removing glass beads from the tare side. A glove
bag was then taped to the front of the oven, air was evacuated from the oven with a
vacuum pump and an inert helium atmosphere was maintained during the transfer of the
sample holder from the glass vial to the balance arm. Once the balance was sealed, the
glove bag was removed and the oven was allowed to equilibrate to the experimental
temperature.
16 2.3 Results and Discussion
2.3.1 Buoyancy correction
In gravimetric measurement of adsorption, estimation of the volume of the
adsorbent is one potential source of error. Usually, the volume is obtained from a blank
run using helium since helium adsorption is considered negligible(Sing et al. 1985; Sircar
and Myers 1985). However, recent research(Malbrunot et al. 1997) shows that if the
helium adsorption is ignored, the volume of the adsorbent will be underestimated at low
temperature. Malbrunot et al. measured the helium adsorption isotherm showing the
adsorption of helium is significant at low temperature(Malbrunot et al. 1997). They
recommended the blank run be performed at the regeneration temperature of the
adsorbent. Unfortunately, the upper limit (~100 °C) of the oven used in experiment is
much lower than the pretreatment (regeneration) temperature (500 °C), so the following
method is developed to estimate the adsorbent volume.
Because a two-arm torsional balance, rather than a magnetic suspension balance is
used, only the volume difference between the sample and tare weight (∆V = Vsamp - Vtare)
is required for the buoyancy correction. This method reduces substantially the magnitude
of the buoyancy correction. Excess helium adsorptions at 35°C and pressures of 3.447
MPa (500 psia), 6.895 MPa (1000 psia), 10.34 MPa (1500 psia), 13.79 MPa (2000 psia) and 17.24 MPa (2500 psia) are measured separately. The correction for buoyancy due to the volume difference is shown in Equation 1 where ΓH is the true excess adsorption of
helium, Γm is the measured weight of the sample, and ρH is the density of helium(Angus
and Reuck 1977) at the pressure and temperature of the measurement.
17 0.6
0.58
0.56
0.54 Delt V (ml)
0.52 Extrapolated Value Experimental Value 0.5 0 5 10 15 20 Pressure (MPa)
Figure 2.2. Calculated Volume Difference from Helium Adsorption as a Function of
Pressure.
Γ=Γ+∆H mHV ρ [1]
Then,
Γ−Γ ∆=V H m [2] ρH
If ΓH is assumed zero, then ∆V can be directly obtained from Γm but as shown in
Figure 2.2 the data points do not give a constant value of ∆V at all pressures. Therefore,
18 it is not possible to assume helium is not adsorbed and doing so will lead to increased errors at high pressures. This is consistent with previous studies(Malbrunot et al. 1997) showing that helium adsorption increases with increasing pressure, but that it is low and changes smoothly at low pressure. However, the excess helium adsorption does go to zero in the limit of zero pressure and so:
− Γ ∆V = lim m [3] P→0 ρ H
Therefore, by extrapolating our measured data to zero pressure, the true ∆V at zero pressure should be obtained. This extrapolation is also shown in Figure 2.2.
The measured weight can then be corrected for buoyancy effects at any temperature and pressure (Equation 4).
M = M exp + ∆VρCO2 [4]
Where ρCO2 is determined by a modified Benedict-Webb-Rubin (MBWR)
EOS (Ely et al. 1989; Moriyoshi et al. 1993; Bush 1997). The accuracy of our thermistor is ±0.2°C while the measurement error of the digital pressure gauge is ±0.36%. As a result, the maximum error in the calculated density is ≤1.5%. Since the ∆V is a small value, it is unnecessary to measure the density in-situ to obtain highly accurate data.
2.3.2 Adsorption Isotherms
Both adsorption and desorption isotherms were measured to ensure attainment of equilibrium. Isotherms at 32, 35, 40 and 50°C as a function of pressure are shown in
19 Figure 2.3. Surface excess and bulk fluid densities are shown versus pressure at each temperature.
Figure 2.3 Adsorption-Desorption Isotherms of CO2 on NaY Zeolite, CO2 Density
(line) is Plotted on the Right Axis.
20
Figure 2.4 Excess Adsorption as a Function of CO2 Density showing Anomalous
“Hump” (Open symbols represent data from Hocker et al)
The isotherms behave qualitatively in a similar manner to adsorption on
GAC(Humayun and Tomasko 2000) and other data on zeolites(Hocker et al. 2003) with
the excess increasing sharply at low pressure, a broad maximum at medium pressures and
then dropping as the pressure is raised beyond Pc where the bulk fluid density increases 21 sharply. The zeolite has a stronger affinity for CO2 as adsorption increases more sharply at low pressure compared to carbon. The maximum is broader and flatter indicating that the pores are saturated more quickly, which is consistent with narrow, mostly cylindrical pores.
2.4 Thermodynamic Analysis and Modeling
Figure 2.4 shows the excess adsorption isotherms plotted as a function of density along with data on 13X Zeolite(Hocker et al. 2003). Points with error bars are near- critical adsorption data at 32 and 35°C. The excess adsorption on 13X is significantly larger than that on NaY indicating a higher affinity for the 13X that may be due to the clay matrix. At high pressures the excess adsorption decreases linearly with density. This indicates that both the adsorbed phase volume and the adsorbed phase density become constant at high pressure. From the slope and intercept of the linear region we can obtain the volume and density of the adsorbed phase, and these are listed in Table 2.2.
The volume of the adsorbed phase calculated from the linear parts of the excess isotherm is roughly constant while the density of the adsorbed phase changes slightly with temperature. The density of the adsorbed phase from our data is much higher than the liquid CO2 density at 20 MPa and that calculated from the activated carbon system(Humayun and Tomasko 2000). This is consistent with the stronger adsorption observed in the zeolite cages that are of the order of magnitude of the molecular size of
CO2. The density of the adsorbed phase on 13X is even higher than that in NaY and approaches the close-packed density based on reasonable estimates of the diameter of a
22 CO2 molecule. However, analysis based on single values of adsorbed phase density and volume may not be capture the true nature of the adsorbed phase since the binder in 13X contributes meso- and macro-pores to the adsorbent. Nevertheless, agreement in adsorbed phase volume between the data sets in the high-pressure range is quite good.
At low pressure the slope of adsorption isotherms is much sharper than in the
GAC system because the interaction between CO2 and zeolite is stronger. In Figure 2.4,
the excess adsorption “hump” is readily apparent on both zeolites and the maximum
occurs below the critical density of bulk CO2. This is different than observed previously
on activated carbon where the maximum of the “hump” occurs above the critical
density(Humayun and Tomasko 2000). The “hump” on zeolite is also more conspicuous
than in the activated carbon system(Humayun and Tomasko 2000), meaning the total
adsorption on zeolite has a larger maximum. There are two possible reasons for the
higher maximum. One interpretation is that the adsorbed phase is highly compressed to
solid-like density; from Table 2.2, the density of adsorbed phase is much higher than
liquid density. The second possible reason is that the volume of the adsorbed phase is
increased, as reported in literature(Sun et al. 1998). Perhaps the zeolite undergoes a
sorbate-induced symmetry transformation. This points to the need for further
investigation to clearly describe the molecular behavior inside adsorbent micropores.
2.4.1 Heat of Adsorption
The isosteric heat of adsorption (qst) can be calculated from total adsorption
isotherms as(Valenzuela and Myers 1989):
23 2 ∂ ln P ∂ ln P [5] q st = RT = −R ∂T n ∂()1 / T n
Here, the subscript n represents constant total adsorption. Because of the high resolution of our data, it is possible to evaluate this derivative numerically as:
ln(P1 / P)2 qst = R [6] 1/T 1/T 2 − 1 n
Isosteric heat is calculated with 32°C and 50°C isotherms and compared NaY and
13X zeolite(Hyun and Danner 1982) in Figure 2.5. At the maximum of the adsorption isotherm, the isosteric heat exhibits an infinite asymptote, after the maximum, the heat has a negative value(Salem et al. 1998). The negative value has no physical meaning, so here only low-pressure isosteric heat is calculated. The isosteric heat increases slowly according to total loading, but increases abruptly near maximum adsorption. This tendency is consistent with the literature(Hyun and Danner 1982; Salem et al. 1998). The isosteric heat of adsorption is zero when the pressure is high enough as the total adsorption isotherms converge. Figure 2.6 demonstrates this convergence. The total adsorption has been calculated using the volume and density of adsorbed phase from
Table 2.2.
24
60 13X Zeolite 50 NaY Zeolite 40
30
Q (KJ/mole) 20
10
0 012345 Total Loading (mmole/g)
Figure 2.5 Isosteric Heat of Adsorption Calculated directly from Adsorption Data
25
Figure 2.6 Total adsorption as a Function of CO2 Density (Open symbols represent
data from Hocker et al)
26 Temperature (°C) Volume (cm3/g) Density (mmole/ cm3) This work 32.0 0.175 28.84 35.0 0.171 28.81 40.0 0.167 28.41 50.0 0.168 27.80 Data of Hocker et al (Hocker al. 2003) 31.73 0.2024 36.32 35.03 0.1892 38.24 49.23 0.176 38.33 82.69 0.154 39.86 116.75 0.1496 38.24 Critical density (mole/L) 10.63 Saturated liquid density at 30.5°C (mole/L) 12.78 Liquid density at 20 MPa and 30.5°C (mole/L) 20.19 Density of dry ice (14.7 psia, -78.5ºC) (mole/L) 34.16
Density of close-packed spheres ( 2 σ 3 , σ=3.72) (mole/L) 40.82
Table 2.2 Adsorbed Phase Properties Calculated from High Pressure Adsorption Data
2.4.2 Ono-Kondo Lattice Model
Traditional adsorption models and their modifications cannot be used for adsorption at supercritical conditions(Aranovich and Donohue 1997). Here the lattice
27 model developed by Aranovich & Donohue based on the Ono-Kondo statistical theory is used to calculate multilayer adsorption of supercritical CO2 on zeolite. The Ono-Kondo
Equations of thermodynamic equilibrium for a bulk lattice fluid in equilibrium with the ith layer on the surface of an adsorbent are(Aranovich and Donohue 1996):
ln[]xi ()()1− xb / xb 1− xi + z0 ()xi − xb ε / kT + z2 (xi+1 − 2xi + xi−1 )ε / kT = 0 [7]
i=2,3…and
ln[]x1 ()()1− xb / xb 1− x1 + (z1 x1 + z2 x2 − z0 xb )ε / kT + ε A / kT = 0 [8]
Here xi is the fraction of sites occupied by molecules in layer i while xb is the fraction of sites occupied in the bulk. ε is the interaction energy between two adsorbates and the energy between adsorbate and adsorbent is εA. z0 is the volume coordination
z0 − z1 number (z0=12); z1 is the monolayer coordination number (z1=6), and z = ; k is 2 2
Boltzman’s constant and T is the absolute temperature.
Equation 7 is a finite difference Equation, and was solved using the boundary condition (Equation 8) at the adsorbent surface where i=1. The other boundary condition
for macroporous adsorption is that limxi = xb . With the additional assumption of no i→∞ condensation (small changes in density between adjacent layers), the Gibbs excess adsorption can be described as(Aranovich and Donohue 1996):
x 1 Γ = a b − x [9] m εA kT b xb + (1− xb )e 1−ω1
Where am is the monolayer capacity and(Aranovich and Donohue 1996) 28 2 ω1 = a 2 − a 4 −1 [10]
− kT z a = − 1 [11] z2 xb ()1− xb ε z2
a The Gibbs adsorption has three parameters m , ε / kT and ε A / kT . In this model, the adsorbent is approximated as a flat plate, with a surface area equal to the surface area inside the pores and end effects are neglected. The fluid is attracted to the plate in 1D, and near the critical point the effect of adsorption goes beyond the first layer. As it stands, this model is used to approximate macroporous adsorption. To describe microporous adsorption the number of layers for adsorption is limited by the size of the pore and therefore the value of i will be limited to some maximum value n. The monolayer capacity am in Equation 9 is replaced by a constant K reflecting the capacity for n layers in a slit-like pore:
n n−1 K = 2am (1− ω1 ) (1+ ω1 ) [12] The bulk density of adsorbate is determined from the fraction of occupied lattice sites and in order to relate the calculated results to experiment a function mapping the lattice occupancy to the real density is used This function can be chosen to give some reasonable estimate of the actual density at a fractional occupancy of 1 and should also pass through the critical density (ρC) at a fractional occupancy of 0.5. In this work, the mapping function proposed by Hocker et al(2003) is used with a maximum density represented by the close packed state of CO2 (approximately 44.12 mol/L).
29 am t °C ε (k) εA (k) S2/(S1+S2) (mmole/g)
Global all -202.8 -1832 5.09 0.39
Local 32 -202.6 -1856 5.11 0.05
35 -202.8 -1853 5.03 0.26
40 -202.8 -1835 5.02 0.33
50 -202.8 -1903 4.93 0.49
Table 2.3. Result of fitting all 4 Parameters in the Adsorption Model
The adsorbent is approximated by slit-pore similarity, with the surface area of the
adsorbent divided between two walls. The distance between the two walls corresponds to
the pore diameter. This model is limited when it is adjusted for microporous adsorption
on zeolite, and the root of the limitation likely lies in the use of slit-pore geometry, as this
creates two problems. The first is that the pore sizes of the adsorbent are not uniform.
The second is that the geometry of the micropores is better described as cylindrical rather
than slit-like. The first shortcoming of the model could be overcome by dividing the α
cage into two parts: a cage and a window connecting the cages; the cage is large enough
to house about 3 layers of CO2 while the window is only wide enough for two layers, as 30 shown in Figure 2.7. S1 and S2 are used to describe the contribution to the total surface are from the windows (S1) and cages (S2) of the zeolite respectively. The second problem can in principle be addressed through a more complex model, which will be the subject of future work.
Figure 2.7 Bimodal micropore model of adsorbent
31 εA am t °C ε (k) S2/(S1+S2) (k) (mmole/g)
Global All -1832 5.09 0.347
Local 32 -1856 5.11
35 -202.8 -1853 5.03
40 -1835 5.02
50 -1904 4.93
Table 2.4. Results of fitting only εA and am in the Adsorption Model (2.5 parameters)
The experimental data is fitted with the lattice model by using several protocols to explore the meaning and significance of various parameters. Data were fit locally (i.e. one isotherm at a time) and globally (all temperatures simultaneously). The fitted parameters are shown in Table 2.3 where ε, εA, and am are as described in Equations 5, 6, and 10. The last parameter, S2/(S1+S2) represents the fractional surface area in the adsorbent due to pores of width S2 (representing the cages). The R-square value for the fit is about 0.96. From Table 2.3, the interaction energy between fluid molecules is almost independent of temperature. It is also found that the fractional surface area is close
32 to the value one would calculate based on the crystal structure of NaY zeolite (0.347) with the exception of the 32 °C isotherm near the critical point. This is also where the worst fit occurs because the model with these parameters cannot capture the “hump” near the critical point. By fixing the fluid phase interaction energy to the observed constant value from Table 2.3 and the fractional surface area to the known value, nearly the same values of εA, and am upon optimizing just those two parameters (Table 2.4) are obtained.
This indicates that the model can be reduced to essentially a 2.5 parameter model. The lattice model correlations are compared with the experimental isotherms in Figure 2.8.
The dotted line in this Figure 2.is the adsorption in the windows with 2 layers of CO2 while the dashed line is the adsorption in the cages with 3 layers of CO2. With the parameters in Table 2.3, reasonable adsorption isotherms are obtained. The model fits the experiment data perfectly at low and high pressure but cannot accurately fit the “hump” near the critical point. This calculation is consistent with Hocker et al(Hocker et al.
2003), who predicted adsorption isotherms in one-dimensional slit-pores of different widths. In their work, the adsorption “hump” shows up in meso- and macro- pores, but not in micropores. In fact, they show that the “hump” in the adsorption isotherm is not correlated with a 3-layer micropore but is fit only with the inclusion of a 30 layer macropore ascribed to the clay binder in their zeolite material. Our data still exhibit the
“hump” despite the notable absence of large pores.
As a way of using the model to explore this pore size effect, different pore widths guided by the physical measurement of the pore size distribution are chosen. Specifically, around 2% of the surface area is attributable to pores larger than 26 Å that would fit approximately 7 layers of adsorbate. Calculated adsorption isotherms with fixed ratios of 33
Figure 2.8 Comparison of the lattice model (2 and 3 layers in S1 and S2 respectively)
with experimental data. (Globally fitted)
34
Figure 2.9 Comparison of the lattice model (2 and7 layers in S1 and S2 respectively)
with experimental data. (Globally fitted)
35 pore widths containing 2 and 7 layers of adsorbate are shown in Figure 2.9. Adsorption
“humps” exist for the 7 layer pore that makes the total excess adsorption exhibit the
“hump” even at 40°C and 50°C without capturing the magnitude of the hump at 32 °C.
This pore size distribution is obviously one effect that can lead to this phenomenon.
Another choice is to account for adsorption on the outer surface of the zeolite particles where the number of layers could be very large. Packed zeolite particles may form macropores in the void spaces between particles in the sample. One could argue that the adsorption in these “pores” should be counted. Aranovich & Donohue applied the model to data on nonporous Graphon and coal demonstrating the applicability in the supercritical region on such external surfaces(Aranovich and Donohue 1996). Taking the
2% of surface area in large pores and fitting the model with the 2 micropores as shown in
Figure 2.7 and an unbound number of layers on that 2% gives the results shown in Figure
2.10. This amount of external surface or macropore area is not sufficient to quantitatively capture the hump. An external surface area of approximately 10% of the total surface area would be necessary to provide a significant effect as shown in Figure 2.11.
A conclusion is got that pore size distribution is a likely mechanism leading to observation of this “hump” near the critical point at least from a theoretical point of view.
The analysis of Hocker et al(2003) is quite convincing on this point. However, the data presented here on a crystalline zeolite material with well-defined micropores does not squarely conform to this conclusion and requires further study. In fact, compared with the data presented here, the “hump” on materials containing macropores is less significant.
36
Figure 2.10 Comparison of the lattice model (2% external surface) with experimental
data (Globally fitted)
37
Figure 2.11 Comparison of the lattice model (10% external surface) with experimental
data (Globally fitted)
38 2.5 Conclusions
New adsorption and desorption data of supercritical CO2 on NaY zeolite have been presented. The zeolite is a strong adsorbent for CO2 as shown by the sharper initial slope of the adsorption isotherms and the higher calculated adsorbed phase densities compared to carbon and silica gel. The model of Aranovich & Donohue is used to predict the adsorption and this model successfully predicted the adsorption isotherms at whole pressure range but failed to predict the adsorption “hump” near the critical point with physically meaningful parameters. It is believed that a pore size distribution including meso- or macropores can lead to the observation of the hump but there is no conclusive result from the modeling indicating the mechanism of the hump in a strictly microporous adsorbent.
39
CHAPTER 3
MOLECULAR SIMULARION OF SUPERCRITICAL CO2 ON
ACTIVATED CARBON AND NaY ZEOLITE
3.1 Introduction
Recently, supercritical carbon dioxide (SC-CO2) has received much attention as an environmentally benign solvent. SC-CO2 has liquid-like density and gas-like transport properties, and minor adjustments of pressure and temperature can bring large changes in those properties. Because the diffusion coefficient of a solute in SC-CO2 is higher than that in liquids, SC-CO2 may be a better solvent for adsorption separation than liquid solvents, and is especially attractive for commercial adsorptive separation of complex, difficult-to-separate mixtures such as isomers. Previous research (Iwai et al. 2003) indicates that we can efficiently separate isomers using SC-CO2 as a solvent with a proper adsorbent that possesses suitable surface characteristics.
It is important to account for the behavior of CO2 as a solvent in separations research because the competitive adsorption of CO2 may affect the selection of operation conditions (Iwai 2003). Prior research(Specovius 1980; Hocker 2003; Humayun 2000;
Gao 2004) found that, near the critical point, the adsorption isotherm of supercritical fluid
40 has a significant “hump” if the excess adsorption is plotted as a function of bulk density.
Subsequently, researchers (Aranovich 1996; Hocker 2003; Gao 2004) tried to correlate supercritical adsorption with an adsorption model developed from the Ono-Kondo lattice model. With this model, Hocker et al(2003) qualitatively predicted adsorption on zeolite/clay pellets over a wide pressure range and also the adsorption “hump” near the critical point. Gao et al (2004) correlated the adsorption isotherms of CO2 on NaY zeolite crystals, and compared this lattice model correlation with experimental data. While the model, with four parameters, can quantitatively describe the adsorption behavior over the entire pressure range, it only captures the enhanced adsorption near the critical point with physically unrealistic values of some parameters (Gao et al. 2004).
Understanding the behavior of near-critical CO2 adsorbed on surfaces at a molecular level may reveal a better method of correlating and predicting adsorption isotherms for separations. It may also lend insight into other processes where anomalous behavior near the critical point is observed such as catalytic reaction kinetics (Jessop
1995; Hyde 2002; Gordon 2004) and polymer thin film swelling(Sirard 2002) In particular, it is of interest to identify whether the “hump” can correlate to the macroscopic or “bulk” CO2 properties near the critical point as well as determining whether the “hump” can be represented as an equilibrium property only or if it should be represented as having both equilibrium and kinetic properties. The focus in this work is to gain some initial insight into the equilibrium molecular level behavior of CO2 adsorbed in micropores near the bulk fluid critical point.
In this research, molecular simulation is used to explore the adsorption behavior.
We know that the behavior of molecules adsorbed in the pores of a microporous material 41 is very difficult to directly study by experiment. So simulation is chosen to get this microscopic information. Computer simulation provides a direct path from the statistical mechanics of a system to the macroscopic properties of experimental interest. Molecular simulation is extensively used in supercritical adsorption research (Takaba 1996;
Nicholson 1998; Du1999; Mart 2002; Do2003; Turner 2003; Cao, 2004). Zhou et al
(2000) simulated adsorption of supercritical CO2 on activated carbon slit at 323K and
348K with Gand Canonical Monte Carlo (GCMC) method when the pressure is in the range 2~7 MPa. Pantatosaki et al (2004) simulated adsorption of CO2 in slit and cylindrical pore with GCMC method at 298K and 308K under pressure rang 1~10 bar.
They found the pore size has strong effect on adsorption. Nitta et al (1998) simulated adsorption of benzene and CO2 mixture on a slit pore of graphite plane wall at 313.2K and 323.2K and the pressure is up to 16.0Mpa. They found the competitive adsorption of
CO2 has significant effect on the adsorption of benzene. Hirotani et al (1997) simulated adsorption of CO2 on silicalite and NaZSM-5 with GCMC method at 305K and 392K.
Their simulation results are consistent with experimental data. However, their research mainly focuses on low-pressure adsorption; there are only two data points above critical pressure at each temperature, and they show no data near the critical point. All these papers either focus only on low-pressure adsorption, or generally talk about high-pressure adsorption above critical pressure. No work exists for adsorption simulation of CO2 near the critical point. In this chapter, adsorptions of CO2 on activated carbon and NaY Zeolite are simulated from low pressure to supercritical pressure at supercritical temperatures.
The simulation data is then compared with experimental data. This research mainly focuses on the adsorption behavior near critical point because of the observed adsorption 42 “hump” in this area. It is hypothesized that the molecular behavior near the critical point must be different from that at other conditions. Understanding the molecular behavior inside a micropore is helpful for us to explore the competitive adsorption of CO2 with solutes in future separation research. Also, it helps us to design the operating condition of a separation experiment. We may be able to take advantage of the special properties of
CO2 near critical point to improve the efficiency of separation.
3.2 Algorithm and Methodology
In the first step, pure CO2 fluid is simulated with Grand Canonical Monte Carlo method. The simulated fluid density is adjusted to be consistent with that calculated with an accurate Equation of State (EOS) by changing the energy parameter ε and molecular diameter σ. Since the simulation is going to be compared with experimental data at 32.0
°C, which is only 0.28% above the critical temperature, it is important to get an accurate critical temperature for the simulation system. Usually, GCMC simulation is executed to get the liquid-vapor coexistence line and from that an estimate of the critical temperature, but this method is time-consuming. Here, a transition matrix Monte Carlo method
(TMMC)(Errington 2003) is used. In this method, a simple collection matrix is inserted to collect information about the acceptance probability of Monte Carlo moves. With a bias function, this method can easily pass through the low-probability region. Starting from any state, the relative probability of other states can be calculated. Since the probability distribution is a function of chemical potential, adjusting chemical potential 43 can shift the probability distribution; and the coexistence point can be obtained efficiently and precisely. With the liquid-vapor coexistence line, the critical temperature can be calculated with a scaling law. Then the adsorption simulation is executed at the same reduced temperature as in the experiment.
In simulation of adsorption in slit pores, all molecules are confined inside pore.
Adosorbed phase is unlimited only in two-dimension. So the cut–and-shifted Lennard-
Jones 12-6 potential is used to calculate the intermolecular interaction in adsorption simulation. This potential is also used in simulation of pure fluid.
12 6 σ 12 σ 6 σ σ v(r) = 4ε − - 4ε − r < rc r r r r [1] c c v(r) = 0 r ≥ rc
where v(r) is the interaction function, ε is the energy parameter and σ is the diameter of
CO2 molecule. r is the distance between molecules and rc=4σ is the cut off distance.
Periodic boundary conditions are used in three dimensions.
With TMMC method, the number probability distribution is obtained at a specific temperature and chemical potential. With this number distribution, the isothermal
compressibility Κ T of the pure fluid is calculated at different chemical potentials, as shown in Equation 2.
βV 2 2 Κ T = [ N − N ] [2] N 2
1 where β = , k B is the Boltzmann constant and T is the temperature. V is the k BT volume of simulation system and N is the average of number of molecules. 44 The traditional GCMC method is used to simulate adsorption. A group of slit- shaped pores is used to approximate the actual pore distribution for activated carbon. For adsorption of CO2 on activated carbon, Steele’s 10-4-3 solid-fluid potential (Equation 3) is used for adsorbate-adsorbent interaction.
10 4 2 σ σ σ 4 v(z) = 2πρ ε σ 2 ∆ sf − sf − sf [3] s sf sf 5 z z 3 3∆()0.61∆ + z
-3 Where ρs=114nm is the number density of carbon, z is the normal distance between CO2 and carbon wall. ∆=0.335 is the distance between carbon lattice planes, σsf and εsf are calculated by the Lorentz-Berthelot rule with σss=0.34nm and εss/k=28.0K.In simulation of adsorption, periodic boundary conditions are used in the direction parallel to the carbon wall.
Also, adsorption of CO2 on NaY zeolite is simulated. NaY zeolite has α cages with a diameter of about 12.5 A. α cages are tetrahedrally connected through 12-rings windows(Jaramillo 1999). Sodium cations are usually located on four kinds of sites, as shown in Figure 3.1. A unit cell has 16 sites I, and 32 sites I’, II and II’. The sodium cations are constrained. In simulation, the interaction between adsorbate-adsorbate and adsorbate-framework are calculated with cut-and-shifted L-J potential. The ideal framework is used to describe the NaY zeolite strucuture. The L-J potential parameters of
NaY zeolite framework are shown in Table 3.1. The simulation cell has 8 α cages, and periodic boundary conditions are used.
45
Figure 3.1 Framework of NaY zeolite (Jaramillo et al. 2001)
46 Atom type ε/k (K) σ (Å)
O 22.0 3.04
Si 9.8 0.76
Al 10.1 1.14
Na 8.0 2.98
Table 3.1 Potential parameters for NaY framework
3.3 Results and Discussion
Pure CO2 systems at different temperatures are simulated with TMMC method. In all those simulations, Monte Carlo moves consist of 30% particle displacement, 35% creation and 35% destruction. For TMMC simulation, after first 100×106 Monte Carlo
(MC) steps, the weighting function is updated every 1×106 MC steps. The entire simulation comprises 600×106 MC steps. Initially, a small system with 10σ×10σ×10σ was simulated. In this system, a thermodynamic critical point was identified but the highly compressible behavior of the pure fluid near the critical point was not captured.
Because one or two molecules change the density too much, there is no enough data point near the critical point. It is believed that this highly compressible region is the key to understanding the adsorption “hump” observed in activated carbon and NaY zeolite.
Therefore a larger simulation system is needed to test our hypothesis. A pure fluid on a
47 system of 20σ×20σ×20σ is simulated, and the gas-liquid coexistence curve is calculated, as shown in Figure 3.2. The Coexistence data is fitted with a scaling law, as shown in
Equation 4.
0.355 ρl − ρ g = A(TC − T ) [4]
where ρl and ρ g are simulated density of liquid phase and gas phase, and TC and T are critical temperature and simulation temperature. After fitting, the critical temperature is
TC =1.181 while constant A=0.95. The critical density is 10.42 mole/L, which is a little lower than the real critical density (10.63 mole/L).
With this critical temperature, densities of the simulation system are correlated with those calculated from the Equation of State by adjusting ε and σ. When ε/k=257.51K and σ=3.65A, the simulated density is consistent with that calculated by the EOS, as shown in Figure 3.3.
Figure 3.4 is the compressibility calculated with the number probability distribution from TMMC simulation and that calculated with the EOS at 32.0ºC over a pressure range 0~3000 psia. Compressibility calculated from the simulation is consistent with that calculated by EOS. But the compressibility peak is observed at a lower pressure for simulation system. This indicates that the critical pressure for our simulation system is lower than that of real CO2 fluid.
In order to compare the simulation with experiment, the pore size distribution
(PSD) of activated carbon is measured with BJH adsorption method. Most pores of
48 1.3
20X20X20σ3
1.2
1.1 Temperature
1.0
0.9 0.00.10.20.30.40.50.60.7
Density
Figure 3.2 Gas-liquid coexistence curve
49 25
20
15 Density (mole/L) Density
10
5 Equation of State Simulation
0 0 500 1000 1500 2000 2500 3000
Pressure (psia)
Figure 3.3 Density of CO2 fluid at 32.0˚C
50 0.05
0.04
0.03
0.02 Compressibility (1/psia) Compressibility Simulation Equation of State 0.01
0.00 0 500 1000 1500 2000
Pressure (psia)
Figure 3.4 Compressibility of CO2 fluid at 32.0˚C
51 carbon pore are micropores. In order to simplify the simulation, pores with similar size are combined together and the PSD is redefined, as shown in Figure 3.5.
Since the size of the simulation system affects the simulation of pure fluid, the effect of system size on the adsorption simulation is also checked to make sure that our simulation system is large enough. Adsorption of CO2 in three different systems is simulated at 32.0ºC and 933.6 psia, as shown in Figure 3.6. The density of adsorbate inside a slit pore is directly calculated from simulation using equation 5.
N N ρ = a = a [5] a V HL2 where ρa is the density of CO2 fluid inside pore,
The slit width is 14.73σ and the carbon wall size changes from 15σ×15σ to
20σ×20σ and 30σ×30σ, the density of the adsorbed phase remains nearly constant. This is taken as evidence that a 15σ×15σ system is large enough to obtain reasonable results, and is therefore used in all adsorption simulations.
Adsorptions of CO2 on activated carbon at 32.0ºC with different pore sizes are simulated. Simulated adsorbate densities in selected slit pores are presented in Figure 3.7.
In this and subsequent figures showing simulation results on activated carbon, the slit widths used are those representing the pore size distribution in Figure 3.5. The density in picture is defined by equation 5.
52 700 Measured PSD 600 Aproximated PSD
/g) 500 2
400
300
200 Surface Arear (m
100
0 0 102030405060 Pore Diameter (A)
Figure 3.5 Pore size distribution of activated carbon
53 0.8 <ρ>=0.505 0.6
0.4
0.0 0.8 <ρ>=0.509 0.6
0.4 20σX20σ 0.2
0.0 0.8 <ρ>=0.517 0.6
0.4 30σX30σ 0.2
0.0 0 200 400 600 800 1000
100 MC steps
Figure 3.6 Adsorption simulation on carbon wall with different size (P=933.6 psia,
H=14.73σ, t=32.0 ºC)
54 0.8
0.6
2.72σ 0.4 4.40σ H 7.95σ Density (
0.0 012
Bulk density (ρ/ρc)
Figure 3.7 Adsorbate density inside carbon slit (T*=1.0028)
55 Previous research (Hocker, R. et al. 2003; Gao, W. et al. 2004) found that the
“hump” in excess adsorption is observed in big pores. In order to include a discussion of the adsorption inside large pores, adsorption on carbon slits with width of 20σ and 30σ are also simulated, as shown in Figure 3.7. The adsorbate density in the pore increases smoothly with increasing pressure if the slit width is above 7σ. When the slit width is smaller than 7σ, the pore density saturates quickly at relatively low pressure. When the pressure increases, adsorbate density has a flat plateau in all slit widths. At very high pressure, the adsorbate density increases slightly but the slope is small.
The quantity measured experimentally is the excess adsorption. The excess adsorption in experiment is defined in Equation 6
∞ Q excess = ρ − ρ dz [6] ∫ ()a l 0 where ρl is the density of bulk fluid and z is the distance from solid adsorbent wall.
To get the corresponding value from simulation, we first calculate the excess density in the pore as defined in Equation 7.
N − N ρ excess = ρ − ρ = a L [7] a l HL2 where
56 1.0 >)/V) L 2.72σ 0.8 4.40σ >- 0.4 Excess Density (( 0.2 H 0.0 012 Bulk Density (ρ/ρ ) c Figure 3.8 Excess density inside carbon slit (T*=1.0028) 57 excess adsorption is plotted as a function of bulk density, as shown in Figure 3.9; and the excess adsorption is defined by Γ excess = ρ excess × H [8] Figure 3.9 shows that when the pore size is smaller than 7σ the excess adsorption has a maximum at low pressure, and increasing pressure decreases excess adsorption. When the pore size is greater than 7σ, the slit width has no effect on excess adsorption at low pressure. One reason may be that the adsorbed phase near the middle of the pore has almost the same density as the bulk fluid. But when the pressure is much higher, slit width has strong effect on excess adsorption, especially for adsorption in slits with width of 20σ and 30σ. Excess adsorption increases sharply in these two slits. The simulations in 20σ and 30σ slit widths show excess adsorption is still significant in big pores. Near the critical point, molecules exhibit strong long-range correlations(Eckert 1986; Hernan L. Martinez 1996). This correlation results in cluster formation around solute molecule in dilute binary solution (Eckert 1986; Debenedetti 1987; O'Brien, Randolph et al. 1993). In an adsorption system, this long-range correlation may result in adsorption of more than three layers of CO2 on a carbon wall. In order to check the effect of temperature on excess adsorption, isotherms at T*=1.029 and T*=1.093 are simulated, as shown in Figure 3.10 and Figure 3.11 From Figures 3.9-3.11, temperature has a weak effect on adsorption in small slit widths, but has a strong effect on adsorption in larger slits. For slit widths larger than 7σ, when temperature increases, excess adsorption in different pores converges. It is predictable that at higher temperature, increasing slit width has no contribution to excess adsorption. 58 12 ) 2 2.72σ >)/L 10 L H 4.40σ 7.95σ >- a 14.73σ 8 20σ 30σ 6 4 Excess Adsorption (( 0 012 Bulk Density (ρ/ρc) Figure 3.9 Excess adsorption inside carbon slit (T*=1.0028) 59 12 ) 2 2.72σ >)/L L 10 4.40σ H 7.95σ >- 6 4 Excess Adsorption (( 0 012 Bulk Density (ρ/ρ ) c Figure 3.10 Excess adsorption inside carbon slit (T*=1.029) 60 12 ) 2 >)/L L 10 >- 4 Excess Adsorption (( 0 012 Bulk Density (ρ/ρ ) c Figure 3.11 Excess adsorption inside carbon slit (T*=1.093) 61 Figures 3.9-3.11 show significant excess adsorption in large slit pores. In order to get more detailed information, the density profile is calculated from simulation, as shown in Figure 3.12. The density probability distribution is plotted according to the distance from the carbon wall. In Figure 3.12, there are three significant layers of adsorbed molecules on each carbon wall. Except for these layers, the density of the adsorbed phase is almost uniform. So, the adsorbed phase can be divided into two parts: the fluid near the carbon wall and fluid far from the carbon wall. The region far from the carbon wall is defined as “pore fluid”. In order to check the properties of the pore fluid, a simulation is executed in a big slit with slit width 30σ, as shown in Figure 3.13. The plate in Figure 3.13 represents carbon walls, and the cube represents the pore fluid. All molecules inside the slit are counted to calculate potential force, but only molecules inside the cube are used in the calculation of compressibility and fluid density. As noted previously, compressibility is an important property of the fluid near the critical point and may help our understanding of molecular behavior inside slit pores. Simulation results of compressibility and density are compared with bulk fluid simulation in Figure 3.14 and Figure 3.15 separately. Figure 3.14 shows that, even inside a slit pore, the pore fluid still has high compressibility near the critical point. This high compressibility may be the reason of excess adsorption of pore fluid. As shown in Figure 3.15, near the critical point, the density of pore fluid is higher than that of bulk density. More CO2 molecules are packed inside the pore than expected from the bulk fluid density even though the effect of the carbon wall is weak. A strong potential force forms several layers of CO2 molecules packed on carbon wall, as shown in Figure 3.12. Near the critical point, these CO2 molecules have a strong effect on the molecules near by and through these adsorbate- 62 0.006 O 0.005 t=32.0 C P=930.2 PSI Density 0.004 0.003 0.002 0.001 0.000 -8-6-4-202468 Position (σ) Figure 3.12 Density profile of adsorbate inside carbon slit 63 Figure 3.13 Simulation system of pore fluid 64 0.05 Simulation of bulk fluid Simulation of pore fluid 0.04 Equation of State 0.03 0.02 Compressibility (1/psia) Compressibility 0.01 0.00 0 500 1000 1500 2000 Pressure (psia) Figure 3.14 Compressibility of pore fluid (T*=1.0028) 65 3 C Pore fluid ρ/ρ Bulk fluid 2 1 0 0123 Density of bulk fluid ρ/ρ C Figure 3.15 Density of pore fluid (T*=1.0028) 66 adsorbate correlations; the effect of the carbon wall is transferred to molecules in the pore fluid. In order to check the effect of temperature on the property of pore fluid, a simulation is executed at T*=1.093 with same system described above. As shown in Figure 3.16, the compressibility of the pore fluid at high temperature is lower than that of bulk fluid at T*=1.0028, but higher than that of bulk fluid near the critical pressure at same temperature. This high compressibility forms higher density of pore fluid, as shown in Figure 3.17. So, at high temperature, the excess adsorption of pore fluid is not significant, as shown in Figure 3.11. In Chapter 2, the lattice model shows the excess adsorption “hump” when adsorption on the external surface of the adsorbent particles is counted. In simulation, a large slit pore with width of 60σ is used to approximate mesopores between particles. The excess adsorption isotherms in that pore at different temperatures are shown in Figure 3.18. Excess adsorptions in Figure 3.18 have significant maximum at pressure below the critical pressure, this maximum moves to higher pressure when temperature increases. Also, temperature has weak effect on excess adsorption at low pressure and very high pressure, but has strong effect at median pressure. Excess adsorptions inside different carbon slit are combined together according to PSD, as shown in Figure 3.19. Compared with experimental data, the excess adsorption from simulation is larger than experiment data. The maximum value of excess adsorption shows up at higher pressure in simulation system. The slope of linear part is similar to that of experiment data. At near critical point, simulation does not catch adsorption “hump”. In order to check the adsorption in mesopores, adsorptions in slits with width of 67 0.05 Simulation of pore fluid (T*=1.093) 0.04 Equation of state (T*=1.093) 0.03 0.02 Compressibility (1/psia) Compressibility 0.01 0.00 0 500 1000 1500 2000 2500 3000 Pressure (psia) Figure 3.16 Compressibility of pore fluid (T*=1.093) 68 3 c ρ/ρ Pore fluid Bulk fluid 2 1 0 0123 Density of bulk fluid ρ/ρ c Figure 3.17 Density of pore fluid (T*=1.093) 69 18 ) 2 16 >)/L T*=1.0028 L 14 T*=1.029 T*=1.093 >- a 12 10 8 6 4 2 T Excess adsorption (( Figure 3.18 Excess adsorption of CO2 in carbon slit pore with width of 60σ 70 12 Experiment 10 Simulation Simulation With 2% 30σ Pore Simulation 8 With 2% 60σ Pore 6 4 2 Excess adsorption (mmole/g) Excess 0 012 Bulk Density (ρ/ρ ) c Figure 3.19 Excess adsorption of CO2 on activated carbon with 2% mesopores (32.0ºC) 71 30σ and 60σ are counted in simulation. Figure 3.19 shows the excess adsorption isotherms with 2% 30σ or 60σ pore. The total excess adsorption increases when adsorption in mesopore is counted. Also, the maximum of excess adsorption moves to higher pressure. Figure 3.20 shows the excess adsorption isotherms with 5% 30σ or 60σ pore. The total excess adsorption increases while percentage of mesopore increases. Even with adsorption in mesopores, simulation also doesn’t get adsorption “hump” near the critical point. The simulated excess adsorption of CO2 on activated carbon at 40.0ºC is compared with experiment data, as shown in Figure 3.21. The maximum value of simulation shows up at higher pressure, and the total excess adsorption of simulation is larger than experiment data. The simulated adsorption of CO2 on NaY zeolite at 32.0ºC is compared with experiment data, as shown in Figure 3.22. Since α cage dominates according to pore size measurement (Gao et al. 2004), only adsorption inside ideal framework was simulated. Simulated excess adsorption is larger than experiment data. Since α cage is large enough to house three layers of CO2 molecules only, excess adsorption maximum is reached at very low pressure. As observed for adsorption on activated carbon, the simulation system does not catch the adsorption “hump”. 72 14 Experiment 12 Simulation Simulation With 5% 30σ Pore 10 Simulation With 5% 60σ Pore 8 6 4 2 Excess adsorption (mmole/g) 0 012 Bulk Density (ρ/ρ ) c Figure 3.20 Excess adsorption of CO2 on activated cabron with 5% mesopores (32.0ºC) 73 10 Experiment 8 Simulation 6 4 2 Excess adsorption (mmole/g) adsorption Excess 0 012 ρ/ρ c Figure 3.21 Excess adsorption of CO2 on activated carbon (40.0ºC) 74 Figure 3.22 Excess adsorption of CO2 on NaY zeolite (32.0ºC) 75 3.4 Conclusions Transition Matrix Monte Carlo simulations of pure CO2 fluid and Grand Canonical Monte Carlo simulations of CO2 adsorption on activated carbon and NaY zeolite have been presented. The effect of carbon slit width on adsorption at different temperatures is compared. Temperature has a significant effect on adsorption inside large slit widths while the effect is very weak inside small slit widths. The excess adsorption is significant even in a meso-pore slit, and this excess adsorption decreases as temperature increases. The compressibility and density of adsorbate in the mid-region of a large slit pore is compared with that of bulk fluid. Simulated adsorptions of CO2 on activated carbon and NaY zeolite are compared with experiment data. Results indicate the adsorption in simulation is larger than experiment. The simulation failed to show the adsorption “hump” near the critical point with the interaction parameters calculated from simulation of pure CO2 fluid. 76 CHAPTER 4 SUMMARY AND RECOMMENDATIONS 4.1 Summary This work has focused on adsorption of CO2 onto microporous adsorbents near the bulk fluid critical point. Near-continuous adsorption/desorption isotherms were measured for CO2 on NaY zeolite. Then a lattice model was used to correlate adsorption isotherms at different temperatures. In the last part, adsorptions of CO2 on activated carbon and NaY zeolite were simulated with Grand Canonical Monte Carlo method. Adosrption/desorption isotherms on NaY zeolite were measured at 32.0ºC, 35.0ºC, 40.0ºC and 50.0ºC with a gravimetric microbalance. With that balance, the CO2 pressure was increased slowly but continuously, and high-resolution adsorption/desoprtion isotherms were obtained, especially near the critical point where prior data were very sparse. Isotherms show that CO2 has strong affinity with NaY zeolite and micropore saturates quickly at low pressure. At 32.0ºC and 35.0ºC, adsorption isotherms have significant “hump” under near critical pressure if the excess adsorption is plotted as a function of bulk density. The model of Aranovich & Donohue was used to correlate the adsorption isotherms with 2.5 parameters. This model successfully predicted the adsorption 77 isotherms at whole pressure range but failed to predict the adsorption “hump” near the critical point with physically meaningful parameters. It is believed that a pore size distribution including meso- or macropores can lead to the observation of the hump but there is no conclusive result from the modeling indicating the mechanism of the hump in a strictly microporous adsorbent. A Transition Matrix Monte Carlo simulation was executed to obtain the gas-liquid coexistence curve. From that curve, a scaling law was used to get the critical temperature of our simulation system. The simulation density was fitted to that from an accurate Equation of state to get potential function parameters of CO2 fluid. With these parameters, adsorption of CO2 onto activated carbon and NaY zeolite was simulated, and the result is compared with experiment data. The simulated excess adsorption is larger than experiment data, and simulation does not catch adsorption “hump” near the critical point. Also, the effect of carbon pore size on the adsorption of CO2 on activated carbon was investigated. At low and very high pressure, increasing pore width does not increase excess adsorption when the width is larger than 7σ. At near critical temperature, because of the long-rang correlations, the density of the adsorbed phase in the middle of wide pores is still higher than that of bulk density. At high temperature, excess adsorption in the middle of the pore decreases significantly. 4.2 Recommendations The excess adsorption “hump” shows up within the bulk density range of 8mole/L~13mole/L. Since the compressibility of CO2 is very high near the critical point, the pressure increases only by 18psia while bulk density increases from 8mole/L to 78 13mole/L. In the experiment, the system pressure instead of bulk density was measured; the measurement of pressure is critical. An ISCO syringe pump is used to increase pressure of system with speed of 0.5psia/min near the critical point. The working mechanism of syringe pump is more like trial-and-error method. The pump adjusts pressure frequently to get the expected pressure. This mechanism could introduce some measurement error for both pressure and adsorption. Although I have enough confidence of the existence of adsorption “hump”, it is still worth attempting to verify the magnitude of the “hump” with another method. Since our group is going to obtain a Rubotherm magnetic suspension balance and magnetic balance system can be used to measure density of fluid directly, the adsorption of CO2 onto activated carbon and NaY zeolite at 32.0ºC and 35.0ºC could be re-measured with measured bulk density. Focusing on the adsorption of CO2 on solid adsorbent, we want to take advantage of special properties of supercritical CO2 adsorption in separation research. The density of SC-CO2 fluid has a strong effect on solvating power, and the density of adsorbed phase has an abnormal “hump” near the critical point. It could be interesting to explore the adsorption of solute from CO2 carrier fluid. Since the adsorption “hump” exists within a narrow pressure range, it is easy to change the density of adsorbed phase by adjusting system pressure. A simple volumetric adsorption system is sufficient for this experiment. The volumetric set-up includes an adsorption chamber, a syringe pump, a pressure gauge, and heat bath. The adsorption chamber is connected to an analysis system. Activated carbon and NaY zeolite is used as solid adsorbent. Phenol, xylene or ethylbenzene could be used as solute, since a lot of solute-CO2 solution data have been published. For each solution, the system pressure could be shifted in and out of the “hump” range, and 79 measure the concentration of solute in CO2 bulk fluid. Through this research, the effect of adsorption “hump” on adsorption separation could be checked. Since solvent density has effect on transport properties, the adsorption “hump” may affect the diffusion of solute in solvent and the transport between solvent and solid matrix. From experiments of adsorption of CO2 on to activated carbon and NaY zeolite, it is found that the adsorbent has irreversible adsorption even when the purity of the CO2 is very high. Dead adsorption means the adsorbed cannot be desorbed even the system is vacuumed. That implies, after the first run of adsorption and adsorption, there is still some adsorbate inside solid adsorbent. This decreases the real adsorption of following runs. In adsorption separation with supercritical CO2 fluid, it is important to recycle the solid adsorbent. The existence of irreversible adsorption decreases the adsorption capacity of recycled adsorbent. Also, it may affect the adsorption/deaorption of solute in adsorption separation. Seeking the reason of irreversible adsorption would deepen our understanding of gas-solid interaction. Also, if the irreversible adsorption also exists for solute, then it could affect the separation operation. The investigation of irreversible adsorption is recommend as a part of future research. 80 APPENDIX A LATTICE MODEL PROGRAM 81 %This Is the main M-file for the program that finds the amount of excess %adsorption on an adsorbent for an INDIVIDUAL temperature. The fitting has several different functions to %choose from, and more details on how they fit can be found in the %individual funtions. clf; hold on; clear; global k R Na pi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Menu Functions to allow different variables to be chosen %For speed, It may be easier to set the value of one or several in the file, and only %change the others. ADSORBENT=2%Menu('Select Adsorbent','Calgon 400','Zeolite'); PACKING=1%Menu('Choose Packing Type','Cubic','Hexagonal'); CHOICEFIT=1%Menu('Optimize Parameters','Yes', 'No (Quick)'); POROUSTYPE=9%Menu('Adsorption type','Macroporous', 'Microporous','Both', 'Multi Size Pores','Multi Size Pores (N Constant)', %'Multi Size Pores (N and e Constant)','Multi Size Pores (N, e, x Constant)','Distribution (N, e, x Constant)','Distribution with Mapping (N, e, x Constant)'; MAPPING=2%Menu('Mapping type','One term','Two Term'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 82 if ADSORBENT==1 %The GAC Surface CHOICE=Menu('GAC: Choose Temperature','30.5','32','36','40','45','All'); Tmp={'30.5','32', '36','40','45'}; %Temperature (dC) of data points %Properties of Surface S=850; %BET surface area (m2/g) V=.66e-6; %Volume of adsorption (m3/g) Dpore=17e-10; %Average pore radius (m) qst=23000; %Isosteric heat of adsorption (J/mol) Linear approximation from Reich et. al. Dpart=10e-6; %Particle size (m) Mass=0.223; %Mass (g) DescriptA='CO_2 on GAC at'; s=700; %Surface of micropores (m2/g) v=.37e-6; %Volume of micropores (m3/g) elseif ADSORBENT==2 CHOICE=Menu('ZEO: Choose Temperature','32','35','40','50','All'); Tmp={'32', '35','40','50'}; %Temperature (oC) of data points %Properties of Surface S=900; %Manufacturer surface area (m2/g) Dpore=[0.7e-9, 1.8e-9]; %Pore diameter (m) (0.7nm and 1.3nm) qst=25000; %Isosteric heat of adsorption Dpart=1.82e-6; %Average particle size (m) Outerarea=6/Dpart/1.548e6; 83 DescriptA='CO_2 on Zeolite at'; xsmall = .347; %Area Fraction of small to larger micropore xdistribution = [xsmall, 1-xsmall, Outerarea/S]; %[Small Channel, Large Channel, Outer surface] xdistribution = xdistribution/sum(xdistribution); %Normalized distribution of pore sizes xmicro = sum(xdistribution(1:2)); %Fraction inside pores end if CHOICE>length(Tmp) CHOICE=1:length(Tmp); figure; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Properties of Carbon Dioxide Tc = 304.2; rhoc = 10.6; Pc = 73.8e5; %Critical Properties: Temperature (K) Density (mol/l) Pressure (N/m2) d0 = 3.36e-10; %minimum diameter molecule (m) e0 = 426.42e-23; %minimum energy molecule (J) MW = 44.01; %Molecular Weight (g/mol) qvap = 10000; %Heat of vaporization at 0dC (J/mol) %Fundamental Physical Constants k=1.3806503e-23; %J K-1 84 Na=6.02214199e23; %mol-1 R=8.314472; %J mol-1 K-1 pi=3.14159265359; %pi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Packing parameters inputed. These are used for the initial guesses of %parameters if PACKING==1 % If Cubic z0 = 6; % Volume coordination number f0 = 6/pi; % Volume packing factor z1 = 4; % Monolayer coordination number f1 = 4/pi; % Monolayer packing factor DescriptP=' Cubic '; elseif PACKING==2 % If Hexagonal z0 = 12; % Volume coordination number f0 = 9/(2*pi); % Volume packing factor z1 = 6; % Monolayer coordination number f1 = 2*sqrt(3)/pi; % Monolayer packing factor DescriptP=' Hexagonal '; end z2 = (z0-z1)/2; % Definition %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %The Fitting begins for COUNTER=CHOICE 85 %Read in the data from the excel file clear Gamma P rho G B if ADSORBENT==1 filename=char(Tmp(COUNTER)); r=xlsread('CO2 adsorption data on GAC for MATLAB', filename); P=r(:,1); % Pressure in MPa rho=r(:,2); % Density in mole/L. Calculated by MBWR EOS G=r(:,3); % Excess Adsorption (mmole/g) B=r(:,4); % Bouyancy Correction elseif ADSORBENT==2 filename=strcat(char(Tmp(COUNTER)),'a'); r=xlsread('CO2 adsorption data on Zeolite for MATLAB', filename); rho=r(:,1); % Density in g/ml. rho=rho/MW*1000;%Density in mole/L G=r(:,2); % Excess Adsorption (mmole/g) end Tcelsius=str2num(char(Tmp(COUNTER))); % Temp in oC Tkelvin=Tcelsius+273.15; % Temp in K %Guesses for the parameters of the Equation ****Model Parameter e = -4*Tc*k/z0; %**** Energy of adsorbate-adsorbate interactions eA = -(qst-qvap)/Na; %**** Interaction energy for adsorbate molecules on adsorbent surface 86 d = d0/(1-(1+e/e0)^(1/2))^(1/6);% Diameter of molecule by Kihara potential am = S/(f1*d^2*Na)*1000; %**** Monolayer capacity (mmol/g) if MAPPING==1 rhomax=2*rhoc; %Linear Mapping through (0,0) and critical point elseif MAPPING==2 rhomax = 6/(pi*d^3*Na*f0)/1000; % Max Density: This number is about 44 mol/L, end N = round(Dpore/d); %**** Number of layers in slit e_k = e/k; eA_k = eA/k; %Energy in temperature units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Guesses for parameters in Equation for macromicro adsorption %x0 is initial guess, LB is Lower Bound, UB is Upper Bound options=[]; if POROUSTYPE == 1 x0 = [e_k, eA_k, am ]; LB = [2*e_k, 2*eA_k, .5*am]; UB = [-4*Tc/z0, .5*eA_k, 2*am]; DescriptO=' Macroporous '; %myfun=@macropore4_7_03 elseif POROUSTYPE == 2 N = mean(N); x0 = [e_k, eA_k, am, N ]; LB = [2*e_k, 2*eA_k, .5*am, .5*N]; 87 UB = [-4*Tc/z0, .5*eA_k, 2*am, 2*N]; DescriptO=' Microporous '; %myfun=@micropore4_7_03 elseif POROUSTYPE == 3 N = mean(N); x0 = [e_k, eA_k, am, N, xmicro ]; LB = [2*e_k, 2*eA_k, .5*am, .5*N, 0]; UB = [-4*Tc/z0, .5*eA_k, 2*am, 2*N, 1]; DescriptO=' Combination of Porous Types'; %myfun=@macromicropore4_7_03 elseif POROUSTYPE == 4 x0 = [e_k, eA_k, am, N, xsmall ]; LB = [2*e_k, 2*eA_k, .5*am, .5*N, 0]; UB = [-4*Tc/z0, .5*eA_k, 2*am, 2*N, 1]; DescriptO=' Multi size Pore '; %myfun=@dualmicropore4_7_03 elseif POROUSTYPE == 5 x0 = [e_k, eA_k, am, xsmall ]; LB = [2*e_k, 2*eA_k, .5*am, 0]; UB = [-4*Tc/z0, .5*eA_k, 2*am, 1]; DescriptO=' Multi size Pore (Fixed N)'; %myfun=@dualmicropore5_13_03 elseif POROUSTYPE == 6 88 x0 = [eA_k, am, xsmall ]; LB = [2*eA_k, .5*am, 0]; UB = [.5*eA_k, 2*am, 1]; DescriptO=' Multi size Pore (Fixed N,e)'; %myfun=@dualmicropore5_22_03 elseif POROUSTYPE == 7 x0 = [eA_k, am]; LB = [2*eA_k, .5*am]; UB = [.5*eA_k, 2*am]; DescriptO=' Micropore and Outer Surface (Fixed N,e,x)'; %myfun=@dualmicropore5_23_03 elseif POROUSTYPE == 8 x0 = [eA_k, am]; LB = [2*eA_k, .5*am]; UB = [.5*eA_k, 2*am]; DescriptO=' Multi size Pore and Outer Surface(Fixed N,e,x)'; %myfun=@distribution5_23_03 elseif POROUSTYPE == 9 x0 = [eA_k, am, rhomax]; LB = [2*eA_k, .5*am, .4*rhomax]; UB = [.5*eA_k, 2*am, 1.1*rhomax]; DescriptO=' Multi size Pore and Outer Surface (Fixed N,e,x, Change rhomax)'; %myfun=@distribution5_27_03 89 end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Call function to fit selected Parameters if CHOICEFIT==2 x=x0; elseif CHOICEFIT==1 if POROUSTYPE~=9 if MAPPING==1 xbexp=rho./(rhomax); DescriptM= ' One Term '; elseif MAPPING==2 xbexp=1./(1+(rhoc*(rhomax-rho))./((rhomax-rhoc)*rho)); DescriptM = ' Two Terms '; end if xbexp(1) == 0; xbexp(1) = eps; end end if POROUSTYPE==1 x = lsqcurvefit(@macropore4_7_03,x0,xbexp,G,LB,UB,options,Tkelvin,z0,z1); elseif POROUSTYPE==2 x = lsqcurvefit(@micropore4_7_03,x0,xbexp,G,LB,UB,options,Tkelvin,z0,z1); elseif POROUSTYPE==3 x = lsqcurvefit(@macromicropore4_7_03,x0,xbexp,G,LB,UB,options,Tkelvin,z0,z1); 90 elseif POROUSTYPE==4 x = lsqcurvefit(@dualmicropore4_7_03,x0,xbexp,G,LB,UB,options,Tkelvin,z0,z1); elseif POROUSTYPE==5 x = lsqcurvefit(@dualmicropore5_13_03,x0,xbexp,G,LB,UB,options,Tkelvin,z0,z1,N); elseif POROUSTYPE==6 x = lsqcurvefit(@dualmicropore5_22_03,x0,xbexp,G,LB,UB,options,Tkelvin,z0,z1,N,e_k); elseif POROUSTYPE==7 x = lsqcurvefit(@dualmicropore5_23_03,x0,xbexp,G,LB,UB,options,Tkelvin,z0,z1,N,e_k,xs mall); elseif POROUSTYPE==8 x = lsqcurvefit(@dualmicropore5_23_03,x0,xbexp,G,LB,UB,options,Tkelvin,z0,z1,N,e_k,xd istribution); elseif POROUSTYPE==9 x = lsqcurvefit(@dualmicropore5_27_03,x0,rho,G,LB,UB,options,Tkelvin,z0,z1,N,e_k,xdistr ibution,MAPPING,rhoc); end end 91 x=real(x); p0(COUNTER,:)=x0; p(COUNTER,:) =x; %Parameters solved %Plot the Results if length(CHOICE)>=5 & length(CHOICE)<=6 subplot(2,3,COUNTER) elseif length(CHOICE)>=3 & length(CHOICE)<=4 subplot(2,2,COUNTER) else figure(COUNTER) end stepsize=.01; xb=0 : stepsize : 1.0-stepsize; xb(1)=eps; %Fraction of sites occupied in bulk (concentration) if MAPPING==1 rhobtheory = xb*rhomax; elseif MAPPING==2 rhobtheory = (rhomax*rhoc*xb)./(rhomax*(1-xb)-rhoc*(1-2*xb)); end if POROUSTYPE == 1 Gamma = macropore4_7_03(x, xb, Tkelvin, z0, z1); 92 elseif POROUSTYPE == 2 Gamma = micropore4_7_03(x, xb, Tkelvin, z0, z1); elseif POROUSTYPE == 3 %Gamma = macromicropore4_7_03(x, xb, Tkelvin, z0, z1); xinside(1:2) = x(1:2); xinside(3)=x(3)*x(5); xinside(4)=x(4); xoutside(1:2) = x(1:2); xoutside(3)=x(3)*(1-x(5)); Gamma1 = micropore4_7_03(xinside, xb, Tkelvin, z0, z1); Gamma2 = macropore4_7_03(xoutside, xb, Tkelvin, z0, z1); Gamma = Gamma1 + Gamma2; insideis = 'Microporous Adsorption'; outsideis = 'Macroporous Adsorption'; elseif POROUSTYPE == 4 %Gamma = dualmicropore4_7_03(x, xb, Tkelvin, z0, z1); xinside(1:2) = x(1:2); xinside(3)=x(3)*x(6); xinside(4)=x(4); xoutside(1:2) = x(1:2); xoutside(3)=x(3)*(1-x(6)); xoutside(4)=x(5); Gamma1 = micropore4_7_03(xinside, xb, Tkelvin, z0, z1); Gamma2 = micropore4_7_03(xoutside, xb, Tkelvin, z0, z1); Gamma = Gamma1 + Gamma2; insideis = 'Adsorption in Smaller Channels'; outsideis = 'Adsorption in Bigger Channels'; elseif POROUSTYPE == 5 %Gamma = dualmicropore5_13_03(x, xb, Tkelvin, z0, z1); xinside(1:2) = x(1:2); xinside(3)=x(3)*x(4); xinside(4)=N(1); 93 xoutside(1:2) = x(1:2); xoutside(3)=x(3)*(1-x(4)); xoutside(4)=N(2); Gamma1 = micropore4_7_03(xinside, xb, Tkelvin, z0, z1); Gamma2 = micropore4_7_03(xoutside, xb, Tkelvin, z0, z1); Gamma = Gamma1 + Gamma2; insideis = 'Adsorption in Smaller Channels'; outsideis = 'Adsorption in Bigger Channels'; elseif POROUSTYPE == 6 %Gamma = dualmicropore5_22_03(x, xb, Tkelvin, z0, z1, N, e_k); xinside(1) = e_k; xinside(2) = x(1); xinside(3)=x(2)*x(3); xinside(4)=N(1); xoutside(1) = e_k; xoutside(2)=x(1); xoutside(3)=x(2)*(1-x(3)); xoutside(4)=N(2); Gamma1 = micropore4_7_03(xinside, xb, Tkelvin, z0, z1); Gamma2 = micropore4_7_03(xoutside, xb, Tkelvin, z0, z1); Gamma = Gamma1 + Gamma2; insideis = 'Adsorption in Smaller Channels'; outsideis = 'Adsorption in Bigger Channels'; elseif POROUSTYPE == 7 %Gamma = dualmicropore5_23_03(x, xb, Tkelvin, z0, z1, N, e_k, xsmall); xinside(1) = e_k; xinside(2) = x(1); xinside(3)=x(2)*xsmall; xinside(4)=N(1); xoutside(1) = e_k; xoutside(2) = x(1); xoutside(3)=x(2)*(1-xsmall); xoutside(4)=N(2); Gamma1 = micropore4_7_03(xinside, xb, Tkelvin, z0, z1); Gamma2 = micropore4_7_03(xoutside, xb, Tkelvin, z0, z1); 94 Gamma = Gamma1 + Gamma2; insideis = 'Adsorption in Smaller Channels'; outsideis = 'Adsorption in Bigger Channels'; elseif POROUSTYPE == 8 | POROUSTYPE==9 %Gamma = distribution5_23_03(x, xb, Tkelvin, z0, z1, N, e_k, xdistribution); xsmaller(1) = e_k; xsmaller(2) = x(1); xsmaller(3)=x(2)*xdistribution(1); xsmaller(4)=N(1); xlarger(1) = e_k; xlarger(2) = x(1); xlarger(3)=x(2)*xdistribution(2); xlarger(4)=N(2); xoutside(1) = e_k; xoutside(2) = x(1); xoutside(3)=x(2)*xdistribution(3); Gamma1 = micropore4_7_03(xsmaller, xb, Tkelvin, z0, z1); Gamma2 = micropore4_7_03(xlarger, xb, Tkelvin, z0, z1); Gamma3 = macropore4_7_03(xoutside, xb, Tkelvin, z0, z1); Gamma = Gamma1 + Gamma2 + Gamma3; smalleris = 'Adsorption in Smaller Channels'; largeris = 'Adsorption in Bigger Channels'; outsideis = 'Adsorption on Outer Surface'; end %Before Plotting, this for loop makes sure that all of Gamma is real. %If it is not, it is an indication that the fluid is not stable at that %temperature and pressure (sub critical fluid). for count = 1:length(Gamma) 95 if imag(Gamma(count)) > .00001 Gamma(count) = 100; end end %Axis Variable for Plotting rhomax=rho(length(rho)); Gmax=1.5*max(G); %Plot if POROUSTYPE==1 | POROUSTYPE==2 plot(rho,G,'r.',rhobtheory,Gamma,'b-',rhoc*[1,1],[0,Gmax],'k:') legend('Experiment', 'Theory') elseif POROUSTYPE==3 | POROUSTYPE==4 | POROUSTYPE==5 | POROUSTYPE==6 | POROUSTYPE==7 plot(rho,G,'r.',rhobtheory,Gamma,'b- ',rhobtheory,Gamma1,'b:',rhobtheory,Gamma2,'b--',rhoc*[1,1],[0,Gmax],'k:') legend('Experiment', 'Theory',insideis,outsideis) elseif POROUSTYPE==8 | POROUSTYPE==9 plot(rho,G,'r.',rhobtheory,Gamma,'b- ',rhobtheory,Gamma1,'b:',rhobtheory,Gamma2,'b--',rhobtheory,Gamma3,'b- .',rhoc*[1,1],[0,Gmax],'k:') legend('Experiment', 'Theory',smalleris,largeris,outsideis) end 96 ylabel ('Gamma (mmol/g)') xlabel ('Density (mole/L)') axis([0,rhomax,0,Gmax]) Titlestring=strcat(DescriptA,num2str(Tcelsius),'^oC'); title(Titlestring) end 97 APPENDIX B ALGORITHM OF SIMULATION 98 Grand Canonical Monte Carlo method The Grand Canonical Monte Carlo (GCMC) method is used in the pure fluid simulation. In the Grand Canonical ensemble, chemical potential µ, system volume V, and temperature T are fixed while density ρ fluctuates. The partition function of the grand canonical ensemble in classic mechanics form is βµN 1 e −βE()N.R( N ) ()N Ξ()µ,V, β = .... e dR [1] ∑ 3N ∫∫V N N N! Λ where N is the number of molecules, R is the three dimensional vector representing the spatial configuration of the N molecules, β = 1/ k BT (k B is the Boltzmann constant and T is the temperature), and Λ is the thermal de Broglie wavelength. 1/ 2 h 2 Λ = [2] 2πmk BT And the average value of any thermodynamic property A is βµN 1 1 e −βE()N ,R( N ) ()N < A >= .... A()N,R e dR [3] ∑ 3N ∫∫V N Ξ N N! Λ In simulation, the integration is solved by the “importance sampling technique”. In this technique, a sequence of system configurations (or microstates) is chosen randomly from a distribution (Allen 1987). A Markov chain of states is used to construct that sequence. Then the average of property A is (Equation 3) is calculated using the configurational mean of that chain of states. 99 In the simulation, the cut–and-shifted Lennard-Jones 12-6 potential (Equation 4) is used to calculate the intermolecular interaction. 12 6 σ 12 σ 6 σ σ v(r) = 4ε − - 4ε − r < rc r r r r [4] C C v(r) = 0 r ≥ rc where v(r) is the interaction function, ε is the interaction parameter, σ is the diameter of molecule, r is the distance between molecules, and rc = 4σ is the cut-off distance. Configurational energy is calculated by summing up the intermolecular interaction of each pair of molecules. Periodic boundary conditions are used in three dimensions and the minimum image convention (Allen 1987) is used in calculation of the configurational energy. The simulation starts with a microstate consisting of 5 molecules (here, 5 is arbitrarily selected). The initial positions of these 5 molecules are randomly selected within the simulation box. The GCMC approach includes three possible moves: a molecule is displaced, a molecule is created, and a molecule is destroyed (Allen 1987). After any move, a hypothetical microstate n is generated from the original microstate m. If the move is accepted, then n is a new microstate; otherwise n is the same as m. The move is conditionally accepted with a probability shown in Equation 5. p(n) p()m → n = min1, [5] p()m where p(n) and p(m) represent probability of observing microstates n and m, and are calculated for each possible move as discussed in the following paragraphs. 100 In a displacement move, a molecule is randomly selected and moved to a new site within a distance of δrmax. If the configuration energy of the new microstate m, vm , is less than or equal to that of microstate n ( vm ≤ vn ), the move is accepted. If δvnm = vm − vn > 0 , then the move is accepted with a probability p()n exp()− βvn = = exp()− βδvnm [6] p()m exp()− βvm p()n / p (m ) is compared with a random number ξ between (0,1). If p()n / p (m )> ξ , then the move is accepted; otherwise, the move is denied. In the creation move, a molecule is created at a randomly selected site with the same criteria as in the displacement move except the acceptance probability is p()n = exp()− βδv + ln()zV /()N +1 [7] p()m nm where z is Λ−3 exp()βµ . In the destruction move, a randomly selected molecule is removed and the acceptance probability is: p(n) = exp()− βδv + ln()N / ZV [8] p(m) nm In the pour fluid simulation, all independent variables are used in reduced forms * * 3 3 T = k BT /ε , V = V /σ , and z* = zσ = exp(βµc ), where µc is the configurational chemical potential, and ε and σ are set to 1 in order to simplify the calculation. The average property A is calculated with Equation 9 as below: 101 1 s A = ∑ A(t) [9] s t=1 where s is the total number of configurations, and A(t ) is the value of property A for configuration t . From the molecular simulation, the density, pressure, and compressibility of the simulation system can be obtained. First, after the simulation, the average number of molecules in the simulation system is equal to the configurational average number Thus the density of fluid is < N > ρ = [10] V Also, we can calculate the pressure directly from the simulation. The intermolecular pair virial function w(r) is calculated in the simulation (Equation 11): 12 6 dv(r) σ σ w(r) = r = −4ε 12 − 6 [11] dr r r and the internal virial W is(Allen 1987) 1 W = − ∑∑w(rij ) [12] 3 iij > With the internal virial, the pressure of the system can be calculated as shown in Equation 13. PV =< N > k BT + < W > [13] 102 After each simulation, we obtain a number distribution of the simulation system at a specific temperature and chemical potential. The compressibility of the fluid can be calculated directly from the number distribution, using Equation 14. −1 ∂V KT = [14] V ∂P T ,N From Equation 14, we get, 1 ∂V ∂(1/ ρ) 1 ∂ρ KT = − = −ρ = [15] V ∂P T ,N ∂P T ρ ∂P T where ρ is a function of T and µ in the grand canonical ensemble. For isothermal conditions, we have ∂ρ dρ = dµ [16] ∂µ T Since ∂ρ ∂ρ ∂µ 1 ∂ρ = = [17] ∂P T ∂µ T ∂P T ρ ∂µ T We have 1 ∂ρ V 2 1 ∂N V ∂N K = = = [18] T 2 2 2 ρ ∂µ T N V ∂µ T ,V N ∂µ T ,V From the Grand Canonical partition function,We obtain ∂ ln Ξ = β < N > [19] ∂µ T ,N 103 and ∂ 2 ln Ξ =< (βN) 2 > − < βN > 2 [20] ∂µ 2 Substituting Equation 19 into 20 yields: ∂ < N > = β (< N 2 > − < N > 2 ) [21] ∂µ Then, the compressibility is, βV K = (< N 2 > − < N > 2 ) [22] T < N > 2 Transition Matrix Monte Carlo method Usually, the GCMC simulation is executed to obtain the liquid-vapor coexistence line and from that an estimate of the critical temperature is calculated, but this method is time- consuming. Here, we use Transition Matrix Monte Carlo (TMMC) method (Errington 2003). The TMMC method is developed from the traditional GCMC and most of its algorithm is the same as that of the GCMC. In the Transition Matrix Monte Carlo Method (TMMC), a collection matrix C(M → N ) is inserted to obtain the macrostate probability (Errington 2003), as shown in Equations 23 and 24. C()()M → N = C M → N + p(m → n) [23] C()()M → M = C M → M +1− p(m → n) [24] 104 Then, the transition probability between macrostates N and M is C(M → N ) P()M → N = [25] C M → M + ∆M ∑∆M () In the calculation of liquid-gas coexistence data, the system goes through a low probability area. In order to sample the whole system efficiently, a bias function (Equation 26) is inserted and Equation 5 is replaced by Equation 27. The bias function is used to calculate acceptance only, the unbiased acceptance is still used to calculate the collection matrix. η(N) = −ln P(N) [26] η(N) p(n) p(M → N) = min1, [27] η(M ) p(m) In the TMMC simulation, we start with zero molecules, and assign a probability of 1 to the arbitrarily selected macrostate N = 5. We can calculate the probability of other macrostates as follows: P(N → N +1) ln P(N +1) = ln P(N) + ln [28] P(N +1 → N) After we get the relative probability distribution of macrostates at a specific chemical potential µ0 , we can shift the distribution by adjusting the chemical potential, as shown in Equation 29. ln P(N : µ) = ln P(N : µ0 ) + β (µ − µ0 )N [29] 105 Shifting the chemical potential causes the peaks in the probability distribution to change relative to each other. The coexistence point for a specific temperature is obtained when the volumes of the two peaks of the bimodal number distribution are the same. With this method, we can calculate the system density and system pressure at any chemical potential. The simulation methodology for adsorption is the same as that for the pure fluid. Only the calculation of configurational energy is different. 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