Investigation of adsorption and phase transition phenomena in porous media

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Content INVESTIGATION OF ADSORPTION AND PHASE TRANSITION
PHENOMENA IN POROUS MEDIA
by
Jiyue Wu
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(PETROLEUM ENGINEERING)
December 2024
Copyright 2024 Jiyue Wu

ii
Acknowledgments
Reflecting on my doctoral journey fills me with profound gratitude as I conclude this
chapter of my life. This period of growth and learning has been the most transformative experience,
where moments of uncertainty were met with unwavering support from my advisors, professors,
colleagues, friends, and family. Their belief in me has been the bedrock of my perseverance and
personal development.
I am immensely grateful to my advisors, Prof. Theo Tsotsis and Prof. Kristian Jessen, for
not only providing me with the opportunity to pursue advanced studies but also for guiding me
toward meaningful research. Their patience and selfless guidance have transformed me from a
person who could not speak up to one who is confident in presenting publicly. Their dedication to
their work and passion for their research astounded me and opened my eyes to a new world,
showing me what an academic role model looks like. Their continued commitment to research has
inspired me during challenging times and instilled in me a deep sense of purpose and determination.
Their mentorship is a gift that I will forever cherish.
I extend my heartfelt thanks to Prof. Doug Hammond for his insightful feedback and
support during my qualifying exam and dissertation. I am also grateful to Prof. Iraj Ershaghi for
his inspiring courses and conversations during my master's studies, which encouraged me to pursue
my doctoral degree. My gratitude also goes to Prof. Birendra Jha for his courses, care, and support,
as well as for serving on my qualifying exam committee.
Additionally, I want to express my appreciation to Prof. Jincai Chang, Shokry Bastorous,
and Tina Silva. Their guidance while I served as a teaching assistant in the lab has been invaluable.

iii
I am also thankful to Tina Silva and Shokry Bastorous for their continuous support in the lab,
enabling me to conduct experiments smoothly and safely.
I am grateful for the financial support and resources provided for my research by the Viterbi
School of Engineering, the Mork Family Department of Chemical Engineering and Materials
Science, the University of Southern California's Center for Advanced Research Computing
(CARC), and the Center for Mechanistic Control of Unconventional Formations (CMC-UF), an
Energy Frontier Research Center funded by the U.S. Department of Energy.
I would also like to express my gratitude to my colleagues and friends, especially Sun Lin,
Sheng Hu, Linghao Zhao, Ye Lyu, Jingwen Gong, Zehan Yu, Jingzhe Zhang, Zhuofan Shi,
Mingyuan Cao, Dongwan Xu, Mohammad Bazmi, Saeed Alahmari, Mohammed S. Raslan, and
all others whose names are not mentioned. Their companionship and care have brought me
constant joy and inspiration. Their encouragement and support have been a vital part of my Ph.D.
journey.
To my beloved parents, Jianping Song and Gaofeng Wu, whose love and support have been
my anchor through the storms of this journey, especially for my Mom, I am forever grateful. Your
belief in me has been a source of unwavering motivation, guiding me through the highs and lows
of this challenging journey. Your selfless love and encouragement have been my safe haven in
times of difficulty. Thank you for always being there for me, for believing in my dreams, and for
instilling in me the courage to pursue them.
To my beloved Zhongqi Liu, your unwavering love, support, and belief in me, even when
I doubted myself, has been a constant source of strength. Thank you for being my rock and my

iv
biggest cheerleader. I am grateful for every moment we have shared and look forward to many
more adventures together.
Lastly, I extend my gratitude to all those who have contributed to this dissertation and
supported me along the way. Your belief in me has made all the difference, and I am deeply
thankful for your support.

v
Table of Contents:
Acknowledgments............................................................................................................. ii
List of Tables:.................................................................................................................. vii
List of Figures: ................................................................................................................ vii
Abstract: ............................................................................................................................ x
Chapter 1: Introduction ............................................................................................................... 1
1.1 Motivation.............................................................................................................. 1
1.2 Organization of the dissertation .......................................................................... 4
1.3 References.............................................................................................................. 7
Chapter 2: A new approach to study adsorption on Shales and other microporous solids
via the thermogravimetric analysis (TGA) technique ............................................................. 15
2.1 Introduction......................................................................................................... 15
2.2 Experimental ............................................................................................................. 20
2.3 Results and discussion .............................................................................................. 23
2.4 Conclusions................................................................................................................ 41
2.5 References.................................................................................................................. 42
2.6 Appendix.................................................................................................................... 45
Chapter 3: Experimental Study of Capillary Condensation in model Mesoporous Silicas
....................................................................................................................................................... 53

vi
3.1 Introduction............................................................................................................... 53
3.2 Experimental ............................................................................................................. 54
3.3 Results and discussion .............................................................................................. 56
3.4 Summary.................................................................................................................... 79
3.5 References.................................................................................................................. 80
3.6 Appendix.................................................................................................................... 80
Chapter 4: Exploring Ethane's Phase Behavior in MCM-41 ................................................. 82
4.1 Introduction............................................................................................................... 82
4.2 Experimental ............................................................................................................. 87
4.3 Simulations ................................................................................................................ 88
4.4 Results and Discussions............................................................................................ 91
4.5 Conclusions.............................................................................................................. 106
4.6 References................................................................................................................ 108
Chapter 5. Summary ................................................................................................................ 119

vii
List of Tables:
Table 3-1. Summary of sample characterization .......................................................................... 55
Table 3-2. C2H6 excess adsorption vs. pressure at 50°C............................................................... 58
Table 3-3. C2H6 excess adsorption vs. pressure at 20°C............................................................... 60
Table 3-4. C2H6 excess adsorption vs. pressure for three isobaric experiments........................... 61
Table 3-5. Comparison between the operating conditions for C2H6 and CO2 .............................. 75
Table 4-1. Lennard-Jones potential parameters used in the simulations...................................... 89
Table 4-2. Phase transition pressures at various temperatures as predicted by the GaugeGEMC simulations...................................................................................................................... 100
List of Figures:
Figure 2-1. Shale cube sample prepared for this study................................................................. 21
Figure 2-2. Schematic of the experimental set-up ........................................................................ 22
Figure 2-3. He adsorption in shale, mapp and pressure vs. time during a pressure step change
from 1 to 2 bar............................................................................................................................... 25
Figure 2-4. He adsorption in shale, mapp vs. ρg,He and pressure at equilibrium from vacuum to
15 bar ............................................................................................................................................ 27
Figure 2-5. Schematic of the mapp and pressure vs. time response ............................................... 28

viii
Figure 2-6. The three different operating configurations of the magnetic suspension balance
(Dreisbach & Lösch, 2000)........................................................................................................... 31
Figure 2-7. The comparison between dynamic Ar adsorption data as corrected by the MSB
software and the proposed new ZP correction method. mapp vs. time during a pressure step
change from 2 to 5 bar .................................................................................................................. 35
Figure 2-8. Ar adsorption in shale, mapp and pressure vs. time during a pressure step change
from vacuum to 2 bar.................................................................................................................... 36
Figure 2-9. Ar adsorption in shale, mapp vs. time in the region near the minimum point............ 38
Figure 2-10. The linear plot of ln(α) vs. (t-t1)............................................................................... 39
Figure 2-11. Comparison of excess adsorption isotherms calculated with different techniques.. 40
Figure 3-1. C₂H₆ phase diagram and measurement conditions for synthetic sample ................... 56
Figure 3-2. C2H6 excess adsorption vs. pressure and density at 50°C.......................................... 58
Figure 3-3. Excess C2H6 adsorption vs. pressure and density at 20 ℃ (red line: saturation
pressure at 20 ℃).......................................................................................................................... 59
Figure 3-4.C2H6 excess adsorption vs. temperature for the three isobaric experiments
(saturation temperature: 35 bar-red, 40 bar-green and 45 bar-blue)............................................. 61
Figure 3-5. Condensation pressure calculated by the Kelvin equation and PR-EOS method for
capillaries with various pore diameters, as indicated on the Figures............................................ 66
Figure 3-11. C2H6 and N2 cryogenic isothermal adsorption and desorption experiments............ 73
Figure 3-15. Multilayer growth..................................................................................................... 78
Figure 4-1. Chemical potential vs. pressure calculated from NPT simulations at various
temperatures.................................................................................................................................. 91

ix
Figure 4-2. C2H6 temperature vs. vapor and liquid density data at saturation pressure were
calculated from the GEMC simulations in the bulk phase from 183 K to 294 K compared to
the NIST database......................................................................................................................... 92
Figure 4-3. C2H6 density vs. pressure in the bulk phase calculated from the GCMC simulations
compared to the NIST database. ................................................................................................... 93
Figure 4-7. The comparison of the GCMC simulations, experimental results, and the data in
the literature at 283.8 K and 303.2 K............................................................................................ 97
Figure 4-8. The comparison between the Gauge-GEMC and GCMC simulations at 174.2 K,
184.2 K, 214.2 K, and 235 K. ....................................................................................................... 98
Figure 4-9. The comparison between the Gauge-GEMC and GCMC simulations at 244.2 K,
264.75 K, 273.2 K, and 283.8 K. .................................................................................................. 99
Figure 4-10. The comparison between the saturation pressure in the bulk phase and the phase
transition pressure in the pore ..................................................................................................... 100
Figure 4-11. The comparison between the Gauge-GEMC simulations (Square and circle) and
Sharma et al.’s (Sharma et al., 2023) extrapolation results (Solid lines).................................... 102
Figure 4-12. The molecular distribution in the pore at various temperatures. The top and
bottom curves in each subfigure represent the molecular distribution profile after and before
the phase transition pressure in the pore..................................................................................... 104

x
Abstract:
The importance of shale reservoirs in oil and gas production has made it necessary to have
a deeper understanding of the mechanisms governing the behavior of hydrocarbons within these
unconventional resources. This dissertation explores the mechanism behind the adsorption,
desorption, and phase behavior in the context of shale gas extraction and production, aiming to
enhance the efficiency and accuracy of the estimation of the gas in place and contribute to the
acquisition of more accurate parameters for use in reservoir simulators.
Shale gas reservoirs, characterized by their fine-grained sedimentary rock formations and
diverse range of pore sizes and compositions, present unique challenges in oil and gas production.
The hydrocarbons in these reservoirs typically exist in two distinct states: free gas occupying
confined pore spaces and gas adsorbed on the pore surfaces or bound to the organic matter within
the shale matrix. Understanding the dynamics of adsorption and desorption are crucial for
optimizing the production capacity of shale gas reservoirs, as these processes significantly
influence the total gas content and its subsequent release during production.
This dissertation focuses on two different but interrelated research topics: the study of
adsorption and desorption processes in porous media and phase transition phenomena in such
materials. Under the first topic, this dissertation introduces a novel approach for measuring gas
adsorption in shales, that does not require the traditional use of helium (He) for determining the
skeletal volume of the microporous solid. We apply the technique to accurately measure the
adsorption of argon (Ar), as a model gas, within the nanopores of Marcellus Shales by the
gravimetric method. Additionally, the dissertation also presents a new technique for directly
calculating the volume of the adsorbed layer (Va), a yet intractable challenge in the adsorption

xi
area, based on dynamic adsorption data collected using volumetric adsorption measurements. The
technique is then applied in the study of ethane (C2H6) and carbon dioxide (CO2) in the wellcharacterized porous material, namely, MCM-41. The experimental Va data are compared with
the findings from molecular simulations of the material.
For the second topic, this dissertation concentrates on an experimental exploration of
capillary condensation within mesoporous silica materials. Specifically, it studies the sorption
behavior of three gases: nitrogen (N₂), C₂H₆, and CO₂. The investigation revolves around two
varieties of porous silica, each with an average diameter of about 4 nm and 7 nm. The study is
reinforced by thermodynamic calculations and employs a geometric pore filling model to interpret
the observed experimental results.
Under the second topic, the dissertation investigates the phase transition behavior of C2H6,
within the nanopore structure of MCM-41. We employ a combination of molecular simulation
methods including Grand Canonical Monte Carlo (GCMC), Gauge Cell Gibbs Ensemble Monte
Carlo (Gauge-GEMC) and isobaric-isothermal ensemble Monte Carlo (NPT). The study predicts
the conditions under which phase transitions take place within these materials and also calculates
the fluid distribution within the nanopores, thus contributing to a better understanding of fluid
properties under pore confinement, which is essential for optimizing reservoir simulations in the
oil and gas industry. Unfortunately, experiments with C2H6 adsorption in the same materials failed
to unambiguously detect the presence of phase transitions. We attribute this to the inevitable poresize distribution of the sample. The reason is that we observe that the capillary condensation step
on the experimental isotherms is gradual, which is expected for a phase transition in a system
consisting of numerous pores with inevitable variations in pore width.

xii
To summarize, this dissertation provides valuable insights into the adsorption and phase
transition phenomena in porous materials, with significant implications for enhancing the
efficiency of shale gas production operations. The findings contribute to the development of more
precise models for predicting gas storage capacities in shale formations and offer potential
improvements for adsorption-based processes in porous materials for various other technical areas.

1
Chapter 1: Introduction
1.1 Motivation
1.1.1. The importance of shale reservoirs in the oil and gas production.
Unconventional shale gas and oil reservoirs, have played a crucial role in oil and gas
production in the United States. Tight shale oil and gas have become a significant energy source
over the past few years. Shale formations with substantial accumulations of natural gas (NG) and
oil are distributed among 30 states in the U.S. The Barnett Shale in Texas , for example, has been
a major source of NG production for over a decade now, serving as a foundational source of
technological knowledge for the development of other shale plays. Currently, the Marcellus shale
play, spanning the states of Ohio, Pennsylvania, and West Virginia in the Appalachian Basin, is
the largest supplier of NG among all known US shale formations.
The development of hydraulic fracturing and horizontal drilling, resulting in significantly
increased production capacity has made it possible to take advantage and utilize the valuable shale
resources. The U.S. Energy Information Administration (EIA) projects a 15% increase in U.S. NG
production between 2022 and 2050, with production expected to reach 42.1 trillion cubic feet (TCF)
by 2050. According to projections in the Annual Energy Outlook 2023 (AEO 2023), in 2050 up to
38.74 TCF of U.S. dry NG production is expected to come from shale and tight gas resources,
equivalent to about 92% of the estimated total U.S. dry NG production.
1.1.2. The role of adsorption/desorption during oil and gas production from shale
In shale gas reservoirs, the hydrocarbon content typically exists in two distinct states: free
gas occupying the pore spaces and gas adsorbed on the pore surfaces or found in the organic matter
within the shale matrix. This dual storage mechanism significantly influences the total gas content,

2
with adsorbed gas potentially constituting approximately 40-50% of the total gas in place (Cipolla
et al., 2010). However, current technological limitations restrict the efficient production of the
adsorbed gas. Factors that impact the productive capacity of adsorbed gas include high flowing
bottom hole pressure, the ultra-tight nature of the matrix rock, and a desorption profile that
necessitates reduced pressures for substantial volumes of adsorbed gas to be produced
(Arogundade & Sohrabi, 2012).
The process of desorption plays a critical role in the dynamics of gas flow through the
fracture network of shale gas reservoirs. It involves the release of adsorbed gas and its subsequent
diffusion towards the matrix-fracture interface. This phenomenon becomes increasingly
significant later in the life of the reservoir and can contribute to approximately 5-15% of the total
produced gas volume (Cipolla et al., 2010). Gas desorption, typically, accounts for about 17% of
the Expected Ultimate Recovery (EUR), and helps enhance the final recovery (Thompson et al.,
2011). Factors known to affect the degree of desorption include the reservoir's pressure and
temperature conditions, as well as the properties of the shale matrix and organic matter
(Arogundade & Sohrabi, 2012).
1.1.3. The role of phase behavior during oil and gas production from shale
Shale is a fine-grained sedimentary rock formed from the compaction of silt and clay-sized
mineral particles. It is known for its tendency to break into thin, parallel layers. Black shale, in
particular, contains organic material capable of generating oil and natural gas, which become
trapped within the pores in the rocks. The reservoir rocks of shale formations are characterized by
a diverse range of pore sizes, spanning from nanometers to micrometers. This wide range of pore
sizes lead to complex fluid phase behavior. Macropores and fractures, in the micrometer range,

3
result in the fluid existing in a bulk phase, whereas, in nanopores, the phase behavior is
significantly different from the bulk due to pore confinement effects.
Despite the substantial reserves of hydrocarbons in shale and tight formations, their
production is hindered by several nanoscopic characteristics. These include fine grain sizes, the
presence of nanopores, low porosities ranging from 2% to 10%, very low permeabilities, as well
as complex mineral compositions (Arogundade & Sohrabi, 2012; Du & Chu, 2012; Jin et al., 2013;
Kuila & Prasad, 2013; Nojabaei et al., 2013; Passey et al., 2010; Zhang et al., 2013). Such
properties challenge conventional reservoir evaluation methods, leading to difficulties in
accurately estimating the original hydrocarbons in place, and thus the ultimate recovery potential.
This, in turn, affects the ability to predict a reservoir's economic return. A notable example is the
finding that a detailed understanding of fluid phase behavior in small pores is needed to achieve a
satisfactory history matching for oil production from wells in the middle Bakken formation
(Alharthy et al., 2013; Chen et al., 2012).
Current industrial fluid characterization practices rely on laboratory measurements in a
pressure/volume/temperature (PVT) cell using the bulk fluid to tune the equations of state (EOS)
for simulations. However, conventional bulk PVT cell tests cannot accurately measure the phase
behavior of shale reservoir fluids, as a significant portion of the fluids in shale occupies the nanoscale pore space, and its behavior is likely to be quite different from that of the bulk fluid. Therefore,
investigating the phase behavior of hydrocarbons under nanopore confinement is crucial for
bridging the gap between conventional PVT measurements and actual reservoir fluid behavior in
shale formations (Wang et al., 2016).

4
Because of its significant practical importance, several studies in recent years have
explored the fluid phase thermodynamic behavior in nanopores. The interaction between the fluid
and solid surfaces within these pores leads to confinement effects, thus causing deviations from
bulk fluid phase behavior. Two primary approaches are commonly used to describe these effects.
The first approach treats confinement as a capillarity effect, characterized by capillary pressure,
which is the pressure difference between the liquid and vapor phases. This method often employs
the Young-Laplace equation to express capillary pressure, assuming an ideally smooth and
homogeneous interface between the liquid and vapor phases (Alharthy et al., 2013; Wang et al.,
2016; Zhang et al., 2017). The second approach entails molecular simulations, and offers a more
realistic model of phase behavior under pore confinement (Jin et al., 2017; Jin & Nasrabadi, 2016;
Q. Yang et al., 2019). In this dissertation we make primarily use of the second method. To describe
phase behavior.
1.2 Organization of the dissertation
This dissertation involves the study of adsorption and phase transition phenomena in
porous materials. The dissertation is organized in four Chapters (including this Introduction
Chapter). A brief description of the remaining four Chapters is provided below:
1.2.1 Chapter 2: A new approach to study adsorption on Shales and other microporous solids
via the thermogravimetric analysis (TGA) technique
Measuring the adsorption of gases in microporous solids like the shales requires accurate
knowledge of the solid’s skeletal volume. Helium (He) is commonly used to determine the
sample’s skeletal volume based on the assumption that it does not adsorb in these porous media.
The validity of such an assumption for microporous solids has been questioned in recent years,

5
and in this Chapter, we show that it is not applicable for the shale sample studied. We present a
new method to measure the adsorption of shale-gas components in shales, which does not require
the use of He to measure the solid’s skeletal volume. Since the proposed analysis method relies on
the use of dynamic adsorption data, we also propose here a new zero-point correction method for
the magnetic suspension balance, which is better suited for the analysis of such data. We employ
the new technique to the study of Argon (as a model gas) adsorption in Marcellus Shales.
1.2.2 Chapter 3: Experimental Study of Capillary Condensation in model Mesoporous Silica
This study investigates capillary condensation phenomena in mesoporous materials using
two types of porous silica: AGC-40, a commercial material, and a synthetic silica prepared in our
laboratory, focusing on the sorption behaviors of nitrogen (N2), C2H6, and CO2. Employing both
static adsorption systems and thermogravimetric analysis, the research explores phase transitions
at varying pressures and temperatures. Despite expectations of observing capillary condensation,
the experiments demonstrated no abrupt phase transitions in the tested conditions on the synthetic
silica. However, evaporation phenomena were observed in the nitrogen and C2H6 experiments
under cryogenic conditions. Theoretical calculations using the Kelvin equation and PengRobinson equation of state further supported these findings, suggesting that the gap between the
predicted phase transition line and the saturation pressure line is not wide enough for the synthetic
sample. To better understand and demonstrate the layer filling and emptying process, a geometric
pore filling model was also used to interpret the experimental results.

6
1.2.4. Chapter 4: Exploring Ethane's Phase Behavior in MCM-41 Confinement: Insights from
GCMC and Gauge-GEMC Simulations
Investigating the phase behavior of ethane (C2H6) within nanopore structures is essential
for optimizing production processes in the oil and gas industry. Accounting for phase behavior
changes under pore confinement enhances the accuracy of simulation models for fluid properties.
Therefore, gaining a detailed understanding of the mechanism behind phase transitions using
model materials with regularly shaped and arranged pores is important. However, the conditions
for the phase transition of C2H6 from vapor to liquid uer pore confinement are not completely
understood. This Chapter investigates the phase behavior of C2H6 in MCM-41, a type of ordered
mesoporous silica, using the gravimetric method. Additionally, molecular simulations, including
Gauge-GEMC and GCMC simulations, are employed to predict the adsorption and phase transition
behavior of C2H6 within the MCM-41 structure. The results show that the Van der Waals loop of
the investigated fluid-pore system gradually shrinks with increasing temperature, eventually
vanishing at around 235 K, suggesting that critical conditions are reached under confinement. The
study also incorporates the low-energy barrier theory, explaining that in small the activationenergy barrier for phase transitions is extremely, resulting in spontaneous and seamless transitions
that lack distinct thermodynamic signatures and appear as smooth adsorption processes. This
chapter further provides detailed molecular distribution profiles within the pores under various
pressure and temperature conditions, offering insights into the formation of C2H6 layers and their
dependence on operating conditions. The phase transition pressures predicted by the simulations
align well with data from existing literature, offering valuable validation and additional insights
from a simulation perspective.

7
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15
Chapter 2: A new approach to study adsorption on Shales and other
microporous solids via the thermogravimetric analysis (TGA)
technique1
2.1 Introduction
In the past decade, or so, US shale-gas resources have received great attention due to the
potential for supplying an immense amount of energy for the nation (Lee & Kim, 2016). It is
estimated, for example, that by the year 2040 around two-thirds of the total US natural gas (NG)
production will be from such shale-gas resources (Kenomore et al., 2018). Shale, in which the
shale-gas is stored, is a fine-grained sedimentary rock, that is composed of mud, clays, minerals,
such as quartz and calcite, and organic matter (kerogen) (Blatt et al., 2006). NG in shale-gas
reservoirs is stored as free gas in macropores, in natural and induced fractures, and as adsorbed
gas in the micropores of clays, minerals and organic matter. In addition, gas may also be dissolved
in residual oil and water in the formation although it, typically, represents a small fraction of the
overall NG stored in shales (Curtis, 2002; Tan et al., 2014; Zhang et al., 2013). In the major shale
gas-plays in North America, adsorbed gas represents the major fraction of NG stored there (Hill
& Nelson, 2001). For example, adsorbed gas accounts for 40–60%, 50%, 70–75%, 40–60%, and
60–85%, of the gas stored in the Barnett Shale, the Ohio Shale, the Antrim Shale, the New Albany
Shale and the Lewis Shale (Ratnakar & Dindoruk, 2019), respectively. Therefore, understanding
of the adsorption processes that take place during the production of shale-gas from these and other
1 Part of this section is quoted from our manuscript: Wu, J., Sun, L., Jessen, K., & Tsotsis, T. (2022). A new approach to
study adsorption on shales and other microporous solids via the thermogravimetric analysis (TGA) technique. Chemical
Engineering Science, 247, 117068.

16
shale formations is important in order to determine the total gas in place, and the most efficient
means to transport this gas from the source rock to the wellbore (Rexer et al., 2013).
To measure gas adsorption in solids, one can employ either indirect or direct methods.
Volumetric or manometric methods constitute the indirect route, for which the amount of gas
adsorbed is estimated from the change in gas volume or pressure via an appropriate equation of
state (EOS). The direct route, in contrast, involves directly measuring the change in mass of the
solid due to adsorption by employing a microbalance (Maggs et al., 1960). The indirect methods
require simpler hardware, but they suffer from omnipresent gas leaks, and they are generally less
accurate than the direct methods for measurements over a broad range of pressures. Both
techniques measure the excess adsorption, defined as 𝑚𝑒 = 𝑚𝑎 − 𝑉𝑎 × 𝜌𝑔, where 𝑚𝑎 and 𝑉𝑎 are
the mass and volume of the adsorbed layer, and 𝜌𝑔 the bulk-phase gas density at the experimental
pressure and temperature conditions.
Both methods require knowledge of the so-called “impenetrable” volume (also known as
the skeletal volume) of the solid adsorbent, but accurate measurement of the skeletal volume of
solid adsorbents continues to be a technical challenge. Helium (He) is, typically, used to determine
the impenetrable solid volume (and, thus, the true solid density) under the following assumptions:
(1) That the He molecules are not adsorbed on the solid at the relevant pressure/temperature
measurement conditions; (2) that the volume of solid inaccessible to He and to the gas of interest
are the same. There is mounting evidence today, including the He adsorption data presented here
for the particular shale sample studied, that for several types of solids assumption #1 is not valid.
Schwabe et al. (Maggs et al., 1960), who used He to determine the true solid density of various
carbons employing a volumetric sorption apparatus to measure the so-called dead space

17
(comprised of the free space in the sample cell and pore space within the sample not occupied by
the adsorbate layer) under different temperatures. They found the dead space to decrease with
temperature and interpreted this (Maggs et al., 1960) to be indicative that He adsorption is
occurring. To determine micropore size distributions, Kaneko et al. (Kaneko et al., 1994) measured
gravimetrically He and N2 adsorption isotherms on activated carbon fiber (ACF) materials at 4.2
K and 77 K, respectively. The comparison between the two isotherms showed that the adsorbed
He could assess a larger number of micropores, especially ultra-micropores (pore width <0.7nm).
This indicates that the two gases probe different regions of the pore space at these conditions.
Gumma and Talu (Gumma & Talu, 2003) carried out He adsorption experiments on
silicalite using a magnetic suspension balance (MSB), over a wide range of temperatures and
pressures. They reported that He was adsorbed under certain conditions. Helium is commonly used
for determining the “dead volume” of the sample cell (i.e., the cell volume minus the sample’s
skeletal volume) during adsorption experiments. Subsequently, the sample cell is evacuated for
the experiment to begin. However, Silvestre-Albero et al. (Silvestre-Albero et al., 2013) reported
that He molecules remained adsorbed in micropores at 77.4 K after evacuation, which they
surmised would, likely, influence the subsequent isotherm measurements. Keller and Göbel
(Keller & Göbel, 2015) carried out gravimetric experiments in which they exposed carbon samples
to He over a 26-hour period. They observed a distinct increase of the apparent sample weight,
which they interpreted as a strong indicator that adsorption of He is occurring in the micropores
of the carbon samples.
In view of mounting evidence that He cannot be viewed as a non-adsorbing gas, at least
not under all circumstances, there have been a few efforts in recent years aiming to address the

18
challenge of accurately estimating the impenetrable solid volume. These efforts can be classified
into two categories: Those that try to avoid altogether the need to use He, and efforts that continue
to employ He to determine the impenetrable solid volume, but try to analyze the data, without
resorting to the assumption that He is non-adsorbing gas. Keller and Göbel (Keller & Göbel, 2015)
proposed an oscillometric-volumetric method to measure pure gas adsorption on porous solids.
They demonstrated their method by measuring at near ambient conditions He, CO2, N2 and Ar
adsorption on activated carbon. They employed an elaborate volumetric set-up, in which the
sample cell is connected via a sensitive relief valve to a free-standing cylinder mounted
perpendicularly on the top of the adsorption chamber. A constant flow of gas was fed to the
chamber which increases the pressure to the point that forces the relief valve to open, gas to be
released and the cylinder to move upwards; subsequently as the pressure in the adsorption chamber
drops, the relief valve closes, the cylinder moves downwards and, simultaneously, adsorbed gas
gets released. Then the pressure in the chamber builds-up again, which causes the relief valve to
open again, the cylinder to move upwards, and the cycle to repeat. By monitoring the amplitude
and frequency of the cylinder oscillations via optical techniques, Keller and Göbel (Keller & Göbel,
2015) were able to estimate the quantity of gas adsorbed without needing to assume that He is nonadsorbing. However, the device has only been applied near ambient pressure conditions and has
yet to be demonstrated for the high-pressure conditions relevant to shale-gas formations. Herrera
et al. (Herrera et al., 2011) suggested a new definition of excess adsorption by identifying a new
Gibb’s dividing surface which is a zero-potential surface. This surface separates the accessible and
inaccessible volume to the fluid molecules, and can be, potentially, calculated via molecular

19
simulations for well-defined solid surfaces, but it is unlikely to be applicable to complex porous
solids like the shales.
In the second class of methods, falls the approach proposed by Sircar (Sircar, 2001) that
built upon ideas discussed earlier by Suzuki et al. (Suzuki et al., 1987). Sircar’s paper (Sircar, 2001)
represents a first effort to utilize He adsorption data in order to determine the so-called true Gibb’s
dividing surface. However, his approach relies on the assumption that a temperature can be found
for which He does not adsorb on the material. Gumma and Talu (Gumma & Talu, 2003)
subsequently formulated a self-consistent method, based on the method of Sircar (Sircar, 2001),
which relaxes the need for finding a temperature for which the He does not adsorb. Arami-Niya et
al. (Arami-Niya et al., 2017) used He adsorption and the analysis method of Gumma and Talu
(Gumma & Talu, 2003) to measure the true impenetrable solid volume of natural clinoptilolite and
of various synthetic molecular sieves. They compared the solid volumes determined by employing
the Gumma and Talu (Gumma & Talu, 2003) technique with those generated by the standard He
pycnometry analysis method that is based on the assumption that He is non-adsorbing. The analysis
method of Gumma and Talu (Gumma & Talu, 2003) provided skeletal volumes which were 10-
15% larger than the volumes calculated by the standard technique. Arami-Niya et al. (Arami-Niya
et al., 2017) estimated that such a difference would, likely, cause a 2.6 - 28% uncertainty in the
measurement of the equilibrium adsorption capacities.
The method proposed by Gumma and Talu (Gumma & Talu, 2003) provides a systematic
approach to using He data to estimate the solid’s skeletal volume. However, it requires
experimenting over a broad range of temperatures, which is effort-intensive particularly for
microporous solids for which the times required to reach equilibrium can be quite lengthy

20
(Gasparik et al., 2015). For materials like the shales, that contain organic inclusions, studying them
over a broad range of temperatures over lengthy periods is not always feasible. Lorenz and
Wessling (Lorenz & Wessling, 2013) introduced, most recently, a method that utilizes He data to
calculate the skeletal volume of a polymer solid without needing extensive experimentation over
a broad range of temperatures. The estimated volume was then utilized to calculate CO2 sorption
in the solid. Lorenz and Wessling (Lorenz & Wessling, 2013) noted that through the use of the
new approach they managed to calculate positive sorption capacities, while negative values
resulted when employing the standard approach.
In this paper, we propose a new method for the study of gas adsorption in shales and other
similar microporous materials via the gravimetric technique. The method builds on the idea
originally found in the paper of Lorenz and Wessling (Lorenz & Wessling, 2013), but unlike them
we do not make use of He for the determination of the solid’s impenetrable volume. We present
here an example of the application of the technique for the study of Ar (as a model surrogate gas
for methane, C2H6 and other shale gas components) adsorption in shale utilizing a high-pressure
microbalance. The approach is equally applicable, however, for the study of gas adsorption
employing volumetric/manometric experimental methods. In what follows, we first describe the
materials used followed by a description of the experimental system and procedures. We then
present the experimental data and describe the proposed analysis method.
2.2 Experimental
2.2.1 Materials
The rock sample in this study was extracted from the Marcellus shale formation in the
Appalachian Basin from a depth of 2395.7 m (7860 ft). Further details about the sample’s

21
characteristics have been published elsewhere (Dasani et al., 2017). The particular sample used in
this work is shown in Fig. 2-1, and it is a ~1 cm3
cube that was cut from a larger core via a
mechanical saw. Using such a regularly shaped sample facilitates the modeling of dynamic
adsorption data (Dasani et al., 2017), which, however, is not the key focus of this paper.
Figure 2-1. Shale cube sample prepared for this study
2.2.2 Experimental procedure
A magnetic suspension microbalance (MSB; Rubotherm IsoSORP®, Germany) is used to
study adsorption on shale via thermogravimetric analysis (TGA). This instrument consists of five
main components: The balance mechanism, an electromagnet, a suspension magnet, the measuring
cell where the sample is placed, and the electronic control unit. The electromagnet is attached to
the bottom of the balance and is used to lift the suspension magnet during the measurements. The
suspension magnet consists of a permanent magnet, a sensor core and a measuring load decoupling
device. The electromagnet is controlled by the electronic control unit and maintains the suspension
magnet in a freely-suspended state. Since its balance mechanism is not in direct contact with the
gas atmosphere, the MSB system allows sorption measurements under high pressure and

22
temperature in a variety of gas atmospheres, which presents an advantage over the conventional
microbalances.
The experiments begin by placing the sample (the shale cube in Fig. 2-1 in this case) in the
sample cell and evacuating at a select temperature (50 oC is used here) until the sample cell reaches
30 mTorr in order to remove any impurities that may be adsorbed on the sample (for a schematic
of the overall experimental set-up see Fig. 2-2). This step is considered complete if the balance
weight reading stays constant (less than 10 μg change) for over 1 hour.
Figure 2-2. Schematic of the experimental set-up
Then the system is pressurized by allowing the gas (He or Ar in this study) to flow into the
sample cell till the desired pressure is reached. For supercritical gases like He and Ar, the gas flow
is controlled by a mass flow controller (MFC, Brooks 5850). For more condensable gases (e.g.,
CO2 and C2H6) for measurements at high pressures, we utilize an ISCO pump (Teledyne 260D)

23
with a heated cylinder to attain the desired pressures. During flow experiments, the pressure is
controlled by an electronic back-pressure regulator (BPR, Brooks 5820). The temperature of the
sample is maintained constant (+-0.1 oC) by a circulating oil bath. The adsorption measurements
reported here are carried out by increasing the gas phase pressure in a stepwise manner. During
each sorption experiment, the weight of the sample along with the temperature and pressure in the
sample cell are measured and recorded at pre-determined intervals (every 6 s in this study). Further
details about the MSB operation and method of data analysis are reported below.
2.3 Results and discussion
2.3.1 Measuring adsorption in shale
As noted in Sect. 2.2, the TGA method is used in this study to measure adsorption in shale.
The MSB instrument we employ enables sorption measurements with an accuracy of ~10 μg (the
readability (resolution) of the balance is 1 μg). The balance weighs the sample placed in a cell, as
it is being exposed to various gas atmospheres, by means of magnetic suspension coupling that
transmits the weight force to the microbalance mechanism through a permanent magnet installed
in the measuring cell. This instrument design separates the measuring cell from the balance
mechanism which, in turn, enables the MSB instrument to work under a wide range of pressures
and temperatures and gas atmospheres that could otherwise be detrimental to conventional TGA
instruments.
During an experiment, the MSB instrument records the apparent weight (mass) of the
sample, 𝑚𝑎𝑝𝑝, which is described by the following Eqn. 2-1
𝑚𝑎𝑝𝑝 = 𝑚𝑏 + 𝑚𝑠 + 𝑚𝑎 − 𝜌𝑔
(𝑉𝑏 + 𝑉𝑠 + 𝑉𝑎
) = 𝑚𝑏 + 𝑚𝑠 + 𝑚𝑒 − 𝜌𝑔
(𝑉𝑏 + 𝑉𝑠
) , (2 − 1)

24
where 𝑚𝑏 and 𝑉𝑏 are the mass (weight) and volume of the sample container (termed the
basket), 𝑚𝑠 and 𝑉𝑠 are the mass and (skeletal) volume of the sample itself, 𝑚𝑎 and 𝑉𝑎 are the mass
and volume of the adsorbed layer, and 𝑚𝑒 = 𝑚𝑎 − 𝑉𝑎 × 𝜌𝑔 is the excess adsorption mass. From
Eqn. 2-1, upon rearranging, one can calculate the excess adsorption, assuming that (𝑚𝑏 + 𝑚𝑠
) and
(𝑉𝑏 + 𝑉𝑠
) are known, from the following Eqn. 2.
𝑚𝑒 = 𝑚𝑎𝑝𝑝 − (𝑚𝑏 + 𝑚𝑠
) + 𝜌𝑔
(𝑉𝑏 + 𝑉𝑠
) . (2 − 2)
(𝑚𝑏 + 𝑚𝑠
) can be directly measured by the MSB itself under vacuum conditions but
measuring (𝑉𝑏 + 𝑉𝑠
) is more challenging. As noted in Sect. 2-1, traditionally, this is done by
exposing the sample to He, on the premise that it is a non-adsorbing gas, and using the following
variant of Eqn. 2 to calculate (𝑉𝑏 + 𝑉𝑠
).
𝑚𝑎𝑝𝑝 = 𝑚𝑏 + 𝑚𝑠 − 𝜌𝑔,𝐻𝑒(𝑉𝑏 + 𝑉𝑠
) . (2 − 3)
Typically, one measures the apparent sample weight (𝑚𝑎𝑝𝑝) for a number of He pressures,
spanning the expected range of adsorption measurements, and then plots 𝑚𝑎𝑝𝑝 vs. 𝜌𝑔,𝐻𝑒. Linearity
of the resulting plot is interpreted as validation of the assumption that no He adsorption takes place.
Though this may be true for some materials, for others like the Marcellus shale studied here, as it
will be discussed later in this paper, this is not the case, however. Further, as Dreisbach et al.
(Dreisbach et al., 2002) noted previously, for supercritical, lightly adsorbing gases like He, the
adsorbed amount at equilibrium is, likely, itself to have a linear or near-linear dependence on He
pressure (density), and linearity of the 𝑚𝑎𝑝𝑝 vs. 𝜌𝑔,𝐻𝑒 plot in that case does not necessarily imply
that adsorption is not taking place.
To make this point clear, we present here some of our own measurements with the
Marcellus shale sample. Fig. 2-3 shows 𝑚𝑎𝑝𝑝 and the He pressure as a function of time in an

25
experiment, in which the shale sample was originally equilibrated at a pressure of 1 bar and then
the He pressure was changed in a near-linear fashion to 2 bar. Note that, as expected, initially due
to the increased buoyancy (because of the increase in the bulk fluid density) 𝑚𝑎𝑝𝑝 decreases.
However, after the pressure levels off, 𝑚𝑎𝑝𝑝 instead of also leveling off, as one would have
expected if no adsorption took place, it continues to gradually increase, indicating that He
adsorption does, indeed, take place.
Figure 2-3. He adsorption in shale, mapp and pressure vs. time during a pressure step change from 1 to 2 bar
In Fig. 2-4 we plot 𝑚𝑎𝑝𝑝 at “equilibrium” for a multi-step run vs. the corresponding 𝜌𝑔,𝐻𝑒
at “equilibrium”. Each point (during linear regression) represents an “aggregate” of 500 data points

26
taken at “equilibrium” over a period of one hour during which the apparent weight reading from
the balance changed less than 10 μg. Note that the 𝑚𝑎𝑝𝑝 vs. 𝜌𝑔,𝐻𝑒 relationship appears to be quite
linear providing no indication that He adsorption is, indeed, taking place, as Fig. 2-3 indicates.
Ignoring, however, that He adsorption takes place, and using Eqn. 2-3 to calculate (𝑉𝑏 + 𝑉𝑠
) has
serious implications, in that one ends-up often calculating negative excess adsorption, especially
in the region of higher pressures (Silvestre-Albero et al., 2013), which barring unusual
circumstances (e.g., pore mouth “plugging”) is an implausible finding. This is, indeed, the case in
this paper as well, as it is further detailed later. Finally, it should be noted that even if He does not
adsorb in a microporous solid, it is not entirely clear that what behaves like “impenetrable solid
volume” being subjected to a buoyancy force in a He fluid phase, will also behave as a
“impenetrable solid volume” in a fluid phase consisting of a gas with a larger kinetic diameter (e.g.,
Ar or C2H6), as these two molecules may have access, to different porous regions of the
microporous solid. In fact, we noted in Sect. 2-1 the study by Kaneko et al. (Kaneko et al., 1994),
where this clearly is not the case for He and N2 adsorption under cryogenic conditions on ACF.

27
Figure 2-4. He adsorption in shale, mapp vs. ρg,He and pressure at equilibrium from vacuum to 15 bar
The approach utilized here, instead, for measuring the adsorption of supercritical, lightadsorbing gases in microporous solids, like the shales, is based, as noted in Sect. 2-1, on a method
originally proposed by Lorenz and Wessling (Lorenz & Wessling, 2013). These authors used the
TGA technique to study absorption in polymeric materials. For the study of absorption in dense
solids, like polymers, Eqn. 2-1 applies equally well. However, for this case the 𝑉𝑎 rather than being
the volume of the adsorbate layer, as in the case of microporous solids, it now represents the change
in the volume of the solid due to swelling as a result of sorption. Again the balance measures
𝑚𝑎 − 𝑉𝑎 × 𝜌𝑔 via Eqn. 2-3, and for being able to calculate 𝑚𝑎, if swelling is indeed present, one
must be able to measure 𝑉𝑎 via a different technique. We note that for solids like the shale studied

28
here that contain organic inclusions, 𝑉𝑎 likely reflects both the volume of the adsorbate layer in the
clay and mineral fraction and swelling due to sorption in the organic phase.
Figure 2-5. Schematic of the mapp and pressure vs. time response
Lorenz and Wessling (Lorenz & Wessling, 2013) noted that for some polymers, He
sorption behavior during a step change in between two different pressure levels looks as in Fig. 2-
5, which is quite similar to the He adsorption behavior we observed with the Marcellus shale (see
Fig. 2-3). They, then, made the assumption that no He was sorbed prior to the minimum point in
the weight vs. time response. The analysis to follow is adapted from their original paper (Lorenz
& Wessling, 2013).
Since, to begin with, one measures (𝑚𝑏 + 𝑚𝑠) under vacuum, the initial step in the

29
adsorption experiment involves pressurizing the sample cell from vacuum to the first pressure level
chosen. At the minimum point of the weight vs. time response (see Fig. 2-5), the apparent sample
mass 𝑚𝑎𝑝𝑝,1, based on the assumption that no gas adsorption has taken place prior to that point, is
described by Eqn. 2-4
𝑚𝑎𝑝𝑝,1 = (𝑚𝑏 + 𝑚𝑠
) − (𝑉𝑏 + 𝑉𝑠
) × 𝜌𝑔,1m, (2 − 4)
where 𝜌𝑔,1𝑚 denotes the density of the gas at the first minimum point.
After reaching the equilibrium state at the first pressure level, the apparent weight, 𝑚𝑎𝑝𝑝,1
𝑓
,
is given by Eqn. 2-5.
𝑚𝑎𝑝𝑝,1
𝑓 = (𝑚𝑏 + 𝑚𝑠
) + 𝑚𝑎,1 − (𝑉𝑏 + 𝑉𝑠 + 𝑉𝑎,1) × 𝜌𝑔,1𝑓， (2 − 5)
where 𝑚𝑎,1 is the equilibrium absolute adsorption mass, 𝑉𝑎,1 the volume of the adsorbate
layer at equilibrium (which may also, potentially, encompass the swelling of the organic phase of
the shale, as noted previously). 𝜌𝑔,1𝑓 is the density of gas at equilibrium at the first pressure step,
which may differ from 𝜌𝑔,1𝑚 due to the slight changes of temperature. Subtracting Eqn. 2-4 from
Eqn. 2-5 and rearranging:
𝑚𝑒,1 = 𝑚𝑎𝑝𝑝,1
𝑓 − 𝑚𝑎𝑝𝑝,1 − (𝑉𝑏 + 𝑉𝑠
) × (𝜌𝑔,1m − 𝜌𝑔,1𝑓) , (2 − 6)
where 𝑚𝑒,1 = 𝑚𝑎,1 − 𝑉𝑎,1 × 𝜌𝑔,1𝑓 is the excess equilibrium adsorption mass at the
conclusion of the first pressure change step.
During the second pressure change step, the apparent mass, 𝑚𝑎𝑝𝑝,2, at the second minimum
point, is given by the following equation
𝑚𝑎𝑝𝑝,2 = (𝑚𝑏 + 𝑚𝑠
) + 𝑚𝑒,1 − (𝑉𝑏 + 𝑉𝑠
) × 𝜌𝑔,2𝑚 + 𝑉𝑎,1 × (𝜌𝑔,1𝑓 − 𝜌𝑔,2𝑚) , (2 − 7)

30
where 𝜌𝑔,2𝑚 is the density of the gas at the second minimum pressure point. After reaching
equilibrium at the second pressure point, the apparent weight, 𝑚𝑎𝑝𝑝,2
𝑓
, is given as
𝑚𝑎𝑝𝑝,2
𝑓 = (𝑚𝑏 + 𝑚𝑠
) + 𝑚𝑒,2 − (𝑉𝑏 + 𝑉𝑠
) × 𝜌𝑔,2𝑓 , (2 − 8)
where 𝜌𝑔,2𝑓 is the gas density at equilibrium during the second pressure change step.
Combining Eqns. 2-4 and 2-8:
𝑚𝑒,2 = 𝑚𝑎𝑝𝑝,2
𝑓 + (𝑚𝑏 + 𝑚𝑠
) × (
𝜌𝑔,2f
𝜌𝑔,1𝑚
− 1) − 𝑚𝑎𝑝𝑝,1 × (
𝜌𝑔,2f
𝜌𝑔,1𝑚
) . (2 − 9)
Following a similar procedure, one then gets:
𝑚𝑒,𝑛 = 𝑚𝑎𝑝𝑝,n
𝑓 − 𝑚𝑎𝑝𝑝,1 × (
𝜌𝑔,𝑛𝑓
𝜌𝑔,1m) − (𝑚𝑏 + 𝑚𝑠
) × (1 −
𝜌𝑔,𝑛𝑓
𝜌𝑔,1m) , (2 − 10)
where 𝑚𝑒,𝑛 is the excess adsorption at equilibrium at the conclusion of the nth pressure
change step and 𝜌𝑔,𝑛𝑓 the corresponding gas density. Assuming that one can accurately locate the
𝑚𝑎𝑝𝑝,1 and measure the 𝑚𝑎𝑝𝑝,n
𝑓
for each of the pressure change steps, one can utilize Eqn. 2-10 to
determine the excess adsorption isotherm for He or for any other gas (we use Ar as a model gas in
this study) for which such data can be generated.
2.3.2 A new method for TGA zero-point correction
The accurate determination of 𝑚𝑎𝑝𝑝,1 is quite challenging, however, because it requires the
availability of accurate dynamic sorption data. The MSB, as is the case for all balances, must be
tared/calibrated at regular intervals to assure measurement accuracy. The procedure by which the
MSB instrument carries out such taring, a task known as zero-point (ZP) calibration, is shown
schematically in Fig. 2-6.

31
Figure 2-6. The three different operating configurations of the magnetic suspension balance (Dreisbach & Lösch, 2000)
The MSB operates in three different configurations: In configuration 1, the ZP position,
the sample container (basket) and the sample when inside the basket, are decoupled from the
balance mechanism, and only the permanent magnet and the linkage system are lifted and their
combined weight measured. In configuration 2, the measurement position, the sample container is
engaged by the raised permanent magnet, and their total weight (basket, sample, magnet and
linkage) is measured. Configuration 3 is utilized to measure in situ the density of the bulk gas. This
is accomplished by also engaging a metal (Ti) piece of known mass, referred to as the sinker, and
then by measuring again the total weight (basket, sample, magnet, linkage, and sinker). Since for
both He and Ar their densities are well-known, in this study we do not utilize the data from
configuration 3. In all three configurations, the basket, sample, magnet, linkage, and sinker are all
immersed in the gas atmosphere. The ZP calibration can be set via the MSB software to take place
at pre-selected, fixed time intervals (tZP). For experiments involving a step pressure change, it is
important that the ZP measurements take place before the pressure change is initiated and then

32
after the pressure has leveled-off to its final intended value in order to avoid loss of dynamic
sorption data.
At the ith ZP measurement, the apparent weight of the magnet and the linkage
system, 𝑀0,𝑖,
sensed by the TGA balance is given by Eqn. 2-11
𝑀0,𝑖 = 𝑚𝑚 − 𝜌𝑔,𝑖 × 𝑉𝑚 ， (2 − 11)
where 𝑚𝑚 is the true weight of the permanent magnet and linkage system (i.e., the weight
measured by the balance under vacuum), 𝑉𝑚 is the (impenetrable) volume of the permanent magnet
and the linkage system, and 𝜌𝑔,𝑖
is the density of the gas surrounding the magnet and linkage
system during the ith ZP measurement.
In reality, the balance software does not report 𝑀0,𝑖
, but reports, instead, a difference, 𝑚𝑧𝑃,𝑖
,
between 𝑀0,𝑖
and an unspecified reference weight for the magnet and the linkage system.
(However, as explained below, exact knowledge of this weight is not needed to carry out the ZP
correction). If one defines this reference weight as 𝑀𝑠
, then 𝑚𝑧𝑃,𝑖
is given by Eqn. 2-12
𝑚𝑧𝑃,𝑖 = 𝑀0,𝑖 − 𝑀𝑠 = 𝑚𝑚 − 𝑀𝑠 − 𝜌𝑔,𝑖 × 𝑉𝑚 . (2 − 12)
The MSB instrument in between the two ZP measurements reports, the so-called corrected
weight, 𝑚𝑀𝑃,𝑖,𝐶𝑜𝑟𝑟(𝑡) , and 𝑚𝑀𝑃,𝑖
(𝑡) , the so-called uncorrected weight measured in operating
configuration 2, described by Eqn. 2-13 (in Eqn. 2-13, to simplify notation, 𝑚𝑠
and 𝑣𝑠
include
both the mass and volume of the solid as well as the mass and volume of the adsorbate layer).
𝑚𝑀𝑃,𝑖
(𝑡) = (𝑚𝑚 + 𝑚𝑠 + 𝑚𝑏
) − (𝑉𝑚 + 𝑣𝑠 + 𝑣𝑏
)𝜌𝑔
(𝑡) − 𝑀𝑠
. (2 − 13)
𝑚𝑀𝑃,𝑖,𝐶𝑜𝑟𝑟(𝑡), and 𝑚𝑀𝑃,𝑖
(𝑡) relate to each other via the following Eqn. 14
𝑚𝑀𝑃,𝑖
(𝑡) = 𝑚𝑀𝑃,𝑖,𝐶𝑜𝑟𝑟(𝑡) + 𝑚𝑍𝑃(𝑡). (2 − 14)

33
In the absence of mechanical drift in the balance mechanism in between the two ZP
measurements, 𝑚𝑍𝑃(𝑡) (signifying the instantaneous apparent weight of the magnet and linkage)
is described by Eqn. 2-12 (with 𝜌𝑔,𝑖 being replaced by 𝜌𝑔
(𝑡)). Combining Eqns. 2-12, 2-13 and
2-14 then one gets:
𝑚𝑀𝑃,𝑖,𝐶𝑜𝑟𝑟(𝑡) = (𝑚𝑠 + 𝑚𝑏
) − (𝑣𝑠 + 𝑣𝑏
)𝜌𝑔
(𝑡) . (2 − 15)
Unfortunately, the MSB does not measure 𝑚𝑍𝑃(𝑡) continuously, instead, as noted above,
measuring and recording the 𝑚𝑧𝑃,𝑖 at fixed time intervals (at the points where the 𝑚𝑍𝑃(𝑡) are
directly measured and known, then Eqn. 2-15 applies identically). In between consecutive ZP
measurements, an approximate method must, therefore, be used to estimate 𝑚𝑍𝑃(𝑡). The MSB
software employs a linear approximation. Specifically, it utilizes Eqn. 2-16 (in lieu of Eqn. 2-14)
to calculate 𝑚𝑀𝑃,𝑖,𝐶𝑜𝑟𝑟(𝑡)
𝑚𝑀𝑃,𝑖,𝐶𝑜𝑟𝑟(𝑡) = 𝑚𝑀𝑃,𝑖
(𝑡) − 𝐾[𝑡 − (𝑖 − 1)𝑡𝑍𝑃] − 𝑚𝑍𝑃,𝑖
, (2 − 16)
where
𝐾 =
𝑚𝑍𝑃,𝑖+1 − 𝑚𝑍𝑃,𝑖
𝑡𝑍𝑃
. (2 − 17)
This approach is reasonable at constant density, where changes in the 𝑚𝑍𝑃(𝑡) in the time
interval 𝑡𝑍𝑃 (in between two successive ZP measurements) are due to either mechanical drift or
minor random fluctuations in pressure and temperature, particularly absent detailed knowledge on
the nature of these perturbations. In the experiments presented here (we have used a 𝑡𝑍𝑃= 20 min ),
we have discovered that for measurements under constant temperature and pressure the differences
in between two consecutive 𝑚𝑧𝑃,𝑖 values are generally small, typically, 10 - 20 µg, which further
justifies the use of the linear approximation described by Eqns. 2-16 and 2-17. However, this

34
approach is not accurate when measuring adsorption dynamics, like in Fig. 2-2, where the two ZP
measurements “bracket” a step pressure (density) change. For that, absent any significant
mechanical drift (see the Supplementary Materials section on how to augment the proposed
approach in the presence of a mechanical drift), we propose determining 𝑚𝑍𝑃(𝑡) by Eqn. 2-18
𝑚𝑧𝑃(𝑡) = 𝑚𝑧𝑃,𝑖 + (𝜌𝑔𝑎𝑠,𝑖 − 𝜌𝑔𝑎𝑠(𝑡)) × 𝑉𝑚 , (2 − 18)
where
𝑉𝑚 =
(𝑚𝑧𝑃,𝑖+1 − 𝑚𝑧𝑃,𝑖)
(𝜌𝑔𝑎𝑠((𝑖−1)𝑡𝑍𝑃) − 𝜌𝑔𝑎𝑠((𝑖)𝑡𝑍𝑃)
)
. (2 − 19)
Fig. 2-7 shows the comparison between dynamic Ar adsorption data calculated using the
proposed new ZP correction and the corresponding data employing the MSB instrument software
for correction (as reported by the MSB system). These data are recorded in the time interval
“bracketing” the pressure change step. It is clear that substantial differences exist between the two
approaches. The new method generates 𝑚𝑀𝑃,𝑖,𝐶𝑜𝑟𝑟(𝑡) data that are qualitatively similar to the
𝑚𝑀𝑃,𝑖
(𝑡) directly recorded by the balance, as one may have expected, Also note, that the
conventional ZP correction approach indicates that the 𝑚𝑀𝑃,𝑖,𝐶𝑜𝑟𝑟(𝑡) increases immediately after
the first ZP measurement, while the pressure (Ar density) remains constant, which is implausible.
So, for the data reported in this paper, during the dynamic period we employ the new ZP correction
method, while for the rest of the data, past the ZP correction after the completion of the pressure
change step, for which density remains constant we utilize the ZP correction suggested by the MSB
software, i.e., we use the 𝑚𝑀𝑃,𝑖,𝐶𝑜𝑟𝑟(𝑡) as reported by the balance (and calculated from Eqns. 2-16
and 2-17).

35
Figure 2-7. The comparison between dynamic Ar adsorption data as corrected by the MSB software and the proposed new ZP
correction method. mapp vs. time during a pressure step change from 2 to 5 bar
2.3.3 A modified Lorenz-Wessling method
The method of analysis of the sorption data described in Sect. 2.3.1 above is based on the
assumption that no adsorption has taken place at the minimum point of the 𝑚𝑎𝑝𝑝 vs. time response.
Though this may be true for dense polymeric materials, it is clearly not truly applicable for Ar
adsorption on the sample shale studied (see Fig. 2-8) where ~4.3% of the total excess adsorption
at equilibrium for that step already takes place before the minimum point. Below we present a
modified analysis method that relaxes such an assumption, and we use it to analyze Ar adsorption
data on the shale sample. We then compare the Ar isotherm data generated with the corresponding

36
isotherm based on the original assumption of no adsorption taking place at the minimum point of
the 𝑚𝑎𝑝𝑝 vs. time response.
Figure 2-8. Ar adsorption in shale, mapp and pressure vs. time during a pressure step change from vacuum to 2 bar
From the definition of 𝑚𝑎𝑝𝑝 , Eqn. 2-20 below, and from the relationship 𝑚𝑒 =
𝑚𝑎 (1 −
𝜌𝑔
𝜌𝑎
) with 𝜌𝑎 being the density of the adsorbate layer
𝑚𝑎𝑝𝑝(𝑡) = (𝑚𝑏 + 𝑚𝑠
) + 𝑚𝑒 − (𝑉𝑏 + 𝑉𝑠
) × 𝜌𝑔 , (2 − 20)
and assuming that 𝜌𝑔 and 𝜌𝑎 are constant for t > t1, see Fig. 4, setting A=(1 −
𝜌𝑔
𝜌𝑎
) we find
𝑑𝑚𝑎𝑝𝑝(𝑡)
𝑑𝑡 =
𝑑𝑚𝑒
𝑑𝑡 =
𝑑𝑚𝑎
𝑑𝑡 𝐴 . (2 − 21)

37
Define 𝑚𝑎𝑠 to be the total adsorption capacity of the shale, and the fractional sorption
capacity as 𝜃 =
𝑚𝑎
𝑚𝑎𝑠
. For a range of t near 𝑡1, we assume that the rate of change in fractional
sorption capacity follows a linear (driving-force) type rate, i.e., Eqn. 2-22 below (this assumption
will be validated later by the Ar sorption data, see discussion to follow)
𝑑𝜃
𝑑𝑡 = 𝑘𝑎𝜌𝑔
(1 − 𝜃) (2 − 22)
For 𝑡 > 𝑡1, 𝜌𝑔 is constant, i.e., 𝜌𝑔
(𝑡) = 𝜌𝑔,1
𝜃 = 1 − (1 − 𝜃1
)𝑒𝑥𝑝[−𝑘𝑎𝜌𝑔,1
(𝑡 − 𝑡1
)] , (2 − 23)
where 𝜃1 is the fractional sorption capacity at 𝑡 = 𝑡1. Substituting into Eqn. 2-22 gives
𝑑𝜃
𝑑𝑡
= 𝑘𝑎𝜌𝑔,1
(1 − 𝜃1
)𝑒𝑥𝑝[−𝑘𝑎𝜌𝑔,1
(𝑡 − 𝑡1
)] . (2 − 24)
Then from Eqn. 2-21
𝑑𝑚𝑎𝑝𝑝(𝑡)
𝑑𝑡
= 𝐴
𝑑𝑚𝑎
𝑑𝑡
= 𝐴𝑚𝑎𝑠 (1 − 𝜃1
)𝑘𝑎𝜌𝑔,1𝑒𝑥𝑝[−𝑘𝑎𝜌𝑔,1
(𝑡 − 𝑡1
)]. (2 − 25)
Integrating Eqn. 2-22 for 𝑡 < 𝑡1
𝜃 = 1 − (1 − 𝜃1
)𝑒𝑥𝑝 [𝑘𝑎 ∫ 𝜌𝑔(𝑡
′
)
𝑡1
𝑡
𝑑𝑡′
] (2 − 26)
𝑑𝑚𝑎
𝑑𝑡
= 𝑚𝑎𝑠 (1 − 𝜃1
)𝑘𝑎𝜌𝑔
(𝑡)𝑒𝑥𝑝 [𝑘𝑎 ∫ 𝜌𝑔
(𝑡)
′
𝑡1
𝑡
𝑑𝑡′
] (2 − 27)
Integrating Eqn. 2-27 between 𝑡𝑖 and 𝑡1, and since at 𝑡 = 𝑡𝑖
there is no adsorption
𝑚𝑎
(𝑡1
) − 𝑚𝑎
(𝑡𝑖
) = 𝑚𝑎
(𝑡1
) = 𝑚𝑎𝑠 (1 − 𝜃1
)𝑘𝑎𝜌𝑔,1 ∫
𝜌𝑔
(𝑡)
𝜌𝑔,1
𝑒𝑥𝑝 [𝑘𝑎 ∫ 𝜌𝑔
(𝑡
′
)
𝑡1
𝑡
𝑑𝑡′
]
𝑡1
𝑡𝑖
𝑑𝑡
(2 − 28)
Combining Eqns. 2-25 and 2-28

38
𝑚𝑒
(𝑡1
) =
𝑑𝑚𝑎𝑝𝑝(𝑡)
𝑑𝑡 |
𝑡1
∫
𝜌𝑔
(𝑡)
𝜌𝑔,1
𝑒𝑥𝑝 [𝑘𝑎 ∫ 𝜌𝑔
(𝑡)
′
𝑡1
𝑡
′′
𝑑𝑡′
]
𝑡1
𝑡𝑖
𝑑𝑡 . (2 − 29)
In Fig. 2-9, for the data in Fig. 2-8, we plot 𝑚𝑎𝑝𝑝(𝑡) vs. time for a range of times (𝑡 > 𝑡1)
in the neighborhood of the minimum point. The data are fitted, with a relatively high degree of
accuracy, to a 3rd order polynomial, as indicated in the plot.
Figure 2-9. Ar adsorption in shale, mapp vs. time in the region near the minimum point
Defining 𝜶 =
(
𝒅𝒎𝒂𝒑𝒑(𝒕)
𝒅𝒕 )
(
𝒅𝒎𝒂𝒑𝒑(𝒕)
𝒅𝒕 )
𝒕𝟏
= 𝒆𝒙𝒑[−𝒌𝒂𝝆𝒈,𝟏
(𝒕 − 𝒕𝟏
)], we plot in Fig. 2-10, ln 𝜶 vs.
(𝒕 − 𝒕𝟏).

39
Figure 2-10. The linear plot of ln(α) vs. (t-t1)
The linearity of the plot in Fig. 2-10 validates the use of Eqn. 2-22. From the slope of the
plot, one calculates 𝑘𝑎, and once 𝑘𝑎 is known, from Eqn. 2-29 one calculates 𝑚𝑒
(𝑡1
). The algebra
is simplified, because for the Ar data in Fig. 2-8 𝜌𝑔
(𝑡) between 𝑡𝑖 and 𝑡1 is a linear function of
time. Once 𝑚𝑒
(𝑡1
) is known the excess adsorption isotherm can be calculated by the following
revised form of Eqn. 2-10.
𝑚𝑒,𝑛 = 𝑚𝑎𝑝𝑝,n
𝑓 − (𝑚𝑎𝑝𝑝(𝑡1
) − 𝑚𝑒
(𝑡1
)) × (
𝜌𝑔,𝑛𝑓
𝜌𝑔,1m) − (𝑚𝑏 + 𝑚𝑠
) × (1 −
𝜌𝑔,𝑛𝑓
𝜌𝑔,1m) (2 − 30)

40
Fig. 2-11 shows the comparison between the Ar adsorption isotherms calculated employing
the original approach of assuming that no adsorption takes place at the minimum point, and the
revised procedure described here, whereby the original raw 𝑚𝑀𝑃,𝑖
(𝑡) data were corrected
employing the new technique discussed in Sect. 2.3.2. As one notes from Fig. 2-11, accounting for
the fact that adsorption has already taken place at the minimum point, has an impact on the
calculation of the isotherm, with the difference between the two isotherms increasing with
increasing equilibrium pressure. In Fig. 2-11, we also plot the two isotherms for the case for which,
when correcting the raw 𝑚𝑀𝑃,𝑖
(𝑡) data, we also take into account the potential presence of
mechanical drift, following the procedure described in the Supplementary Materials Section.
Correcting for the mechanical drift has a relatively minor effect.
Figure 2-11. Comparison of excess adsorption isotherms calculated with different techniques

41
In addition, in Fig. 2-11 we plot the Ar isotherm employing the conventional technique
(termed the traditional method in Fig. 2-11) in which we use He experiments (here the data in Fig.
2-4) on the assumption that it is not adsorbing on the shale sample. In the calculation of the Ar
isotherm, the skeletal volume of the sample is calculated from the mapp vs. He density plot (Fig. 2-
4), and the excess adsorption is calculated according to Eqn. 2-2. Notice, that the conventional
approach gives lower excess adsorption values, which in fact become negative at around 30 bar.
2.4 Conclusions
Measuring the adsorption of gases in microporous solids like the shales requires accurate
knowledge of the solid’s skeletal volume. Traditionally, both in TGA as well as in
volumetric/manometric measurements, He has been used to accomplish this goal. Analysis of the
He data to determine the solid’s skeletal volume makes the assumption that the He does not adsorb
on the solid. Though this may be true for some solids, it is not applicable for others, including the
Marcellus shale sample studied in this paper.
In recent years, methods have been proposed to analyze He data to measure the skeletal
volume even for solids for which He adsorbs in their structure. They require extensive
experimentation over a broad range of temperatures, which is quite a time-consuming undertaking
particularly for a slow adsorbing gas as He, and may not be even feasible at all for solids like the
shales that contain organic inclusions which may be temperature-sensitive. As a result, these
methods have not found widespread use during adsorption measurements with such materials, for
which He is still used to estimate their impenetrable solid volume under the assumption that it does
not adsorb on the solid. As noted by Lorenz and Wessling (Lorenz & Wessling, 2013), however,
such an assumption may lead to inaccurate adsorption isotherm calculations, and in some instances

42
may even result in implausible negative excess adsorption values, as is the case with Ar adsorption
in Marcellus shale discussed here.
In this paper, we have presented a method for measuring the adsorption of light gases in
microporous solids that does not rely on using He to measure the solid’s impenetrable volume.
Application of the method requires being able to acquire accurate dynamic sorption data. A new
zero-point correction that is different from the one presently employed by the TGA instrument’s
software that, we believe, is better suited for dynamic sorption data was proposed and applied to
the analysis of such data. Our method is based on the concept originally presented by Lorenz and
Wessling (Lorenz & Wessling, 2013). We relax, however, the requirement that a point in the
dynamic sorption weight vs. time line must be identified prior to which no adsorption has taken
place on the sample. We apply the method to a Marcellus Shale sample using Ar as a model gas
and safe surrogate for the gas components in shale-gas. We calculate the Ar adsorption isotherm
employing both the original as well as the modified method and compare the resulting isotherms.
For the low pressures, isotherms are similar but substantial differences are found for the higher
pressures.
2.5 References
Arami-Niya, A., Rufford, T. E., Birkett, G., & Zhu, Z. (2017). Gravimetric adsorption
measurements of helium on natural clinoptilolite and synthetic molecular sieves at pressures up
to 3500 kPa. Microporous and Mesoporous Materials, 244, 218-225.
https://doi.org/10.1016/j.micromeso.2016.10.035

43
Blatt, H., Tracy, R., & Owens, B. (2006). Petrology: igneous, sedimentary, and metamorphic.
Macmillan.
Curtis, J. B. (2002). Fractured shale-gas systems. AAPG bulletin, 86(11), 1921-1938.
Dasani, D., Wang, Y., Tsotsis, T. T., & Jessen, K. (2017). Laboratory-scale investigation of
sorption kinetics of methane/ethane mixtures in shale. Industrial & engineering chemistry
research, 56(36), 9953-9963.
Dreisbach, F., & Lösch, H. (2000). Magnetic suspension balance for simultaneous measurement
of a sample and the density of the measuring fluid. Journal of thermal analysis and calorimetry,
62(2), 515-521.
Dreisbach, F., Lösch, H., & Harting, P. (2002). Highest pressure adsorption equilibria data:
measurement with magnetic suspension balance and analysis with a new adsorbent/adsorbatevolume. Adsorption, 8, 95-109.
Gasparik, M., Gensterblum, Y., Ghanizadeh, A., Weniger, P., & Krooss, B. M. (2015). Highpressure/high-temperature methane-sorption measurements on carbonaceous shales by the
manometric method: experimental and data-evaluation considerations for improved accuracy.
SPE Journal, 20(04), 790-809.
Gumma, S., & Talu, O. (2003). Gibbs dividing surface and helium adsorption. Adsorption, 9, 17-
28.
Herrera, L., Fan, C., Do , D. D., & Nicholson, D. (2011). A revisit to the Gibbs dividing surfaces
and helium adsorption. Adsorption, 17(6), 955-965. https://doi.org/10.1007/s10450-011-9374-y
Hill, D. G., & Nelson, C. (2001). ABSTRACT: Geologic Characteristics of 567 Frontier Coalbed
Methane Plays in North America. AAPG bulletin, 85, 568.
Kaneko, K., Setoyama, N., & Suzuki, T. (1994). Ultramicropore Characterization by He
Adsorption. In Characterization of Porous Solids III (pp. 593-602).
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44
Keller, J., & Göbel, M. (2015). Oscillometric—volumetric measurements of pure gas adsorption
equilibria devoid of the non-adsorption of helium hypothesis. Adsorption Science & Technology,
33(9), 793-818.
Kenomore, M., Hassan, M., Dhakal, H., & Shah, A. (2018). Economic appraisal of shale gas
reservoirs. SPE Europec featured at EAGE Conference and Exhibition?,
Lee, K. S., & Kim, T. H. (2016). Integrative understanding of shale gas reservoirs. Springer.
Lorenz, K., & Wessling, M. (2013). How to determine the correct sample volume by gravimetric
sorption measurements. Adsorption, 19(6), 1117-1125. https://doi.org/10.1007/s10450-013-
9537-0
Maggs, F., Schwabe, P., & Williams, J. (1960). Adsorption of helium on carbons: influence on
measurement of density. nature, 186(4729), 956-958.
Ratnakar, R. R., & Dindoruk, B. (2019). A new technique for simultaneous measurement of
nanodarcy-range permeability and adsorption isotherms of tight rocks using magnetic
suspension balance. SPE Journal, 24(06), 2482-2503.
Rexer, T. F., Benham, M. J., Aplin, A. C., & Thomas, K. M. (2013). Methane adsorption on shale
under simulated geological temperature and pressure conditions. Energy & Fuels, 27(6), 3099-
3109.
Silvestre-Albero, J., Silvestre-Albero, A. M., Llewellyn, P. L., & Rodríguez-Reinoso, F. (2013).
High-Resolution N2 Adsorption Isotherms at 77.4 K: Critical Effect of the He Used During
Calibration. The Journal of Physical Chemistry C, 117(33), 16885-16889.
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Sircar, S. (2001). Measurement of Gibbsian surface excess. AIChE Journal, 47(5), 1169-1176.
Suzuki, I., Kakimoto, K., & Oki, S. (1987). Volumetric determination of adsorption of helium
over some zeolites with a temperature‐compensated, differential tensimeter having symmetrical
design. Review of scientific instruments, 58(7), 1226-1230.

45
Tan, J., Weniger, P., Krooss, B., Merkel, A., Horsfield, B., Zhang, J., Boreham, C. J., van Graas,
G., & Tocher, B. A. (2014). Shale gas potential of the major marine shale formations in the
Upper Yangtze Platform, South China, Part II: Methane sorption capacity. Fuel, 129, 204-218.
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Conference and Exhibiti
2.6 Appendix
The MSB instrument is quite stable during measurements under constant pressure and
temperature, with ZP drift due to mechanical and other environmental (i.e., minor fluctuations in
temperature and pressure) factors in between two consecutive ZP corrections, generally, averaging
10 – 20 µg. During the dynamic phase, in which one transitions from one pressure level to the next
that is not always the case, however, with the balance appearing to undergo a mechanical
perturbation, as can be seen from Figs. S1 – S5 that show the 𝑚𝑎𝑝𝑝vs. t for the Ar adsorption
experiments. These perturbations appear to be, generally, small (though that is not always the case),
and the balance appears to recover soon after the initial perturbation. The analysis presented in the
body of the paper for calculating 𝑚𝑎𝑝𝑝 from the raw data that the balance generates does not

46
account for such perturbations of mechanical/electronic nature. In this section, we present a
modified method for calculating 𝑚𝑎𝑝𝑝 from the raw data in the presence of such disturbances. We
use this modified correction method to calculate the Ar sorption isotherms and compare them with
those calculated with the method that does not take into account these
Figure S1: Ar adsorption in Shale, 𝑚𝑎𝑝𝑝 and pressure vs. time from vacuum to 2 bar (left), and 2 bar to 5 bar (right)
balance perturbations. As can be seen from Fig. 2-10, the differences between the two
methods are relatively small.
Figure S2: Ar adsorption in Shale, 𝑚𝑎𝑝𝑝 and pressure vs. time from 5 bar to 10 bar (left), and 10 bar to 20 bar (right)

47
Figure S3: Ar adsorption in Shale, 𝑚𝑎𝑝𝑝 and pressure vs. time from 20 bar to 30 bar (left), and 30 bar to 40 bar (right)
Figure S4: Ar adsorption in Shale, 𝑚𝑎𝑝𝑝 and pressure vs. time from 40 bar to 50 bar (left), and 50 bar to 60 bar (right)

48
Figure S5: Ar adsorption in Shale, 𝑚𝑎𝑝𝑝 and pressure vs. time from 60 bar to 70 bar
𝑚𝑧𝑝,1 at the first zero point, prior to the initiation of the experiments and the first step
change in pressure, is described by the following equation
𝑚𝑧𝑝,1 = 𝑚𝑚 − 𝑀𝑆 − 𝜌𝑔,1 × 𝑉𝑚 . (𝑆1)
After the pressure is changed, in some instances (see Figs. S1 - S5), mechanical
perturbation/drift seems to take place. Assuming that it varies with time and setting it to be equal
to 𝐷(𝑡), then 𝑚𝑍𝑃(𝑡) is described by the following equation:
𝑚𝑍𝑃(𝑡) = 𝑚𝑚 − 𝑀𝑆 − 𝜌𝑔
(𝑡) × 𝑉𝑚 + 𝐷(𝑡) . (𝑆2)
𝑚𝑍𝑃,2 at the second ZP measurement is described by the following equation (𝑡𝑧𝑝 being the
time elapsing in between the two consecutive ZP measurements equal to 20 min in the Ar
experiments reported here):
𝑚𝑍𝑃,2 = 𝑚𝑚 − 𝑀𝑆 − 𝜌𝑔(𝑡𝑧𝑝) × 𝑉𝑚 + 𝐷(𝑡𝑧𝑝) , (𝑆3)
where 𝐷(𝑡𝑧𝑝) is the cumulative mechanical perturbation/drift at that point.

49
The method for calculating 𝑚𝑍𝑃(𝑡) described in the body of the paper does not consider
the mechanical drift. Instead, we calculate 𝑉𝑚 from
𝑉𝑚
′ =
𝑚𝑧𝑃,2 − 𝑚𝑧𝑃,1
𝜌𝑔,1 − 𝜌𝑔,2
, (𝑆4)
and then calculate
𝑚𝑍𝑃(𝑡) = 𝑚𝑚 − 𝑀𝑆 − 𝜌𝑔
(𝑡) ×
𝑚𝑧𝑃,2 − 𝑚𝑧𝑃,1
𝜌𝑔,1 − 𝜌𝑔,2
. (𝑆5)
In reality, when mechanical drift takes place, 𝑉𝑚 is given by
𝑉𝑚
𝑡 =
𝑚𝑧𝑃,2 − 𝑚𝑧𝑃,1 − 𝐷(𝑡𝑧𝑝)
𝜌𝑔,1 − 𝜌𝑔,2
. (𝑆6)
The balance at every point measures 𝑚𝑀𝑃(𝑡), the apparent total weight (sample container
+ magnet + sample). If no mechanical drift is present (in Eqn. S7 and those to follow, to simplify
notation, 𝑚𝑠
and 𝑣𝑠
include both the mass and volume of the solid as well as the mass and volume
of the adsorbate layer), 𝑚𝑀𝑃(𝑡) is described by Eqn. S7.
𝑚𝑀𝑃(𝑡) = (𝑚𝑚 + 𝑚𝑏 + 𝑚𝑠
) − (𝑉𝑚 + 𝑉𝑏 + 𝑉𝑠
) × 𝜌𝑔
(𝑡) − 𝑀𝑆
(𝑆7)
= (𝑚𝑏 + 𝑚𝑠
) − (𝑉𝑏 + 𝑉𝑠
) × 𝜌𝑔
(𝑡) − 𝑀𝑆 + [𝑚𝑍𝑃(𝑡) + 𝑀𝑆
]
= 𝑚𝑀𝑃,𝑐𝑜𝑟𝑟(𝑡) + 𝑚𝑍𝑃(𝑡) => 𝑚𝑀𝑃,𝑐𝑜𝑟𝑟(𝑡) = 𝑚𝑀𝑃(𝑡) − 𝑚𝑍𝑃(𝑡) .
If mechanical drift is present,
𝑚𝑀𝑃(𝑡) = (𝑚𝑚 + 𝑚𝑏 + 𝑚𝑠
) − (𝑉𝑚 + 𝑉𝑏 + 𝑉𝑠
) × 𝜌𝑔
(𝑡) − 𝑀𝑆 + 𝐷(𝑡) =
= (𝑚𝑏 + 𝑚𝑠
) − (𝑉𝑏 + 𝑉𝑠
) × 𝜌𝑔
(𝑡) + 𝐷(𝑡) − 𝑀𝑆 + [𝑚𝑚 − 𝑉𝑚𝜌𝑔
(𝑡)]
= (𝑚𝑏 + 𝑚𝑠
) − (𝑉𝑏 + 𝑉𝑠
) × 𝜌𝑔
(𝑡) + 𝐷(𝑡) − 𝑀𝑆 + [𝑚𝑍𝑃(𝑡) + 𝑀𝑆 − 𝐷(𝑡)]
= (𝑚𝑏 + 𝑚𝑠
) − (𝑉𝑏 + 𝑉𝑠
) × 𝜌𝑔
(𝑡) + 𝑚𝑍𝑃(𝑡) . (𝑆8)

50
So,
𝑚𝑀𝑃,𝑐𝑜𝑟𝑟(𝑡) = 𝑚𝑀𝑃(𝑡) − 𝑚𝑍𝑃(𝑡) . (𝑆9)
In order to calculate 𝑚𝑍𝑃(𝑡) when mechanical drift is present, we combine Eqns. S1, S2
and S6
𝑚𝑍𝑃(𝑡) = 𝑚𝑧𝑝,1 +
𝜌𝑔,1 − 𝜌𝑔
(𝑡)
𝜌𝑔,1 − 𝜌𝑔,2
[𝑚𝑧𝑃,2 − 𝑚𝑧𝑃,1 − 𝐷(𝑡𝑧𝑝)] + 𝐷(𝑡) . (𝑆10)
So,
𝑚𝑀𝑃,𝑐𝑜𝑟𝑟(𝑡) = 𝑚𝑝
(𝑡) − 𝑚𝑧𝑝,1 −
𝜌𝑔,1−𝜌𝑔(𝑡)
𝜌𝑔,1−𝜌𝑔,2
[𝑚𝑧𝑃,2 − 𝑚𝑧𝑃,1 − 𝐷(𝑡𝑧𝑝)] − 𝐷(𝑡).
(S11)
When mechanical drift takes place, but one fails to recognize that and carries out the ZP
correction without taking the mechanical drift into account, i.e., by calculating 𝑚𝑍𝑃(𝑡) from Eqn.
S5 (defined in this case as 𝑚𝑧𝑝
𝐷 (𝑡)) , then the corrected weight 𝑚𝑀𝑃,𝑐𝑜𝑟𝑟
𝐷 (𝑡) one calculates,
described by Eqn. S12 below, is not accurate, because the 𝑉𝑚 calculated is not accurate:
𝑚𝑀𝑃,𝑐𝑜𝑟𝑟
𝐷 (𝑡) = 𝑚𝑀𝑃(𝑡) − 𝑚𝑧𝑝
𝐷 (𝑡)
= (𝑚𝑏 + 𝑚𝑠 + 𝑚𝑚) − (𝑉𝑏 + 𝑉𝑠 + 𝑉𝑚) × 𝜌𝑔
(𝑡) + 𝐷(𝑡) − 𝑀𝑆 − 𝑚𝑧𝑝
′
(𝑡) =
= (𝑚𝑏 + 𝑚𝑠
) − (𝑉𝑏 + 𝑉𝑠
) × 𝜌𝑔
(𝑡) + [𝑚𝑚 − 𝑉𝑚𝜌𝑔
(𝑡)] + 𝐷(𝑡) − 𝑀𝑆 − 𝑚𝑧𝑝
′
(𝑡)
= 𝑚𝑀𝑃,𝑐𝑜𝑟𝑟(𝑡) + [𝑚𝑍𝑃(𝑡) + 𝑀𝑆 − 𝐷(𝑡)] + 𝐷(𝑡) − 𝑀𝑆 − 𝑚𝑧𝑝
𝐷 (𝑡)
= 𝑚𝑀𝑃,𝑐𝑜𝑟𝑟(𝑡) + 𝑚𝑍𝑃(𝑡) − 𝑚𝑧𝑝
𝐷 (𝑡). (𝑆12)
In the above Eqn. 𝑆12, 𝑚𝑍𝑃(𝑡) is given by Eqn. S10 while 𝑚𝑧𝑝
𝐷 (𝑡) is given by the
following Eqn.
𝑚𝑧𝑝
𝐷 (𝑡) = 𝑚𝑧𝑝,1 +
𝜌𝑔,1 − 𝜌𝑔
(𝑡)
𝜌𝑔,1 − 𝜌𝑔,2
(𝑚𝑧𝑃,2 − 𝑚𝑧𝑃,1). (𝑆13)

51
So,
𝑚𝑍𝑃(𝑡) − 𝑚𝑧𝑝
𝐷 (𝑡) = −
𝜌𝑔,1 − 𝜌𝑔
(𝑡)
𝜌𝑔,1 − 𝜌𝑔,2
× 𝐷(𝑡𝑧𝑝) + 𝐷(𝑡). (𝑆14)
Then,
𝑚𝑀𝑃,𝑐𝑜𝑟𝑟(𝑡) = 𝑚𝑀𝑃,𝑐𝑜𝑟𝑟
𝐷 (𝑡) +
𝜌𝑔,1−𝜌𝑔(𝑡)
𝜌𝑔,1−𝜌𝑔,2
× 𝐷(𝑡𝑧𝑝) −
𝐷(𝑡) (S15)
From Eqn. S15, when 𝜌𝑔
(𝑡) = 𝜌𝑔,2, 𝐷(𝑡) = 𝐷(𝑡𝑧𝑝) => 𝑚𝑀𝑃,𝑐𝑜𝑟𝑟
𝐷 (𝑡) = 𝑚𝑀𝑃,𝑐𝑜𝑟𝑟(𝑡)
Rearranging Eqn. S15,
𝑚𝑀𝑃,𝑐𝑜𝑟𝑟
𝐷 (𝑡) = (𝑚𝑏 + 𝑚𝑠
) −
𝜌𝑔,1
𝜌𝑔,1−𝜌𝑔,2
× 𝐷(𝑡𝑧𝑝) − [(𝑉𝑏 + 𝑉𝑠
) −
𝐷(𝑡𝑧𝑝)
𝜌𝑔,1−𝜌𝑔,2
] × 𝜌𝑔
(𝑡) + 𝐷(𝑡)
(𝑆16)
Typically, what one observes is that when mechanical drift is present it usually occurs at the time
when the pressure change takes place and things gradually settle down after that. In fact, after a
certain time 𝑡
∗ one observes a linear correlation between 𝑚𝑀𝑃,𝑐𝑜𝑟𝑟
𝐷 (𝑡) vs. 𝜌𝑔
(𝑡), see Figs. S1 – S5;
this would then indicate that for 𝑡 > 𝑡
∗
, 𝐷(𝑡) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝐷(𝑡𝑧𝑝).
The straight-line portion of the 𝑚𝑐𝑜𝑟𝑟
′
(𝑡) 𝑣𝑠. 𝜌𝑔
(𝑡) plot has an intercept equal to
(𝑚𝑏 + 𝑚𝑠
) −
𝜌𝑔,1
𝜌𝑔,1 − 𝜌𝑔,2
× 𝐷(𝑡𝑧𝑝) + 𝐷(𝑡𝑧𝑝)
= (𝑚𝑏 + 𝑚𝑠
) −
𝜌𝑔,2
𝜌𝑔,1 − 𝜌𝑔,2
× 𝐷(𝑡𝑧𝑝) (𝑆17)
and a slope is given as:
− [(𝑉𝑏 + 𝑉𝑠
) −
𝐷(𝑡𝑧𝑝)
𝜌𝑔,1 − 𝜌𝑔,2
]. (𝑆18)

52
Since (𝑚𝑏 + 𝑚𝑠
) is known, from the intercept one can calculate 𝐷(𝑡𝑧𝑝) . From Eqns. S12
and S14, for 𝑡 > 𝑡
∗
, one can the calculate 𝑚𝑀𝑃,𝑐𝑜𝑟𝑟(𝑡).

53
Chapter 3: Experimental Study of Capillary Condensation in model
Mesoporous Silicas2
3.1 Introduction
This study focuses on the phenomenon of capillary condensation within mesoporous silica
materials, particularly, examining the sorption behavior of three gases: N₂, C₂H₆, and CO₂. This
investigation centers on two types of porous silica. The first is a commercial AGC-40 porous glass
purchased from Advanced Glass & Ceramics, LLC. It has an unimodal pore size distribution (see
Fig. A-1 in Appendix A) with an average diameter of approximately 4 nm (Table 3.1). The second
is a silica sample prepared in our lab (for further details about the preparation method, see Sect.
3.2.1) with a larger average pore diameter of about 7 nm (Table 3.1).
To study the adsorption behavior in these materials, we employed two experimental setups:
(i) a static adsorption system (Micromeritics ASAP 2010) that was utilized for experiments at
ambient pressure conditions under cryogenic conditions, and (ii) a TGA instrument (a magnetic
suspension microbalance (MSB) from Rubotherm IsoSORP®, Germany) that was used for the
high-pressure experiments. The selection of these experimental conditions (i.e., cryogenic and high
pressure) was motivated by classical thermodynamic predictions, which indicates that they may
be conducive to capillary condensation behavior.
In the following sections, we describe the key experimental findings and their implications
for understanding capillary condensation in mesoporous materials.
2 Part of this section is quoted from our manuscript: Simeski, F., Wu, J., Hu, S., Tsotsis, T., Jessen, K., & Ihme, M.
(2023). Local Rearrangement in Adsorption Layers of Nanoconfined Ethane. The Journal of Physical Chemistry C 2023 127 (34),
17290-17297 DOI: 10.1021/acs.jpcc.3c04869

54
3.2 Experimental
3.2.1 Experimental samples
As noted in Sect. 3.1, we investigate in this study two different solid porous silica samples,
a commercial porous glass (AGC-40, supplied by Advanced Glass & Ceramics, LLC), and a homemade mesoporous silica sample prepared in our lab, by Dr. Sheng Hu.
The mesoporous silica sample was prepared by utilizing monodisperse nano-silica particles
(Nanocym), oleic acid (Tokyo Chemical Industry, > 85 % purity), and ethanol (Koptec, 200 proof,
anhydrous). The synthesis procedure involves, as a first step, mixing a quantity of silica particles
with ethanol in a clean beaker. Oleic acid is then added to the mixture, with silica-to-oleic acid
mass ratio of 5:1. The suspension is then sonicated for 30 min, and the ethanol is allowed to
vaporize naturally. The mixture of silica particles and oleic acid is loaded into a stainless-steel die
with a diameter of 1.27 cm, which is then pressurized at approximately 680 atm for 3 min to create
a disc-shaped green body. This green body is sintered in a tubular furnace under flowing air with
a volume flowrate of 3 cm3/min. More details can be found in a recent publication (Simeski et al.,
2023).
The AGC-40 sample, purchased from Advanced Glass & Ceramics, LLC, is made by acidleaching of a borosilicate glass containing 95% SiO₂ and 5% B₂O₃. Both samples were
characterized using BET with N₂ at 77 K using the static adsorption instrument. The results of
these measurements are summarized in Table 3-1.

55
Table 3-1. Summary of sample characterization
Synthetic sample AGC 40
BET surface area (m2/g) 42.09 187.29
Average pore diameter (nm) 7.0 4.08
3.2.2 Experimental procedure
In the measurements with the MSB instrument, the glass sample, after being placed in the
instrument’s cell, was heated to 80 oC, and was then evacuated (30 mTorr) for as long as it takes
for its weight to stabilize, indicating no further desorption of adsorbed impurities from its structure
(the cell temperature is regulated using an oil bath (Julabo F25) connected to the MSB). We have
carried out both batch and flow experiments. For the batch experiments, after evacuation the TGA
cell is pressurized with the gas to be studied at the selected initial pressure and heated to the desired
temperature. After adsorption equilibrium is reached (indicated by a weight change of less than 10
μg over one hour), the system sample was pressurized to the next pressure point, for the additional
adsorption to take place, until the desired upper limit of temperatures was reached. Additional
experimental details about the batch as well as the flow experiments are provided in Section 3.3.
For the measurements in the static adsorption system, initially, the sample was degassed
under high vacuum (10 mTorr) for 12 hours. During the adsorption experiments, the gas was
introduced stepwise into the system’s sample cell from vacuum (< 1 mTorr) up to 1 atm to generate
the adsorption isotherm. The desorption branch was generated by stepwise venting of the gas from
atmospheric pressure at the end of the adsorption phase to vacuum at the end of the desorption
phase.

56
3.3 Results and discussion
3.3.1 Experimental results for the home-made silica sample
Five different C₂H₆ adsorption experiments, two under isothermal and three under isobaric
conditions, were conducted with the home-made silica sample using the MSB instrument. Figure
3-1 illustrates the measurement conditions of these experiments alongside the saturation pressure
line for bulk C₂H₆. In the figure, each point marks the temperature and pressure at which
equilibrium was measured. The two vertical lines in the figure denote the isothermal experiments,
while the horizontal lines indicate the isobaric experiments.
Figure 3-1. C₂H₆ phase diagram and measurement conditions for synthetic sample
In the first experiment, we measured the isotherm at 50 °C by varying the pressure from
30 to 55 bar in increments of ~2 bar. For this series of measurements, after placing the silica sample

57
in the TGA cell, the cell was first evacuated at a temperature of 80 oC until the apparent weight
(real weight minus buoyancy) reported by the balance became constant (less than 10 μg change
over a period of 1 hour). Then the sample cell was pressurized with C₂H₆ to a pressure of 30 bar.
This experiment was performed under flow conditions, with the sample cell pressure being
controlled by an electronic back-pressure regulator (BPR, Brooks 5820). An ISCO pump,
Teledyne 260D, with a heated cylinder was utilized to provide the C2H6 flow at the desired feed
pressures and temperature. After the desired pressure was reached and stabilized (less than 0.1 bar
of change), the pressure was maintained constant (<0.1 bar of change) for 90 min during which
the apparent sample weight was monitored. This was then followed by pressurizing the sample
cell with C₂H₆ in a stepwise manner, from 30 to 32 bar etc. crossing into the supercritical region
(see Fig. 3-1) up to a maximum pressure of 55 bar. For each equilibrium point, the excess
adsorption was measured and is reported in Fig. 3-2 (The experimental data are also listed in Table
3-2). The measurements show a gradual “filling” of the pore structure with an inflection point
around a density value of 0.07 g/cm3
(42 bar). However, no abrupt phase transitions were observed
under these conditions.

58
Figure 3-2. C2H6 excess adsorption vs. pressure and density at 50°C
Table 3-2. C2H6 excess adsorption vs. pressure at 50°C
Pressure
(bar)
Mex
(mg/g)
30.7 8.6
32.3 9.2
34.7 9.9
36.8 10.6
38.6 11.3
41.0 12.2
42.7 12.6
45.0 13.4
47.2 14.3
48.5 15.0
50.6 16.2
52.9 17.6
55.1 19.5

59
A second isothermal flow experiment was conducted at a temperature of 20 °C. In this
experiment, we incrementally increased the system pressure from 33 bar to 36.9 bar in steps of
approximately 1 bar, carefully avoiding crossing the gas-liquid (G-L) equilibrium line. At each
pressure increment, both temperature and pressure were allowed to stabilize with variations of less
than 0.1°C and 0.1 bar, respectively. The balance was monitored until the measured excess
adsorption changed by less than 1% over a period of one hour. The results of this isothermal
experiment are illustrated in Figure 3-3, and the detailed data are tabulated in table 3-3. Once again,
the excess adsorption gradually increased with the increase in pressure, and no sudden
condensation was observed, although the operation approached close to the G-L saturation line, as
shown in Figure 3-1.
Figure 3-3. Excess C2H6 adsorption vs. pressure and density at 20 ℃ (red line: saturation pressure at 20 ℃)

60
Table 3-3. C2H6 excess adsorption vs. pressure at 20°C
Pressure (bar) Mex (mg/g)
33.2 36.86
34.5 39.23
35.5 42.44
37.0 46.60
Three isobaric flow experiments were then conducted at pressures of 35 bar, 40 bar, and
45 bar. In each experiment, the temperature was systematically reduced from 45 °C to 20 °C, in
steps of approximately 5 °C. At each temperature step, the system temperature and pressure were
allowed to stabilize, with changes of less than 0.1 °C and 0.1 bar, respectively. At each temperature,
the experiment was allowed to continue until the measured excess adsorption changed by less than
1% over a period of one hour. The C₂H₆ excess adsorption data from these experiments are shown
in Figure 3-4, and the detailed experimental data are listed in Table 3-4.
From Fig. 3-4, one observes that the excess adsorption gradually increased as the
temperature decreased, with no sudden condensation noted, despite the proximity of the
experiments to the saturation line, as depicted in Figure 3-1. Additionally, the isobaric data at 35
bar with the home-made sample were compared with Grand Canonical Monte Carlo (GCMC)
simulations. These simulations were conducted using a model quartz pore with a pore diameter of
6 nm (for further details, see (Simeski et al., 2023). The excess adsorption values derived from the
simulations closely agree with the experimental results, also confirming the absence of capillary
condensation, and underscoring the consistency of both methodologies.

61
Figure 3-4.C2H6 excess adsorption vs. temperature for the three isobaric experiments (saturation temperature: 35 bar-red, 40 bargreen and 45 bar-blue).
Table 3-4. C2H6 excess adsorption vs. pressure for three isobaric experiments
Temperature
(°C)
Pressure
(bar)
Mex
(mg/g)
34.4 35 28.16
30.1 35 31.14
26.4 35 33.32
21.1 35 38.15
41.7 40 28.91
36.3 40 32.28
30.8 40 37.97
25.1 40 46.13
44.7 45 30.48
40.4 45 33.94
35.8 45 39.26
30.6 45 52.98

62
3.3.2 Thermodynamic calculations
We have explored the condensation behavior of the C2H6/mesoporous silica system using
continuum theories, focusing on predictions derived from the classical Kelvin Equation and a more
generalized Equation of State (EOS) approach. This latter method allows us to relax some of the
assumptions inherent in the Kelvin equation, specifically the ideal gas behavior and that of an
incompressible liquid phase residing in the pore. Below, we detail the specific procedures
employed in these calculations.
4.3.2.1 The Kelvin equation
The derivation of the Kelvin equation is based on the classical thermodynamic description
of vapor/liquid phase equilibrium, with the assumption that the vapor phase is an ideal gas and that
the liquid phase is an incompressible fluid. To derive the Kelvin equation, one assumes that the
porous medium is represented by a bundle of parallel, straight capillary tubes, each with the same
diameter. One, further, assumes that the capillary pressure Pc, which is the difference between the
pressure of the vapor phase outside and the liquid phase pressure inside the capillary, is represented
by the Young-Laplace equation in the following form:
𝑃𝑐 = 𝑃
𝑣 − 𝑃
𝑙 =
2𝜎 cos 𝜃
𝑟𝑐
. (3 − 1)
Here, 𝜃 is the contact angle, 𝑟𝑐
the capillary radius, 𝜎 the interfacial tension, and 𝑃
𝑣
and 𝑃
𝑙
are the
pressure of the vapor and liquid phases, respectively. We assume complete wetting of the liquid,
i.e., 𝜃 =0 and Eqn, 3-1 then takes the simpler form:
𝑃𝑐 = 𝑃
𝑣 − 𝑃
𝑙 =
2𝜎
𝑟𝑐
. (3 − 2)
The interfacial tension is computed by the following equation (Schechter & Guo, 1998):

63
𝜎 = [𝜒(𝜌
𝑙 − 𝜌
𝑣
)]
𝐸
, (3 − 3)
where E is the scaling exponent, 𝜒 is the parachor constant for C2H6, and 𝜌
𝑙
and 𝜌
𝑣
are the densities
of liquid and vapor phases at the experimental temperature T. In this study, we set E=4, and
𝜒=112.91 according to the study by Schechter et al. (1998). We note that the calculated values of
the surface tension by this approach agrees well with the experimental values reported by NIST
webbook. In what follows, we first derive the Kelvin equation.
The chemical potential 𝜇 of a given species is defined as the partial molar Gibbs function,
and is described by the following equation:
𝑑𝜇 = 𝑉𝑚 𝑑𝑃 − 𝑆𝑑𝑇 , (3 − 4)
where 𝑉𝑚 is the molar volume, S is the molar entropy, P is the pressure, and T is the temperature.
From Eqn. 3-4, at constant temperature T, one has:
(
𝜕𝜇
𝜕𝑃)
𝑇
= 𝑉𝑚 (3 − 5)
∫ 𝑑𝜇
𝜇
𝜇0
= ∫ 𝑉𝑚 𝑑𝑃
𝑃
𝑃0
, (3 − 6)
where 𝑃
0
is a reference pressure, taken here as the saturation pressure for the pure substance, and
𝜇
0
is the chemical potential at the reference pressure. For a substance with a molar volume that is
independent of pressure (incompressible liquid), the chemical potential is given as:
𝜇
𝑙
(𝑇, 𝑃
𝑙
) = 𝜇
𝑙
(𝑇, 𝑃
0
) + 𝑉𝑚
𝑙
(𝑃
𝑙 − 𝑃
0
), (3 − 7)
where 𝜇
𝑙
is the chemical potentials of the liquid, and 𝑉𝑚
𝑙
is the bulk liquid molar volume at T and
𝑃
0
. If the substance is an ideal gas, we have instead
𝑉𝑚 =
𝑅𝑇
𝑃
, (3 − 8)

64
where R is the gas constant. At constant temperature T, Eqn. 3-6 then becomes:
∫ 𝑑𝜇
𝜇
𝜇0
= 𝑅𝑇 ∫
𝑑𝑃
𝑃
𝑃
𝑃0
, (3 − 9)
and the chemical potential for the ideal gas is given as:
𝜇
𝑣
(𝑇, 𝑃
𝑣
) = 𝜇
𝑣
(𝑇, 𝑃
0
) + 𝑅𝑇 ln (
𝑃
𝑣
𝑃0
) , (3 − 10)
where 𝜇
𝑣
is the chemical potentials of the gas (vapor phase). For a pure liquid in a capillary in
equilibrium with its own vapor at a given temperature T, the following relationship applies.
𝜇
𝑣
(𝑇, 𝑃
𝑣
) = 𝜇
𝑙
(𝑇, 𝑃
𝑙
) (3 − 11)
Combining Eqn. 3-7, Eqn. 3-10, and Eqn. 3-11 (since the standard potentials for pure bulk
liquid and vapor at their saturated pressure are equal) one gets:
𝑅𝑇 ln (
𝑃
𝑣
𝑃0
) = 𝑉𝑚
𝑙
(𝑃
𝑙 − 𝑃0
) . (3 − 12)
Substituting 𝑃
𝑙
in Eqn. 3-12 from Eqn. 3-2, we find:
𝑅𝑇 ln (
𝑃
𝑣
𝑃0
) = −𝑉𝑚
𝑙 𝑃𝑐 + 𝑉𝑚
𝑙
(𝑃
𝑣 − 𝑃0
) = −𝑉𝑚
𝑙
2𝜎
𝑟𝑐
+ 𝑉𝑚
𝑙
(𝑃
𝑣 − 𝑃0
) . (3 − 13)
In the classical derivation of the Kelvin equation, the assumption is made that the capillary
pressure 𝑃𝑐
is much larger than the pressure difference between the vapor pressure and bulk
saturation pressure, so the last term on the right-hand side of Eqn. 3-13 is omitted. Then, Eqn. 3-
13 is simplified to the following classical form of the Kelvin equation:
𝑅𝑇 ln (
𝑃
𝑣
𝑃0
) = −𝑉𝑚
𝑙
2𝜎
𝑟𝑐
. (3 − 14)
However, in the calculations presented here, we make use of Eqn. 3-13.

65
3.3.2.2 Modeling pore condensation using the Peng-Robinson equation of state
In terms of fugacities, Eqn. 3-11 can be expressed as follows (Sandoval et al., 2016):
ln 𝐹
𝑣
(𝑇, 𝑃
𝑣
) − ln 𝐹
𝑙
(𝑇, 𝑃
𝑙
) = 0. (3 − 15)
To describe equilibrium between a pure liquid in a capillary with its vapor, one additional Eqn.
that correlates the capillary pressure with the liquid and vapor pressures is required
𝑃
𝑙 − 𝑃
𝑣 + 𝑃𝑐
(𝑇, 𝑃
𝑣
, 𝑃
𝑙
) = 0. (3 − 16)
Eqns. 3-15 and 3-16, assuming that the temperature is known, form a system of two
Equations with two unknown variables 𝑃
𝑙
, 𝑃
𝑣
. Solving these two equations one can calculate the
value of 𝑃
𝑣
for which condensation takes place to compare with experimental data.
To solve the equations, one substitutes in Eqn. 3-15 𝐹
𝑙 = 𝑃
𝑙𝜑
𝑙
, and 𝐹
𝑣 = 𝑃
𝑣𝜑
𝑣
, where
𝜑
𝑙
and 𝜑
𝑣
are the corresponding fugacity coefficients. The fugacity coefficient is calculated by
using the Peng-Robinson equation of state (PR-EOS). For the capillary pressure, we again utilize
the Young-Laplace Eqn. 3-1, and the detailed solution approach is given by Sandoval et al. (2016)
Figure 3-5 reports the calculation results from the Kelvin equation (Eqn. 3-13) and the
corresponding results employing the PR-EOS. In Fig. 3-5 we plot the G-L equilibrium line
calculated based on the PR-EOS, the 𝑃
𝑣
calculated by the Kelvin method, noted as 𝑃
𝑣
(𝐾𝑒𝑙𝑣𝑖𝑛),
and the 𝑃
𝑣
calculated by the PR EOS, noted as 𝑃
𝑣
(𝑃𝑅 − 𝐸𝑂𝑆). The results in Fig. 3-5 are for three
different capillaries with a diameter of 2 nm, 3 nm, and 7 nm. Note that the 𝑃
𝑣
(𝐾𝑒𝑙𝑣𝑖𝑛) and
𝑃
𝑣
(𝑃𝑅 − 𝐸𝑂𝑆) results are quite close with each other, which is indicative of the fact that the
assumptions of ideal gas and impressible liquid are reasonable for C2H6. Note, also that for the
large 7 nm pore capillary, the 𝑃
𝑣
lines are quite close to the G-L line (slightly below), which may
explain the fact that we have been unable to observe an abrupt condensation phenomenon in our

66
experiments. As the capillary diameter decreases, the 𝑃
𝑣
lines begin to separate from the G-L line.
Figure 3-5. Condensation pressure calculated by the Kelvin equation and PR-EOS method for capillaries with various pore
diameters, as indicated on the Figures.
3.3.3 Experimental results for AGC-40
3.3.3.1 C2H6 experiments at higher temperatures.
In Section 3.3.2, thermodynamic calculations indicated that the lack of observed abrupt
condensation phenomena in our experiments could be attributed to the narrow gap between the
calculated phase transition pressure line and the saturation pressure line. Consequently, we shifted
our focus to investigate a model silica material with a smaller pore diameter. The chosen material,

67
AGC-40 porous glass, features a fairly unimodal pore size distribution (see Figure 3-6) with an
average pore diameter of approximately 4 nm.
Figure 3-6 BJH analysis with N2 for AGC-40 at 77 k.
The experimental studies were conducted using both the static adsorption system and the
MSB instrument, following the detailed experimental procedures described in Section 3.2.2. The
static adsorption system was used for experiments under cryogenic and from vacuum to ambient
pressure conditions, while the MSB set-up was utilized for high-pressure and near-room
temperature conditions. These experimental conditions were selected based on classical
thermodynamic calculations detailed in Section 3.3.2, which suggest they may be conducive to

68
observing capillary condensation behavior.
Figure 3-7. C2H6 adsorption and desorption experiments at 31 bar.
Figure 3-7 presents the results from a series of adsorption/desorption experiments
conducted on the porous silica sample at 31 bar. Initially, the sample temperature was set at 20 °C
and gradually reduced to 12 °C. At each temperature step, we allowed the sample reach
thermodynamic equilibrium, characterized by a change in excess adsorption of less than 1% over
a 1-hour period, with temperature and pressure staying constant within 0.1 °C and 0.1 bar,
respectively. Following this, the temperature was incrementally raised to 52 °C. Despite crossing
the condensation line as predicted by classical thermodynamics (see further discussion below), no

69
hysteresis or abrupt phase transitions were observed.
Following up the isobaric C2H6 experiments conducted at a pressure of 31 bar, a subsequent
isothermal experiment was performed using the same sample, with the data shown in Figure 3-8.
This experiment began at a lower pressure of 25 bar, with the temperature held constant at 12 °C.
The pressure was incrementally raised to 31 bar. At each pressure increment, the sample was
allowed to stabilize at thermodynamic equilibrium, as indicated by a change in excess adsorption
of less than 1% over an hour. Throughout the experiment, sample temperature and pressure were
maintained constant, with minimal fluctuations not exceeding 0.1 °C and 0.1 bar, respectively.
Figure 3-8. C2H6 excess adsorption vs. pressure at 12°C

70
Figure 3-9. G-L equilibrium line in the bulk phase and the condensation P-T line from classical thermodynamics. The points
indicate the experimental conditions in Fig. 3-6 and Fig. 3-7.
Continuum thermodynamic calculations were also performed using the Peng-Robinson
Equation of State (PR-EOS) for C2H6 to define the gas-liquid (G-L) coexistence line and the
condensation line for a cylindrical pore with a diameter of 4 nm, as detailed in Section 3.3.2. These
theoretical lines are illustrated in Figure 3-9, which also depicts the paths followed during the
experiments in Figures 3-7 and 3-8 above. As Figure 3-8 clearly shows, during the cooling phase
of the adsorption cycle in Figure 3-7, we crossed the condensation pressure-temperature (P-T) line,
and this crossing occurred again during the heating phase. The condensation P-T line was also

71
crossed during the isothermal experiment. Despite these crossings, which typically suggest phase
transitions, no abrupt changes in behavior or hysteresis were observed, echoing the unexpected
findings noted earlier in our experiments. The inevitable variation in pore size distribution could
potentially obscure any abrupt changes in the excess adsorption. Additionally, in Figure 3-7, a
noticeable change in the slope of the excess adsorption versus temperature line occurs at around
20°C. This suggests that the rate of change in ethane's excess adsorption accelerates in response to
changes in pressure and temperature, implying that ethane may enter the supercritical region under
the operational conditions
3.3.3.2 N2 and C2H6 experiments at cryogenic conditions
In addition to the ethane experiments, we conducted isothermal N2 adsorption/desorption
experiments using the same AG-40 silica sample in the static adsorption system under cryogenic
conditions (77 K). The results of these experiments are displayed in Figure 3-10. During these
experiments, N2 was introduced stepwise into the static system’s sample cell, starting from vacuum
and increasing to 760 mmHg to generate the adsorption branch. The desorption branch was created
by gradually venting the nitrogen out of the cell, reducing the pressure stepwise from atmospheric
at the end of the adsorption run to vacuum at the end of the desorption cycle. The
adsorption/desorption isotherm shown in Figure 3-10 indicates a clear hysteresis between the
adsorption and desorption branches, along with evidence of abrupt behavior during the desorption
phase, a phenomenon we term capillary "evaporation."
We also explored the phase behavior of C2H6 under cryogenic conditions, by conducting
an isothermal experiment at ~195 K, near its normal boiling point (184.6 K). The C2H6 excess
adsorption versus pressure data for this experiment is shown in Figure 3-11. Similar to the results

72
obtained with N2, the C2H6 isotherm at 195 K exhibits an abrupt phase transition in the desorption
branch, indicative of capillary evaporation phenomena. There is also notable hysteresis between
the adsorption and desorption branches as is the case with N2. This observation supports the
presence of distinct phase behaviors under conditions approximating the substances' normal
boiling points.
Figure 3-10. Nitrogen isothermal adsorption and desorption experiments at 77 K.

73
Figure 3-6. C2H6 and N2 cryogenic isothermal adsorption and desorption experiments
3.3.3.3 CO2 experiments
A CO2 isotherm study on a natural sandstone sample at a temperature of 50 oC, previously
performed in our Group, displayed an abrupt increase in excess adsorption when the sample
pressure exceeded the critical pressure of CO2 (73.8 bar), as illustrated in Figure 3-12, clearly
indicating a phase transition within the sample. From the BET analysis of the sandstone sample,
whose weight was 2.86 g, the total pore volume of the sample for pores with diameters less than
50 nm was determined to be 0.015 cc. From the adsorption data, on the assumption that these pores
upon condensation are completely filled with liquid-like CO2, its density within the pores is
estimated to be 0.953 g/cc. Comparatively, the density of CO2 in the bulk phase at 50 ℃ and 80

74
bar (from the NIST database) is 0.219 g/cc, indicating that the CO2 density within the pores is over
four times greater than that of the bulk phase. Research by Melnichenko and Wignall (2009)
studied the density and volume fraction of an adsorbed CO2 phase in aerogels through a
methodology that utilizes independent measurements from neutron transmission and small-angle
neutron scattering of fluid-saturated absorbers. They found that CO2, under similar conditions to
our study, has a density of about 1 g/cc, supporting the conclusion that CO2 indeed condensed
under the experimental conditions of 50 ℃ and 80 bar.
Figure 3-12. CO2 excess adsorption vs. pressure up to 100 bar at 50 ℃ with a sandstone sample
According to the principle of corresponding states, substances exhibit similar
thermodynamic behaviors when they are at equivalent reduced temperatures and pressures. To test
this hypothesis, we conducted isotherm adsorption experiments with CO2 with the AG-40 silica
sample under the same reduced temperatures as those used for the C2H6 isotherm experiment (see

75
Fig. 3-8). The critical properties for both gases are shown in Table 3-5, with Pc and Tc representing
the critical pressure and temperature, respectively. The same Table also lists the saturation pressure
P0 corresponding to the experimental temperature .
Figure 3-13 shows the CO2 and C2H6 isothermal adsorption experiments at a reduced
temperature, Tr, of 0.4. Both gases show qualitatively similar behavior with excess adsorption
increasing with pressure, but with no abrupt changes observed, even as the experimental pressure
approached the saturation pressure P0 for both gases (corresponding to Pr=0.65 for CO2 and
Pr=0.66 for C2H6). The absence of abrupt phase transitions for both these gases at ambient
temperature conditions indicates that such phenomena are also absent in similar mesoporous
siliceous materials for temperature conditions related to shale gas operations for which Tr > 1.
Table 3-5. Comparison between the operating conditions for C2H6 and CO2
PC (bar) TC (°C) P0 @ operating T
(bar)
Tr
CO2 73.8 31 47.9 0.4
C2H6 48.7 32 32.2 0.4

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Figure 3-13. CO2 and C2H6 excess adsorption vs. reduced pressure
3.3.4 Geometric pore filling model
To interpret qualitatively the observed adsorption behavior, we have employed a simple
geometric pore filling model. In this model, we assume the pore is cylindrical, and the adsorbed
gas molecules are modeled as spheres, represented by their kinetic diameters (for N₂ the kinetic
diameter is reported to be 3.64 Å, and for C₂H₆ 4.43 Å). In the model, the molecules are arranged
within a 4 nm pore space, filling one layer at a time. This process assumes that the gas molecules
manage to achieve full packing for each layer according to a close-packing pattern, attaining a
packing density of ~74% (see Figure 3-14). After one layer is completely filled, another layer
begins to adsorb until the entire pore space is occupied. This sequential filling process for the
initial layer and the subsequent layers is shown in Figure 3-15.

77
Figures 3-16 and 3-17 interpret the experimental data from Figures 3-8 and 3-9,
respectively, using the pore-filling model. Figure 3-16 distinctly shows that during the adsorption
part of the cycle, the rate of pore filling accelerates as the temperature increases. Conversely,
during the desorption phase, there is a noticeable acceleration in the emptying of the pore as the
temperature continues to rise. Figure 3-17 demonstrates similar behavior with accelerated rates of
pore emptying observed.
Both figures indicate an abrupt phase transition during the pore emptying process. This
transition points to a rapid change in the state or arrangement of the adsorbed N2 or C2H6 molecules,
likely reflecting significant thermodynamic or structural shifts within the pore environment.
Figure 3-14. Close-packing structure

78
Figure 3-7. Multilayer growth
Figure 3-16. C2H6 Adsorption/Desorption in Porous SiO2 (dp=4 nm): Data interpretation with a pore-filling model
Monolayer Second layer Third layer

79
Figure 3-17. N2 Adsorption/Desorption in Porous SiO2 (dp=4 nm): Data interpretation with a pore-filling model
3.4 Summary
This section presents a detailed experimental study on phase transition phenomena,
including capillary condensation and evaporation, using two model samples: a home-made silica
sample with an average pore diameter (Dav) of ~7 nm, and a commercial silica (AGC-40) silica
sample with a Dav of ~4 nm. The experiments, conducted with C2H6, CO2, and N2 gases, and
included both isothermal and isobaric tests. The operating conditions were chosen to be nearby the
G-L to favor phase transitions and pore condensation phenomena. Despite this, no abrupt changes
in excess adsorption were observed in the experiments during adsorption even under conditions
for which the condensation line predicted from continuum thermodynamic calculations was
crossed (see Figure 3-9).

80
The only time an abrupt change in excess adsorption was observed in this study, is with
the AGC-40 sample during desorption under cryogenic isothermal conditions (see Figures 3-10
and 3-11), a phenomenon which we termed capillary evaporation. It is unclear though that this
interesting behavior reflects a true phase transition or is simply the manifestation of other complex
(e.g., percolation-type) phenomena taking place. Additional studies are, therefore, needed
particularly involving complementary techniques such as DSC and NMR spectroscopy to gain
additional insight into these phenomena and to validate that they indeed represent true phase
transitions.
3.5 References
Melnichenko, Y. B., & Wignall, G. D. (2009). Density and volume fraction of supercritical CO 2
in pores of native and oxidized aerogels. International Journal of Thermophysics, 30, 1578-
1590.
Sandoval, D. R., Yan, W., Michelsen, M. L., & Stenby, E. H. (2016). The phase envelope of
multicomponent mixtures in the presence of a capillary pressure difference. Industrial &
engineering chemistry research, 55(22), 6530-6538.
Schechter, D., & Guo, B. (1998). Parachors based on modern physics and their uses in IFT
prediction of reservoir fluids. SPE Reservoir Evaluation & Engineering, 1(03), 207-217.
Simeski, F., Wu, J., Hu, S., Tsotsis, T. T., Jessen, K., & Ihme, M. (2023). Local Rearrangement in
Adsorption Layers of Nanoconfined Ethane. The Journal of Physical Chemistry C, 127(34),
17290-17297.
3.6 Appendix
Appendix A Experimental data for ethane experiments

81
Table A-1 C2H6 experimental data for the synthetic sample at 50°C
Pressure(bar) Weight (g)
30 5.899853
32 5.897187
34 5.892936
36 5.889074
38 5.885487
40 5.880522
42 5.876466
44 5.870851
46 5.865273
48 5.861587
50 5.855421
52 5.847689
55 5.839566
Table A-2 C2H6 excess adsorption vs. pressure at 20°C
Pressure (bar) Weight (g)
33 5.897226
34 5.894409
35 5.892095
37 5.886608
Table A-3 C2H6 excess adsorption vs. pressure for three isobaric experiments
Temperature (℃) Pressure (bar) Weight (g)
34 35 5.894601
30 35 5.893451
26 35 5.892876
21 35 5.892077
42 40 5.885145
36 40 5.882316
31 40 5.879145
25 40 5.875582
45 45 5.873445
40 45 5.870041
36 45 5.865463
31 45 5.858498

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Chapter 4: Exploring Ethane's Phase Behavior in MCM-41
4.1 Introduction
Understanding phase transitions is foundational for many scientific and industrial
applications. Phase transitions are classified based on the derivative of the thermodynamic
potential (Binder, 1987). A first-order phase transition, such as melting and boiling, is
characterized by a discontinuity in the first order derivative of the Helmholtz free energy. Secondorder phase transition, typically observed at critical points in phase diagrams, involves continuous
first order but discontinuous second order derivative. In confined systems, such as within
nanoporous materials, the phase behavior can differ significantly from that in the bulk phase due
to the strong interactions between the molecules and the pore walls. These molecule-pore wall
interactions lead to unique phase behaviors that are not observed in bulk systems, emphasizing the
importance of understanding confinement effects.
Phase transitions in mesoporous media are of importance in many fields, including drug
delivery (Felice et al., 2014), membrane-based gas separations (Bernardo et al., 2009), gas storage,
and natural gas recovery (Chai et al., 2019; Krishna et al., 2022), among others. In the oil and gas
industry, understanding phase transitions is of importance for accurate reservoir simulations and
fluid-in-place calculations. This is particularly true for unconventional reservoirs like the shales,
which are characterized by a large fraction of micropores in their structure (Luo et al., 2020) , by
some accounts constituting more than 20% of the total porosity. Consequently, the phase behavior
of hydrocarbons in these materials can differ significantly from bulk-phase behavior due to
confinement effects and interactions with the pore walls. An important departure from bulk
behavior that can be observed in the small pores of shale reservoirs is capillary condensation,

83
where saturation pressures in the pores (defined as the system pressure for which the pores are
filled with a liquid-like phase) can be more than 50% lower than that under bulk conditions
(Barsotti, 2019). This highlights the potential for significant inaccuracies in phase behavior
predictions, if the phenomena related to pore confinement is not accounted for. Despite its
importance, the fundamental understanding of capillary condensation phenomena in shales is
rather limited, primarily due to their complex and heterogeneous nature (Barsotti et al., 2016).
Molecular simulation like molecular dynamics (MD) and Monte Carlo (MC) simulations
offer atomic-level insight into the behavior of molecules in porous materials. Adsorption,
desorption, and phase behavior can be simulated by accounting for the interactions between gas
molecules and the pore surfaces. Molecular simulations provide detailed information on the
behavior of individual molecules, allowing for a better understanding of the underlying
mechanisms of the various phenomena, including phase transitions. Additionally, simulation
methods allow one to explore regions of temperature and pressure conditions not always readily
accessible by the experimental methods, thus allowing one to gain a more comprehensive
understanding of the behavior of a given system.
A number of molecular simulation methods that have been applied to simulate
adsorption/desorption and phase behavior in porous media. They include techniques such as Grand
canonical Monte Carlo (GCMC), Gibbs ensemble Monte Carlo (GEMC), Gauge cell Gibbs
ensemble Monte Carlo (Gauge-GEMC), and Gauge cell Grand canonical Monte Carlo (GaugeGCMC). The GCMC method is particularly well-suited for simulating adsorption phenomena
because its ensemble mimics the typical isothermal experimental conditions, by fixing the
chemical potential (µ), volume (V), and temperature (T) of the simulation box. The method

84
involves randomly inserting gas molecules into the porous structure (i.e., the simulation box),
exchanging molecules with an imaginary gas reservoir at the same thermostat, and allowing them
to interact with the material until equilibrium is established, i.e., a pre-determined value of µ is
reached. The GCMC method is commonly used to generate adsorption/desorption isotherms and
to study hysteresis phenomena for both pure and multicomponent systems in mesopores and
micropores (Rowley et al., 1975, Coasne et al., 2005, Libby & Monson, 2004). Jin and Nasrabadi
(2016) added a gauge box to the original GCMC simulation method and proposed the GaugeGCMC method to improve the efficiency of the GCMC simulation for multicomponent fluid
systems.
One weakness of the GCMC method is that it cannot predict the gas-liquid phase
coexistance line accurately. In contrast, the GEMC simulation method introduced by
Panagiotopoulos (1987) offers a more advanced approach for modeling vapor-liquid equilibrium
under bulk conditions. The use of the two simulation boxes distinguishes the GEMC method from
other methods, with each phase (liquid or gas) represented by a given box. The use of the two
boxes improves the efficiency of computing the phase equilibrium (Ungerer et al., 2007). However,
when applying the GEMC method to a confined system, the determined coexistence point is
sensitive to the initial configuration (Neimark & Vishnyakov, 2000). Therefore, based on
Panagiotopoulos’s work, Neimark and Vishnyakov (2000) proposed a gauge cell Gibbs ensemble
Monte Carlo method (Gauge-GEMC), which extended the capabilities of the GEMC method. The
finite size of the gauge box in the Gauge-GEMC method allows the simulation to access unstable
states. The Gauge-GEMC simulations are performed starting from a low gas density towards a
higher density by increasing the total number of molecules in the two boxes while the total volume

85
remains constant. This approach yields the Van der Waals loop, which correlates the chemical
potential and density under the entire range of relevant pressure conditions at the specified
temperature. The Gauge-GEMC method has a higher computational efficiency compared to other
methods, and it is the prevalent method to study phase transitions of fluids under confinement.
Although this chapter does not focus on binary or ternary systems, the Gauge-GEMC
method's applicability to such systems merits mention. For example, Jin and Nasrabadi (2016)
presented a modified version of the Gauge-GEMC method to study the simulation of binary
systems. They also combined the Gauge-GEMC and GCMC methods to develop a new method
(Gauge-GCMC method) to investigate binary and ternary (C1/C3/C5) systems in bulk and a slit
pore with a 4 nm pore diameter. In subsequent work, Jin (Jin et al., 2017) used their proposed
Gauge-GCMC method to study the effect of pore size distribution (PSD) on the confined phase
behavior of hydrocarbons. They not only conducted the simulations in the single pores with pore
diameters ranging from 4 to 10 nm but also applied the simulations in a multi-pore structure to
further investigate the PSD effect on the phase behavior.
Turning our focus to C2H6, one of the more condensable gases, found in the natural gas
(Patankar et al., 2016), is mainly used to produce ethylene and for power generation, and as a
feedstock to make plastics. Despite its significance, research on C2H6 adsorption in mesoporous
media has been limited. Yun et al. (2002) explored the adsorption of mixtures of methane with
C2H6 at 264.55 K and C2H6 with carbon dioxide by using a volumetric method (experiments) and
the GCMC simulation approach to predict the adsorption equilibrium in a pure-silica MCM-41
material with a pore width of 4.1 nm. They compared the results with the ideal adsorbed solution
theory (IAST) model. IAST provides accurate predictions in terms of multicomponent adsorption

86
equilibrium, while GCMC gives accurate predictions for both pure-gas and binary adsorption. He
and Seaton (2003) studied C2H6 adsorption in MCM-41 experimentally and using GCMC
simulation at a temperature between 264 K and 303 K and pressure up to 3 MPa. The GCMC
simulations were carried out using three structure models of MCM-41 to represent different
degrees of surface heterogeneity. Among these structural models, the model with an amorphous
structure provides a better prediction for C2H6 adsorption over the whole pressure range. Patankar
et al. (2016) investigated the role of confinement on adsorption and dynamics of C2H6 and
C2H6/CO2 mixture in mesoporous controlled pore glass (CPG) with diameters of 41.5nm and
11.1nm by using gravimetric adsorption and quasi-elastic neutron scattering (QENS) methods.
Elola and Rodriguez (2019) studied the structural and dynamical properties of C2H6 confined with
cylindrical silica nanopores of 3.8 nm and 1nm diameters with and without another species, CO2,
by using molecular simulations. In addition to our own studies reported in this Thesis, Yang et al.
(2022) quantified the capillary condensation of ethane in MCM-41, examining conditions ranging
from 206 K and 1.1 bar up to the pore critical point (PCP) region, which they identified at a
temperature of 275 K and a pressure of 14.3 bar, using an isochoric cooling procedure with
differential scanning calorimetry (DSC). They characterized capillary condensation through the
appearance of a relatively small heat flow peak on the thermogram, in contrast to the larger peak
corresponding to bulk condensation. However, they did not observe the same peak at the
conditions above the PCP. They concluded that a gradual phase transition of ethane exists, and
experimentally demonstrated it by employing a three-step DSC method comprising isothermal
compression, isochoric cooling, and a polytropic process.

87
However, the mechanisms underpinning C2H6 sorption dynamics and potential phase
transitions, such as capillary evaporation/condensation phenomena, under pore confinement is still
unclear and merrits further investigation. In this work, we conduct C2H6 adsorption experiments
on a MCM-41 sample with a pore diameter of 3.4 nm. We also utilize GCMC and Gauge-GEMC
simulation methods to investigate the sorption behavior for a wide range of pressure and
temperature conditions. Using the Gauge-GEMC method allowed us to access the metastable and
unstable states of the isotherms and to construct the vapor-liquid coexistence curve via the
application of Maxwell’s rule of equal areas. Furthermore, the simulations allowed us to construct
the local density profiles in the pore, which helps to improve our understanding of the observed
sorption behavior.
4.2 Experimental
The C2H6 adsorption experiments were carried out with MCM-41 (ACS Material®), which
has a regular honeycomb-type pore structure with an average pore diameter of 3.4 nm. The
adsorption experiments were conducted via the gravimetric method using a magnetic suspension
microbalance (MSB, Rubotherm IsoSORP®, Germany). The measurements commenced by
placing the MCM-41 material in the sample cell and evacuating it under a high vacuum at 80 ℃
until the apparent weight (real weight minus buoyancy) stabilized (less than 10 μg change observed
over a period of 1 hour). Subsequently, the sample cell was pressurized with pure C2H6 to the
selected initial pressure. After adsorption equilibrium was established (as indicated by the fact that
the weight of the sample did not change by more than 10 μg over 1 hour period), the sample cell
was pressurized to the next pressure level. Throughout these measurements, the temperature was
maintained constant using an oil bath (Julabo F25) connected to the MSB instrument.

88
4.3 Simulations
The Gauge-GEMC and GCMC methods were employed to investigate the adsorption and
phase transitions of C2H6 within the MCM-41 structure, and a brief description of both techniques
is provided below.
4.3.1 GCMC simulations
GCMC simulations are commonly utilized for the study of adsorption/desorption
phenomena as they operate within the grand-canonical (µVT) ensemble, allowing for a direct
correlation between the chemical potential (µ) and pressure. Similarly, the isotherms derived from
the adsorption/desorption experiments are conducted under specified temperature conditions, thus
enabling GCMC simulations to efficiently replicate the conditions of these experiments. In our
study, we have performed complementary MC simulations within the isobaric-isothermal (NPT)
ensemble to establish the relationship between pressure and chemical potential, further facilitating
a comprehensive interpretation of both experimental and simulation results.
We utilize a structural model for the MCM-41 developed previously by Ugliengo et al
(2008). The interaction between the fluid and solid pore walls is described by the Lennard-Jones
12-6 potential augmented with an additional term describing the Coulombic electrostatic
contribution:
𝑈𝑖𝑗(𝑟) = 4𝜀𝑖𝑗 [(
𝜎𝑖𝑗
𝑟𝑖𝑗)
12
− (
𝜎𝑖𝑗
𝑟𝑖𝑗)
6
] +
𝑞𝑖𝑞𝑗
4𝜋𝜀0𝑟𝑖𝑗
， (4 − 1)
Here 𝑈𝑖𝑗(𝑟) is the LJ intermolecular energy, 𝑟𝑖𝑗 is the distance between the centers of particles i
and j, 𝜀𝑖𝑗 is the depth of the potential well of the minimum interaction energy, and 𝜎𝑖𝑗 is the
separation distance of particles i and j when the LJ interaction is zero, and 𝑞𝑖
is the electrostatic

89
charge of the atom i. For the interaction between unlike sites i and j, the Lorentz-Berthelot rules
are applied:
𝜎𝑖𝑗 =
𝜎𝑖𝑖 + 𝜎𝑗𝑗
2
,
𝜀𝑖𝑗 = √𝜀𝑖𝑖𝜀𝑗𝑗. (4 − 2)
The parameter values utilized in this study are summarized in Table 4-1.
Table 4-1. Lennard-Jones potential parameters used in the simulations
Molecule Site σ (Å) ε/kB (k)
MCM-41 Si 3.804 70
O 3.033 40
C2H6 CH3 3.75 98
4.3.2 Gauge-GEMC method
The Gauge-GEMC method is used to simulate the phase behavior of pure fluids under pore
confinement (Vishnyakov & Neimark, 2001). The method employs two simulation boxes, one of
which is a gauge meter with a finite volume and the other represents the confined fluid system (i.e.,
the pores). The total volume of the two simulation boxes, the total number of molecules, and the
system temperature all remain constant during the Gauge-GEMC simulation. The Gauge-GEMC
method connects the vapor-like and the liquid-like spinodal points by a continuous path to
construct a comprehensive phase diagram in the form of a Van der Waals loop, that includes, stable,
metastable, and unstable states. This is possible as the restricted capacity of the gauge box limits
density fluctuations within the fluid system, therefore allowing the fluid to remain in a metastable
and even unstable state. The vapor-liquid equilibrium in porous media is influenced by pore size,
temperature, and pore surfaces. To initialize a simulation, the temperature and box size(s) are fixed,

90
and only the number of C2H6 molecules is changed to obtain a complete isotherm Van der Waals
loop. Under each operation condition, five million MC (Monte Carlo) steps were applied, and an
additional million MC steps was carried out for the statistical calculations. The system equilibrium
is reached when the total Helmholtz free energy in the boxes is minimized, and the chemical
potentials of the gauge cell and the confined system become equal. Phase transitions are possible
in between the spinodal points, whereby the grand thermodynamic potentials (Ω) are equal in the
vapor-like state on the adsorption branch and the liquid-like on the desorption branch:
Ω𝑎
(𝜇𝑒
) = Ω𝑑
(𝜇𝑒
). (4 − 3)
In the above equation, Ω𝑎 represent the grand potential of the adsorption branch, Ω𝑑 represents
the grand potential of the desorption branch, and the 𝜇𝑒
represents the chemical potential when the
equilibrium state is achieved. Furthermore, the adsorption isotherm satisfies the Gibbs adsorption
equation:
𝑁 = − (
𝜕𝛺
𝜕𝜇)
𝑇
, (4 − 4)
where N represents the number of molecules. The determination of phase coexistence conditions
from Eqn. 4-3 is achieved by applying integration across the entire loop, following to Maxwell's
principle of equal areas:
∫ 𝑁𝑎𝑑𝜇
𝜇𝑠𝑣
𝜇𝑒
− ∫ 𝑁𝑢𝑑𝜇
𝜇𝑠𝑣
𝜇𝑠𝑙
+ ∫ 𝑁𝑑𝑑𝜇
𝜇𝑠𝑙
𝜇𝑒
= 0, (4 − 5)
where subscripts sv and sl represent the vapor-like and the liquid-like spinodal points, respectively,
𝑁𝑎 is the number of molecules on the vapor-like (adsorption) branch, 𝑁𝑑 the number of molecules
on the liquid-like (desorption) branch, and 𝑁𝑢 represents the number of molecules at the unstable
state.

91
4.4 Results and Discussions
4.4.1 Model validation
The chemical potential serves as the input for the GCMC simulations but also as the output
for the Gauge-GEMC simulations. Thus, NPT simulations are needed to correlate the chemical
potential with the pressure in real experiments. In NPT simulations, the number of particles (N),
the pressure (P), and the temperature (T) of the system remain constant. In this work, NPT
simulations were performed in a cubic simulation box at various temperatures from 174.2 K to
303.2 K. For each pressure and temperature condition, 10 million MC moves were performed, and
the data from the last 1 million MC moves were averaged to calculate the value of pressure
corresponding to any given chemical potential. Fig. 4-1 shows the chemical potential and the
pressure relationship provided by the NPT simulations for C2H6.
Figure 4-1. Chemical potential vs. pressure calculated from NPT simulations at various temperatures.

92
The GEMC method was employed to model the C2H6 bulk phase behavior and to compare
it with the available experimental data prior to applying the method to study C2H6 adsorption in
MCM-41. For that, C2H6 densities for both the gas and liquid phases in the bulk were calculated
via GEMC for temperatures ranging from 183 K to 294 K. The calculated vapor and liquid phase
C2H6 densities at saturation for various temperatures are shown in Fig. 4-2. They are also compared
with the experimental data from the National Institute of Standards and Technology (NIST)
database. The densities calculated via GEMC show good agreement with the experimental density
data, showing that the TRAPPE forcefield is suitable for predicting the bulk phase behavior. The
gas phase C2H6 density was also calculated via the GCMC simulation method and is compared
with the NIST data in Fig. 4-3 demonstrating, again, a good agreement and thus validating the use
of the TRAPPE forcefield.
Figure 4-2. C2H6 temperature vs. vapor and liquid density data at saturation pressure were calculated from the GEMC simulations
in the bulk phase from 183 K to 294 K compared to the NIST database.

93
Figure 4-3. C2H6 density vs. pressure in the bulk phase calculated from the GCMC simulations compared to the NIST database.
4.4.2 Adsorption experiments on MCM-41
This section describes the experimental results observed for the MCM-41 sample. We first
performed isothermal experiments at 184 K up to 1 bar, observing the behavior of ethane (C2H6).
As illustrated in Figure 4-4, the excess adsorption on MCM-41 gradually increase with pressure
without any abrupt phase transitions. The absence of noticeable phase changes suggests that either
the phase transition does not occur under these specific conditions, or it is not detectable with our

94
experimental setup. This could potentially be attributed to the pore-size distribution of the material,
as also discussed by Neimark and Vishnyakov (2000).
Figure 4-4. C2H6 isothermal adsorption and desorption experiments on MCM-41 at 184 K.
Subsequently, we conducted isobaric experiments at 31 bar from 283.8 K to 323 K, and an
isothermal experiment at temperature of 283.8 K from 10 bar to 31 bar. Figures 4-5 and 4-6 report
the excess adsorption, normalized to the surface area determined by a separate BET analysis.
During these experiments, we note a gradual increase in excess adsorption with no abrupt changes
observed. Although the pressure and temperature approached the phase saturation line, no phase
transitions were detected within the pore confines of MCM-41 under these conditions.

95
Figure 4-5. C2H6 excess adsorption on MCM 41 normalized to the surface area at 283.8 K from 10 bar to 31 bar.
Figure 4-6. C2H6 excess adsorption on MCM-41 normalized to the surface area at 31 bar from 283.8 K to 323 K.

96
4.4.3 Comparison of experimental and simulation results.
Fig. 4-7 shows a comparison of the GCMC simulations with experimental measurements
at 283.8 K in terms of the C2H6 excess adsorption as a function of pressure. On the same figure,
we also report data from the literature (He and Seaton,(2003); Yun et al.,(2002)) taken at 303.2 K.
The excess adsorption (𝑀𝑒𝑥) from the GCMC simulations was calculated as follows:
𝑀𝑒𝑥 = 𝑁𝑡 − 𝜌𝐵𝑢𝑙𝑘 × 𝑉
𝑝
, (4 − 6)
where 𝑁𝑡
(mol) is the total of moles residing inside the pore at equilibrium, 𝜌𝐵𝑢𝑙𝑘 is the bulk
density at the corresponding pressure and temperature conditions (from the NIST database), and
𝑉
𝑝 = 0.74 cc/g is the total pore volume per mass of adsorbent. The details about the calculation
of the pore volume for this simulation model are provided in Chapter 3. Further details on the
experimental measurements can be found in Chapter 4 and also in (Wu et al., 2022). The relatively
good agreement between the GCMC simulation results and the experimental findings suggest that
the DREIDING forcefield is capable of accurately predicting the adsorption behavior of C2H6
within the MCM-41 structure.

97
Figure 4-4. The comparison of the GCMC simulations, experimental results, and the data in the literature at 283.8 K and 303.2 K.
4.4.4 Comparison of Gauge-GEMC and GCMC simulations.
Fig. 4-8 and Fig. 4-9 report the calculated, GCMC and Gauge-GEMC simulations, volumeaveraged C2H6 density in the pore versus the chemical potential in a range of temperatures from
174.2 K to 283.8 K. The choice of the lower T value in this range is to incorporate the condition
where we observed capillary evaporation phenomena (See Chapter 4, Figure 4-10). During the
simulation, we attempted to approach the temperature at where the S shape of the Van der Waals
loop vanishes, so we operate at the temperature up to 283.8 K. In the Gauge-GEMC simulations,
below a certain temperature, see Fig. 4-8, one observes clearly the presence of the S-shaped Van
der Waals loop. Above that temperature, see Fig 4-9, there is good agreement between the GCMC
and the Gauge-GEMC simulations. In the region of temperatures for which the Van der Waals

98
loop is present, the GCMC and the Gauge-GEMC simulations overlap at conditions for which the
stable and some of the metastable states prevail. However, the GCMC simulations spontaneously
condense into a liquid-like state before the Gauge-GEMC reaches the vapor-like spinodal point. It
is well known that the S shape of the Van der Waals loop disappears at the critical point for bulk
phase conditions. Therefore, the disappearance of the S shape shown in Fig. 4-9 could be
interpreted as the C2H6 molecules under such pore confinement have exceeded the critical
conditions. This explains the absence of abrupt phase transitions at higher temperatures even
though the operating pressure is approaching the saturation pressure.
Figure 4-5. The comparison between the Gauge-GEMC and GCMC simulations at 174.2 K, 184.2 K, 214.2 K, and 235 K.

99
Figure 4-6. The comparison between the Gauge-GEMC and GCMC simulations at 244.2 K, 264.75 K, 273.2 K, and 283.8 K.
4.4.5 The predicted equilibrium phase transition pressure.
Table 4-2 lists the predicted phase transition (condensation) pressures for four different
temperatures in the MCM-41 material calculated based on Maxwell’s principle of equal areas as
well as the corresponding bulk-phase saturation pressures taken from the NIST database. Fig. 4-
10 shows the same data plotted versus temperature, to visually emphasize the difference between
the phase transition pressure under pore confinement in MCM-41 and the saturation pressure in
the bulk phase. The reduction in the phase transition pressure due to pore confinement decreases
with a 60 K increase in temperature, from 77.5 % to 65.3 %. Comparing the results in Table 4-2

100
with the BET measurements, presented in Figure 4-4, the adsorbed amount in the BET
measurements exhibits a rapid increase starting at approximately 0.2 P/P0 (0.197 bar), which is
about 10% lower than the phase transition pressure predicted by the Gauge-GEMC simulation.
Table 4-2. Phase transition pressures at various temperatures as predicted by the Gauge-GEMC simulations.
Temperature
(K)
Phase transition pressure
(bar)
Psat in the bulk phase
(bar)
Reduction of the phase
transition pressure
(%)
174.2 0.13 0.56 77.5
184.2 0.22 0.99 75.3
214.2 1.32 3.94 67.8
235.0 2.86 8.26 65.3
Figure 4-7. The comparison between the saturation pressure in the bulk phase and the phase transition pressure in the pore

101
This discrepancy can be partially attributed to the temperature control employed during the
BET measurements. Specifically, we used a cooling bath of n-butanal and liquid nitrogen to
maintain the temperature at 184 K throughout the experiment. However, potential temperature
fluctuations during the measurements cannot be entirely excluded, and may contribute to the
observed difference between BET and simulation results. Additionally, the BET measurements
reveal a gradual change in the pressure range from 0.2 P/P0 to 0.4 P/P0, rather than the sharp
transition typically expected for a condensation phenomenon. This difference is likely due the pore
size distribution of the material: A similar observation was reported by Neimark and Vishnyakov
(2000), who noted that discrepancies between Gauge-GEMC results, and experimental data were
due to the pore size distribution in MCM-41 samples.
Our simulation effort is the first to predict the ethane saturation pressure within an MCM41 sample with a 3.4 nm pore diameter using the Gauge-GEMC method, and we proceed to
compare this prediction with available experimental observations. For example, Sharma et al.
(2023) conducted an experimental study on the confined phase behavior of C2H6 using a
gravimetric approach. They identified an inverse linear relationship between the capillary
condensation pressure and the pore size of MCM-41 samples over a temperature range of 208 K
to 253 K, with pore sizes ranging from 7 nm to 12.5 nm. They extended the linear relationship to
a broader pore size range up to 0.3 (1/nm), as demonstrated in Fig. 4-8 for temperatures of 208.15
K and 214.15 K. In the same figure, we report our Gauge-GEMC simulation results for the MCM41 material tested in this study (symbols), and observe that the results align well with the findings
of Sharma et al. (2023).

102
Figure 4-8. The comparison between the Gauge-GEMC simulations (Square and circle) and Sharma et al.’s (Sharma et al., 2023)
extrapolation results (Solid lines).
To further understand the C2H6 thermodynamic behavior under pore confinement, the C2H6
molecular density distribution profiles are shown in Fig. 4-12 for various temperatures. In Fig. 4-
12 we plot the molecular density (# of molecules per unit volume) versus the position (i.e., distance
from the pore wall) where such density is calculated, showing only the molecular distribution for
half of the pore due to its symmetrical nature. The molecular distribution is determined by counting
the number of methyl (CH3) groups along with their corresponding coordinates. We then filter
pairs of CH3 groups belonging to the same ethane (C2H6) molecule. Subsequently, the midpoint of
each pair of CH3 groups is calculated, and the number of C2H6 molecules is the reported versus
the coordinates of these midpoints. The top curve (orange) in each plot shows the molecular
distribution profile for the liquid-like fluid within the pore immediately after reaching the phase
transition pressure, and the bottom curve (blue) indicates the molecular distribution profile of the

103
vapor-like phase within the pore just before the phase transition pressure is reached. At all
temperature conditions, the molecular distribution profiles for both the liquid-like and the vaporlike phases indicate a high-density first layer near the pore wall as well as the presence of other
lower-density layers. The difference in the molecular distribution profiles corresponding to the
states before and after capillary condensation has taken place is that the density of the additional
(central) layers is much larger in the liquid-like phase (post-condensation) than in the vapor-like
phase. In both instances, however, the density at the pore center remains smaller than the bulkphase liquid density at the corresponding saturation pressure. As the temperature increases the
differences between the molecular distribution profiles for the pre-and post-condensation states
are reduced, indicating the approach to conditions where abrupt phase transitions are no longer
feasible.

104
Figure 4-9. The molecular distribution in the pore at various temperatures. The top and bottom curves in each subfigure represent
the molecular distribution profile after and before the phase transition pressure in the pore.
4.4.6 Effect of Pore Size on Capillary Condensation Phenomena
The study of capillary condensation in porous materials like MCM-41 reveals complex
interactions between fluid molecules and pore walls, which vary significantly depending on pore
size. In our experiments with ethane (C2H6) adsorption in MCM-41, with an average pore diameter
of 3.4 nm, no clear capillary condensation phase transition was observed. This experimental
outcome contrasts with our Gauge-GEMC and GCMC simulations, which clearly indicate the
onset of capillary condensation.

105
The discrepancy between experimental and simulation results can be understood by
considering the findings by Auti et al. (2024), who developed a statistical model that combines the
3D Ising framework with nanothermodynamics to analyze phase transitions in fluids confined
within mesoporous materials, such as metal-organic frameworks (MOFs). Their study emphasizes
the pore size plays a critical role in determining phase behavior. In smaller pores, the activationenergy barrier for phase transitions is extremely low (~0.01𝑘𝐵𝑇 ), enabling a spontaneous and
continuous transition from a gas-like phase to a liquid-like condensed phase without a need for a
significant thermal or external energy input. Consequently, these transitions lack distinct
thermodynamic signatures and appear as smooth and seamless processes that blend into the overall
adsorption behavior. Applying their theory to our system explains why the adsorption isotherms
from our experiments do not show abrupt changes in the adsorbed amount, while the simulation
clearly captures phase transitions. The reason is that in small pores, the energy barrier for phase
transitions is so low that it becomes indistinguishable from adsorption phenomena when analyzing
the isotherms. Moreover, experimental techniques like BET analysis, which measures adsorption
isotherms as a function of pressure, inherently average the adsorption behavior across all pores.
This averaging prevents BET measurements from effectively capturing localized phenomena, such
as adsorption layering or confined phase transitions, resulting in the smoothing of subtle transitions
and a reduced resolution for detecting complex phase behavior in confined systems, thereby
highlighting the significance of simulations in providing deeper insights into these phenomena.
A similar discrepancy between simulation and experiments was observed by Neimark and
Vishnyakov (2000) in their study of nitrogen adsorption in MCM-41 with a 3.55 nm pore diameter
at 77.4 k. They noted that the capillary condensation step in experimental isotherms often appears

106
rounded, which they attributed to variations in pore width within the sample. Simulated isotherms,
which typically consider uniform pore sizes show sharper transitions. Neimark and Vishnyakov
conclude that this discrepancy was primarily due to the inevitable pore size distribution within the
MCM-41 samples. However, their comparison between samples with pore diameters of 4.42 nm
and 3.55 nm demonstrated that larger pore diameters result in sharper transitions and better
agreement with simulation results, despite the difference in pore diameter being less than 1 nm.
While we acknowledge the possible influence of pore size distribution within our samples, the low
energy barrier theory provides a more comprehensive and mechanistic explanation for this type of
discrepancy. In our experiments with MCM-41, the absence of observable capillary condensation
is therefore attributed to a combination of the small pore size and the potential pore size distribution
within the material, both of which contribute to smoothing out sharp transitions and result in an
observed continuous phase change that is challenging to detect experimentally.
4.5 Conclusions
This study investigated the adsorption and phase transition behavior of C2H6 within the
mesoporous material MCM-41, with an average pore diameter of 3.4 nm. The experimental work
applied gravimetric measurements of C2H6 adsorption at various pressures and temperatures,
employing a magnetic suspension microbalance, as well as BET measurements of C2H6 adsorption
under cryogenic conditions. While these experiments provided detailed observations of the
adsorption process, no distinct capillary condensation was observed over the range of test
conditions.

107
In addition to the experimental work, comprehensive simulations were conducted using
both Gauge-GEMC and GCMC methods to model the phase behavior and adsorption of C2H6
within the MCM-41 structure. The Gauge-GEMC method, in particular, provided a robust
framework for simulating phase transitions in confined systems, allowing us to capture the
complete isotherm, including metastable and unstable states. The GCMC simulations
complemented this by replicating the experimental conditions closely, providing for a direct
comparison with the experimental adsorption isotherms.
One key advantage of the simulation approach is the ability to predict phase transition
behavior in confined systems where experimental detection can be masked by pore size variations.
The simulations accurately predicted the phase transition pressures for ethane in MCM-41 across
a range of temperatures, using Maxwell’s principle of equal areas to determine equilibrium
conditions. These results not only highlight the impact of confinement on the equilibrium between
vapor and liquid phases but also provide a deeper mechanistic insight into the smooth and
continuous transitions observed experimentally. This capability is critical for advancing our
understanding of phase transitions in porous media and serves as a benchmark for both future
experimental validation and theoretical developments.
In conclusion, the combination of experimental gravimetric measurements and advanced
simulation techniques in this study promote a more comprehensive understanding of the adsorption
and phase transition behavior of C2H6 in MCM-41. While the experimental work revealed the
challenges in observing distinct phase transitions due to the combined effects of small pore sizes
and pore size distribution, the simulation results provide essential insights into the underlying

108
mechanisms, emphasizing the significance of using complementary approaches to study confined
fluid behavior.
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Chapter 5. Summary
This dissertation presents a comprehensive investigation of the adsorption and phase
transition phenomena occurring within porous media, with a particular emphasis on the
complexities associated with shale reservoirs and synthetic porous materials. The primary goal of
this research is to enhance our understanding of the mechanisms that dictate the phase behavior of
hydrocarbons in unconventional resources, and to explore new approaches for the study of
adsorption phenomena. By achieving this, this dissertation aims to improve the accuracy of
estimating gas in place and to provide more precise parameters for reservoir simulation models,
which are crucial for optimizing the production of shale gas and understanding its storage
mechanisms.
In Chapter 2, the dissertation introduces a novel approach for measuring the adsorption of
gases with low adsorption affinities in shale, without utilizing the traditional method which using
helium for determining the skeletal volume of microporous solids. This new method provides
valuable insights into the adsorption behavior in a model adsorptive gas, with a specific focus on
argon adsorption in Marcellus Shales. The findings from this chapter contribute to a more accurate
way to quantify the adsorbed amount of gases in shale formations, which is essential for future
accurate resource estimation and effective production strategies.
Chapter 3 explores capillary condensation in mesoporous materials, specifically AGC-40
and a synthetic silica sample, analyzing the sorption of nitrogen, C2H6, and carbon dioxide. Using
static adsorption systems and thermogravimetric analysis via the MSB setup. This research
investigates phase transitions across various pressures and temperatures. Although no abrupt phase
transitions were observed in the synthetic silica, evaporation phenomena in nitrogen and C2H6

120
experiments were observed under the cryogenic conditions. Theoretical analyses with the Kelvin
equation and Peng-Robinson equation supported these observations, indicating insufficient
disparity between the predicted phase transition and saturation pressure lines. Additionally, a
geometric pore filling model helped elucidate the processes of layer filling and emptying.
Chapter 4 delves into the phase transition behavior of C2H6 within the nanoporous
structures of synthetic model materials, MCM-41. Besides the experimental measurements of the
ethane isotherms at various pressure and temperature conditions, this chapter also employs a
combination of molecular simulation methods, including GCMC and Gauge-GEMC simulations,
this research predicts phase transition pressures under varying pressure and temperature conditions.
The insights gained from this chapter enhance our understanding of fluid properties in pore
confinement, which is crucial for developing more accurate simulation models for reservoirs in
the oil and gas industry.
In summary, this dissertation provides a comprehensive examination of adsorption and
phase transition phenomena in porous materials, with significant implications for enhancing the
accuracy and efficiency of shale gas production. The research contributes to the development of
more precise models for predicting gas storage capacities in shale formations and offers insights
into the complex interactions between hydrocarbons and porous media. The findings from this
study have the potential to inform future research and technological advancements in the field of
unconventional hydrocarbon recovery and storage.

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Asset Metadata

Creator Wu, Jiyue (author)

Core Title Investigation of adsorption and phase transition phenomena in porous media

Contributor Electronically uploaded by the author (provenance)

School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Petroleum Engineering

Degree Conferral Date 2024-12

Publication Date 01/23/2025

Defense Date 01/26/2024

Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital)

Tag adsorption,MCM-41,OAI-PMH Harvest,phase transition,porous media,SHALE

Format theses (aat)

Language English

Advisor Jessen, Kristian (committee chair), Tsotsis, Theo (committee member)

Creator Email jiyuewu@usc.edu,jyjywyy@gmail.com

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Document Type Dissertation

Format theses (aat)

Rights Wu, Jiyue

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Abstract (if available)

Abstract The importance of shale reservoirs in oil and gas production has made it necessary to have a deeper understanding of the mechanisms governing the behavior of hydrocarbons within these unconventional resources. This dissertation explores the mechanism behind the adsorption, desorption, and phase behavior in the context of shale gas extraction and production, aiming to enhance the efficiency and accuracy of the estimation of the gas in place and contribute to the acquisition of more accurate parameters for use in reservoir simulators.
Shale gas reservoirs, characterized by their fine-grained sedimentary rock formations and diverse range of pore sizes and compositions, present unique challenges in oil and gas production. The hydrocarbons in these reservoirs typically exist in two distinct states: free gas occupying confined pore spaces and gas adsorbed on the pore surfaces or bound to the organic matter within the shale matrix. Understanding the dynamics of adsorption and desorption are crucial for optimizing the production capacity of shale gas reservoirs, as these processes significantly influence the total gas content and its subsequent release during production.
This dissertation focuses on two different but interrelated research topics: the study of adsorption and desorption processes in porous media and phase transition phenomena in such materials. Under the first topic, this dissertation introduces a novel approach for measuring gas adsorption in shales, that does not require the traditional use of helium (He) for determining the skeletal volume of the microporous solid. We apply the technique to accurately measure the adsorption of argon (Ar), as a model gas, within the nanopores of Marcellus Shales by the gravimetric method. Additionally, the dissertation also presents a new technique for directly calculating the volume of the adsorbed layer (Va), a yet intractable challenge in the adsorption area, based on dynamic adsorption data collected using volumetric adsorption measurements. The technique is then applied in the study of ethane (C2H6) and carbon dioxide (CO2) in the well-characterized porous material, namely, MCM-41. The experimental Va data are compared with the findings from molecular simulations of the material.
For the second topic, this dissertation concentrates on an experimental exploration of capillary condensation within mesoporous silica materials. Specifically, it studies the sorption behavior of three gases: nitrogen (N₂), C₂H₆, and CO₂. The investigation revolves around two varieties of porous silica, each with an average diameter of about 4 nm and 7 nm. The study is reinforced by thermodynamic calculations and employs a geometric pore filling model to interpret the observed experimental results.
Under the second topic, the dissertation investigates the phase transition behavior of C2H6, within the nanopore structure of MCM-41. We employ a combination of molecular simulation methods including Grand Canonical Monte Carlo (GCMC), Gauge Cell Gibbs Ensemble Monte Carlo (Gauge-GEMC) and isobaric-isothermal ensemble Monte Carlo (NPT). The study predicts the conditions under which phase transitions take place within these materials and also calculates the fluid distribution within the nanopores, thus contributing to a better understanding of fluid properties under pore confinement, which is essential for optimizing reservoir simulations in the oil and gas industry. Unfortunately, experiments with C2H6 adsorption in the same materials failed to unambiguously detect the presence of phase transitions. We attribute this to the inevitable pore-size distribution of the sample. The reason is that we observe that the capillary condensation step on the experimental isotherms is gradual, which is expected for a phase transition in a system consisting of numerous pores with inevitable variations in pore width.
To summarize, this dissertation provides valuable insights into the adsorption and phase transition phenomena in porous materials, with significant implications for enhancing the efficiency of shale gas production operations. The findings contribute to the development of more precise models for predicting gas storage capacities in shale formations and offer potential improvements for adsorption-based processes in porous materials for various other technical areas.