X-ray Scattering Study Of Capillary Condensation In Mesoporous Silica
A thesis presented to
the faculty of
the College of Arts and Sciences of Ohio University
In partial fulfillment
of the requirements for the degree
Master of Science
Mayur Sundararajan
May 2013
© 2013 Mayur Sundararajan. All Rights Reserved.
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This thesis titled
X-ray Scattering Study Of Capillary Condensation In Mesoporous Silica
by
MAYUR SUNDARARAJAN
has been approved for
the Department of Physics and Astronomy
and the College of Arts and Sciences by
Gang Chen
Assistant Professor of Physics and Astronomy
Robert Frank
Dean, College of Arts and Sciences
3
ABSTRACT
SUNDARARAJAN, MAYUR., M.S., May 2013, Physics and Astronomy
X-ray Scattering Study Of Capillary Condensation In Mesoporous Silica
Director of Thesis: Gang Chen
The capillary condensation deforms a nanoporous material due to the capillary
force generated by the fluid inside the pores. In-situ small and wide angle x-ray
scattering(S/WAXS) were used to study the deformation with respect to relative vapor
pressure of the fluid. Periodic mesoporous silica such as MCM-41 and SBA-15 were
synthesized and used as the samples with water as the capillary condensation agent. The
gas sorption method and SAXS were used to extract the pore parameters of the samples.
The stresses acting on the silica scaffold due to the presence of water in the pores were
deduced by careful analysis of various forces acting on it. The Poisson’s ratio and elastic
moduli of the two samples and their annealed forms were estimated and compared
quantitatively. Our study demonstrates a novel WAXS-based technique for calculating
the mechanical properties of nanoporous materials with much wider applicability than the
previously reported SAXS technique.
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DEDICATION
I dedicate this thesis to my parents M. Sundararajan and S. Vasuki, my brother S. Arvind,
my friend K. Maheswari, my wife S. Poorani and everybody who thinks I am
AWESOME!!
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ACKNOWLEDGMENTS
I am grateful to my advisor Dr. Gang Chen for his teaching, guidance and support
from the initial to the final stages of the research. I wish to thank my colleague
Chandrasiri A. Ihalawela for all the long discussions and the help with the experiments. I
must thank Dr. Xiaobing Zuo of Argonne National Laboratory for his help to set up the
experiment conducted there. I thank Dr. Alexander Govorov and Dr. David F.J. Tees for
serving in my committee.
I take this opportunity to thank all the faculty and staff of the Department of
Physics and Astronomy for aiding in the growth of both professional and personal aspects
of my life in the last couple of years. I thank all my graduate friends for the wonderful
time and especially Chandrasiri, Sneha, Meenakshi, Bijay, Binay for supporting and
helping me in various difficult situations.
On a personal note, I thank my father M. Sundararajan (M.S. EE) for treading this
path 27 years ago, which I followed to reach here. I thank my mother S. Vasuki for her
encouragement and love. I also thank my friend K. Maheswari for all the support.
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TABLE OF CONTENTS
Page
Abstract ............................................................................................................................... 3
Dedication ........................................................................................................................... 4
Acknowledgments............................................................................................................... 5
List of Tables ...................................................................................................................... 8
List of Figures ..................................................................................................................... 9
Chapter 1: Introduction ..................................................................................................... 13
Chapter 2: Material Synthesis ........................................................................................... 17
2.1. MCM41.................................................................................................................. 17
2.1.1. Synthesis mechanism ...................................................................................... 17
2.1.2. MCM-41 Synthesis ......................................................................................... 18
2.2. SBA-15 .................................................................................................................. 19
2.2.1. Synthesis mechanism ...................................................................................... 19
2.2.2. Synthesis of SBA-15....................................................................................... 21
Chapter 3: Material characterization................................................................................. 23
3.1. Introduction............................................................................................................ 23
3.2. Gas sorption Method.............................................................................................. 23
3.2.1. Adsorption....................................................................................................... 24
3.2.2. Capillary Condensation................................................................................... 25
3.2.3. Capillary action and the formation of meniscus ............................................. 26
3.2.4. Capillary pressure ........................................................................................... 28
3.2.5. Isotherms......................................................................................................... 29
3.2.6. Kelvin Equation and BJH method .................................................................. 33
3.3. X-ray scattering method......................................................................................... 40
3.3.1. Mechanical parameters ................................................................................... 41
3.3.2. Capillary action of water as stress .................................................................. 43
3.3.3. X-ray scattering............................................................................................... 49
Chapter 4: Experiment ...................................................................................................... 56
4.1. Gas sorption experiment ........................................................................................ 56
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4.2. X-ray scattering experiment................................................................................... 57
4.2.1 SAXSess .......................................................................................................... 57
4.2.2. Synchrotron ..................................................................................................... 59
4.2.3. Comparison between SAXSess and synchrotron data .................................... 60
Chapter 5: Results and Discussion.................................................................................... 64
5.1. Gas-sorption method:............................................................................................. 64
5.2. X-ray scattering technique: .................................................................................... 65
Chapter 6: Conclusion....................................................................................................... 92
References......................................................................................................................... 94
Appendix A: Error analysis .............................................................................................. 97
Appendix B: Kelvin Equation........................................................................................... 98
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LIST OF TABLES
Page
Table 1: Physical pore parameters extracted from gas-sorption method………………...64
Table 2: Summary of the stresses acting on the porewall in each plane…………………81
Table 3: The poreload modulus estimated by SAXS method……………………………90
Table 4: The modulus estimated by WAXS……………………………………………..90
Table 5: Compilation of the results obtained from Gas-sorption, SAXS & WAXS
methods…………………………………………………………………………91
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LIST OF FIGURES
Page
Figure 1. Schematic of the steps in the formation of MCM-4116..................................... 18
Figure 2. a) MCM41 after calcinations b) The finely ground powder and apart of the
pressed pellet of MCM-41................................................................................ 19
Figure 3. The presence of PEO in the walls of the mesopores in SBA-1518.................... 21
Figure 4. a) The filtered part of the aged solution before calcination b) The finely ground
powder after calcination and a part of the pellet of SBA-15............................ 22
Figure 5. Adsorption and Capillary condensation. ........................................................... 26
Figure 6. Capillary action and the pressure at different points......................................... 27
Figure 7. Types of adsorption isotherms: adsorption (green), desorption (red); if there is
no change in desorption line from adsorption line then it is not represented
separately.......................................................................................................... 29
Figure 8. Shape of hysteresis in isotherms and the corresponding pore shapes 27 ............ 31
Figure 9. Isotherm of SBA-15 with nitrogen as adsorptive.............................................. 32
Figure 10. Isotherm of MCM-41 with nitrogen as adsorptive.......................................... 33
Figure 11. The meniscus with radii r1 and r2 inside the pore forming the core volume; the
thin layer t of adsorbate (liquid) on the inner wall ........................................... 35
Figure 12. The table of BJH method calculation (top) for SBA-15, The plot pore volume
vs pore width (bottom) showing the pore size distribution for SBA-15. ......... 38
Figure 13. The plot of pore volume vs pore width showing pore size distribution for
MCM-41........................................................................................................... 39
Figure 14. Stresses and strains in the three planes. .......................................................... 42
Figure 15. Various levels of water in the pore with the increase of RH .......................... 44
10
Figure 16. The forces with dashed arrows are on solid due to liquid; Tangential
component(red), Normal component(yellow),Force due to Laplace
pressure(maroon);The direction of surface tension on each interface (red solid
arrow);Effective forces on the solid(right). ...................................................... 48
Figure 17. The scattering of x-rays from a planar arrangement of particles37 ................. 51
Figure 18. Comparison of SAXS patterns of MCM-41 at 54% and 86%. ....................... 53
Figure 19. Comparison of WAXS patterns of MCM-41 at 54% and 86%....................... 55
Figure 20. Micromeritics degas system (left); Micromeritics Tristar Surface area and
Porosity(right)37................................................................................................ 56
Figure 21. SAXSess instrument and the related devices (left); The raw 2D data (right top)
and the converted 1 D data (right bottom). ...................................................... 59
Figure 22. The 1D scattering patterns from synchrotron; the arrow indicating the
intensity of the SAXS peak (top);The arrow indicate the intensity of the
WAXS peak bottom). ...................................................................................... 62
Figure 23. The 1D scattering pattern from the SAXSess indicating the intensity of the
part of SAXS peak (top arrow) and the WAXS peak (bottom arrow). ............ 63
Figure 24. Top left: SAXS peak position vs RH plot of MCM-41 ‘AS’ ;Top right: SAXS
peak position vs RH plot of MCM-41 ‘AN’; Bottom left: SAXS peak position
vs RH plot of SBA-15 ‘AS’; Bottom right: SAXS peak position vs RH plot.. 65
Figure 25. The schematic of the hexagonal arrangement, from the first Bragg’s peak in
SAXS the interpore distance is calculated by using the perpendicular triangle.
The area filled with red represents the porewall. ............................................. 66
Figure 26. WAXS peak position vs RH plot of MCM-41 ‘AS’; Top right: WAXS peak
position vs RH plot of MCM-41 ‘AN’; Bottom left: WAXS peak position vs
RH plot of SBA-15 ‘AS’; Bottom right: WAXS peak position vs RH plot..... 67
Figure 27. The amplitude of the first Bragg’s peak of SAXS vs RH in MCM-41 ‘AS’.
The line represents that the rate of water loss during capillary evaporation. ... 69
Figure 28. The SAXS intensity and the SAXS strain of MCM-41 ‘AN’ are plotted
together to deduce the data points that are used to calculate the elastic modulus.
.......................................................................................................................... 70
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Figure 29. The SAXS intensity and the WAXS strain of MCM-41 ‘AN’ are plotted
together to deduce the data points that are used to calculate the elastic modulus.
.......................................................................................................................... 71
Figure 30. The SAXS intensity and the SAXS strain of SBA-15 ‘AN’ are plotted together
to deduce the data points that are used to calculate the elastic modulus. ......... 72
Figure 31. The SAXS intensity and the WAXS strain of SBA-15 ‘AN’ are plotted
together to deduce the data points that are used to calculate the elastic modulus.
.......................................................................................................................... 73
Figure 32. The amplitude of SAXS first bragg peak vs RH plot for SBA-15 ‘AS’. The
lines represent that the rate of water loss has two different rates during the
capillary evaporation. ....................................................................................... 74
Figure 33. The strain in the x-y plane extracted from the SAXS plotted against ln(RH) for
MCM-41‘AS’. The trendline corresponds to data points(black) before the
capillary evaporation. The inset shows the slope and the error in the trendline.
.......................................................................................................................... 76
Figure 34. The strain in the x-y plane extracted from the SAXS plotted against ln(RH) for
MCM-41‘AN’. The trendline corresponds to data points(black) before the
capillary evaporation. The inset shows the slope and the error in the trendline.
.......................................................................................................................... 77
Figure 35. The strain in the x-y plane extracted from the SAXS plotted against ln(RH) for
SBA-15‘AS’. The trendline corresponds to data points(black) before the
capillary evaporation. The inset shows the slope and the error in the trendline.
.......................................................................................................................... 78
Figure 36. The strain in the x-y plane extracted from the SAXS plotted against ln(RH) for
SBA-15‘AN’. The trendline corresponds to data points(black) before the
capillary evaporation. The inset shows the slope and the error in the trendline.
.......................................................................................................................... 79
Figure 37. The bold blue circles are the stress free configuration of pores and the broken
circles represent the change at high RH in x-y plane (SAXS). The pore wall
(right) is drawn for cubic arrangement of pores for simplicity. ....................... 82
Figure 38. The bold blue circles are the stress free configuration of pores and the broken
circles represent the change at RH before capillary condensation. The pore wall
(right) is drawn for cubic arrangement of pores for simplicity. ....................... 83
12
Figure 39. The strain calculated from FSDP vs the ln(RH) for MCM-41 ‘AS’. The
trendline corresponds to the data points(black) before the capillary evaporation.
The inset shows the slope and the error in the trendline. ................................. 84
Figure 40. The strain calculated from FSDP vs the ln(RH) for MCM-41 ‘AN’. The
trendline corresponds to the data points(black) before the capillary evaporation.
The inset shows the slope and the error in the trendline. ................................. 85
Figure 41. The strain calculated from FSDP vs the ln(RH) for SBA-15 ‘AS’. The
trendline corresponds to the data points(black) before the capillary evaporation.
The inset shows the slope and the error in the trendline. ................................. 86
Figure 42. The strain calculated from FSDP vs the ln(RH) for SBA-15 ‘AN’. The
trendline corresponds to the data points(black) before the capillary evaporation.
The inset shows the slope and the error in the trendline. ................................. 87
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CHAPTER 1: INTRODUCTION
In the advent of nanotechnology, nanomaterials and their properties promise a
wide range of applications. The physical properties of bulk materials change as the
material is scaled down to the nanoscale. This change can be attributed to parameters
such as surface area and surface tension which were neglected in the bulk state begin to
dominate on the nanoscale. Even the dominant forces acting on the material varies: at the
nanoscale, surface effects such as adhesion and cohesion take precedence over gravity
and mass effects. A bulk material in its powdered form shows variation in some of its
physical properties due to the significant increase in the surface area. If the powder
particles has crevices or pores then the increase in surface area is large, so the surface
effects define its physical characteristics. These materials are broadly known as
nanoporous materials.
These nanoporous materials are classified into three categories with respect to
their pore sizes: microporous (<2nm), mesoporous (2-50nm) and macroporous (>50nm)1.
The microporous materials such as zeolites and active carbons have been studied as early
as mid-twentieth century by Amberg and McIntosh2. Some examples of mesoporous
materials are mesoporous silica and active carbon. Ceramics are a good example of
macroporous materials. The micro and mesoporous materials have attracted a lot of
interest due to their wide range of applications such as catalysts, sorption media and
molecular sieves. The works in mesoporous material and in particular mesoporous silica
began to bloom after the discovery of MCM-41 in the early 1990s3,4. MCM is an
acronym for ‘Mobil Crystalline Material’. MCM-41 and its variant MCM-48 are
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mesoporous silica which were synthesized using a template self assembly mechanism by
researchers at Mobil Oil Corporation. The interesting characteristics of these materials
are the ability to control the size of the pores and the ordered arrangement of pores,
which were missing in zeolites and other microporous materials. This tunability and high
degree of order, improved it as a molecular sieve and paved the way for new applications
such as drug delivery, chemical sensor etc. In the late 1990s, researchers from University
of California, Santa Barbara discovered a mesoporous silica material with micropores in
its mesopore wall and this was named as SBA-155. SBA is an acronym for ‘Santa
Barbara Amorphous’ material. This material also attracted lot of interest due to its
interconnected pores, which diverges its characteristics from MCM-41. In the 2000s,
many researchers began studying the deformation effects in mesoporous silica due to
adsorption and found interesting results deviating from microporous materials. In this
study, MCM-41 and SBA-15 were synthesized and used as the samples.
The phenomenon of inducing deformation on a solid by adsorption has been
studied as early as 1927 by F.T.Meehan6. This was followed by similar studies by
Bangham and Fakhoury7 and they related the deformation to the decrease in surface
energy due to adsorption. The later works were more on the deformation of micro and
mesoporous materials due to adsorption. Some mesoporous materials such as active
carbons8, zeolites8 and mesoporous silica9,10 exhibit varying deformations (i.e. expansion
and contraction). The major difference between adsorption on a typical solid and a
nanoporous solid is the phenomenon of capillary condensation due to the presence of
pores. This phenomenon stimulated the interest in researchers to study its effects on these
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materials. The detailed discussion of this phenomenon will be presented later. Similar
interest drives this study of physical effects on mesoporous silica material due to
capillary condensation.
Capillary condensation in the pores fills the pore with liquid, which instigates
capillary action1. Water rising against gravity in a narrow tube immersed in water is an
example of capillary action. In nature this allows transportation of water to the leaves at
the top of the tree by the roots, blotting up by a towel, transportation of fluids in our body
and much more. This capillary action has very interesting effects on nanosized
capillaries. When the tubes (capillaries) are on the nanoscale the height the fluid reaches
and the capillary pressure are humongous. For instance, consider a capillary of width 1
nanometer in water: it exerts a capillary pressure of 14MPa and the height reached by the
water is 14km. The pressure is comparable to the pressure at the bottom of the Mariana
trench. In this study, capillary action plays a vital role and it is used as the stress that
deforms the porous material to measure its strength. A detailed analysis of capillary
action will be discussed.
The physical parameters such as surface area, pore volume, pore size distributions
are essential to comprehend and explain the results. There are at least 6 methods to
extract the pore parameters: Gas sorption, Mercury porosimetry, Transmission Electron
Microscopy, Scanning Electron Microscopy, x-ray scattering and neutron scattering11. In
this study the gas sorption method was used to extract the pore parameters of the
mesoporous silica samples. This method is used because of its simplicity and versatility.
X-ray scattering is used to measure the strain on the silica scaffold of the mesoporous
16
silica due to capillary condensation of water in the porous material. The small angle x-ray
scattering technique has been used for the estimation of that strain9, 10 but in this study a
novel technique using wide angle x-ray scattering was developed.
The primary aim of this study is the development of the new wide angle x-ray
scattering technique to estimate that strain in the mesoporous silica by capillary action of
water and calculate its Poisson’s ratio and mechanical strength. In the path towards that,
certain ambiguities and missing links in the earlier studies9, 10 were rectified. The samples
of MCM-41 and SBA-15 used in this study were synthesized in the lab. Each sample was
studied in two different forms, ‘as synthesized (-AS)’ and ‘annealed (-AN)’, so the
effects of annealing on the mechanical properties can also be estimated.
The theory and the material synthesis are discussed in the 2 nd chapter. The theory
of each method and the important phenomenon used in them are discussed in the 3rd
chapter. The experiment is detailed in the 4th chapter and its results are presented with
discussion in the 5th chapter.
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CHAPTER 2: MATERIAL SYNTHESIS
The mesoporous silica materials were synthesized by the template self-assembly
mechanism. The pore width is controlled by the dimensions of the template, which
usually is a surfactant or a block polymer. The silica attaches to the template and forms
the scaffold of the material. The template is removed by calcination leaving the silica
scaffold. The silica in the scaffold is amorphous, which is characteristic of both MCM-41
and SBA-15. The structure of SBA-15 is more complex than MCM-41 due to the
presence micropores connecting the mesopores. In this section the self-assembly
mechanism and synthesis of MCM-41 and SBA-15 are briefly presented. MCM-41 and
SBA-15 were synthesized in the lab by referring to earlier works 19.
2.1. MCM41
2.1.1. Synthesis mechanism
The synthesis of MCM-41 can be described in three steps. The main constituents
of the synthesis are surfactant (Cetyl trimethylammonium bromide (CTAB)) and the
silica precursor (tetra ethyl orthosilicate (TEOS)). In the first step, the surfactants
spontaneously form a rod like arrangement known as a micelle and the micelles become
hexagonally ordered12,15. The silicate anions from the precursor interact electrostatically
with the surfactant cations and form a layer12,15. The diameter of the enveloping silica
layer is about 50 nm with a number of micelles encapsulated inside them at the beginning
of the reaction13,15. In the second step, the number of surfactants decreases resulting in a
smaller pore size (encapsulation)12,14,15. The hydrolysis and the condensation of the
18
silicate precursor take place on the surface of the micelle. The second step begins even
before the completion of the first step after around three minutes 12. In the third step, the
silica layer after hydrolysis and condensation each encapsulate one micelle and thus the
pore size is approximately 5 nm and become more ordered 12,13,15. Later the surfactants are
evaporated by calcination at 550°C for 4hours.
Figure 1. Schematic of the steps in the formation of MCM-4116.
2.1.2. MCM-41 Synthesis
The synthesis of MCM-41 is a mixture of two separately prepared solutions as
follows:
Solution 1: 1.6g CTAB + 25ml H2O + 38ml Ethanol + 16ml Ammonium Hydroxide
Solution 2: 2.55ml TEOS + 5ml Ethanol
Solution1 and Solution2 were stirred separately at 40°C for 15 minutes before mixing and
then the mixed solution was stirred for 15 minutes. The solution was aged for 48 hrs at
19
60°C. The aged solution was filtered and the sediment was spread into a thin layer on a
ceramic plate. The sediment on the ceramic plate was air dried and was then calcined in
an oven. The calcination process was carried out as follows: The sediment was initially
heated to 90°C at 1°C/min and kept at that temperature for 4 hours. It was then heated to
500°C at 1°C/min and kept at that temperature for four hours before it was naturally
cooled down to 40°C. The calcined sample forms a dry layer on the ceramic plate. The
layer was scrapped and ground into a fine powder, some of which was later pressed into a
13mm diameter pellet to be used in x-ray scattering experiment.
a
b
Figure 2. a) MCM41 after calcinations b) The finely ground powder and a part of the
pressed pellet of MCM-41.
2.2. SBA-15
2.2.1. Synthesis mechanism
The block polymer in this synthesis is Pluronic (poly (ethylene glycol)-poly
(propylene glycol)-poly (ethylene glycol) (EOyPOxEOy)) and the silica precursor is
TEOS. The formation of the long-range hexagonally-ordered mesoporous structure is by
20
the self-assembly template mechanism. The block polymer forms micelles at the
beginning of the synthesis. After the hydrolysis of silicate, it is attracted by van der
Waals force to the micelles. The long range hexagonal order and the attraction of
hydrolysed silicate attraction happen at the same time17. These steps are similar to the
ones discussed for MCM-41. Micropores are created by poly ethylene oxide (PEO) in the
surfactant, which interacts with the silicate in the mesopore wall as shown in the figure
below. Poly propylene oxide (PPO) is the part of the polymer which forms the template
for the mesopores. During calcination along with the rest of the polymer, the PEO
molecules in the walls evaporate leaving open pores in the mesopore wall 18.
21
Figure 3. The presence of PEO in the walls of the mesopores in SBA-1518
2.2.2. Synthesis of SBA-15
The SBA-15 synthesis began with the preparation of 1.7 M concentrated HCl. 2g
of P123 (Pluronic P123 (Mav = 5800), EO20PO70EO20, (Aldrich)) was added to 11.9g of
the prepared conc.HCl and 62.9g of water mixture in a suitable beaker. The beaker was
sealed and the contents were stirred for 1 hour at room temperature. Four grams of TEOS
solution was added drop by drop into the above solution. The beaker was sealed again
and the contents were stirred for 4 hours at 40°C. This was followed by aging the solution
for 40 hours at 60°C. The rest of the processes (filtering, calcinations) were the same as
22
in MCM-41. The sample powder was pressed into pellets for the x-ray scattering
experiment.
Figure 4. a) The filtered part of the aged solution before calcination b) The finely ground
powder after calcination and a part of the pellet of SBA-15.
23
CHAPTER 3: MATERIAL CHARACTERIZATION
3.1. Introduction
The pore parameters that are necessary for this study are surface area, pore width,
micropore area, pore size distribution of the sample and strain on the sample due to the
capillary action. A properly synthesized MCM-41 and SBA-15 has hexagonally ordered
porous structure as shown in Figure 1. The quality of the synthesized sample was
determined by observing the parameters in the gas sorption method and SAXS. In this
study, the above mentioned physical pore parameters were necessary to explain the
changes in the strain between different samples. In this chapter, the method of extracting
the pore parameters in the gas sorption method and strain on the sample by x-ray
scattering methods is discussed. In general, the gas sorption method is used for various
measurements and the ones used in this study are discussed briefly. Similarly in x-ray
scattering the necessary theory is discussed briefly with the applications in the study.
3.2. Gas sorption Method
This method is simple and economical to extract the structural parameters of
porous material. This analysis is done with the aid of computer and complex equipment,
which reduces the work to placing the sample and pushing the button. This section has a
brief discussion on the basic principles of this technique and the methods of data
extraction used in this study.
24
3.2.1. Adsorption
Adsorption has been used practically by humans for thousands of years. Scientific
study of adsorption was extensive during the late 19th century like Kayser20 in 1881but it
has been known even hundred years before that. The large capacity of porous solids to fill
up gases has been known as early as 1771 by Fontana 21. In early and mid 19th century, the
role of surface area and pores were known and established to explain the greater capacity
of porous solids by Saussere22 and Mitscherlich23. In the early 1916, Langmuir24
proposed a semi-empirical model with isotherms. Later in 1938, the existing Langmuir
theory was improved to BET theory25, which has been successful and improved later.
Adsorption phenomenon forms the basis of this whole study as it induces the
capillary condensation and capillary action. Adsorption is a surface based process in
which a thin film of fluid is created on a surface of solid due to the van der Waals force
between the atoms of the fluid and surface. It differs from absorption as the latter
involves the whole volume of the solid to dissolve inside the fluid rather than only the
surface. There are two types of adsorption, physical adsorption (physi-orption) and
chemical adsorption (chemi-sorption).When the force between the atoms of the fluid and
the surface are due to van der Waals force then it is physical adsorption and if the force is
due to chemical bonding then it is chemical adsorption. The adsorbing fluid is known as
adsorptive24 before adsorption and adsorbate after adsorption. The complementary
phenomenon is known as desorption, which is the return of the adsorbate to adsorptive.
This phenomenon along with a few others is used to extract the structural pore
parameters.
25
3.2.2. Capillary Condensation
Capillary condensation is the process by which multilayer adsorption of vapor
into the pore fills the pore space with condensed liquid. The capillary of the pore induces
condensation at a lower vapor pressure than the saturated vapor pressure of the pure
liquid. The lower vapor pressure is due to the high van der Waals force on the adsorptive
due to the pore (cylindrical here) structure of the adsorbent. Van der Waals force is
inclusive of adhesive and cohesive forces. The attraction between the adsorptive and the
surface is specifically adhesive force. The adhesive force is the intermolecular attraction
between two different molecules and when the intermolecular attraction is between like
molecules it is called as cohesive force. On a planar adsorbent the direction of the
adhesive force between the adsorptive (liquid) and adsorbent (solid surface) is along the
closest distance between them as shown in Figure 5. Inside a nanopore the adsorptive
experiences the adhesive force on all directions as shown in the Figure. The high
attractive force decreases the energy of the adsorptive lower than the energy of the pure
liquid at the same vapor pressure. Thus the adsorptive condenses at a lower vapor
pressure inside a nanopore. During adsorption multilayer formation fills up the whole
space of the pore with sufficient vapor pressure of the adsorptive. During desorption the
layer recedes in a different degree due to the presence of meniscus. The curved top
surface of the liquid is called as meniscus, which is discussed in the next section.
26
Figure 5. Adsorption and Capillary condensation.
3.2.3. Capillary action and the formation of meniscus
When a capillary is immersed in a liquid, the liquid in the capillary rises with
curved top surface above the surface of the immersed liquid as shown in the Figure 6.
Capillary action can be explained by adhesive force, cohesive force and surface tension.
Surface tension is the force per unit length required to pierce through the surface of the
liquid. Surface tension of water at room temperature is 7.12 N/cm, which means a denser
1 cm long body applying less than 7.12 N on the surface will float on water.
The top curved surface of the liquid as shown in the Figure 6 experiences both
adhesion and cohesion. The surface molecules experience cohesive force only on the
downward direction as there are no molecules above it. This imbalance results in a net
27
force downwards, which causes the surface tension. When the adhesive forces between
the capillary and the liquid are greater than the cohesive forces, the molecules near the
walls of the capillary move upwards forming a curved top surface in the liquid called the
meniscus (as shown in the Figure). The upward pull does not breach the surface tension
of the meniscus hence the liquid under the meniscus is pulled up to height h. The height h
depends on the weight of the liquid that can be lifted by the surface tension.
P1
P1
Pw
P
Pw θ
P2
P
Figure 6. Capillary action and the pressure at different points.
The upward force due to the surface tension is Fup= T (2π r), where T is the
vertical component of surface tension γ.
The downward force due to the pressure is Fd= ρgh(π r2), as the pressure due to
the water in the capillary is ρgh.
28
At equilibrium the height risen can be calculated by equating the two forces,
which results as h=2T/r.
The height of the liquid raised is inversely proportional to the radius, hence
smaller capillary rises higher. The liquid rising through the capillary against gravity is
known as the capillary action. The height of the liquid creates capillary pressure.
3.2.4. Capillary pressure
Capillary pressure is the pressure difference on the meniscus between two
immiscible fluids. It is the compensation in the pressure to keep the interface between the
fluids intact. The forces on the meniscus at equilibrium are as shown in the Figure 6 then
the force balance equation is as follows.
(
)=
(
) + (2
)
T is the vertical component of surface tension γ.
(1)
Rearranging the above equation we get,
=
−
=
2
=
2
This equation is called as the Law of Laplace.
(2)
Also consider the various pressures P marked in the Figure 6,
P1 = P2, since both are atmospheric pressure.
P2 =P, since there is no capillary action in the interface. It implies P 1 = P = Pw.
P = Pw + ρgh.
As P = P1, Capillary pressure Pc = P1 – Pw = ρgh.
So, if the surface tension is γ ,angle of contact between the liquid and the capillary is θ
and height risen is h, as shown in the Figure 6 then the capillary pressure is Pc= 2γcosθ/r=
29
ρgh = 2T/r. The capillary condensation pressure and capillary (pore) radius are correlated
in a function known as Kelvin Equation, which will be discussed later. The discussions
(3.2.1-3.2.4) were referred from (Refs. 1,26).
3.2.5. Isotherms
It was noted earlier that during adsorption (desorption) the adsorptive forms a
layer on the solid surface. The amount of adsorptive which turns to adsorbate at a
particular vapor pressure reveals a great deal of information. This leads to adsorption and
desorption isotherms. An isotherm is a plot of quantity (volume) of gas adsorbed
(desorbed) at a constant temperature by a solid surface as a function of relative vapor
pressure. The relative vapor pressure is the ratio of the actual vapor pressure of the
adsorptive (gas) to the saturated vapor pressure of the adsorbate (liquid). The shape of the
plots reveals a great deal of information about the adsorption system. The following six
forms of plots describe pores of different sizes and structure.
Figure 7. Types of adsorption isotherms: adsorption (solid line), desorption (broken line);
if there is no change in desorption line from adsorption line then it is not represented
separately.
30
This study is mainly based on mesopores so only the Type IV isotherm is
discussed. The nonporous isotherm (Type II) has the same general shape as a porous
isotherm but the intermediate rise is sharp. The adhesive forces experienced by the
adsorptive atoms increase when they are attracted by the adsorbent in more than one
direction such as inside a pore as discussed earlier. The increase in attraction decreases its
energy, which results in capillary condensation. Due to the capillary condensation, the
amount of liquid inside the pore is more even at a lower relative vapor pressure. It
explains the sharper intermediate for porous isotherm than non porous isotherms. The
stark difference between Type IV isotherms and others is the hysteresis. The rest of the
isotherm types except Type V have overlapping adsorption and desorption isotherms.
Hysteresis in an isotherm is a significant characteristic, which ascertains the
presence of mesopores. An isotherm of a nonporous material will not have hysteresis but
an isotherm without a hysteresis does not prove that it is from a nonporous material. The
hysteresis obviously indicates that the capillary evaporation is different from capillary
condensation in the mesoporous material. The shape of the hysteresis reveals information
about the shape of the mesopores as shown in the Figure 8.
31
Figure 8. Shape of hysteresis in isotherms and the corresponding pore shapes27
The enhanced adsorption in pores forms the basic principle for the gas sorption
method. A real porous material may have more than one particular size of pores, which
varies the isotherms from the ideal but useful information can still be elicited.
32
Figure 9. Isotherm of SBA-15 with nitrogen as adsorptive.
The isotherm plot of SBA-15 shows interesting features, which reveals
information about the sample. The presence of hysteresis proves that the sample has
mesopores. The shape of the hysteresis reveals the pore shape to be cylindrical from the
Figure 5, which is characteristic to SBA-15. The relative vapor pressure at which
capillary condensation and evaporation occurs is known by reading x axis of the plot. The
hysteresis is larger in SBA-15, which indicates that it takes lower relative vapor pressure
for capillary evaporation during desorption. This can be attributed to the presence of
micropores, which needs a lower vapor pressure for evaporation.
33
Figure 10. Isotherm of MCM-41 with nitrogen as adsorptive.
Similarly isotherm of MCM-41 also reveals some information about it. It is obvious that
it has cylindrical mesopores due to the presence of hysteresis and its shape. The
hysteresis is smaller, due to the presence of only mesopores. Another difference is the
relative vapor pressure at which the capillary condensation and evaporation begins, which
is due to the difference in the pore size between the samples. The above discussion was
based on (Refs. 1, 11, 31).
3.2.6. Kelvin Equation and BJH method
The BJH method is used for calculating surface area, pore width and pore size
distribution in this study. The BJH method is based on Kelvin equation and it is popular
34
for mesopore analysis. This method was described by Barrett, Joyner and Halenda, hence
known as BJH method. The micropore area is found by using t-plot method and it is
exclusive of the surface area calculated by BJH method.
3.2.6.1. Kelvin Equation
The phenomenon of capillary condensation is always observed in mesopores as
discussed earlier. The function which correlates pore radius and the capillary
condensation vapor pressure is known as Kelvin equation(Appendix B), which is shown
below.
∗
= −
(3)
P* is the critical condensation pressure for the radius r m, Po is the saturated vapor pressure
of the fluid, V is the molar volume of the condensate, rm is the mean capillary radius, R is
the gas constant and T is the temperature of the adsorptive.

When the angle of contact θ < 90° for a mean capillary radius of r m, the
condensation will occur if the vapor pressure of the adsorptive is greater than the
critical condensation pressure.

Another view is that the pore radius determines whether the condensation can
occur for the particular relative vapor pressure of the adsorptive.
The second point is critical for the BJH method, for a particular relative vapor pressure
the necessary pore radius for capillary condensation to occur can be found. The Kelvin
equation sheds some light on the hysteresis seen on the isotherms discussed earlier. The
mean capillary radius for a pore, which is open at both ends, is given by two radii r1 and
35
r2 as shown in the Figure. The materials used in the study has open ended pore, in these
pores the condensation is nucleated on the inner wall.
1
=
1
2
1
+
1
(4)
r2
r1
Figure 11. The meniscus with radii r1 and r2 inside the pore forming the core volume; the
thin layer t of adsorbate (liquid) on the inner wall
During condensation the condensate builds layers inward to fill the pore, which
implies that rm = 2r1 because r2 is infinity as shown in the Figure. Similarly during
evaporation r2 = r1 = rm, which is less than 2r1. The difference in mean radius changes the
critical capillary condensation/evaporation and thus there is a hysteresis in filling and
emptying of the pores.
In addition to the mean capillary radius, the thickness of the film of adsorbate on
the pore must be accounted for. A pore will have a thin film of adsorbate on the pore wall
irrespective of filling or emptying. This thickness of the layer t is calculated using one of
the three expressions shown below developed with varying level of complexity. The
volume of the adsorbate inside the thin film filling up the pore is known as core volume
as shown in the Figure 11.
36
So,
=
+
Where rp is the pore radius, rm is the core radius and t is the thickness of the film.
= 3.54
=
Å
13.99
0.034 − log
= 3.54
−5
ln
/
∗
/
∗
[24]
(5)
(6)
[24]
(7)
(8)
3.2.6.2 BJH method
The extraction of the mentioned pore parameter is a complex series of
calculations with some assumptions. In the BJH method, the pore shape is assumed to be
cylindrical which is good for this study but would skew the results for pores of any other
shape. The other assumptions are the values of surface tension and molar volume in the
Kelvin equation, which might vary due to the presence of only few molecules. The
deviations due to these assumptions are only small in the final result and thus this method
is still popular.
The calculation is usually in the form of a table as shown in Figure 12. The
desorption cycle of the isotherm is usually used for this calculation. The isotherm gives
out two columns relating volume of gas adsorbed and P/P 0, the difference between
subsequent data points would give the volume of gas adsorbed(∆Vg) for that particular
37
decrease in P/P0. The product of (∆Vg) with molar volume of that liquid (here nitrogen)
would give the change in volume of liquid (∆Vl) between that change in P/P0. The mean
radius for each P/P0 can be calculated using the Kelvin equation and using the constants
for nitrogen. Similarly the thickness of the layer can be calculated for each P/P 0 by using
one of the equations (6, 7 and 8). Now by using the equation (5), the core radius can be
calculated. The thickness decrease ∆t for each P/P0 can be found by the difference of
thickness of the layer calculated for subsequent P/P 0. This volume lost due to this
thickness is the product of surface of the film ∆S and the thickness decrease ∆t. Now, the
volume of liquid lost can be calculated as,
∆
=
+∆ ∆
(9)
It is obvious that,
=
(10)
By substituting the value of l from equation9 to equation10, the pore volume can be
calculated.
=
(∆
− (∆ ∆ ))
(11)
Now with volume of pore, it is simple to calculate the surface area S using mensuration
formula for a cylinder.
38
Figure 12. The table of BJH method calculation (top) for SBA-15, The plot pore volume
vs pore width (bottom) showing the pore size distribution for SBA-15.
39
In the path towards the calculation of volume the pore radius (rp) has been calculated. The
pore size distribution is estimated by plotting pore volume and pore radius. The details of
the process are explained in [29].
Figure 13. The plot of pore volume vs pore width showing pore size distribution for
MCM-41.
40
3.2.6.3 t-plot method
A plot of volume adsorbed Va and thickness of the adsorbed layer t is known as t-plot.
This plot varies between materials and microporous materials show a unique shape. This
uniqueness is used to estimate the micropore area in the material. This method is based
on BET theory. The details of the process are explained in [29].
The discussion in sections 3.2.6.1-3 were based on (Refs. 1, 11, 28).
3.3. X-ray scattering method
In this study, an innovative method has been devised by combining the wide angle
x-ray scattering (WAXS) method and capillary action of mesoporous material to
calculate the elastic modulus. The small angle x-ray scattering (SAXS) data has already
been used to extract the modulus in earlier works9,10. A similar experimental method for
SAXS was used here but the interpretation of the extracted strain and the calculation of
the modulus were modified to get a more accurate estimation. The changes suggested to
the earlier works were validated by examining the various forces due to capillary
condensation on the porous material, which has been discussed in the following section.
Extracting the strain by WAXS and calculating the modulus of the porous material has
not been used before. The theory of x-ray scattering with respect to this study will be
briefly discussed in this section along with the use of capillary action as stress. The
section begins with a brief explanation about the mechanical parameters used in this
study.
41
3.3.1. Mechanical parameters
The mechanical properties that are extensively used in this study are elastic
modulus and Poisson’s ratio. The elastic modulus of a material is defined as the ratio of
the applied stress P to the strain ε in the same direction as in equation12. Stress is the
force applied per unit area and strain is the deformation of the material due to that stress.
Modulus has the same dimensions as the pressure and it is measured in the units of
gigapascals (GPa) in this study. In a macroscopic material it is possible to apply a known
stress and directly measure the deformation of the material optically or any other suitable
method and calculate the modulus. In a nanoporous material it is difficult to do the same,
so the capillary action of water is used as the stress and the x-ray scattering method is
used to measure the strain.
=
(12)
Usually a solid stretched in one plane will contract in the other perpendicular
planes and vice versa. Poisson’s ratio is the ratio between the strains in the plane
perpendicular to the applied stress and the direction of the applied stress. The Poisson
ratio is the factor by which the perpendicular strain can be estimated from the strain in the
direction of the applied stress. If the material is not isotropic the poisson’s ratio will have
three different values for different sets of planes. This porous silica scaffold in these
samples is considered to be isotropic with the same poisson’s ratio as the bulk silica
(ν=0.17). This assumption was arrived at after studying the same for honeycomb
mechanics and eliminating that due to the difference in the direction and the points at
42
which the forces are acting here29. Hence the silica scaffold is similar to a thin silica
structure so it can be considered as bulk silica.
=−
(
(
)
)
(13)
When the poisson’s ratio is isotropic then the strain in each direction can be expressed as,
=
=
=
1
1
1
(
−
+
)
(
− (
+
))
(
−
+
)
(14)
E is the young’s modulus and it is the same in all direction if the material is
isotropic, σ is the stress in the direction of the subscript and ν is the poisson’s ratio. These
expression will be used later to express the measured strain and calculate the young’s
modulus of the material. This section was referred from (Refs. 29, 30).
Figure 14. Stresses and strains in the three planes.
43
3.3.2. Capillary action of water as stress
The usual method for calculating this modulus for a macroscopic material is to
apply a known stress on the material in one direction and measuring the strain on the
same direction due to the stress. It is difficult to apply stress on a nanopore, so the
capillary action of water in a pore was used to simulate the applied stress.
The relative vapor pressure (P*/Po) of water is called as relative humidity (RH). In
this document the RH is specific term used in the place of relative vapor pressure. The
RH around the sample is varied to control the amount of water inside the pore, which
varies the magnitude of the stress. During the adsorption, when the RH is increased from
0%, water begins to adsorb to the inner surface of the pore wall. The thickness of this
layer of water increases as the RH is further increased until a critical value. At this
critical value, the thick layer fuses in the lengthwise middle of the pore to form a
meniscus and this meniscus moves rapidly to the pore entrance with increase in RH. This
rapid increase in the amount of water in the pore is due to capillary condensation which
was discussed earlier. A further increase in the RH decreases the curvature of the
meniscus with very little change in the amount of water inside the pore. In this study the
converse of the above is used in the x-ray scattering experiments, which is desorption and
capillary evaporation.
Note:- RH and the P/Po are interchangeable as the fluid used here is water and both the
notations are used here.
44
Figure 15. Various levels of water in the pore with the increase of RH
The amorphous silica forming the porous structure is referred to as a silica
scaffold. The forces applied on the silica scaffold by water are due to the surface tension
and the Laplace pressure. Surface tension has been discussed earlier and the action of
surface tension on the pore will be discussed here. There are three different surface
tension components in this system, solid-liquid, liquid-vapor, solid-vapor as shown in
Figure 16. It is intuitive to consider the γSL(surface tension solid-liquid) as the influence
of tangential force but it is the combination of all the surface tension which influences it.
The tangential force on the silica scaffold by water is the adhesive force between them.
The normal force arises from the liquid vapor interface of the meniscus. The tangential
component is γLV (1+cosθ), which is derived below. The subscripts L, V, S represents
liquid, vapor, solid respectively.
45
Consider a bulk liquid and solid, each of which are stripped into two separate
parts in vacuum. The energy necessary to strip these two bulk entities are the adhesive
forces ALL (liquid-liquid) and ASS (solid-solid) respectively. The new surfaces will have
surface tension 2γLV=ALL (factor 2 is due to the two separated parts) and 2γSV=ASS and
when the separated liquid and solid part are joined together the adhesive forces between
solid and liquid will reduce the surface energy. So,
=
+
−
(15)
The force on the solid is ASL,as discussed previously, can be found by rearranging
the above equation and using Young’s law for the equilibrium contact angle.
=
=
+
−
−
(Young’s law)
=
(1 +
)
(16)
(17)
Thus the tangential force on the solid arises from the interaction between the solid and
liquid which is given by adhesive force ASL which has been derived to be γLV (1+cosθ).
This relation for tangential component has been confirmed with DFT calculation33. The
normal component γLV sinθ arises from the reaction of the solid to the Laplace pressure
and has been confirmed by DFT calculation33. The resultant force due to these
components is always into the liquid in the direction of half of the contact angle θ. It is to
be noted these forces on the solid are due to the liquid in the core volume, (i.e.) the liquid
excluding the layer of liquid in the wall. Hence the components on the solid due to liquid
in the core volume are γLV (1+cosθ) and γLV sinθ. This normal component γLV sinθ acts
only at the meniscus so it is neglected considering the other forces.
46
The force exerted by a flat film of liquid on the solid will be in the normal
direction to the surface. This force arises due to the curvature of the porewall 33. This
force can also be explained as the molecular interaction at the solid-liquid interface.
There will be an attractive force experienced by the solid molecules. This attractive force
will be balanced by repulsive force inside the liquid far from the interface. At the curved
solid-liquid interface the repulsive forces between the molecules become zero and the
unbalanced attractive forces pull the solid towards the liquid due to pressure difference of
γLV (1+cosθ)/r32. This stress is in the normal direction and towards the center of the pore.
This stress is present along the length of the pore and hence it is considered in estimating
the total stress.
The other force on the solid by liquid is the force due to Laplace pressure of the
meniscus. It is critical to include this force into the estimation of the total force on the
solid. The curved interface between two phases means that there is a pressure difference
between them. At equilibrium this pressure difference is balanced by the surface tension
of the interface34. This can be expressed as,
−
=
2
(18)
The radius of curvature is r and the subscripts v and l denote vapor and liquid
respectively. If the radius of curvature of interface is non-spherical then the curvature part
of the equation will change to two radii in perpendicular planes as seen in the Figure 11.
−
= (
1
+
1
)
(19)
47
When the relative humidity is 100% it could be thought that there is no different
phase and the pressure difference as zero. The decrease in relative humidity increases the
curvature of the interface and pressure difference between increases. The direction of the
pressure is perpendicular to the surface of the meniscus. This force applies to the whole
length of the pore. According to Pascal’s law, the pressure exerted on a confined liquid
transmits equally on all direction and because of this phenomenon the Laplace pressure
on the interface spreads throughout the liquid. This force on the solid is outwards and
normal throughout the length of the pore until the presence of meniscus in the pore. After
the disappearance of the meniscus the film of water on the inner surface of the wall
contributes to the force in the normal direction but it is now due to the surface tension
only as discussed earlier.
In the previous section, the Kelvin equation(Appendix B) has been discussed with
respect to the gas sorption. The Kelvin equation also represents the Laplace pressure due
to the presence of a meniscus.
=
;
1
=
1
2
1
+
1
(20)
The variables represent the same as before. It is evident from the above equations that the
Kelvin equation relates the Laplace pressure to the natural logarithm of relative vapor
pressure. This relation provides an easier way to estimate than measuring the surface
tension and wetting angle in experiment. By rearranging, the equation for Laplace
pressure PL becomes as follows.
48
=
=
The above expression is vital and will be used in the calculation of modulus in the results
section.
Figure 16. The forces with dashed arrows are on solid due to liquid; Tangential
component(red), Normal component(blue),Force due to Laplace pressure(yellow);The
direction of surface tension on each interface (red solid arrow);Effective forces on the
solid(right).
These effective stresses in each direction will be used in the calculation of the
modulus. The discussion in this section was based on (Refs. 32, 33, 34)
49
3.3.3. X-ray scattering
This microscopic technique works under the principle of interference of x-rays
and scattering of X-rays. This varies from optical microscopy in the reconstruction of the
image after interaction of the incident waves with the sample and the structure detail. The
lens system used on the reconstruction in optical microscopy is replaced with
mathematical methods. In simple terms, it is the beam of collimated x-rays incident on
the sample and scattered due to the electron density contrast of the sample onto the
detector.
The scattered waves are recorded as such and reconstructed by mathematical
methods. This mathematical reconstruction has phase loss due to the way of recording of
the scattered waves. Hence the retrieval of the shape and size distribution together is not
possible. The details of the sample are average rather than unique. The scattering data of
structures needs some information of the sample from other methods for proper
interpretation. Though there are some shortcomings, this method is preferred due to the
flexibility in sample preparation such as in-situ and in-vivo observations and for the
average detail of the whole sample. The sample preparations are usually none or simple,
and the sample are not damaged so this method is considered non-destructive. In this
study the effect of capillary condensation on the pores is measured in-situ, thus x-ray
scattering is a relevant technique.
The two primary interactions of x-rays with matter are absorption and scattering.
The absorption is the process in which the energy of the incident x-ray photon is used up
by the atom to bump out an electron and a fluorescent radiation emitted by the atom to
50
restore the original configuration. The absorption depends on the sample and the
wavelength of the x-ray, and it must be minimized for a good scattering data. There are
two types of scattering, Compton (Inelastic) and Rayleigh (Elastic) scattering. In the
inelastic scattering the incident x-ray photon loses some energy during collision with
electrons in the sample. In the elastic scattering the incident photon collides with
strongly-bound electrons and excites them to emit coherent waves. These coherent waves
interfere and produce the scattering pattern on the detector. The inelastic scattering
produces incoherent waves, which cannot produce any pattern so it would form a
background noise.
The constructive and destructive interference of the scattered x-rays depends on
the angle of observation in the detector with respect to the incident x-ray, distance
between the atoms and the orientation of the configuration of atoms. The scattered x-rays
form a pattern of intensity variation in 2θ scale. The interference pattern of particular
arrangement of particles will create identical pattern. If there is more number of that
particular arrangement than the other then the interference pattern will have a higher
intensity. The distances in the pattern are measured with the quantity q which is
and also known as the scattering vector. The quantity q represents a length
in reciprocal space so its dimension is the inverse of length. It is derived from the Bragg’s
law
=2
and
=
, where d is the distance between two consecutive planes in
the arrangement, θ is the angle of incidence and λ the wave-length of the radiation as
shown in the Figure.
51
Figure 17. The scattering of x-rays from a planar arrangement of particles37
The equation of scattering vector implies that the size of the structure probed by
the x-rays inversely depends on the scattering angle. As q is in reciprocal space, smaller q
value means a larger value in real space and vice versa. Hence larger interplane distance
d has peaks in the smaller q values which correspond to the small scattering angle and
smaller interplane distance have peaks at larger scattering angle. This angle dependence
results in two types of x-ray scattering technique, small angle x-ray scattering (SAXS)
and wide angle x-ray scattering (WAXS). When the scattering angle is from 0° to 10° it is
SAXS and this enables in probing structures (interplane distances) that are nanometer to
micrometer dimension. The detector is kept at a distance farther from the sample
depending upon the necessary resolution. When the scattering angle is larger than 10° it
is WAXS and this enables probing structures smaller than that studied in SAXS. The
detector is near the sample and the distance depends on the necessary resolution. The
necessary theory for the scope of this study has been discussed in (Ref. 36).
52
When the interplane distances are ordered then it influences the intensity pattern
to peak at the 2θ angle of the respective distance. This peak is known as Bragg peak. The
maximum of the peak gives that distance in q space and it is the inverse of the interplane
distance in real space (
.The SAXS patterns of the sample for two different
values of relative humidity (RH) are combined in the Figure below. The first peak in the
SAXS represents the ‘10’ plane of the hexagonal arrangement of the pores in the
reciprocal space as seen in the Figure 19. This position of the maximum of the peak
corresponds to the interpore distance in real space. The real space distance can be
calculated from the position of the peak as discussed earlier. The Bragg peak position of
RH-54% shifts towards the right, which is increasing in reciprocal space. In real space the
distance decreases, which means that pores are closer at RH54% than RH86%. The
implication of closer pores is the compression of the porewall (silica scaffold). Thus the
shift in peak position can measure the strain. The detailed discussion is presented in later
chapters.
53
Figure 18. Comparison of SAXS patterns of MCM-41 at 54% and 86%.
The electron density contrast between the sample and its background must be
significant for a good data. The intensity of scattering pattern also depends on this density
contrast. The scattering pattern of the background is usually collected before introducing
the sample and this pattern is subtracted from the pattern of the sample. It is always good
to have as much high intensity as possible, which implies that the intensity of the sample
should be greater than the background. In in-situ and in-vivo observations the intensity
variations can be used to detect the quantitative change in the composition of the sample.
The x-rays collected on the detector are scattered from the electrons in the sample. The
intensity of the pattern is given by,
(22)
P(q) is the form factor, S(q) is the structure factor, v volume of a particle, ∆ρ is the
density contrast and I is the scattering intensity of one electron. The form factor reveals
the shape of the particle and it is the pattern occurring due to the atoms in a particle. The
54
structure factor reveals the distances between the particle planes and it is the pattern
occurring due to inter-particle distances.
Another interesting factor in Figure 18 is the change in the intensity between the
two patterns. This intensity difference is attributed to the change in the amount of water
in the pore. It can be inferred from equation 22 that the intensity depends on the density
contrast. There are three media in the system, silica, water and air. The density contrast
between silica and air is obviously greater than it is between silica and water. Hence if
there is more water in the system then the overall density contrast will be lower. The
amount of water will definitely be greater at RH86% than RH54%. This intensity
variation will be later used in determining the data points that must be used to calculate
the modulus.
The structures probed in the WAXS are smaller and usually on the order of a
atomic scale. The form factor mentioned in the above discussion becomes the atomic
form factor at these scales. Some amorphous glass materials and liquids exhibit a peak in
WAXS and it is known as ‘first sharp diffraction peak’ (FSDP). FSDP represents an
intermediate range order in the system. The corresponding real space structure is still not
clearly determined.
The WAXS pattern is shown below for two different values of RH, with the fit
model as inset. Similar to SAXS, FSDP of WAXS is also used to measure the strain. The
difference in this strain is that it measures the strain of the whole silica skeleton rather
than the one plane measured in SAXS. The peak position in WAXS is not as obvious as
SAXS because the FSDP is a combination of two peaks. The two peaks represent
55
presence of silica and water in the system. The FSDP is fit with two asymmetric pseudo
voigt (APV) functions to model the presence of silica and water as shown in the inset of
the Figure. The algorithm for the fit is modeled so as to simulate the change in the
amount of water for each RH. The shift of the peak of silica APV is used to measure the
strain. These strain measurements will again be discussed in the following chapters.
Figure 19. Comparison of WAXS patterns of MCM-41 at 54% and 86%.
The experiments in this study were conducted using an in-house x-ray scattering
source and also a synchrotron source. The in-house scattering apparatus uses an x-ray
tube with copper target to produce x-rays. The production of x-ray in these is typical:
incidence of electron on the target. Synchrotron sources use a different technique, where
x-rays are produced by accelerating electrons to relativistic speeds in a circular path. The
brief discussion of x-ray scattering in this section was referred from (Ref. 36).
56
CHAPTER 4: EXPERIMENT
4.1. Gas sorption experiment
Micromeritics Tristar II Surface Area and Porosity System (Micromeritics
Instrument Corporation, USA) was employed to extract structural pore parameters such
as surface area, pore width, micropore area, pore size distribution and sorption isotherms.
A test tube with about 0.2g of the synthesized sample was loaded to the Micromeritics
degas system. During the degassing, sample was heated to 400°C and flushed with
nitrogen. After degassing, the mass of the degassed sample was accurately obtained and
the test tube was loaded into Micromeritics Tristar II Surface Area and Porosity System.
The characteristics of the sample were given as input into the application in the computer,
which controls the whole process. The sample was held in a liquid nitrogen bath and the
data were determined by the adsorption of nitrogen and helium. Three samples can be
analyzed simultaneously and it takes about 15 hours for the whole process.
Figure 20. Micromeritics degas system (left); Micromeritics Tristar Surface area and
Porosity (right)37.
57
4.2. X-ray scattering experiment
4.2.1 SAXSess
SAXS and WAXS data were used to extract the strain on the pores due to
capillary action of water as discussed earlier. SAXSess is a table-top x-ray scattering
system manufactured by Anton Paar GmbH, Austria. This was employed to get the
SAXS/WAXS pattern with a homemade sample chamber which had provisions to control
and monitor the relative humidity (RH) around the sample pellet. The RH was controlled
by a humidity generator with distilled water as the fluid and the actual RH in the chamber
was monitored with the sensor inside the chamber. There was a small difference between
the set RH and the actual RH in the sample chamber and the actual RH value was always
considered for the data. The preliminary adjustments in SAXSess were made and it was
calibrated to the particular sample holder. The sample pellet was placed in the pellet
holder inside the sample chamber with ample surface exposed for the scattering. The
sample chamber was then loaded into the SAXSess in the appropriate position. The
SAXSess was evacuated to 1atm with the vacuum generator which ran throughout the
experiment to maintain the vacuum. After the vacuum was attained the x-rays were
generated to impinge on the sample. The relative humidity inside the chamber was varied
from 100% to 0% in steps of 5% with 1 hour for each step. The scattered x-rays were
collected using an imaging plate. The imaging plate records the x-rays by locking the
electrons in it to a metastable state and it was read by illuminating with visible light in a
reader. Hence, after recording each step the imaging plate was carefully transferred into
58
its reader in the dark. The data from the imaging plate transfers to a computer as a 2D
data and it is converted to 1D by an application. In SAXSess only the WAXS data was
used to estimate the strain. The SAXS pattern was acquired separately for 10 min at RH
of the room and was used to determine the quality of the sample and the interpore
distance. During the recording of the WAXS a lead block was used to protect the imaging
plate from over exposure. The exposure of high intensity of the SAXS for one hour
would damage the imaging plate. The lead block prevented the high intensity part of the
pattern from reaching imaging plate. The WAXS peak was fit using WinXAS and its
parameters was extracted for further analysis. The WinXAS is developed by Thorsten
Ressler for x-ray adsorption spectroscopy studies. The wavelength of x-rays used was
1.542Å which is the k-alpha line of copper. There was approximately a 20 min gap
between setting the RH for the particular step and the beginning of the exposure of x-rays
which gave enough time for RH to stabilize inside the chamber.
The 1 D data from the computer was fit with WinXAS software to extract usable
parameters. The maxima of WAXS peaks were extracted by fitting functions in the 1D
data. An algorithm was created for each sample and it was run for the dataset of each
sample. The maxima were not directly read because the peaks in the raw 1D data are
combinations of more than one function. In WAXS, a pseudo voigt function representing
silica and a higher degree polynomial representing water are fit to extract the proper
position of maximum of the peaks due to silica. In SAXS data the maximum of the first
Bragg’s peak was directly found and used to estimate the interpore distance.
59
After the synthesis of the sample, the SAXS pattern was collected in SAXSess to
assess sample quality. If the pattern shows the proper peaks and ordering then the sample
is subjected to the gas sorption method to extract the pore parameters. After that the strain
due to capillary action was measured using the WAXS pattern as discussed above.
Figure 21. SAXSess instrument and the related devices (left); The raw 2D data (right top)
and the converted 1 D data (right bottom).
4.2.2. Synchrotron
Apart from using SAXSess, the same study was conducted in the synchrotron at
Argonne National Laboratory to justify and confirm the results with better quality of data.
The sample pellets used were similar to the ones used in SAXSess. The pellet was cut
into a smaller piece which would fit in the synchrotron sample holder. The sample holder
had an inlet and outlet for the circulation of air with the controlled RH. The same
humidity generator mentioned above was connected to the inlet and the same humidity
sensor was placed at the outlet to measure the actual RH. Due to the high energy of the
60
synchrotron x-rays the exposure time was reduced to 10 seconds for each step. The steps
were similar to the SAXSess experiment, RH was varied from 100% to 0% with 5% steps
and noting the actual RH for the data. The wait time for the RH to stabilize was set to 10
min after observing no difference between the patterns of 10 and 15 min stabilizing time.
The SAXS and WAXS were obtained at the same time and sample’s parameters were
extracted in a similar way as above using the WinXAS. The wavelength of the x-rays in
synchrotron was 0.2034Å.
The fitting of this SAXS and WAXS was different. In SAXS data, an exponential
function representing the decrease of intensity with the increase of q, a polynomial
function representing the background due to the inelastic collision of x-rays and
significantly the asymmetric pseudo voigt (APV) function representing the peaks were fit
to extract the proper position of the peaks. Sometimes the range of q considered for the
data is small, so the exponential function is avoided to create a better algorithm. In
WAXS, two APV functions representing silica and water were fit to extract the proper
position of maximum of the peak due to silica. The position of the APV representing
water was fixed at the value near to the peak of pure water. In the fit algorithm for the
WAXS data, the amplitude of the APV representing water reduced with decrease of RH
and this automatically simulates the decrease of water in the system with decrease of RH.
4.2.3. Comparison between SAXSess and synchrotron data
The x-ray scattering experiment was performed with two different sources as
mentioned in the experimental section. The quality difference between the SAXSess and
synchrotron data was significant due to the energy of the source, collimation and the
detector. The detector used in SAXSess was an imaging plate, which traps electrons
61
excited by incident x-rays in it to a metastable state known as F-trap38. The trapped
electrons were brought to the ground state by illuminating with visible light. The energy
released by the electrons is radiated as fluorescent radiation which is collected and
processed by a computer algorithm to give a 2d plot. In the synchrotron, the detector was
a wire detector called as Pilatus. This basically has a matrix of electronics on the screen
representing each pixel. The wire detector has shorter data retrieval time and can acquire
higher intensity without any damage. Hence the exposure time has no constraint. The
imaging plate tends to reach a limit for the intensity it can record.
The source in the synchrotron is a point source and the source in SAXSess is a
line source. In a point source, only a small part of the sample is exposed and the 2d
pattern has concentric circles. For the line source a large portion of the sample is exposed
as the x-ray beam is confined in only one direction. The 2d pattern has broader concentric
ovals38. The broadening is the result of the larger area of exposure of the sample and it
causes smearing of the pattern38. The larger area exposure decreases the exposure time
for a particular intensity with same energy. Thus the point collimation was not used in the
SAXSess instrument. In the synchrotron the energy of the x-ray is higher than in
SAXSess and is sufficient for point collimation to produce a higher intensity than the
SAXSess without smearing.
The intensity difference between the SAXS peak and WAXS peak is an indication
for the quality of the pattern. The intensity difference between them must be larger for a
better pattern. In the Figures below, the pattern from the SAXSess and the synchrotron
has been shown. The synchrotron pattern has a larger difference between SAXS and
WAXS. Although the data collected from the SAXSess enabled in deducing the trend of
62
the strains of the sample, the error in the data was large. All of the data presented in the
work are from synchrotron, which was more reliable for the actual calculations.
Figure 22. The 1D scattering patterns from synchrotron; the arrow indicating the
intensity of the SAXS peak (top); The arrow indicate the intensity of the WAXS peak
(bottom).
63
Figure 23. The 1D scattering pattern from the SAXSess indicating the intensity of the
part of SAXS peak (top arrow) and the WAXS peak (bottom arrow).
64
CHAPTER 5: RESULTS AND DISCUSSION
The data and the interpretation of the each experiment are discussed in separate
sections and finally summarized.
5.1. Gas-sorption method:
The specific surface area, pore width, micropore area, pore size distributions and
isotherms are the data extracted from this method. These data are essentially used to
determine the quality of the sample before undertaking the X-ray scattering experiments.
Table 1
Physical pore parameters extracted from gas-sorption method
Sample
Surface
Area (m2/g)
Pore
width (Å)
Porewall
thickness (Å)
Micro pore
Area (m2/g)
MCM-41 ‘AS’
1082.83
26
15
0
MCM-41 ‘AN’
1132.88
24
16
0
SBA-15 ‘AS’
367.90
60
48
147.90
SBA-15 ‘AN’
368.44
59
48
130.19
The surface area, micropore area, pore width are directly extracted from the gassorption method as discussed earlier. The pore wall thickness is the difference between
the interpore distance at stress free configuration calculated from SAXS and the pore
65
width. These data will be used later to explain the elastic modulus variation between the
samples.
5.2. X-ray scattering technique:
There were two scattering patterns produced by these samples, SAXS and WAXS
as discussed earlier. The position of the first Bragg’s peak of SAXS, found by fitting the
data, corresponds to the interpore distance. It was plotted against RH to view the
variation of that distance. It must be noted that the y axis is in reciprocal space.
Figure 24. Top left: SAXS peak position vs RH plot for MCM-41 ‘AS’ ;Top right: SAXS
peak position vs RH plot for MCM-41 ‘AN’; Bottom left: SAXS peak position vs RH
plot for SBA-15 ‘AS’; Bottom right: SAXS peak position vs RH plot for SBA-15’AN‘.
66
The Bragg’s peak position actually gives the distance between the ‘10’ planes, by
using simple geometry the interpore distance can be calculated as shown in the Figure.
Figure 25. The schematic of the hexagonal arrangement, from the first Bragg’s peak in
SAXS the interpore distance is calculated by using the perpendicular triangle. The area
filled with red represents the porewall.
These interpore distances are in the x-y plane as shown in the Figure 25. It must be noted
that the position is in the units of inverse of length. In MCM-41, at high RH the interpore
distance is higher than it is at stress free interpore distance of the lowest RH, which
implies that the pore wall is stretched at those RH. The lowest RH with almost no water
in the pore is considered as the stress free configuration. Gradually the inter pore distance
decreases with reducing RH, which implies that the pore wall is compressed. The
67
compression continues beyond the stress free configuration. Further reduction in RH the
pore wall relaxes to the stress free configuration.
In SBA-15, it is more complicated than the above explanation. There are two
series of compression and tension similar to the one observed in MCM-41 as explained
above. The first one occurs when the meniscus is at the entrance and the second occurs
during the capillary evaporation. The second one can be attributed to the effect of
micropores due to the continued presence of water in them even when the mesopores are
emptying.
Figure 26. WAXS peak position vs RH plot of MCM-41 ‘AS’; Top right: WAXS peak
position vs RH plot of MCM-41 ‘AN’; Bottom left: WAXS peak position vs RH plot of
SBA-15 ‘AS’; Bottom right: WAXS peak position vs RH plot.
68
The WAXS data corresponds to the distance in atomic scale. Similar to the
SAXS plot, the FSDP position of WAXS after the fit is plotted against RH. The FSDP
peak position distances are on the atomic scale and their variation represents the average
change in those distances in all planes.
In MCM-41, the distance at high RH is almost same as the stress free distance at
the lowest RH. The whole sample begins to compress with reducing RH. At high RH the
interpore distances in x-y plane is at compression but the atomic scale distances in the
whole material is stress free, which is a very interesting result. This would imply that the
cumulative atomic distances must extend in the z plane in the same amount as the
cumulative compression in the atomic distances due to the compression of porewall in xy plane (observed in SAXS). The highest compression occurs after the beginning of the
capillary evaporation. Further reduction in RH, the distances expand to the stress free
configuration. In this study a new technique is developed to use this distance variation to
estimate the strain in all planes.
The Kelvin equation and Laplace pressure hold true only till the meniscus is
present at the pore entrance during desorption. The elastic modulus of the sample is
calculated using the Kelvin equation hence the strain must be extracted only with the data
before the capillary evaporation. The variations in density contrast correlate the amount
of water present in the pore as discussed earlier. The amplitude of the Bragg’s peak is
plotted against the RH to visualize the change in amount of water. These plots can be
studied to determine the data that must be used to estimate the strain to calculate the
elastic modulus using Kelvin equation.
69
Figure 27. The amplitude of the first Bragg’s peak of SAXS vs RH in MCM-41 ‘AS’.
The line represents that the rate of water loss during capillary evaporation.
70
Intensity_SAXS
Strain_SAXS
0.003
3.5
0.002
0.001
2.5
0.000
2.0
-0.001
1.5
-0.002
1.0
0.5
Strain_SAXS
Intensity(arb.units)
3.0
-0.003
0.0
0.5
1.0
-0.004
RH
Figure 28. The SAXS intensity and the SAXS strain of MCM-41 ‘AS’ are plotted
together to deduce the data points that are used to calculate the elastic modulus.
71
Intensity_SAXS
Strain_SAXS
3.0
0.002
0.001
2.0
0.000
1.5
-0.001
Strain_SAXS
Intensity(arb.units)
2.5
-0.002
1.0
-0.003
0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
RH (%)
Figure 29. The SAXS intensity and the SAXS strain of MCM-41 ‘AN’ are plotted
together to deduce the data points that are used to calculate the elastic modulus.
72
Intensity_SAX S
Strain_SAX S
35
0.0015
30
0.0010
0.0005
20
0.0000
15
10
-0.0005
5
-0.0010
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Strain_SAXS
Intensity_SAXS
25
1.0
RH
Figure 30. The SAXS intensity and the SAXS strain of SBA-15 ‘AS’ are plotted together
to deduce the data points that are used to calculate the elastic modulus.
73
Intensity _SAXS
Strain_WAXS
0.002
30
0.001
25
0.000
20
-0.001
15
-0.002
10
-0.003
5
-0.004
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Strain_WAXS
Intensity(arb.units)
35
1.0
RH
Figure 31. The SAXS intensity and the WAXS strain of SBA-15 ‘AN’ are plotted
together to deduce the data points that are used to calculate the elastic modulus.
74
30
S A X S _ In t e n sity
25
20
15
10
5
0.0
0 .1
0.2
0 .3
0 .4
0.5
0 .6
0.7
0 .8
0.9
1 .0
RH
Figure 32. The amplitude of SAXS first Bragg’s peak vs RH plot for SBA-15 ‘AS’. The
lines represent that the rate of water loss has two different rates during the capillary
evaporation.
The strain for the samples were plotted together to enable better comprehension of
the discussion. The estimation and plot of the strain is discussed next in this section. The
relatively flat portion of the curve in the plots at high RH corresponds to little or no
change in the amount of water. This region marks the presence of the meniscus at the
pore entrance. The strain is extracted from this range of RH to calculate the elastic
modulus. Beyond that region the meniscus moves inside the pore with the reducing RH.
This movement of meniscus reduces the amount of water inside the pore and it also the
sign of capillary evaporation. In SBA-15 the range of RH at which capillary condensation
occurs has two different slopes unlike the MCM-41 which has only one as shown in the
75
above Figures. This is attributed to the presence on micropores, during capillary
condensation in mesopores the micropores are still filled with water. The rate of loss of
water is different for micropores. It takes a lower RH for the water in micropores to
evaporate.
After fitting the curves and extracting the position of the maxima, the strain from
both SAXS and WAXS can be calculated by the following formula by using the peak
shift.
=
(
( )
)
−1
(23)
ε is the strain, p(0) is the position of the maximum for zero stress which is the
lowest humidity data point considered as reference state, p(RH) is the position of the
maximum for every other RH data points. The strain is then plotted against the natural
logarithm of the RH as shown in the graphs below.
76
0.004
0.003
Strain_SAXS
0.002
0.001
Equation
y = a + b*x
W eight
No W eighting
9.32953E-8
Residual Sum
of Squares
Pearson's r
0.99864
Adj. R-Square
0.99695
Value
Strain_SAXS
Strain_SAXS
Intercept
Slope
0.00415
0.00921
Standard Error
8.23879E-5
1.69818E-4
0.000
-0.001
-0.002
-0.003
-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00
ln(RH)
Figure 33. The strain in the x-y plane extracted from the SAXS plotted against ln(RH) for
MCM-41‘AS’. The trendline corresponds to data points(black) before the capillary
evaporation. The inset shows the slope and the error in the trendline.
77
0.004
0.003
Strain_SAXS
0.002
0.001
Equation
y = a + b*x
W eight
No W eightin
Residual Sum
of Squares
3.09091E-8
Pearson's r
0.99935
Adj. R-Square
0.99856
Value
Strain_SAXS Intercept
Strain_SAXS Slope
Standard Erro
0.00228
3.7304E-5
0.0063
7.572E-5
0.000
-0.001
-0.002
-0.003
-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00
ln(RH)
Figure 34. The strain in the x-y plane extracted from the SAXS plotted against ln(RH) for
MCM-41‘AN’. The trendline corresponds to data points(black) before the capillary
evaporation. The inset shows the slope and the error in the trendline.
78
0.003
Strain_SAXS
0.002
Equation
y = a + b*x
W eight
No W eighting
Residual Sum
of Squares
9.46304E-7
Pearson's r
0.95019
Adj. R-Square
0.87857
Value
Strain_SAXS
Intercept
Strain_SAXS
Slope
Standard Error
0.0038
4.71597E-4
0.01108
0.00182
0.001
0.000
-0.001
-0.002
-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00
ln(RH)
Figure 35. The strain in the x-y plane extracted from the SAXS plotted against ln(RH) for
SBA-15‘AS’. The trendline corresponds to data points(black) before the capillary
evaporation. The inset shows the slope and the error in the trendline.
79
0.003
0.002
Equation
y = a + b*x
W eight
No W eighting
Residual Sum
of Squares
7.28482E-7
Pearson's r
0.86028
Adj. R-Square
0.65344
Strain_SAXS
Value
0.001
Strain_SAXS
Intercept
Strain_SAXS
Slope
Standard Error
0.00223
5.51429E-4
0.0073
0.0025
0.000
-0.001
-0.002
-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00
ln(RH)
Figure 36. The strain in the x-y plane extracted from the SAXS plotted against ln(RH) for
SBA-15‘AN’. The trendline corresponds to data points(black) before the capillary
evaporation. The inset shows the slope and the error in the trendline.
Similar to the earlier discussion with distances of the peaks, the strain also reveals
the same compression and tension at various stages. In MCM-41 the strain is positive at
high RH and changes to negative with the decrease of RH to its lowest value just before
capillary evaporation. The physical interpretation is, the porewall is at highest tension at
the highest RH and begins to relax during desorption to the reference stress free
configuration. It contracts below the reference stress free configuration and reaches the
highest compression just before capillary evaporation. During the capillary evaporation
80
the pore relaxes to stress free configuration. After the breaking of the meniscus the pore
contracts to the stress free configuration.
In SBA-15 at high RH it is similar to MCM-41 but the lowest compression is
reached at relatively higher RH than MCM-41. During capillary the evaporation, the
porewall begins to stretch to a point and again compresses to the lowest compression.
Further reduction in RH relaxes the porewall to the stress free configuration. The
variation in the strain during capillary evaporation is due to the presence of water in the
micropores at these ranges of RH and they emulate a similar mechanism of mesopores
before capillary evaporation.
The strain extracted from the SAXS data is in the plane perpendicular to the
channel of the pore. According to the earlier works by Prass9 and Gor10, the strain was
presented as
=
∗
1
(24)
and
=
1
(−
+
∗
+(
−
∗ ))
(25)
respectively. All the symbols represent the same as before and P/P0=RH. It is evident
from the earlier arguments and the literature32,33 there are forces acting on the solid other
than the force due to Laplace pressure which is the natural logarithm term in the above
expression (also discussed earlier). The forces arising from the surface tension effects
were not considered in equation24. In the equation 25, the effect of the tangential
component of surface tension was not considered.
81
According to the earlier discussion it can be inferred that there are other stresses
involved in this system. The forces are summarized here based on the earlier arguments.
Table 2
Summary of the stresses acting on the porewall in each plane
Stress Direction
Stress
Stress term
Normal (X-Y Plane)
σx, σy
Tangential (Z Plane)
σz
2
−
+
+
Due to the ordering of the pores and the symmetry of the forces here, the stress on
both x and y direction could be considered same (σx=σy). The equation 14 represents the
total strain in each plane in relation to the stress and poisson’s ratio. Now by substituting
the stresses into that equation we get,
=
∗
(2 − 3 ) +
(26)
At high RH the pores are farther apart from each other in x-y plane (normal to the
pore channel), which also mean the porewall is stretched. The stress due to the surface
tension of thin layer of water on the inner wall pulls the pore wall toward the center of the
pore. The tangential compression also contributes the stretching of the pore wall in x-y
plane as shown in Figure 37.
82
Figure 37. The bold blue circles are the stress free configuration of pores and the broken
circles represent the change at high RH in x-y plane (SAXS).
As the RH is reduced the curvature of the meniscus increases and the magnitude
of the stress due to Laplace pressure begins to increase. When its magnitude is more than
that of the stress due to surface tension, the porewall begins to compress as shown in the
Figure. This explains the variation in the strain direction during desorption before
capillary evaporation. This deformation is represented in Figure 38.
83
Figure 38. The bold blue circles are the stress free configuration of pores and the broken
circles represent the change at RH before capillary condensation.
The equation 26 is a straight line equation with variables ε x and ln(P/Po). The
modulus can be represented from the slope extracted from the strain vs ln(RH) plot of
SAXS data (P/P0=RH). This is the elastic modulus of the material.
=
(2 − 3 )
R=8.314 J/Kg/K-1; T=294K; V=18 Χ 10-6; ν=0.17, Slope= ε / ln(RH).
(27)
Apart from the estimation of strain in x-y plane in SAXS, the strain from all
planes is estimated from the FSDP of the WAXS, which is the primary objective of this
84
study. The strain is estimated with FSDP peak position in the same way as the SAXS
and it is also plotted against the natural logarithm of RH.
0.000
S train_W A X S
-0.001
-0.002
-0.003
-0.004
-0.005
E quation
y = a + b*x
W eight
N o W eighting
R esidual S um
of S quares
5.31695E -7
P earson's r
0.97963
A dj. R -S quare
0.95464
V alue
-0.006
S train_W A X S
Intercept
S train_W A X S
S lope
S tandard E rror
1.35433E -4
1.96682E -4
0.00559
4.05401E -4
-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25
0.00
ln(R H )
Figure 39. The strain calculated from FSDP vs the ln(RH) for MCM-41 ‘AS’. The
trendline corresponds to the data points(black) before the capillary evaporation. The inset
shows the slope and the error in the trendline.
85
0.000
Strain_WAXS
-0.001
-0.002
-0.003
-0.004
-0.005
Equation
y = a + b*x
W eight
No W eighting
Residual Sum
of Squares
4.5218E-7
Pearson's r
0.97764
Adj. R-Square
0.95087
Value
-0.006
W AXS_Strain(p Intercept
W AXS_Strain(p Slope
Standard Error
-0.00227
1.42682E-4
0.00404
2.89617E-4
-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00
ln(RH)
Figure 40. The strain calculated from FSDP vs the ln(RH) for MCM-41 ‘AN’. The
trendline corresponds to the data points(black) before the capillary evaporation. The inset
shows the slope and the error in the trendline.
86
0.002
0.001
Equation
y = a + b*x
W eight
No W eighting
4.11006E-7
Residual Sum
of Squares
Pearson's r
0.88091
Adj. R-Square
0.70134
Strain_WAXS
Value
0.000
Strain(p)
Intercept
Slope
Standard Error
-1.62817E-4
4.14194E-4
0.00605
0.00188
-0.001
-0.002
-0.003
-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00
ln(RH)
Figure 41. The strain calculated from FSDP vs the ln(RH) for SBA-15 ‘AS’. The
trendline corresponds to the data points(black) before the capillary evaporation. The inset
shows the slope and the error in the trendline.
87
0.002
0.001
Strain_WAXS
0.000
Equation
y = a + b*x
W eight
No W eighting
Residual Sum
of Squares
3.25657E-7
Pearson's r
0.81946
Adj. R-Square
0.56203
Value
Standard Error
Strain(p)
Intercept
-0.0014
3.68689E-4
Strain(p)
Slope
0.00414
0.00167
-0.001
-0.002
-0.003
-0.004
-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00
ln(RH)
Figure 42. The strain calculated from FSDP vs the ln(RH) for SBA-15 ‘AN’. The
trendline corresponds to the data points(black) before the capillary evaporation. The inset
shows the slope and the error in the trendline.
The WAXS strain corresponds to the averaged variation of distances between
atomic scales. The biggest contrast between the WAXS and SAXS strain is that the
former is always compression i.e. negative strain. At high RH, the whole material is
under compression and reaches the highest compression with reducing RH. In MCM-41
the highest compression is reached at the RH in which the capillary evaporation begins.
In SBA-15 the highest compression is reached at a RH during the capillary evaporation.
88
Further decrease in RH relaxes the material gradually to the stress free configuration in
both samples.
The new technique developed in this study is extracting the strain from WAXS
by a similar approach as SAXS. The WAXS peak represents the whole solid silica
scaffold in all planes. The strain calculated using the shift of the peak as earlier will yield
the averaged strain on all planes. Using equation 14 and substituting the appropriate
stresses from table 2 we get,
=
(2 − 3 ) +
=
(28)
The strain in the x-y planes have been discussed earlier in the SAXS method.
=
(1 − 4 ) −
(1 + 2 )
(29)
(30)
The above equation represents the strain in the z direction due to the stresses. At
the highest RH the strain is negative, which implies that it is compressed in z plane. As
the RH decreases, the first term becomes positive and eventually greater than the other
two terms. This consequently changes the direction of the strain from compression to
tension. This change occurs after the capillary evaporation in MCM-41 and at the
beginning of the capillary evaporation in SBA-15 samples. This argument is arrived by
plugging the constants in the equation 30 and setting εz to zero. (γLV=0.072 N/m; V=18 Χ
10-6; ν=0.17; R=8.314 J/Kg/k-1-; T=294K; r1 from the table 1 for each sample.
The average strain can be expressed as,
1
= (
3
+
+
)
(31)
89
After substitutions,
=
3
(5 − 10 )
+
3
(1 − 2 )
(32)
The above equation is a straight line equation with variables ε x and ln(P/Po). Now
the modulus can be represented from the slope extracted from the strain vs ln(RH) plot of
WAXS data(P/P0=RH). The modulus can be calculated using,
=
(5 − 10 )
3×
R=8.314 J/Kg/k-1; T=294K; V=18 Χ 10-6 ; ν=0.17, Slope= εavg / ln(RH).
(33)
The moduli E estimated by the two methods must represent the same quantity, as
it is measured from the same material. Hence the equation 27 and equation 33 can be
equated to estimate the Poisson’s ratio of the material. Surprisingly, the poisson’s ratio
was found to be closer to the value of a metal than the bulk silica(=0.17). After the
estimation of the Poisson’s ratio, it can be used to estimate the modulus of the material
using equation 27. This is a very significant advantage of the novel technique of
estimating the strain using the WAXS.
90
Table 3
The Poisson’s ratio estimated by WAXS/SAXS
Sample
Poisson’s ratio
MCM-41 ‘AS’
0.30 ± 0.02
MCM-41 ‘AN’
0.27 ± 0.02
SBA-15 ‘AS’
0.33 ± 0.1
SBA-15 ‘AN’
0.33 ± 0.12
Table 4
The modulus estimated by SAXS method
Sample
Modulus(GPa)
MCM-41 ‘AS’
16.2 ± 1.1
MCM-41 ‘AN’
25.6 ± 1.9
SBA-15 ‘AS’
12.4 ± 5.0
SBA-15 ‘AN’
18.8 ± 7.4
The elastic modulus of the ‘AS’ is less than ‘AN’ of each sample. In MCM-41,
according to the data from the gas-sorption method (table1) the pore width decreases with
annealing. The decrease in the pore width increases the curvature, which internal stress
built on the pore wall increases. The increase in internal stress will reduce the effective
strain due to the applied stress. Thus the elastic modulus of the ‘AN’ sample is greater
than the ‘AS’.
91
In SBA-15, the data from gas sorption shows that the pore width decreases with
the annealing. The greater elastic modulus of the ‘AN’ sample is the result of the increase
in pore width due to annealing. In both samples the modulus of the annealed form is
nearly 50% stronger which is very significant then the difference in the strength between
the materials. Thus it is inferred that the internal stress plays a vital role in the strength of
the material than the structure of the pore.
The difference between the MCM-41 and SBA-15 is due to the micropores. The
micropores in SBA-15 reduce the internal stress due to the absence of the material. The
lesser internal stress of SBA-15 decreases the elastic modulus of SBA-15 than MCM-41.
Table 5
Compilation of the results obtained from Gas-sorption, SAXS and WAXS methods
Properties/Sample
MCM41-‘AS’
MCM41-‘AN’
SBA-15 ‘AS’
SBA-15 ‘AN’
Pore width (Å)
26
24
60
59
Porewall thickness (Å)
15
16
48
48
Surface Area (m2/g)
Micropore Area (m2/g)
Poisson’s ratio
Modulus -SAXS (GPa)
1082.23
1132.88
367.9
368.44
0
0
147.9
130.19
0.3 ± 0.02
0.27±0.02
0.33±0.1
0.33±0.12
16.2 ± 1.1
25.6 ± 1.9
12.4 ± 5.0
18.8 ± 7.4
The error analysis of the SAXS and WAXS methods are given in Appendix A.
92
CHAPTER 6: CONCLUSION
In this study, a novel x-ray technique for the estimation of the strain of porous
silica by capillary condensation using WAXS was developed. The earlier techniques
using SAXS estimates the strain in the plane perpendicular to the pore channel. The
combination of these two techniques will enable the estimation of the Poisson’s ratio of
the material which was unprecedented till now. The Poisson’s ratio of MCM-41 and
SBA-41 were found to be closer to the value of a metal than the value for silica. This is
the most interesting result of this study and this result would offer more control over the
mechanical properties of these materials in applications.
The forces acting on the porous structure due to the capillary action was studied
closely. After careful contemplation the factors ignored in the previous methods have
been identified and the corrections were suggested for accurate results. It was found that
the modulus of MCM-41 was greater than SBA-15 in both ‘annealed’ and ‘as
synthesized’ forms. The higher modulus of MCM-41 seems to be due to the absence of
micropores in it. The difference in the modulus between these two nanoporous materials
was already known earlier by SAXS method but was reiterated here with the suggested
corrections. The ‘annealed’ forms have higher modulus than the ‘as synthesized’ forms,
due to the contraction of pores during annealing process which increases the internal
stress. This difference is more significant, annealing make the materials nearly 50%
stronger. This offers another dimension which can be controlled in synthesis to produce
materials with requisite mechanical properties.
93
The accuracy of the new WAXS technique is highly related to the quality of the
data. The obvious limitation of this method is the existence of FSDP for the scattering
pattern of the examined nanoporous material. The new technique increases the range of
the materials that can use x-ray scattering technique to estimate the strain as the WAXS
method does not require the ordered arrangement of pores like the existing SAXS
method. The presence of micropores in SBA-15 necessitates a better strategy of analysis
of its strain, which might reduce the error experienced in this study. This technique
provides more information on the physical parameters of porous silica than before.
94
REFERENCES
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S. Ruthstein, V. Frydman, S. Kababya, M. Landau and D. Goldfarb J. phys. Chem. B 107,
1739-1748 (2003).
19
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H. Kayser, Wied. Ann. 14, 451 (1881).
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F. Fontana, Memorie Mat. Fis. Soc. Ital. Sci. I, 679 (1777).
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E. Mitscherlich, Pogg. Ann. 59, 94 (1843).
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I. Langmuir, J.Am. Chem. Soc. 40, 1631 (1918).
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26
G.P. Wilhite, Waterflooding (SPE Textbook Series, Vol. 3, 1986).
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P. A. Webb and C. Orr, Analytical Methods in Fine Particle Technology (Micromeritics, 1997).
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Lorna J. Gibson and Michael Ashby, Cellular Solids-Structure and Properties (Cambridge,
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M. Vable, Mechanics of Materials (2002).
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A. Grosman and C. Ortega, Phys. Rev. B 78, 0854333 (2008).
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97
APPENDIX A: ERROR ANALYSIS
The formula for calculating the modulus can be generalized from equation 27 and 33 as,
=
×
ln (
)
‘k’ represents the constants in those equation. The slope extracted from the strain ε vs
ln(RH) plot gives the modulus E. Those two factors are plotted with the assumption of
negligible deviation (precision error) in the relative humidity (RH). The measured RH is
precise and it is varied constantly by 5% in the experiment. Hence the error in RH is
independent of strain. The standard error in the slope of a trendline always gives the error
of the term in the y axis39. In this plot it is the strain ε so its error (∆ε) is the standard
error of the slope. The accuracy error in RH due to the instrument error is 2%. The 2%
error is the deviation in each change of 5% in the experiment.
The error in E is given by the root of the sum of the squares of each error39. Hence the
error in the modulus (∆E) becomes:
∆ =
×
∆
+
∆
ln (
)
98
APPENDIX B: KELVIN EQUATION
Figure: Spherical liquid-gas interface in a capillary of radius rm.
The figure represents a curved interface between the vapor phase α and liquid phase β,
hence Laplace equation is expressed as,
=
−
=
=
(1)
=
The chemical equilibrium with chemical potential μ can be expressed as,
(2)
Now if the equilibrium is shifted with small changes in p α , pβ ,rm and μi, the expressions
become,
−
=
= 2
=
1
(3)
(4)
99
By using the first law of thermodynamics and Maxwell’s relation at constant temperature,
constant volume Vα , Vβ and constant number of molecules in each phase, the equation 4
becomes,
=
=
Using equation 5 in equation 3 with the assumption that volume of the gas phase Vβ is
(5)
greater than the volume of the liquid phase Vα,
1
2
=
(
)
(6)
−
/
(7)
ln (
)
(8)
Integrating equation 6 from a flat surface to curvature r m and using equation 4, we get
2
1
=
The vapor is assumed to be ideal gas, so
−
=
Using equation 8 in equation 7,
=
2
∗
=
(9)
The terms pα and pβ are replaced by the corresponding terms, the vapor pressure of the
gas P* and the saturated vapor pressure of the liquid P o . The volume of the liquid Vα is
replaced by molar volume V.
From the figure, r = rm / cosθ,
=
2
=
∗
(10)
This is equation is known as Kelvin equation. This derivation was based on [refs.34, 11].
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