Sundararajan, Mayur 12-13-12(word2003)

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.

2

This thesis titled


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


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 .................................................................. 34

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............................................................................................... 48

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………………...67

Table 2: Summary of the stresses acting on the porewall in each plane…………………85

Table 3: The poreload modulus estimated by SAXS method……………………………94

Table 4: The modulus estimated by WAXS……………………………………………..94

Table 5: Compilation of the results obtained from Gas-sorption, SAXS & WAXS

methods…………………………………………………………………………95

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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.......................................................................................................... 30

Figure 8. Shape of hysteresis in isotherms and the corresponding pore shapes27 ............ 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

11 Figure 29. The SAXS intensity and the WAXS strain of MCM-41 ‘AN’ are plotted


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


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


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


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


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


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


Figure 41. The strain calculated from FSDP vs the ln(RH) for SBA-15 ‘AS’. The


Figure 42. The strain calculated from FSDP vs the ln(RH) for SBA-15 ‘AN’. The


13

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

14 acronym for ‘Mobil Crystalline Material’. MCM-41 and its variant MCM-48 are

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

15 pores. This phenomenon stimulated the interest in researchers to study its effects on these

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.

16 X-ray scattering is used to measure the strain on the silica scaffold of the mesoporous

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 2nd 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 works19.

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 minutes12. 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 ordered12,13,15. 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 wall18.

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 Fontana21. 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.

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.

θ

P1

P2

Pw

P P

Pw

P1

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.

T is the vertical component of surface tension γ.

Rearranging the above equation we get,

This equation is called as the Law of Laplace.

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 P1 = 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.

30

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.

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

31 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.

Figure 8. Shape of hysteresis in isotherms and the corresponding pore shapes27

32 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.

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

33 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.

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

34 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

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.

P* is the critical condensation pressure for the radius rm, 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 rm, 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.

35 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

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.

r1

r2

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

36 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.

So,

Where rp is the pore radius, rm is the core radius and t is the thickness of the film.

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.

37

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/P0, the difference between

subsequent data points would give the volume of gas adsorbed(∆Vg) for that particular

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/P0 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/P0. 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,

It is obvious that,

By substituting the value of l from equation9 to equation10, the pore volume can be

calculated.

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.

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 which the forces

42 are acting here29. Hence the silica scaffold is similar to a thin silica structure so it can be

considered as bulk silica.

When the poisson’s ratio is isotropic then the strain in each direction can be expressed as,

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,

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)

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 porewall33. 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,

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.

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

47 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.

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.

The above expression is vital and will be used in the calculation of modulus in the results

section.

48

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)

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

49 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

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

50 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 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,

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

54 the shape of the particle and it is the pattern occurring due to the atoms in a particle. The

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

55 SAXS because the FSDP is a combination of two peaks. The two peaks represent

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 Pore Porewall Micro pore Area (m2/g) width (Å) thickness (Å) 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 gas-

sorption 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 x-

y 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

0.0 0.5 1.00.5

1.0

1.5

2.0

2.5

3.0

3.5

Intensity_SAXS Strain_SAXS

RH

Inte

nsity

(arb

.uni

ts)

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

Stra

in_S

AXS

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

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.5

1.0

1.5

2.0

2.5

3.0 In tensit y_SAX S St ra in_SA XS

RH (%)

Inte

nsity

(arb

.uni

ts)

-0.003

-0.002

-0.001

0.000

0.001

0.002

Stra

in_S

AXS

Figure 29. The SAXS intensity and the SAXS strain of MCM-41 ‘AN’ are plotted


72

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

5

10

15

20

25

30

35 In ten sity_SA XS St ra in_S AXS

RH

Inte

nsity

_SAX

S

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

Stra

in_S

AXS

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

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

5

10

15

20

25

30

35 Intensity _SAXS Strain_WAXS

RH

Inte

nsity

(arb

.uni

ts)

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002

Stra

in_W

AXS

Figure 31. The SAXS intensity and the WAXS strain of SBA-15 ‘AN’ are plotted


74

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.05

10

15

20

25

30SA

XS_I

nten

sity

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.

(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

-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004S

train

_SAX

S

ln(RH)

Equation y = a + b*x

W eight No W eighting

Res idual Sum of Squares

9.32953E-8

Pearson's r 0.99864

Adj. R-Square 0.99695Value Standard Error

Strain_SAXS Intercept 0.00415 8.23879E-5

Strain_SAXS Slope 0.00921 1.69818E-4


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

-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004S

train

_SAX

S

ln(RH)

Equation y = a + b*xWeight No WeightinResidual Sum of Squares

3.09091E-8

Pearson's r 0.99935Adj. R-Square 0.99856

Value Standard ErroStrain_SAXS Intercept 0.00228 3.7304E-5Strain_SAXS Slope 0.0063 7.572E-5


MCM-41‘AN’. The trendline corresponds to data points(black) before the capillary


78

-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00-0.002

-0.001

0.000

0.001

0.002

0.003

Stra

in_S

AXS

ln(RH)



Residual Sum of Squares

9.46304E-7

Pearson's r 0.95019


Strain_SAXS Intercept 0.0038 4.71597E-4

Strain_SAXS Slope 0.01108 0.00182


SBA-15‘AS’. The trendline corresponds to data points(black) before the capillary


79

-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00-0.002

-0.001

0.000

0.001

0.002

0.003S

train

_SAX

S

ln(RH)

Equation y = a + b*xWeight No W eightingResidual Sum of Squares

7.28482E-7

Pearson's r 0.86028Adj. R-Square 0.65344

Value Standard ErrorStrain_SAXS Intercept 0.00223 5.51429E-4Strain_SAXS Slope 0.0073 0.0025


SBA-15‘AN’. The trendline corresponds to data points(black) before the capillary


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

and

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

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,

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


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


circles represent the change at RH before capillary condensation.

The equation 26 is a straight line equation with variables εx 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.

R=8.314 J/Kg/K-1; T=294K; V=18 Χ 10-6; ν=0.17, Slope= ε / ln(RH).

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.

-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

Stra

in_W

AXS

ln(RH)

E quation y = a + b*x

W eight No Weighting

Residual Sum of S quares

5.31695E -7

P earson 's r 0 .97963

A d j. R-S quare 0.95464Value S tandard Err or

S tra in_W AX S In ter cept 1 .35433E-4 1.96682E-4

S tra in_W AX S S lope 0.00559 4.05401E-4

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

-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0.000S

train

_WAX

S

ln(RH)


Weight No W eighting


4.5218E-7

Pearson's r 0.97764


WAXS_Strain(p Inter cept -0.00227 1.42682E-4

WAXS_Strain(p Slope 0.00404 2.89617E-4

Figure 40. The strain calculated from FSDP vs the ln(RH) for MCM-41 ‘AN’. The



86

-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00

-0.003

-0.002

-0.001

0.000

0.001

0.002S

train

_WAX

S

ln(RH)


W eight No Weighting


4.11006E-7

Pearson's r 0.88091

Adj. R-Square 0.70134Value Standar d Error

Strain(p)Inter cept -1.62817E-4 4.14194E-4

Slope 0.00605 0.00188

Figure 41. The strain calculated from FSDP vs the ln(RH) for SBA-15 ‘AS’. The



87

-2.50 -2.25 -2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002S

train

_WAX

S

ln(RH)




3.25657E-7

Pearson's r 0.81946

Adj. R-Square 0.56203Value Standar d Error

Strain(p) Intercept -0.0014 3.68689E-4

Strain(p) S lope 0.00414 0.00167

Figure 42. The strain calculated from FSDP vs the ln(RH) for SBA-15 ‘AN’. The



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,

The strain in the x-y planes have been discussed earlier in the SAXS method.

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,

89 After substitutions,

The above equation is a straight line equation with variables εx 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,

R=8.314 J/Kg/k-1; T=294K; V=18 Χ 10-6 ; ν=0.17, Slope= εavg / ln(RH).

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

91 strain due to the applied stress. Thus the elastic modulus of the ‘AN’ sample is greater

than the ‘AS’.

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) 1082.23 1132.88 367.9 368.44 Micropore Area (m2/g) 0 0 147.9 130.19 Poisson’s ratio 0.3 ± 0.02 0.27±0.02 0.33±0.1 0.33±0.12 Modulus -SAXS (GPa) 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

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97

APPENDIX A: ERROR ANALYSIS

The formula for calculating the modulus can be generalized from equation 27 and 33 as,

‘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:

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,

Now if the equilibrium is shifted with small changes in pα , pβ ,rm and μi, the expressions

become,

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

greater than the volume of the liquid phase Vα,

Integrating equation 6 from a flat surface to curvature rm and using equation 4, we get

The vapor is assumed to be ideal gas, so

Using equation 8 in equation 7,

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 Po . The volume of the liquid Vα is

replaced by molar volume V.

From the figure, r = rm / cosθ,

This is equation is known as Kelvin equation. This derivation was based on [refs.34, 11].

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Sundararajan, Mayur 12-13-12(word2003)