This thesis titled MAYUR SUNDARARAJAN. has been approved for. the Department of Physics and Astronomy. and the College of Arts and Sciences by

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2 2 This thesis titled X-ray Scattering Study Of Capillary Condensation In Mesoporous Silica by MAYUR SUNDARARAJAN has been approved for the Department of Physics and Astronomy and the College of Arts and Sciences by Gang Chen Assistant Professor of Physics and Astronomy Robert Frank Dean, College of Arts and Sciences

3 3 ABSTRACT SUNDARARAJAN, MAYUR., M.S., May 2013, Physics and Astronomy X-ray Scattering Study Of Capillary Condensation In Mesoporous Silica Director of Thesis: Gang Chen The capillary condensation deforms a nanoporous material due to the capillary force generated by the fluid inside the pores. In-situ small and wide angle x-ray scattering(s/waxs) were used to study the deformation with respect to relative vapor pressure of the fluid. Periodic mesoporous silica such as MCM-41 and SBA-15 were synthesized and used as the samples with water as the capillary condensation agent. The gas sorption method and SAXS were used to extract the pore parameters of the samples. The stresses acting on the silica scaffold due to the presence of water in the pores were deduced by careful analysis of various forces acting on it. The Poisson s ratio and elastic moduli of the two samples and their annealed forms were estimated and compared quantitatively. Our study demonstrates a novel WAXS-based technique for calculating the mechanical properties of nanoporous materials with much wider applicability than the previously reported SAXS technique.

4 4 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!!

5 5 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.

6 6 TABLE OF CONTENTS Page Abstract... 3 Dedication... 4 Acknowledgments... 5 List of Tables... 8 List of Figures... 9 Chapter 1: Introduction Chapter 2: Material Synthesis MCM Synthesis mechanism MCM-41 Synthesis SBA Synthesis mechanism Synthesis of SBA Chapter 3: Material characterization Introduction Gas sorption Method Adsorption Capillary Condensation Capillary action and the formation of meniscus Capillary pressure Isotherms Kelvin Equation and BJH method X-ray scattering method Mechanical parameters Capillary action of water as stress X-ray scattering Chapter 4: Experiment Gas sorption experiment... 56

7 X-ray scattering experiment SAXSess Synchrotron Comparison between SAXSess and synchrotron data Chapter 5: Results and Discussion Gas-sorption method: X-ray scattering technique: Chapter 6: Conclusion References Appendix A: Error analysis Appendix B: Kelvin Equation... 98

8 8 LIST OF TABLES Page Table 1: Physical pore parameters extracted from gas-sorption method...64 Table 2: Summary of the stresses acting on the porewall in each plane 81 Table 3: The poreload modulus estimated by SAXS method 90 Table 4: The modulus estimated by WAXS..90 Table 5: Compilation of the results obtained from Gas-sorption, SAXS & WAXS methods 91

9 9 LIST OF FIGURES Page Figure 1. Schematic of the steps in the formation of MCM Figure 2. a) MCM41 after calcinations b) The finely ground powder and apart of the pressed pellet of MCM Figure 3. The presence of PEO in the walls of the mesopores in SBA 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 Figure 5. Adsorption and Capillary condensation Figure 6. Capillary action and the pressure at different points 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 Figure 8. Shape of hysteresis in isotherms and the corresponding pore shapes Figure 9. Isotherm of SBA-15 with nitrogen as adsorptive Figure 10. Isotherm of MCM-41 with nitrogen as adsorptive 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 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 Figure 13. The plot of pore volume vs pore width showing pore size distribution for MCM Figure 14. Stresses and strains in the three planes Figure 15. Various levels of water in the pore with the increase of RH... 44

10 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) Figure 17. The scattering of x-rays from a planar arrangement of particles Figure 18. Comparison of SAXS patterns of MCM-41 at 54% and 86% Figure 19. Comparison of WAXS patterns of MCM-41 at 54% and 86% Figure 20. Micromeritics degas system (left); Micromeritics Tristar Surface area and Porosity(right) Figure 21. SAXSess instrument and the related devices (left); The raw 2D data (right top) and the converted 1 D data (right bottom) 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) 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) 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 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 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 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

11 11 Figure 29. The SAXS intensity and the WAXS strain of MCM-41 AN are plotted together to deduce the data points that are used to calculate the elastic modulus 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 Figure 31. The SAXS intensity and the WAXS strain of SBA-15 AN are plotted together to deduce the data points that are used to calculate the elastic modulus 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 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 Figure 34. The strain in the x-y plane extracted from the SAXS plotted against ln(rh) for MCM-41 AN. The trendline corresponds to data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline Figure 35. The strain in the x-y plane extracted from the SAXS plotted against ln(rh) for SBA-15 AS. The trendline corresponds to data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline Figure 36. The strain in the x-y plane extracted from the SAXS plotted against ln(rh) for SBA-15 AN. The trendline corresponds to data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline 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 Figure 38. The bold blue circles are the stress free configuration of pores and the broken circles represent the change at RH before capillary condensation. The pore wall (right) is drawn for cubic arrangement of pores for simplicity

12 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 Figure 40. The strain calculated from FSDP vs the ln(rh) for MCM-41 AN. The trendline corresponds to the data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline Figure 41. The strain calculated from FSDP vs the ln(rh) for SBA-15 AS. The trendline corresponds to the data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline Figure 42. The strain calculated from FSDP vs the ln(rh) for SBA-15 AN. The trendline corresponds to the data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline

13 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 acronym for Mobil Crystalline Material. MCM-41 and its variant MCM-48 are

14 14 mesoporous silica which were synthesized using a template self assembly mechanism by researchers at Mobil Oil Corporation. The interesting characteristics of these materials are the ability to control the size of the pores and the ordered arrangement of pores, which were missing in zeolites and other microporous materials. This tunability and high degree of order, improved it as a molecular sieve and paved the way for new applications such as drug delivery, chemical sensor etc. In the late 1990s, researchers from University of California, Santa Barbara discovered a mesoporous silica material with micropores in its mesopore wall and this was named as SBA-155. SBA is an acronym for Santa Barbara Amorphous material. This material also attracted lot of interest due to its interconnected pores, which diverges its characteristics from MCM-41. In the 2000s, many researchers began studying the deformation effects in mesoporous silica due to adsorption and found interesting results deviating from microporous materials. In this study, MCM-41 and SBA-15 were synthesized and used as the samples. The phenomenon of inducing deformation on a solid by adsorption has been studied as early as 1927 by F.T.Meehan6. This was followed by similar studies by Bangham and Fakhoury7 and they related the deformation to the decrease in surface energy due to adsorption. The later works were more on the deformation of micro and mesoporous materials due to adsorption. Some mesoporous materials such as active carbons8, zeolites8 and mesoporous silica9,10 exhibit varying deformations (i.e. expansion and contraction). The major difference between adsorption on a typical solid and a nanoporous solid is the phenomenon of capillary condensation due to the presence of pores. This phenomenon stimulated the interest in researchers to study its effects on these

15 15 materials. The detailed discussion of this phenomenon will be presented later. Similar interest drives this study of physical effects on mesoporous silica material due to capillary condensation. Capillary condensation in the pores fills the pore with liquid, which instigates capillary action1. Water rising against gravity in a narrow tube immersed in water is an example of capillary action. In nature this allows transportation of water to the leaves at the top of the tree by the roots, blotting up by a towel, transportation of fluids in our body and much more. This capillary action has very interesting effects on nanosized capillaries. When the tubes (capillaries) are on the nanoscale the height the fluid reaches and the capillary pressure are humongous. For instance, consider a capillary of width 1 nanometer in water: it exerts a capillary pressure of 14MPa and the height reached by the water is 14km. The pressure is comparable to the pressure at the bottom of the Mariana trench. In this study, capillary action plays a vital role and it is used as the stress that deforms the porous material to measure its strength. A detailed analysis of capillary action will be discussed. The physical parameters such as surface area, pore volume, pore size distributions are essential to comprehend and explain the results. There are at least 6 methods to extract the pore parameters: Gas sorption, Mercury porosimetry, Transmission Electron Microscopy, Scanning Electron Microscopy, x-ray scattering and neutron scattering11. In this study the gas sorption method was used to extract the pore parameters of the mesoporous silica samples. This method is used because of its simplicity and versatility. X-ray scattering is used to measure the strain on the silica scaffold of the mesoporous

16 16 silica due to capillary condensation of water in the porous material. The small angle x-ray scattering technique has been used for the estimation of that strain9, 10 but in this study a novel technique using wide angle x-ray scattering was developed. The primary aim of this study is the development of the new wide angle x-ray scattering technique to estimate that strain in the mesoporous silica by capillary action of water and calculate its Poisson s ratio and mechanical strength. In the path towards that, certain ambiguities and missing links in the earlier studies9, 10 were rectified. The samples of MCM-41 and SBA-15 used in this study were synthesized in the lab. Each sample was studied in two different forms, as synthesized (-AS) and annealed (-AN), so the effects of annealing on the mechanical properties can also be estimated. The theory and the material synthesis are discussed in the 2 nd chapter. The theory of each method and the important phenomenon used in them are discussed in the 3rd chapter. The experiment is detailed in the 4th chapter and its results are presented with discussion in the 5th chapter.

17 17 CHAPTER 2: MATERIAL SYNTHESIS The mesoporous silica materials were synthesized by the template self-assembly mechanism. The pore width is controlled by the dimensions of the template, which usually is a surfactant or a block polymer. The silica attaches to the template and forms the scaffold of the material. The template is removed by calcination leaving the silica scaffold. The silica in the scaffold is amorphous, which is characteristic of both MCM-41 and SBA-15. The structure of SBA-15 is more complex than MCM-41 due to the presence micropores connecting the mesopores. In this section the self-assembly mechanism and synthesis of MCM-41 and SBA-15 are briefly presented. MCM-41 and SBA-15 were synthesized in the lab by referring to earlier works MCM 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 18 silicate precursor take place on the surface of the micelle. The second step begins even before the completion of the first step after around three minutes 12. In the third step, the silica layer after hydrolysis and condensation each encapsulate one micelle and thus the pore size is approximately 5 nm and become more ordered 12,13,15. Later the surfactants are evaporated by calcination at 550 C for 4hours. Figure 1. Schematic of the steps in the formation of MCM 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 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 SBA 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 20 the self-assembly template mechanism. The block polymer forms micelles at the beginning of the synthesis. After the hydrolysis of silicate, it is attracted by van der Waals force to the micelles. The long range hexagonal order and the attraction of hydrolysed silicate attraction happen at the same time17. These steps are similar to the ones discussed for MCM-41. Micropores are created by poly ethylene oxide (PEO) in the surfactant, which interacts with the silicate in the mesopore wall as shown in the figure below. Poly propylene oxide (PPO) is the part of the polymer which forms the template for the mesopores. During calcination along with the rest of the polymer, the PEO molecules in the walls evaporate leaving open pores in the mesopore wall 18.

21 21 Figure 3. The presence of PEO in the walls of the mesopores in SBA 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 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 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 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 Adsorption Adsorption has been used practically by humans for thousands of years. Scientific study of adsorption was extensive during the late 19th century like Kayser20 in 1881but it has been known even hundred years before that. The large capacity of porous solids to fill up gases has been known as early as 1771 by Fontana 21. In early and mid 19th century, the role of surface area and pores were known and established to explain the greater capacity of porous solids by Saussere22 and Mitscherlich23. In the early 1916, Langmuir24 proposed a semi-empirical model with isotherms. Later in 1938, the existing Langmuir theory was improved to BET theory25, which has been successful and improved later. Adsorption phenomenon forms the basis of this whole study as it induces the capillary condensation and capillary action. Adsorption is a surface based process in which a thin film of fluid is created on a surface of solid due to the van der Waals force between the atoms of the fluid and surface. It differs from absorption as the latter involves the whole volume of the solid to dissolve inside the fluid rather than only the surface. There are two types of adsorption, physical adsorption (physi-orption) and chemical adsorption (chemi-sorption).when the force between the atoms of the fluid and the surface are due to van der Waals force then it is physical adsorption and if the force is due to chemical bonding then it is chemical adsorption. The adsorbing fluid is known as adsorptive24 before adsorption and adsorbate after adsorption. The complementary phenomenon is known as desorption, which is the return of the adsorbate to adsorptive. This phenomenon along with a few others is used to extract the structural pore parameters.

25 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 26 Figure 5. Adsorption and Capillary condensation 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 27 force downwards, which causes the surface tension. When the adhesive forces between the capillary and the liquid are greater than the cohesive forces, the molecules near the walls of the capillary move upwards forming a curved top surface in the liquid called the meniscus (as shown in the Figure). The upward pull does not breach the surface tension of the meniscus hence the liquid under the meniscus is pulled up to height h. The height h depends on the weight of the liquid that can be lifted by the surface tension. P1 P1 Pw P Pw θ P2 P Figure 6. Capillary action and the pressure at different points. The upward force due to the surface tension is Fup= T (2π r), where T is the vertical component of surface tension γ. The downward force due to the pressure is Fd= ρgh(π r2), as the pressure due to the water in the capillary is ρgh.

28 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 Capillary pressure Capillary pressure is the pressure difference on the meniscus between two immiscible fluids. It is the compensation in the pressure to keep the interface between the fluids intact. The forces on the meniscus at equilibrium are as shown in the Figure 6 then the force balance equation is as follows. ( )= ( ) + (2 ) T is the vertical component of surface tension γ. (1) Rearranging the above equation we get, = = 2 = 2 This equation is called as the Law of Laplace. (2) Also consider the various pressures P marked in the Figure 6, P1 = P2, since both are atmospheric pressure. P2 =P, since there is no capillary action in the interface. It implies P 1 = P = Pw. P = Pw + ρgh. As P = P1, Capillary pressure Pc = P1 Pw = ρgh. So, if the surface tension is γ,angle of contact between the liquid and the capillary is θ and height risen is h, as shown in the Figure 6 then the capillary pressure is Pc= 2γcosθ/r=

29 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 ( ) were referred from (Refs. 1,26) Isotherms It was noted earlier that during adsorption (desorption) the adsorptive forms a layer on the solid surface. The amount of adsorptive which turns to adsorbate at a particular vapor pressure reveals a great deal of information. This leads to adsorption and desorption isotherms. An isotherm is a plot of quantity (volume) of gas adsorbed (desorbed) at a constant temperature by a solid surface as a function of relative vapor pressure. The relative vapor pressure is the ratio of the actual vapor pressure of the adsorptive (gas) to the saturated vapor pressure of the adsorbate (liquid). The shape of the plots reveals a great deal of information about the adsorption system. The following six forms of plots describe pores of different sizes and structure. Figure 7. Types of adsorption isotherms: adsorption (solid line), desorption (broken line); if there is no change in desorption line from adsorption line then it is not represented separately.

30 30 This study is mainly based on mesopores so only the Type IV isotherm is discussed. The nonporous isotherm (Type II) has the same general shape as a porous isotherm but the intermediate rise is sharp. The adhesive forces experienced by the adsorptive atoms increase when they are attracted by the adsorbent in more than one direction such as inside a pore as discussed earlier. The increase in attraction decreases its energy, which results in capillary condensation. Due to the capillary condensation, the amount of liquid inside the pore is more even at a lower relative vapor pressure. It explains the sharper intermediate for porous isotherm than non porous isotherms. The stark difference between Type IV isotherms and others is the hysteresis. The rest of the isotherm types except Type V have overlapping adsorption and desorption isotherms. Hysteresis in an isotherm is a significant characteristic, which ascertains the presence of mesopores. An isotherm of a nonporous material will not have hysteresis but an isotherm without a hysteresis does not prove that it is from a nonporous material. The hysteresis obviously indicates that the capillary evaporation is different from capillary condensation in the mesoporous material. The shape of the hysteresis reveals information about the shape of the mesopores as shown in the Figure 8.

31 31 Figure 8. Shape of hysteresis in isotherms and the corresponding pore shapes27 The enhanced adsorption in pores forms the basic principle for the gas sorption method. A real porous material may have more than one particular size of pores, which varies the isotherms from the ideal but useful information can still be elicited.

32 32 Figure 9. Isotherm of SBA-15 with nitrogen as adsorptive. The isotherm plot of SBA-15 shows interesting features, which reveals information about the sample. The presence of hysteresis proves that the sample has mesopores. The shape of the hysteresis reveals the pore shape to be cylindrical from the Figure 5, which is characteristic to SBA-15. The relative vapor pressure at which capillary condensation and evaporation occurs is known by reading x axis of the plot. The hysteresis is larger in SBA-15, which indicates that it takes lower relative vapor pressure for capillary evaporation during desorption. This can be attributed to the presence of micropores, which needs a lower vapor pressure for evaporation.

33 33 Figure 10. Isotherm of MCM-41 with nitrogen as adsorptive. Similarly isotherm of MCM-41 also reveals some information about it. It is obvious that it has cylindrical mesopores due to the presence of hysteresis and its shape. The hysteresis is smaller, due to the presence of only mesopores. Another difference is the relative vapor pressure at which the capillary condensation and evaporation begins, which is due to the difference in the pore size between the samples. The above discussion was based on (Refs. 1, 11, 31) Kelvin Equation and BJH method The BJH method is used for calculating surface area, pore width and pore size distribution in this study. The BJH method is based on Kelvin equation and it is popular

34 34 for mesopore analysis. This method was described by Barrett, Joyner and Halenda, hence known as BJH method. The micropore area is found by using t-plot method and it is exclusive of the surface area calculated by BJH method Kelvin Equation The phenomenon of capillary condensation is always observed in mesopores as discussed earlier. The function which correlates pore radius and the capillary condensation vapor pressure is known as Kelvin equation(appendix B), which is shown below. = (3) P* is the critical condensation pressure for the radius r m, Po is the saturated vapor pressure of the fluid, V is the molar volume of the condensate, rm is the mean capillary radius, R is the gas constant and T is the temperature of the adsorptive. When the angle of contact θ < 90 for a mean capillary radius of r m, the condensation will occur if the vapor pressure of the adsorptive is greater than the critical condensation pressure. Another view is that the pore radius determines whether the condensation can occur for the particular relative vapor pressure of the adsorptive. The second point is critical for the BJH method, for a particular relative vapor pressure the necessary pore radius for capillary condensation to occur can be found. The Kelvin equation sheds some light on the hysteresis seen on the isotherms discussed earlier. The mean capillary radius for a pore, which is open at both ends, is given by two radii r1 and

35 35 r2 as shown in the Figure. The materials used in the study has open ended pore, in these pores the condensation is nucleated on the inner wall. 1 = (4) r2 r1 Figure 11. The meniscus with radii r1 and r2 inside the pore forming the core volume; the thin layer t of adsorbate (liquid) on the inner wall During condensation the condensate builds layers inward to fill the pore, which implies that rm = 2r1 because r2 is infinity as shown in the Figure. Similarly during evaporation r2 = r1 = rm, which is less than 2r1. The difference in mean radius changes the critical capillary condensation/evaporation and thus there is a hysteresis in filling and emptying of the pores. In addition to the mean capillary radius, the thickness of the film of adsorbate on the pore must be accounted for. A pore will have a thin film of adsorbate on the pore wall irrespective of filling or emptying. This thickness of the layer t is calculated using one of the three expressions shown below developed with varying level of complexity. The volume of the adsorbate inside the thin film filling up the pore is known as core volume as shown in the Figure 11.

36 36 So, = + Where rp is the pore radius, rm is the core radius and t is the thickness of the film. = 3.54 = Å log = ln / / [24] (5) (6) [24] (7) (8) BJH method The extraction of the mentioned pore parameter is a complex series of calculations with some assumptions. In the BJH method, the pore shape is assumed to be cylindrical which is good for this study but would skew the results for pores of any other shape. The other assumptions are the values of surface tension and molar volume in the Kelvin equation, which might vary due to the presence of only few molecules. The deviations due to these assumptions are only small in the final result and thus this method is still popular. The calculation is usually in the form of a table as shown in Figure 12. The desorption cycle of the isotherm is usually used for this calculation. The isotherm gives out two columns relating volume of gas adsorbed and P/P 0, the difference between subsequent data points would give the volume of gas adsorbed( Vg) for that particular

37 37 decrease in P/P0. The product of ( Vg) with molar volume of that liquid (here nitrogen) would give the change in volume of liquid ( Vl) between that change in P/P0. The mean radius for each P/P0 can be calculated using the Kelvin equation and using the constants for nitrogen. Similarly the thickness of the layer can be calculated for each P/P 0 by using one of the equations (6, 7 and 8). Now by using the equation (5), the core radius can be calculated. The thickness decrease t for each P/P0 can be found by the difference of thickness of the layer calculated for subsequent P/P 0. This volume lost due to this thickness is the product of surface of the film S and the thickness decrease t. Now, the volume of liquid lost can be calculated as, = + (9) It is obvious that, = (10) By substituting the value of l from equation9 to equation10, the pore volume can be calculated. = ( ( )) (11) Now with volume of pore, it is simple to calculate the surface area S using mensuration formula for a cylinder.

38 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 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 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 were based on (Refs. 1, 11, 28) 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 Mechanical parameters The mechanical properties that are extensively used in this study are elastic modulus and Poisson s ratio. The elastic modulus of a material is defined as the ratio of the applied stress P to the strain ε in the same direction as in equation12. Stress is the force applied per unit area and strain is the deformation of the material due to that stress. Modulus has the same dimensions as the pressure and it is measured in the units of gigapascals (GPa) in this study. In a macroscopic material it is possible to apply a known stress and directly measure the deformation of the material optically or any other suitable method and calculate the modulus. In a nanoporous material it is difficult to do the same, so the capillary action of water is used as the stress and the x-ray scattering method is used to measure the strain. = (12) Usually a solid stretched in one plane will contract in the other perpendicular planes and vice versa. Poisson s ratio is the ratio between the strains in the plane perpendicular to the applied stress and the direction of the applied stress. The Poisson ratio is the factor by which the perpendicular strain can be estimated from the strain in the direction of the applied stress. If the material is not isotropic the poisson s ratio will have three different values for different sets of planes. This porous silica scaffold in these samples is considered to be isotropic with the same poisson s ratio as the bulk silica (ν=0.17). This assumption was arrived at after studying the same for honeycomb mechanics and eliminating that due to the difference in the direction and the points at

42 42 which the forces are acting here29. Hence the silica scaffold is similar to a thin silica structure so it can be considered as bulk silica. = ( ( ) ) (13) When the poisson s ratio is isotropic then the strain in each direction can be expressed as, = = = ( + ) ( ( + )) ( + ) (14) E is the young s modulus and it is the same in all direction if the material is isotropic, σ is the stress in the direction of the subscript and ν is the poisson s ratio. These expression will be used later to express the measured strain and calculate the young s modulus of the material. This section was referred from (Refs. 29, 30). Figure 14. Stresses and strains in the three planes.

43 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 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 45 Consider a bulk liquid and solid, each of which are stripped into two separate parts in vacuum. The energy necessary to strip these two bulk entities are the adhesive forces ALL (liquid-liquid) and ASS (solid-solid) respectively. The new surfaces will have surface tension 2γLV=ALL (factor 2 is due to the two separated parts) and 2γSV=ASS and when the separated liquid and solid part are joined together the adhesive forces between solid and liquid will reduce the surface energy. So, = + (15) The force on the solid is ASL,as discussed previously, can be found by rearranging the above equation and using Young s law for the equilibrium contact angle. = = + (Young s law) = (1 + ) (16) (17) Thus the tangential force on the solid arises from the interaction between the solid and liquid which is given by adhesive force ASL which has been derived to be γlv (1+cosθ). This relation for tangential component has been confirmed with DFT calculation33. The normal component γlv sinθ arises from the reaction of the solid to the Laplace pressure and has been confirmed by DFT calculation33. The resultant force due to these components is always into the liquid in the direction of half of the contact angle θ. It is to be noted these forces on the solid are due to the liquid in the core volume, (i.e.) the liquid excluding the layer of liquid in the wall. Hence the components on the solid due to liquid in the core volume are γlv (1+cosθ) and γlv sinθ. This normal component γlv sinθ acts only at the meniscus so it is neglected considering the other forces.

46 46 The force exerted by a flat film of liquid on the solid will be in the normal direction to the surface. This force arises due to the curvature of the porewall 33. This force can also be explained as the molecular interaction at the solid-liquid interface. There will be an attractive force experienced by the solid molecules. This attractive force will be balanced by repulsive force inside the liquid far from the interface. At the curved solid-liquid interface the repulsive forces between the molecules become zero and the unbalanced attractive forces pull the solid towards the liquid due to pressure difference of γlv (1+cosθ)/r32. This stress is in the normal direction and towards the center of the pore. This stress is present along the length of the pore and hence it is considered in estimating the total stress. The other force on the solid by liquid is the force due to Laplace pressure of the meniscus. It is critical to include this force into the estimation of the total force on the solid. The curved interface between two phases means that there is a pressure difference between them. At equilibrium this pressure difference is balanced by the surface tension of the interface34. This can be expressed as, = 2 (18) The radius of curvature is r and the subscripts v and l denote vapor and liquid respectively. If the radius of curvature of interface is non-spherical then the curvature part of the equation will change to two radii in perpendicular planes as seen in the Figure 11. = ( ) (19)

47 47 When the relative humidity is 100% it could be thought that there is no different phase and the pressure difference as zero. The decrease in relative humidity increases the curvature of the interface and pressure difference between increases. The direction of the pressure is perpendicular to the surface of the meniscus. This force applies to the whole length of the pore. According to Pascal s law, the pressure exerted on a confined liquid transmits equally on all direction and because of this phenomenon the Laplace pressure on the interface spreads throughout the liquid. This force on the solid is outwards and normal throughout the length of the pore until the presence of meniscus in the pore. After the disappearance of the meniscus the film of water on the inner surface of the wall contributes to the force in the normal direction but it is now due to the surface tension only as discussed earlier. In the previous section, the Kelvin equation(appendix B) has been discussed with respect to the gas sorption. The Kelvin equation also represents the Laplace pressure due to the presence of a meniscus. = ; 1 = (20) The variables represent the same as before. It is evident from the above equations that the Kelvin equation relates the Laplace pressure to the natural logarithm of relative vapor pressure. This relation provides an easier way to estimate than measuring the surface tension and wetting angle in experiment. By rearranging, the equation for Laplace pressure PL becomes as follows.

48 48 = = The above expression is vital and will be used in the calculation of modulus in the results section. Figure 16. The forces with dashed arrows are on solid due to liquid; Tangential component(red), Normal component(blue),force due to Laplace pressure(yellow);the direction of surface tension on each interface (red solid arrow);effective forces on the solid(right). These effective stresses in each direction will be used in the calculation of the modulus. The discussion in this section was based on (Refs. 32, 33, 34)

49 X-ray scattering This microscopic technique works under the principle of interference of x-rays and scattering of X-rays. This varies from optical microscopy in the reconstruction of the image after interaction of the incident waves with the sample and the structure detail. The lens system used on the reconstruction in optical microscopy is replaced with mathematical methods. In simple terms, it is the beam of collimated x-rays incident on the sample and scattered due to the electron density contrast of the sample onto the detector. The scattered waves are recorded as such and reconstructed by mathematical methods. This mathematical reconstruction has phase loss due to the way of recording of the scattered waves. Hence the retrieval of the shape and size distribution together is not possible. The details of the sample are average rather than unique. The scattering data of structures needs some information of the sample from other methods for proper interpretation. Though there are some shortcomings, this method is preferred due to the flexibility in sample preparation such as in-situ and in-vivo observations and for the average detail of the whole sample. The sample preparations are usually none or simple, and the sample are not damaged so this method is considered non-destructive. In this study the effect of capillary condensation on the pores is measured in-situ, thus x-ray scattering is a relevant technique. The two primary interactions of x-rays with matter are absorption and scattering. The absorption is the process in which the energy of the incident x-ray photon is used up by the atom to bump out an electron and a fluorescent radiation emitted by the atom to

50 50 restore the original configuration. The absorption depends on the sample and the wavelength of the x-ray, and it must be minimized for a good scattering data. There are two types of scattering, Compton (Inelastic) and Rayleigh (Elastic) scattering. In the inelastic scattering the incident x-ray photon loses some energy during collision with electrons in the sample. In the elastic scattering the incident photon collides with strongly-bound electrons and excites them to emit coherent waves. These coherent waves interfere and produce the scattering pattern on the detector. The inelastic scattering produces incoherent waves, which cannot produce any pattern so it would form a background noise. The constructive and destructive interference of the scattered x-rays depends on the angle of observation in the detector with respect to the incident x-ray, distance between the atoms and the orientation of the configuration of atoms. The scattered x-rays form a pattern of intensity variation in 2θ scale. The interference pattern of particular arrangement of particles will create identical pattern. If there is more number of that particular arrangement than the other then the interference pattern will have a higher intensity. The distances in the pattern are measured with the quantity q which is and also known as the scattering vector. The quantity q represents a length in reciprocal space so its dimension is the inverse of length. It is derived from the Bragg s law =2 and =, where d is the distance between two consecutive planes in the arrangement, θ is the angle of incidence and λ the wave-length of the radiation as shown in the Figure.

51 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 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 53 Figure 18. Comparison of SAXS patterns of MCM-41 at 54% and 86%. The electron density contrast between the sample and its background must be significant for a good data. The intensity of scattering pattern also depends on this density contrast. The scattering pattern of the background is usually collected before introducing the sample and this pattern is subtracted from the pattern of the sample. It is always good to have as much high intensity as possible, which implies that the intensity of the sample should be greater than the background. In in-situ and in-vivo observations the intensity variations can be used to detect the quantitative change in the composition of the sample. The x-rays collected on the detector are scattered from the electrons in the sample. The intensity of the pattern is given by, (22) P(q) is the form factor, S(q) is the structure factor, v volume of a particle, ρ is the density contrast and I is the scattering intensity of one electron. The form factor reveals the shape of the particle and it is the pattern occurring due to the atoms in a particle. The

54 54 structure factor reveals the distances between the particle planes and it is the pattern occurring due to inter-particle distances. Another interesting factor in Figure 18 is the change in the intensity between the two patterns. This intensity difference is attributed to the change in the amount of water in the pore. It can be inferred from equation 22 that the intensity depends on the density contrast. There are three media in the system, silica, water and air. The density contrast between silica and air is obviously greater than it is between silica and water. Hence if there is more water in the system then the overall density contrast will be lower. The amount of water will definitely be greater at RH86% than RH54%. This intensity variation will be later used in determining the data points that must be used to calculate the modulus. The structures probed in the WAXS are smaller and usually on the order of a atomic scale. The form factor mentioned in the above discussion becomes the atomic form factor at these scales. Some amorphous glass materials and liquids exhibit a peak in WAXS and it is known as first sharp diffraction peak (FSDP). FSDP represents an intermediate range order in the system. The corresponding real space structure is still not clearly determined. The WAXS pattern is shown below for two different values of RH, with the fit model as inset. Similar to SAXS, FSDP of WAXS is also used to measure the strain. The difference in this strain is that it measures the strain of the whole silica skeleton rather than the one plane measured in SAXS. The peak position in WAXS is not as obvious as SAXS because the FSDP is a combination of two peaks. The two peaks represent

55 55 presence of silica and water in the system. The FSDP is fit with two asymmetric pseudo voigt (APV) functions to model the presence of silica and water as shown in the inset of the Figure. The algorithm for the fit is modeled so as to simulate the change in the amount of water for each RH. The shift of the peak of silica APV is used to measure the strain. These strain measurements will again be discussed in the following chapters. Figure 19. Comparison of WAXS patterns of MCM-41 at 54% and 86%. The experiments in this study were conducted using an in-house x-ray scattering source and also a synchrotron source. The in-house scattering apparatus uses an x-ray tube with copper target to produce x-rays. The production of x-ray in these is typical: incidence of electron on the target. Synchrotron sources use a different technique, where x-rays are produced by accelerating electrons to relativistic speeds in a circular path. The brief discussion of x-ray scattering in this section was referred from (Ref. 36).

56 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 X-ray scattering experiment 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 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 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) 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 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 Å. 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 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 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 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 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 64 CHAPTER 5: RESULTS AND DISCUSSION The data and the interpretation of the each experiment are discussed in separate sections and finally summarized Gas-sorption method: The specific surface area, pore width, micropore area, pore size distributions and isotherms are the data extracted from this method. These data are essentially used to determine the quality of the sample before undertaking the X-ray scattering experiments. Table 1 Physical pore parameters extracted from gas-sorption method Sample Surface Area (m2/g) Pore width (Å) Porewall thickness (Å) Micro pore Area (m2/g) MCM-41 AS MCM-41 AN SBA-15 AS SBA-15 AN The surface area, micropore area, pore width are directly extracted from the gassorption method as discussed earlier. The pore wall thickness is the difference between the interpore distance at stress free configuration calculated from SAXS and the pore

65 65 width. These data will be used later to explain the elastic modulus variation between the samples 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 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 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 68 The WAXS data corresponds to the distance in atomic scale. Similar to the SAXS plot, the FSDP position of WAXS after the fit is plotted against RH. The FSDP peak position distances are on the atomic scale and their variation represents the average change in those distances in all planes. In MCM-41, the distance at high RH is almost same as the stress free distance at the lowest RH. The whole sample begins to compress with reducing RH. At high RH the interpore distances in x-y plane is at compression but the atomic scale distances in the whole material is stress free, which is a very interesting result. This would imply that the cumulative atomic distances must extend in the z plane in the same amount as the cumulative compression in the atomic distances due to the compression of porewall in xy plane (observed in SAXS). The highest compression occurs after the beginning of the capillary evaporation. Further reduction in RH, the distances expand to the stress free configuration. In this study a new technique is developed to use this distance variation to estimate the strain in all planes. The Kelvin equation and Laplace pressure hold true only till the meniscus is present at the pore entrance during desorption. The elastic modulus of the sample is calculated using the Kelvin equation hence the strain must be extracted only with the data before the capillary evaporation. The variations in density contrast correlate the amount of water present in the pore as discussed earlier. The amplitude of the Bragg s peak is plotted against the RH to visualize the change in amount of water. These plots can be studied to determine the data that must be used to estimate the strain to calculate the elastic modulus using Kelvin equation.

69 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 70 Intensity_SAXS Strain_SAXS Strain_SAXS Intensity(arb.units) RH Figure 28. The SAXS intensity and the SAXS strain of MCM-41 AS are plotted together to deduce the data points that are used to calculate the elastic modulus.

71 71 Intensity_SAXS Strain_SAXS Strain_SAXS Intensity(arb.units) RH (%) Figure 29. The SAXS intensity and the SAXS strain of MCM-41 AN are plotted together to deduce the data points that are used to calculate the elastic modulus.

72 72 Intensity_SAX S Strain_SAX S Strain_SAXS Intensity_SAXS RH Figure 30. The SAXS intensity and the SAXS strain of SBA-15 AS are plotted together to deduce the data points that are used to calculate the elastic modulus.

73 73 Intensity _SAXS Strain_WAXS Strain_WAXS Intensity(arb.units) RH Figure 31. The SAXS intensity and the WAXS strain of SBA-15 AN are plotted together to deduce the data points that are used to calculate the elastic modulus.

74 74 30 S A X S _ In t e n 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 75 above Figures. This is attributed to the presence on micropores, during capillary condensation in mesopores the micropores are still filled with water. The rate of loss of water is different for micropores. It takes a lower RH for the water in micropores to evaporate. After fitting the curves and extracting the position of the maxima, the strain from both SAXS and WAXS can be calculated by the following formula by using the peak shift. = ( ( ) ) 1 (23) ε is the strain, p(0) is the position of the maximum for zero stress which is the lowest humidity data point considered as reference state, p(rh) is the position of the maximum for every other RH data points. The strain is then plotted against the natural logarithm of the RH as shown in the graphs below.

76 Strain_SAXS Equation y = a + b*x W eight No W eighting E-8 Residual Sum of Squares Pearson's r Adj. R-Square Value Strain_SAXS Strain_SAXS Intercept Slope Standard Error E E ln(rh) Figure 33. The strain in the x-y plane extracted from the SAXS plotted against ln(rh) for MCM-41 AS. The trendline corresponds to data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline.

77 Strain_SAXS Equation y = a + b*x W eight No W eightin Residual Sum of Squares E-8 Pearson's r Adj. R-Square Value Strain_SAXS Intercept Strain_SAXS Slope Standard Erro E E ln(rh) Figure 34. The strain in the x-y plane extracted from the SAXS plotted against ln(rh) for MCM-41 AN. The trendline corresponds to data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline.

78 Strain_SAXS Equation y = a + b*x W eight No W eighting Residual Sum of Squares E-7 Pearson's r Adj. R-Square Value Strain_SAXS Intercept Strain_SAXS Slope Standard Error E ln(rh) Figure 35. The strain in the x-y plane extracted from the SAXS plotted against ln(rh) for SBA-15 AS. The trendline corresponds to data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline.

79 Equation y = a + b*x W eight No W eighting Residual Sum of Squares E-7 Pearson's r Adj. R-Square Strain_SAXS Value Strain_SAXS Intercept Strain_SAXS Slope Standard Error E ln(rh) Figure 36. The strain in the x-y plane extracted from the SAXS plotted against ln(rh) for SBA-15 AN. The trendline corresponds to data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline. Similar to the earlier discussion with distances of the peaks, the strain also reveals the same compression and tension at various stages. In MCM-41 the strain is positive at high RH and changes to negative with the decrease of RH to its lowest value just before capillary evaporation. The physical interpretation is, the porewall is at highest tension at the highest RH and begins to relax during desorption to the reference stress free configuration. It contracts below the reference stress free configuration and reaches the highest compression just before capillary evaporation. During the capillary evaporation

80 80 the pore relaxes to stress free configuration. After the breaking of the meniscus the pore contracts to the stress free configuration. In SBA-15 at high RH it is similar to MCM-41 but the lowest compression is reached at relatively higher RH than MCM-41. During capillary the evaporation, the porewall begins to stretch to a point and again compresses to the lowest compression. Further reduction in RH relaxes the porewall to the stress free configuration. The variation in the strain during capillary evaporation is due to the presence of water in the micropores at these ranges of RH and they emulate a similar mechanism of mesopores before capillary evaporation. The strain extracted from the SAXS data is in the plane perpendicular to the channel of the pore. According to the earlier works by Prass9 and Gor10, the strain was presented as = 1 (24) and = 1 ( + +( )) (25) respectively. All the symbols represent the same as before and P/P0=RH. It is evident from the earlier arguments and the literature32,33 there are forces acting on the solid other than the force due to Laplace pressure which is the natural logarithm term in the above expression (also discussed earlier). The forces arising from the surface tension effects were not considered in equation24. In the equation 25, the effect of the tangential component of surface tension was not considered.

81 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, = (2 3 ) + (26) At high RH the pores are farther apart from each other in x-y plane (normal to the pore channel), which also mean the porewall is stretched. The stress due to the surface tension of thin layer of water on the inner wall pulls the pore wall toward the center of the pore. The tangential compression also contributes the stretching of the pore wall in x-y plane as shown in Figure 37.

82 82 Figure 37. The bold blue circles are the stress free configuration of pores and the broken circles represent the change at high RH in x-y plane (SAXS). As the RH is reduced the curvature of the meniscus increases and the magnitude of the stress due to Laplace pressure begins to increase. When its magnitude is more than that of the stress due to surface tension, the porewall begins to compress as shown in the Figure. This explains the variation in the strain direction during desorption before capillary evaporation. This deformation is represented in Figure 38.

83 83 Figure 38. The bold blue circles are the stress free configuration of pores and the broken circles represent the change at RH before capillary condensation. The equation 26 is a straight line equation with variables ε x and ln(p/po). The modulus can be represented from the slope extracted from the strain vs ln(rh) plot of SAXS data (P/P0=RH). This is the elastic modulus of the material. = (2 3 ) R=8.314 J/Kg/K-1; T=294K; V=18 Χ 10-6; ν=0.17, Slope= ε / ln(rh). (27) Apart from the estimation of strain in x-y plane in SAXS, the strain from all planes is estimated from the FSDP of the WAXS, which is the primary objective of this

84 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 S train_w A X S E quation y = a + b*x W eight N o W eighting R esidual S um of S quares E -7 P earson's r A dj. R -S quare V alue S train_w A X S Intercept S train_w A X S S lope S tandard E rror E E E ln(r H ) Figure 39. The strain calculated from FSDP vs the ln(rh) for MCM-41 AS. The trendline corresponds to the data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline.

85 Strain_WAXS Equation y = a + b*x W eight No W eighting Residual Sum of Squares E-7 Pearson's r Adj. R-Square Value W AXS_Strain(p Intercept W AXS_Strain(p Slope Standard Error E E ln(rh) Figure 40. The strain calculated from FSDP vs the ln(rh) for MCM-41 AN. The trendline corresponds to the data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline.

86 Equation y = a + b*x W eight No W eighting E-7 Residual Sum of Squares Pearson's r Adj. R-Square Strain_WAXS Value Strain(p) Intercept Slope Standard Error E E ln(rh) Figure 41. The strain calculated from FSDP vs the ln(rh) for SBA-15 AS. The trendline corresponds to the data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline.

87 Strain_WAXS Equation y = a + b*x W eight No W eighting Residual Sum of Squares E-7 Pearson's r Adj. R-Square Value Standard Error Strain(p) Intercept E-4 Strain(p) Slope ln(rh) Figure 42. The strain calculated from FSDP vs the ln(rh) for SBA-15 AN. The trendline corresponds to the data points(black) before the capillary evaporation. The inset shows the slope and the error in the trendline. The WAXS strain corresponds to the averaged variation of distances between atomic scales. The biggest contrast between the WAXS and SAXS strain is that the former is always compression i.e. negative strain. At high RH, the whole material is under compression and reaches the highest compression with reducing RH. In MCM-41 the highest compression is reached at the RH in which the capillary evaporation begins. In SBA-15 the highest compression is reached at a RH during the capillary evaporation.

88 88 Further decrease in RH relaxes the material gradually to the stress free configuration in both samples. The new technique developed in this study is extracting the strain from WAXS by a similar approach as SAXS. The WAXS peak represents the whole solid silica scaffold in all planes. The strain calculated using the shift of the peak as earlier will yield the averaged strain on all planes. Using equation 14 and substituting the appropriate stresses from table 2 we get, = (2 3 ) + = (28) The strain in the x-y planes have been discussed earlier in the SAXS method. = (1 4 ) (1 + 2 ) (29) (30) The above equation represents the strain in the z direction due to the stresses. At the highest RH the strain is negative, which implies that it is compressed in z plane. As the RH decreases, the first term becomes positive and eventually greater than the other two terms. This consequently changes the direction of the strain from compression to tension. This change occurs after the capillary evaporation in MCM-41 and at the beginning of the capillary evaporation in SBA-15 samples. This argument is arrived by plugging the constants in the equation 30 and setting εz to zero. (γlv=0.072 N/m; V=18 Χ 10-6; ν=0.17; R=8.314 J/Kg/k-1-; T=294K; r1 from the table 1 for each sample. The average strain can be expressed as, 1 = ( ) (31)

89 89 After substitutions, = 3 (5 10 ) + 3 (1 2 ) (32) The above equation is a straight line equation with variables ε x and ln(p/po). Now the modulus can be represented from the slope extracted from the strain vs ln(rh) plot of WAXS data(p/p0=rh). The modulus can be calculated using, = (5 10 ) 3 R=8.314 J/Kg/k-1; T=294K; V=18 Χ 10-6 ; ν=0.17, Slope= εavg / ln(rh). (33) The moduli E estimated by the two methods must represent the same quantity, as it is measured from the same material. Hence the equation 27 and equation 33 can be equated to estimate the Poisson s ratio of the material. Surprisingly, the poisson s ratio was found to be closer to the value of a metal than the bulk silica(=0.17). After the estimation of the Poisson s ratio, it can be used to estimate the modulus of the material using equation 27. This is a very significant advantage of the novel technique of estimating the strain using the WAXS.

90 90 Table 3 The Poisson s ratio estimated by WAXS/SAXS Sample Poisson s ratio MCM-41 AS 0.30 ± 0.02 MCM-41 AN 0.27 ± 0.02 SBA-15 AS 0.33 ± 0.1 SBA-15 AN 0.33 ± 0.12 Table 4 The modulus estimated by SAXS method Sample Modulus(GPa) MCM-41 AS 16.2 ± 1.1 MCM-41 AN 25.6 ± 1.9 SBA-15 AS 12.4 ± 5.0 SBA-15 AN 18.8 ± 7.4 The elastic modulus of the AS is less than AN of each sample. In MCM-41, according to the data from the gas-sorption method (table1) the pore width decreases with annealing. The decrease in the pore width increases the curvature, which internal stress built on the pore wall increases. The increase in internal stress will reduce the effective strain due to the applied stress. Thus the elastic modulus of the AN sample is greater than the AS.

91 91 In SBA-15, the data from gas sorption shows that the pore width decreases with the annealing. The greater elastic modulus of the AN sample is the result of the increase in pore width due to annealing. In both samples the modulus of the annealed form is nearly 50% stronger which is very significant then the difference in the strength between the materials. Thus it is inferred that the internal stress plays a vital role in the strength of the material than the structure of the pore. The difference between the MCM-41 and SBA-15 is due to the micropores. The micropores in SBA-15 reduce the internal stress due to the absence of the material. The lesser internal stress of SBA-15 decreases the elastic modulus of SBA-15 than MCM-41. Table 5 Compilation of the results obtained from Gas-sorption, SAXS and WAXS methods Properties/Sample MCM41- AS MCM41- AN SBA-15 AS SBA-15 AN Pore width (Å) Porewall thickness (Å) Surface Area (m2/g) Micropore Area (m2/g) Poisson s ratio Modulus -SAXS (GPa) ± ± ± ± ± ± ± ± 7.4 The error analysis of the SAXS and WAXS methods are given in Appendix A.

92 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 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 94 REFERENCES 1 S.J. Gregg and K.S.W. Sing, Adsorption, Surface Area and Porosity (2 nd Ed., Academic Press, 1982). 2 C.H. Amberg, R. McIntosh, Can.J. Chem. 30, 1012 (1952). 3 C.T. Kresge, M. ELeonowicz, W.J. Roth, J.C. Vartuli, J.S. Beck, Nature 359, (1992). 4 J.S.Beck, J.C. Vartuli, W.J. Roth, M.E. Leonowicz, C.T. Kresege, K.D. Schmitt, C.T. Chu, D.H. Olson, E.W. Sheppard, S.B. McCullen, J.B. Higgins, J.L. Schlenker, J. Am.Chem.Soc. 114, (1992). 5 D.Y.Zhao, J.L. Feng, Q.S. Huo, N. Melosh, G.H. Fredrickson, B.F. Chmelka and G.D. Stucky, Science 279, 548 (1998). 6 F.T. Meehan, Proc. R. Soc. London A 115, (1927). 7 D.H. Bangham, N. Fakhoury, Nature 122, (1928). 8 R.S. Haines, R. McIntosh, J. Chem. Phys , (1947). 9 J.Prass, D. Muter, P. Fratzl and O. Paris, APL 95, (2009). 10 G.Y. Gor and A.V. Neimark Langmuir 26(16), (2010). 11 S.Lowell, Joan E. Shields, Martin A. Thomas and Matthias Thommes, Characterization of Porous Solids and Powders: Surface area, pore size and density (Springer, 2006). 12 C.Y. Chen, S.L. Burkett, H.X. Li, M.E. Davis, Microporous Mater. 2, (1993). 13 O. Regev, Langmuir 12, (1996). 14 H.B.S.Chan, P.M. Budd, T.D. Naylor, J. Mater. Chem. 11, (2001). 15 Z.Nan, M. Wang and B. Yan, J. Chem. Eng. Data 54, (2009). 16 F. Hoffmann, M. Cornelius, J. Morell, and M. Froba, Silica-Based Mesoporous Organic Inorganic Hybrid Materials.

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97 97 APPENDIX A: ERROR ANALYSIS The formula for calculating the modulus can be generalized from equation 27 and 33 as, = ln ( ) k represents the constants in those equation. The slope extracted from the strain ε vs ln(rh) plot gives the modulus E. Those two factors are plotted with the assumption of negligible deviation (precision error) in the relative humidity (RH). The measured RH is precise and it is varied constantly by 5% in the experiment. Hence the error in RH is independent of strain. The standard error in the slope of a trendline always gives the error of the term in the y axis39. In this plot it is the strain ε so its error ( ε) is the standard error of the slope. The accuracy error in RH due to the instrument error is 2%. The 2% error is the deviation in each change of 5% in the experiment. The error in E is given by the root of the sum of the squares of each error39. Hence the error in the modulus ( E) becomes: = + ln ( )

98 98 APPENDIX B: KELVIN EQUATION Figure: Spherical liquid-gas interface in a capillary of radius rm. The figure represents a curved interface between the vapor phase α and liquid phase β, hence Laplace equation is expressed as, = = = (1) = The chemical equilibrium with chemical potential μ can be expressed as, (2) Now if the equilibrium is shifted with small changes in p α, pβ,rm and μi, the expressions become, = = 2 = 1 (3) (4)

99 99 By using the first law of thermodynamics and Maxwell s relation at constant temperature, constant volume Vα, Vβ and constant number of molecules in each phase, the equation 4 becomes, = = Using equation 5 in equation 3 with the assumption that volume of the gas phase Vβ is (5) greater than the volume of the liquid phase Vα, 1 2 = ( ) (6) / (7) ln ( ) (8) Integrating equation 6 from a flat surface to curvature r m and using equation 4, we get 2 1 = The vapor is assumed to be ideal gas, so = Using equation 8 in equation 7, = 2 = (9) The terms pα and pβ are replaced by the corresponding terms, the vapor pressure of the gas P* and the saturated vapor pressure of the liquid P o. The volume of the liquid Vα is replaced by molar volume V. From the figure, r = rm / cosθ, = 2 = (10) This is equation is known as Kelvin equation. This derivation was based on [refs.34, 11].

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