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Nuclear Magnetic Resonance characterization of chiral nematic mesoporous films Manning, Alan P. 2013

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Nuclear Magnetic ResonanceCharacterization of Chiral NematicMesoporous FilmsbyAlan P. ManningBSc, Carleton University, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2013? Alan P. Manning 2013AbstractUsing templation with Nanocrystalline Cellulose (NCC), a mesoporous silica and organosilica filmwith a tunable chiral nematic pore structure and long, narrow pores has recently been developed.This novel material has interesting optical properties: it selectively reflects left-handed polarizedlight and has an iridescent appearance, with its perceived colour controlled by tuning the pitch ofchiral structure. Its possible applications include enantioselective catalysis and filtering, and opticalsensors. In this work, a variety of Nuclear Magnetic Resonance (NMR) spectroscopy experimentswere run to characterize the films and composite systems. 13C and 29Si Magic Angle SpinningNMR spectra confirmed removal of the NCC template via sulphuric acid and showed the processdoes not cleave organosilica bonds. NMR cryoporometry, which uses the signal from absorbedliquid water, relates freezing point depression to pore size. This method was found to be non-destructive, accurate, and more sensitive and precise than nitrogen sorption to determine poresizes. The silica films were found to have a smaller (?3 nm) pore width size distribution than theorganosilica films (?6-9 nm). Using Pulsed Field Gradient (PFG) NMR, the diffusion of absorbedwater was found to be ?2? as fast perpendicular to the surface normal than parallel to it, withdiffusion parallel to the pore axis essentially unrestricted. Silica films had overall slower diffusionthan organosilica films. Finally, a composite system was made by functionalizing an organosilicafilm with n-Octyl, enabling it to absorb 15N-labelled 8CB liquid crystal. Reversible switching ofthe reflective properties was seen upon heating absorbed liquid crystals to the isotropic phase. 15NNMR spectra were taken of the sample with different orientations to the field, showing that at roomtemperature, the 8CB mesogens are on average aligned down the pores, and after melting, theyare isotropic. Large, unexplained magnetic susceptibility effects are seen in the room temperaturespectra. Overall, these experiments will enable further development of these materials and othercomposite systems.iiPrefaceAll the material under study (Chiral Nematic Mesoporous films) was provided to me from membersof the MacLachlan group in UBC Chemistry. Most of the background on this material presentedin Chapter 2 is repeated from their publications. They have also provided me with the nitrogensorption data used in Chapter 3.The remaining data in this thesis is from NMR experiments suggested by my supervisor, C.A.Michal, and implemented and analyzed myself. The model in Chapter 5 was designed and simulatedby me with input from R.Y. Dong and C.A. Michal.This thesis contains original work that has been published in three papers as follows:? The data in Figures 2.9 and 2.12 were included inShopsowitz, K. E., Hamad, W. Y. & MacLachlan, M. J. Flexible and iridescent chiral ne-matic mesoporous organosilica films. Journal of the American Chemical Society 134, 867?870(2012).? The data in Figure 2.10 were included inTerpstra, A. S., Shopsowitz, K. E., Gregory, C. F., Manning, A. P., Michal, C. A., Hamad, W.Y., Yang, J. & MacLachlan, M. J. Helium ion microscopy: a new tool for imaging novel meso-porous silica and organosilica materials. Chemical Communications 49, 1645?1647 (2013).? The parallel and perpendicular spectra of 8CB@CNMO in Figure 5.9 and the isotropic spectrain Figure 5.8 were included inGiese, M. I., De Witt, J. C., Shopsowitz, K. E., Manning, A. P., Dong, R. Y., Michal,C. A., Hamad, W. Y. & MacLachlan, M. J. Thermal switching of the reflection in chiralnematic mesoporous organosilica films infiltrated with liquid crystals. ACS Applied Materials& Interfaces 5, 6854?6859 (2013).iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The Chiral Nematic Mesoporous Films . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Introduction to Porous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Mesoporous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Non-templated materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.3 Templated periodic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.4 Chirality and chiral porous materials . . . . . . . . . . . . . . . . . . . . . . 62.3 Nano Crystalline Cellulose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 Properties and production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Optical properties of the chiral nematic mesophase . . . . . . . . . . . . . . 82.4 The Chiral Nematic Mesoporous Films . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.2 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 NMR of CNMO films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5.1 13C spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5.2 29Si spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15ivTable of Contents3 Characterization of Film Pores with Nitrogen Adsorption and NMR Cryoporom-etry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1 Relevant Theory of Interfacial Phenomena . . . . . . . . . . . . . . . . . . . . . . . 173.1.1 Adsorption on a surface: BET theory . . . . . . . . . . . . . . . . . . . . . . 173.1.2 Wetting: the Young-Dupre? equation . . . . . . . . . . . . . . . . . . . . . . . 183.1.3 Pressure drop across interfaces: the Young-Laplace equation . . . . . . . . . 183.1.4 Capillary condensation: the Kelvin equation . . . . . . . . . . . . . . . . . . 193.1.5 Freezing point depression: the Gibbs-Thomson equation . . . . . . . . . . . 203.2 Hysteresis in Sorption Isotherms and Cryoporometry Isobars . . . . . . . . . . . . . 223.3 Measuring Pore Size Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.1 Nitrogen sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.2 NMR cryoporometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 PFG Diffusion Studies of Absorbed Water . . . . . . . . . . . . . . . . . . . . . . . 314.1 Theory of PFG Diffusion Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 314.1.1 Bulk diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1.2 The PFG pulse sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1.3 Restricted diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.4 Relaxation at pore surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 NMR of an Oriented Absorbed Liquid Crystal . . . . . . . . . . . . . . . . . . . . 405.1 Overview of Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2 The Effect of External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.3 Behaviour of Confined Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . 435.4 The 8CB@CNMO System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.5 Magnetic Susceptibility Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.6 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.8.1 Isotropic spectrum magnetic susceptibility effects . . . . . . . . . . . . . . . 515.8.2 Anisotropic spectrum magnetic susceptibility effects . . . . . . . . . . . . . . 525.8.3 Lineshape simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56vTable of Contents6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60AppendixA Interactions and the Chemical Shift Tensor . . . . . . . . . . . . . . . . . . . . . . 68A.1 Representing Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68A.2 The Chemical Shift Hamiltonian and Tensorial Representation . . . . . . . . . . . . 69A.2.1 Tensor representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69A.2.2 Tensor rotations: the spherical tensor representation . . . . . . . . . . . . . . 70viList of Tables2.1 Intensities of 29Si peaks in Et-CNMO, before and after NCC removal . . . . . . . . . 155.1 8CB magnetic susceptibility tensor values . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Lineshape simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55viiList of Figures2.1 The synthesis of MCM-41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 The pore structure of some non-templated mesoporous materials . . . . . . . . . . . 52.3 NCC crystallites, the chiral nematic phase, and its optical properties . . . . . . . . . 82.4 CNMS and Et-CNMO film photos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Some of the precursors for CN films . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 The UV-Vis spectra for Et-CNMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 HIM of Et-CNMO pores; wide view . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.8 HIM of Et-CNMO pores; close-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.9 13C spectra of Et-CNMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.10 13C spectra of Bz-CNMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.11 29Si species in CNMS and Et-CNMO . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.12 29Si spectra for two CNMS samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.13 29Si spectra for Et-CNMO samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1 Hysteresis in an isolated for gas sorption and cryoporometry . . . . . . . . . . . . . . 233.2 Common gas adsorption isotherm and hysteresis curves . . . . . . . . . . . . . . . . 243.3 A typical cryoporometry melting curve . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Temperature calibration curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5 Gas sorption hysteresis for some CNM films . . . . . . . . . . . . . . . . . . . . . . . 283.6 Gas sorption and cryoporometry pore size distributions of some CNM films . . . . . 293.7 Post-acquisition correction validity and freezing damage tests . . . . . . . . . . . . . 304.1 The PFG diffusion LED pulse sequence . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Bulk and restricted diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 The transverse and longitudinal diffusion directions . . . . . . . . . . . . . . . . . . . 354.4 Echo intensities at various diffusion times . . . . . . . . . . . . . . . . . . . . . . . . 364.5 Mono-exponential fitted transverse and longitudinal diffusion coefficients . . . . . . . 374.6 Mono-exponential fitted transverse and longitudinal RMS displacement . . . . . . . 384.7 Mono-exponential fit residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.1 The HAT6 and 8CB molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 The phases of 8CB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3 Surface anchoring of liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43viiiList of Figures5.4 8CB@CNMO thermal switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.5 The magnetization of an ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.6 Spectral dependence on rectangular geometry . . . . . . . . . . . . . . . . . . . . . . 475.7 The origin of susceptibility effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.8 8CB@CNMO 15N isotropic spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.9 8CB@CNMO anisotropic 15N spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.10 8CB@CNMO 15N isotropic spectra shape magnetic susceptibility effect . . . . . . . . 525.11 Average alignment of 8CB mesogens is down the pore axis . . . . . . . . . . . . . . . 545.12 The derivation of the 8CB molecule angle to field . . . . . . . . . . . . . . . . . . . . 555.13 Lineshape simulation fit to 8CB@CNMO spectra . . . . . . . . . . . . . . . . . . . . 57A.1 PAF ellipsoid for a 15N shielding tensor . . . . . . . . . . . . . . . . . . . . . . . . . 70ixAcknowledgementsFirst and foremost I would like to thank my supervisor, Carl Michal, for his support and wisdom,scholarly and otherwise. I am also hugely appreciative of my collaborators Mark MacLachlan, An-drea Terpstra, Michael Giese, and Kevin Shopsowitz from the MacLachlan group at UBC Chem-istry. Their unlimited patience, assistance, and creativity has made working with them a privilege.I am also thankful of Ronald Dong?s help in the matter of liquid crystals in particular and NMRin general. In addition, my colleagues Han Shu and Alison Michan were always supportive and fullof good ideas. Finally, thanks goes to Aris for his good late-night advice.xDedicationThis work is dedicated to two classy ladies:my grandmothers,Betty Manning and Hilda MeadowsxiChapter 1IntroductionA great deal of research is focussed on creating novel porous materials. Methods to control pore sizeand geometry, the chemical properties of the surface, and the physical properties of the materialare rapidly advancing, enabling novel catalytic, sensor, and optical applications. This work isconcerned with a new porous material that shows promise for use in all three areas. To enable futuredevelopment, its pore surface chemistry, size and geometry, and the behaviour of absorbed waterand liquid crystals, has been explored using Nuclear Magnetic Resonance (NMR) spectroscopy.The synthesis and properties of these novel porous materials, called Chiral Nematic MesoporousOrganosilica (CNMO) films or Chiral Nematic Mesoporous Silica (CNMS) films, are outlined inChapter 2. They have long, narrow pores of diameter ?2-10 nm. They are created using Nanocrys-talline Cellulose (NCC) crystallites as a template, and these form a chiral nematic mesophase whichis retained in the pore structure. Essentially, the pores have a well-defined helical twist. This givesthe films interesting optical properties, such as iridescence and polarizability. NMR is used toconfirm the end result of the film synthesis and study the surface chemistry.In Chapter 3, the pore diameters and some aspects of their geometry are probed using nitrogensorption and NMR cryoporometry. These both take advantage of interfacial forces on absorbedliquid and adsorbed gas. One consequence of having liquid in a small confined pore is that itsfreezing point becomes depressed. This is taken advantage of in cryoporometry where the amountof liquid absorbed water in the films compared to the amount of frozen absorbed water at a giventemperature yields information about the pore size. NMR gives a direct measurement of the liquidportion since the signal has a much longer relaxation time. The precision and accuracy of nitrogensorption and NMR cryoporometry are compared, and the possibility of damage from cryoporometryinvestigated.Next, Chapter 4 deals with measuring the diffusion of absorbed water to determine how thepore shape and size restricts their movement. This is done using Pulsed Field Gradient (PFG)NMR. The diffusion rate along the pore helical axis and perpendicular to it is measured, allowingthe diffusion rate perpendicular and parallel to the pore axis to be approximated.Finally, the alignment of absorbed 8CB liquid crystals inside the pores is investigated in Chapter5. A composite CNMO system is studied in which the reflective properties switch at the meltingtemperature of the absorbed liquid crystals. By labelling the liquid crystal molecules with 15N, theirchemical shift reveals information about their average thermal orientation. A model is created toexplain the orientation. Magnetic susceptibility effects are seen to play a large role in the resultingspectra.1Chapter 1. IntroductionThis thesis does not include a comprehensive treatment of NMR theory, which can be foundin books by Duer [1], Haeberlen [2], and Mehring [3], for example. However, since the model todescribe the spectra in Chapter 5 relies heavily on the chemical shift, Appendix A covers the theoryof this interaction.2Chapter 2The Chiral Nematic MesoporousFilms2.1 Introduction to Porous MaterialsPorous materials are one of the pillars of current research in nanotechnology, as sub-micron scalepores provide many compelling features. Firstly, smaller pores give larger surface areas and porevolumes, over 1000 m2/g and 1 cm3/g respectively [4]. Also, the physical effects of the surface, suchas anchoring and confinement, can dominate over bulk effects of absorbed molecules. Moreover,surface-catalyzed reactions take place throughout the whole volume?as a result, catalytic applica-tions drive much of the research in this field. Another reason is that the pore sizes can be preciselycontrolled, ensuring that only molecules of a particular size can enter the pores. Consequently,filtration is another use, and catalytic reactions can be very selective. Host-guest systems are beingstudied as well. In these, the porous host contains permanent guest molecules in its pores. In mi-cropores, host-guest systems can be used to study quantum confinement [5]. In mesopores, surfaceeffects combine with bulk effects, leading to interesting critical phenomena and applications forchemical sensing. Finally, some pore geometries are on the order of optical wavelengths, yieldinginteresting optical properties.Pore size is the simplest way to categorized these materials: pores that range from 0 to 2 nmare called micropores, those 2 to 50 nm are mesopores, those above 50 nm are macropores [6].The degree of order, or regularity, of the pores is also important, as precise pore distribution givesgreater selectivity in guest molecules [7]. For example, a bimodal distribution is often required forcatalysis of bioreactions, where large macromolecules, such as a proteins, react with smaller organicmolecules. The pores must accommodate both sizes [8]. Regularity is also important in photoniccrystals, where the repeating pore structure is similar to the periodic potential in crystals. Anoptical bandgap, analogous to the electronic bandgap, appears as a result [4]. Generally, orderedmicroporous materials achieve an organized pore structure through chemical self-assembly, andordered mesoporous materials use templation with supramolecular aggregates, such as micelles.The chemical composition of the porous material is also important, but can be limited by themethod of synthesis. Therefore, there are general chemical motifs that appear, particularly inmicroporous materials. One of these is zeolites, which are aluminosiliates with pores of about 1nm. Their intrinsic negative charge is often neutralized by absorbed metal ions [9]. These can beexchanged for other ions in waste separation and remediation applications, where zeolites absorb32.2. Mesoporous Materialsheavy metals, such as arsenic [10]. They can also act as ?superacids? for hydrocarbon cracking [9].As for mesoporous materials, silica or organosilica is common, since their polymerization precursorcompounds are amenable to templating. Mesoporous carbon is also of interest [11]. Recently, chiralnematic mesoporous carbon films were developed by altering the synthesis pathway described inSection 2.4.2 [12].2.2 Mesoporous Materials2.2.1 IntroductionStriving for better control over the physicochemical properties has led to a multitude of syntheticcrystalline microporous materials [10]. To make large-pore zeolites, template molecules, often or-ganic anions, are used to keep pores ?propped open?, forcing the silicate and aluminate ions tocrystallize around them [9]. However, achieving large, regular pores is difficult due to the ther-modynamic instability of large pore surface areas [4, 7]. Irregular mesoporous materials, discussedin Section 2.2.2, have been around since the invention of aerogel in the 1930s [9, 13]. These areoften used to study critical phenomena of confined liquid crystals, much like the system that willbe discussed later in Chapter 5. As for regular mesoporous materials, the ?escape from the < 2 nmzeolite micropore size prison? was achieved in 1992 with MCM-41, the first synthetic mesoporousmaterial. It was fabricated via templation with micelles (Figure 2.1) [4]. Not only are its poreslarger (1.5 - 10 nm), they are regular as well [14, 15]. The wider pores of mesoporous materialsallow larger organic and biological molecules to enter the pore structure, enabling catalysis andseparation of these molecules.Figure 2.1: The synthesis of the mesoporous material MCM-41. It uses the self-assembly of micel-lular templates to achieve a regular pore structure. Adapted with permission from [15]. Copyright1992 American Chemical Society.2.2.2 Non-templated materialsThe pore structure of two common non-templated mesoporous materials, aerogel and controlledpore glass, are shown respectively in Figures 2.2a and 2.2b. Aerogels are fabricated using sol-gel42.2. Mesoporous Materialschemistry, where solid particles in a colloidal suspension, called a sol, form a network, resulting ina gel [16]. With aerogels, the sol is suspended silica particles. The solvent is removed by one oftwo ways to prevent pore collapse from capillary forces. It can be supercritically dried, or the solis coated with methyl groups, allowing it to ?spring back?. The result is so-called ?frozen smoke?since its ratio of pore volume to total volume (the porosity) is about 0.99 [16, 17]. Controlledpore glass is made by a different method. Alkali borosilicate glass is cooled into the desired shape.Phase separation occurs between the silica and the alkali boric oxide phase, the latter which canbe removed with an acid. This leaves behind about 95% pure silica [18]. Vycor is a common typeof controlled pore glass. Some non-templated mesoporous materials can also have regularity, suchas nuclepore membranes, porous silicon, and anopores.(a) (b)Figure 2.2: The pore structure of aerogel and controlled pore glass.The uses for these disordered mesoporous materials are numerous. Aerogel?s high porosity makesit an excellent thermal and acoustic insulator and a good candidate for supercapacitor electrolytes.It was even recently used to trap space dust on NASA?s Stardust probe [17, 19]. Some of controlledpore glass?s typical uses are filtration, membranes, and, by surface modification, catalysis [20].Nucleopore membranes are used extensively for filtration applications [21]. All of these systemsare also used in the study of confinement effects on critical phenomena. The phase transitions ofabsorbed liquid helium, binary liquids, and liquid crystals change, sometimes dramatically. Thisbehaviour will be elaborated upon in Chapter 5.2.2.3 Templated periodic materialsThe supramolecular templating method used to make MCM-41 is now widespread. The tem-plates are often some phase of a surfactant-based lyotropic (concentration-dependent) liquid crystal.MCM-41 is templated with the hexagonal micelle phase; other phases are possible by changing thereaction parameters (pH, concentration, temperature, etc.) By mixing with a suitable precursor,a variety of pore structures, from spherical to even thread-like, are possible [22]. Unfortunately,52.2. Mesoporous Materialstemplating is very sensitive to these reaction parameters, so the exact pore size and geometry isdifficult to know beforehand. Also, with lyotropic liquid crystal templating, the end product oftenprecipitates out as ?m-scale particles, although some monolithic structures have been achieved[4, 23]. Templating can also be done without the use of surfactants, by using block co-polymers,proteins, or even viruses and bacteria [4, 24]! Whatever the material, it must have a surface chem-ically compatible with the material precursor. Carbohydrates have good compatibility, and thefilms used in this work are templated with carbohydrates in the form of cellulose nanocrystals,as explained in Section 2.4. After polymerization, the template is often removed by hydrolysis orpyrolysis. By careful selection of surfactant molecules and hydrolysing agent, certain groups canbe left on the surface of the pores, leaving it functionalized [4].Most mesoporous materials are made from silica or organosilica, since the polymerization re-actions are easy to control and suitable for small structures [24]. Organosilicas, such as those tobe discussed in Section 2.4.2, have organic backbones, like ethane, benzene, or more complicatedmolecules. This provides a way to change the physicochemical properties of the pore surfaces (suchas hydrophobicity) and the overall material properties (such as flexibility), allowing them to betuned for specific applications. These precursors, combined with the regularity offered by templat-ing, have led to a class of materials called Periodic Mesoporous Organosilicas (PMOs), introducedin 1999 and now very common [25]. Still, other non-silaceous materials can be templated, such asmetal frameworks, semiconductors, and organic polymers [4, 26]. These can overcome some of theproblems with silaceous materials, such as their often amorphous microstructure and the difficultyin obtaining templated monolithic structures [27].There are a wide range of uses for PMO materials. Control over the organic bridge leads toapplications in catalysis and filtration and separation, such as High Performance Liquid Chromatog-raphy. By incorporating appropriate guest molecules, optical applications, from light harvestingantennae to luminescence, are possible. Mozoshita et al. list many more examples of novel PMOs,including those incorporating chiral selectivity [25].2.2.4 Chirality and chiral porous materialsChiral molecules are identical in structure, but cannot be superposed?much like a pair of gloves.Chirality abounds in organic and biological molecules. Enantiomers are chiral molecules that aremirror images of each other, and will have identical physical and chemical properties. However,they differ in how they react with other chiral molecules [28]. This is analogous to the equality ofthe gloves until a left or a right hand is presented?then they are different. Therefore, in reactionswith chiral species, it is essential to either have enantiopure reagents, an enantioselective filter,or an enantioselective catalyst (if the reaction proceeds via catalysis). Biological macromoleculespossess chirality as well; in fact, nearly all of life on earth possesses exclusively left-handed aminoacids and right-handed sugars. The source of this biological homochirality is unknown, but must betaken into account in any biochemical reaction [28]. Since filtration, separation, and catalysis area large portion of the applications for mesoporous materials, it has long been a goal to incorporate62.3. Nano Crystalline Cellulosechirality into these materials to make them chiral-sensitive. One common method of doing this isby using chiral linkers in organosilica [25]. Another is use of a chiral surfactant template, whichleaves behind chiral domains when removed [29].Incorporating chirality on a larger scale is also desirable for optical applications. Chiral materialscan produce or filter circularly polarized light. Thin film deposition techniques have been proven tobe effective but slow. Moreover, the film is not porous, so host-guest applications are not possible,and the type of material is limited [30]. A few mesoporous materials with chirality on a largerscale have been made, such as the hexagonal rods synthesized by Che et al. [31]. This was donewith a chiral surfactant, resulting in helical pores running the length of the cylinder about 2.2 nmwide. However, longer length scales were not accessible, and the reaction produced a powder. Now,templation with a form of cellulose has been found to be an effective way of incorporating chiralityinto monolithic structures.2.3 Nano Crystalline Cellulose2.3.1 Properties and productionCellulose is an incredibly abundant renewable material present in plant cell walls, bacteria, andalgae. It is composed of linear polymers made from ?(1?4) linked d-glucose units, which arein turn held together by hydrogen bonding [32, 33]. This gives cellulose overall high crystallinitywith scattered amorphous regions. Cellulose and cellulose-derived products have been used formany years, from paper to rayon fibre [32]. Extracting these highly-crystalline areas yields Nano-Crystalline Cellulose (NCC), which is now produced on an industrial scale. Depending on thesource, these crystallites can be between 5 - 70 nm in width and 100 nm to several ?m long. Thosefrom softwood kraft pulp are typically about 5 - 15 nm wide and 100 - 150 nm long [32, 33]. NCCis produced by the hydrolysis of wood pulp with sulphuric or phosphoric acid, which is quenchedwith de-ionized water. The mixture is allowed to settle and is decanted, followed by centrifugingand dialysis. Before use, the crystallites are dispersed using ultrasound [34, 35]. A TEM image ofthe cellulose nano-crystallites is shown in Figure 2.3a.NCC properties make it suitable for a number of applications. It has a very high tensile strengthand elastic modulus, leading to its use in composite materials like reinforced plastic [34]. Moreinteresting for the research here is its ability to act as a liquid crystal, forming a chiral nematicmesophase (also known as a cholesteric mesophase) when in solution. In this mesophase, displayedin Figure 2.3b, the director of each nematic layer is twisted relative to the one below it. Thus, thereis a helical axis around which the crystallites twist. The pitch P is sensitive to parameters such asion concentration, pH, temperature, and external electric and magnetic fields [36, 37]. Solutionsof NCC can be evaporated to create dried films that retain the chiral nematic structure, as in theimage in Figure 2.3c. From this, it is evident that the chiral nematic mesophase has unique opticalproperties.72.3. Nano Crystalline Cellulose(a) (b) (c)Figure 2.3: (a) TEM image of NCC cyrstallites. (b) The chiral nematic mesophase assumed byNCC when in solution. Half the pitch length P is indicated. (c) Optical image of dried films ofNCC that have retained their chiral nematic structure, leading to an iridescent appearance. Thefringes give an indication of the helical pitch. (a) Adapted from reference [38] with permissionof The Royal Society of Chemistry. (b) Adapted by permission from Macmillan Publishers Ltd:Nature (reference [35]), copyright 2010 (c) Adapted with permission from [36]. Copyright 2012American Chemical Society2.3.2 Optical properties of the chiral nematic mesophaseChiral nematic liquid crystals can demonstrate incredible iridescence. The repeating unit of thehelix is P/2 since the cellulose crystallite mesogens are symmetric. If this is on the order of opticalwavelengths, Bragg diffraction will take place via2(P/2) cos ? = P cos ? = m?. (2.1)Here, ? is the angle between the helical axis and the incident light (normal incidence is ? = 0), m isthe interference order, and ? the wavelength of incident light. In normal incidence, only the m = 1order is present [37]. This interference is the cause of the iridescence in the dried NCC film seen inFigure 2.3c. Unlike X-ray diffraction in a crystal lattice, the Bragg planes are the same no matterthe incident direction for a given ?, so ignoring polarization effects, the situation is the same asmultilayer interference. The chiral nematic structure also acts as a photonic crystal. In analogywith the periodic electronic potential of a crystalline solid, it is possible to find where the opticalbandgaps are. These will be when half the wavelength inside the liquid crystal becomes close tohalf the pitch, ie ?/(2navg) ? P/2, where navg is the average index of refraction. Therefore, thepeak reflected wavelength ?peak is?peak =navgPcos?, (2.2)where ? is again the incident angle relative to the helical axis.The polarization effects of the chiral nematic mesophase are also very striking. The left-handedtwist of the helices causes incident left-handed circularly polarized light to be strongly reflected, and82.4. The Chiral Nematic Mesoporous Filmsright-handed polarized light to be strongly transmitted. The degree of reflection and transmittancedepends on the angle of incidence ?. When the incidence is not normal, the polarizations areelliptical [37]. Iridescence caused by chiral nematic structures is seen in nature, often due to achiral structure of cellulose as well. Some examples are the skins of certain fruits and exoskeletonsof some beetles [39, 40].2.4 The Chiral Nematic Mesoporous Films(a) (b)Figure 2.4: (a) Chiral Nematic Mesoporous Silica (CNMS) films with varying chiral nematic pitches,yielding different colour reflected light. (b) Ethane Chiral Nematic Mesoporous Organosilica (Et-CNMO), showing the iridescence resulting from the chiral nematic structure. (a) Adapted bypermission from Macmillan Publishers Ltd: Nature (reference [35]), copyright 2010 (b) Adaptedwith permission from [41]. Copyright 2011 American Chemical Society.2.4.1 IntroductionThe Chiral Nematic Mesoporous films (CNM films) used in this work were made by the MacLachlangroup from UBC Chemistry, and the properties and synthesis are discussed in more detail in theirpublications [12, 35, 42, 43]. The novel characteristic of this material is that by templation withnanocrystalline cellulose, a chiral nematic pore structure has been achieved. The result is monolithicfilms, and the ones used in this work were typically about 1 - 25 cm2 in size and 50-100 ?mthick. Three different types of films were used: Chiral Nematic Mesoporous Silica (CNMS), EthaneChiral Nematic Mesoporous Organosilica (Et-CNMO), and Benzene Chiral Nematic MesoporousOrganosilica (Bz-CNMO). Figures 2.4a and 2.4b respectively show images of CNMS and Et-CNMO92.4. The Chiral Nematic Mesoporous Filmsfilms. The iridescent property of the chiral nematic structure is evident. The difference betweenthese three types is due to the precursors used during synthesis.2.4.2 SynthesisCNMS, Et-CNMO, and Bz-CNMO are synthesized similarly, the main difference being the type ofprecursor used. CNMS uses tetramethyl orthosilicate (TMOS, shown in Figure 2.5a), Et-CNMOuses 1,2-bis-(trimethoxysilyl)-ethane (BTMSE, shown in Figure 2.5b), and Bz-CNMO uses 1,4-bis-(triethoxysilyl)-benzene, shown in Figure 2.5c. In each case, NCC is obtained as explainedpreviously in Section 2.3 and suspended in aqueous solution to obtain a chiral nematic phase.Then, the silica or organosilica precursor is added, mixed, and allowed to dry at room temperaturein a petri dish. The surface cracks as it dries (limiting the film size), and produces about 40 - 60mg of material films. (Recently, the MacLachlan group has produced non-cracking films by theaddition of polyols [43]). Next is the removal of the NCC. In the case of CNMS, this is accomplishedby pyrolysis. However, Et-CNMO and Bz-CNMO degrade under high temperatures, so alternativemethods are used. With Et-CNMO, it was found that the cellulose could be removed by sulphuricacid hydrolysis. Bz-CNMO used HCl and silver-activated hydrogen peroxide.(a) (b) (c)Figure 2.5: The precursors used for film synthesis. (a) Tetramethyl orthosilica (TMOS), used tomake CNMS. (b) 1,2-bis-(trimethoxysilyl)-ethane (BTMSE), used to make Et-CNMO. (c) 1,4-bis-(triethoxysilyl)-benzene, used to make Bz-CNMO.2.4.3 PropertiesThe most obvious feature of the films, evident in Figures 2.4a and 2.4b, is their striking iridescence,similar to that seen in the film of dried NCC in Figure 2.3c. The Helium Ion Microscopy imagesin Figures 2.7 and 2.8 respectively show the edge of an Et-CNMO film and a closeup of its porestructure. These illustrate the helical nature of the pores, proving that the left-handed chiralnematic structure of the NCC is successfully templated onto the organosilica. As a result, circulardichroism spectra show incredibly intense peaks. Since the pitch of the helices can be tuned by,for example, the ratio of the precursor to the NCC, films can be made that have a different peakreflected wavelength, according to equation 2.2. Since the reflected wavelengths determine thecolour of the film, films of different colour can be created without the need for dyes. The circular102.5. NMR of CNMO filmsdichroism and UV-Vis spectra for some films of varying pitch are displayed in Figure 2.6.As for the physicochemical properties, the films are hydrophilic and readily absorb water andother liquids. Because the navg is close to that of water, many absorbed liquids cancel the irides-cence and circular polarization. In this situation, the film looks homogenous and isotropic to thelight. Upon evaporation of the liquid, the film regains its initial optical properties. Physically,the flexibility of the films varies depending on the precursor used. CNMS films are quite brittle,Et-CNMO are quite flexible, and Bz-CNMO are in between. Finally, the pore diameter of the films,as measured by nitrogen adsorption, typically give diameters in the range of 3.5-4 nm for CNMS, 5nm for Bz-CNMO, and 8-9 nm for Et-CNMO. However, these can vary depending on the synthesisconditions. Pore size measurements will be the focus of Chapter 3.Figure 2.6: UV-Vis transmittance spectra (solid lines) and CD spectra (dashed lines) for fourEt-CNMO samples, with film colour represented by line colour. By varying the helical pitch ofthe NCC template, different coloured films can be made without the use of dyes. Reprinted withpermission from [41]. Copyright 2012 American Chemical Society.2.5 NMR of CNMO filmsNMR provides a powerful method of both confirming the complete removal of the NCC template (by13C spectra) and determining to some extent how the silica or organosilica precursors polymerize (by29Si spectra). 13C spectra have been taken of Et-CNMO and Bz-CNMO, and 29Si spectra compareCNMS and Et-CNMO. All spectra were collected on a Varian Unity Inova NMR spectrometer witha 400 MHz magnetic and a high-powered solids cabinet. A Varian/Chemagnetics 4 mm T3 MagicAngle Spinning (MAS) probe was used with a spinning rate of 5 kHz.112.5. NMR of CNMO filmsFigure 2.7: Helium Ion Microscopy Image of an Et-CNMO film surface and edge. The bands onthe edge parallel to the surface are half the pitch (P/2) of the chiral nematic helix. Image fromKevin Shopsowitz.Figure 2.8: Helium Ion Microscopy Image closeup of the pores on an Et-CNMO film edge. Acartoon of the NCC crystallite orientation prior to removal has been added. The layers at the topand bottom of the measuring bar for half the helical pitch (P/2) are pores running left-right. Inthe middle are pores running in and out of the page. Image from Kevin Shopsowitz.122.5. NMR of CNMO films2.5.1 13C spectraNatural abundance 13C CP/MAS spectra were collected of Et-CNMO and Bz-CNMO before andafter NCC removal. A clear spectrum (bottom of Figure 2.9) was obtained on Et-CNMO/NCCcomposite, clearly showing the cellulose spectra between 60 and 120 ppm and the single carbonon the ethylene bridge (SiCH2CH2Si) of Et-CNMO at about 5 ppm. After acid hydrolysis, onlythe organosilica signal remains, showing complete removal of the NCC (top of Figure 2.9). Thesespectra used 3000 scans, a recycle delay of 2 s, and with 100 Hz of Gaussian line broadening.Spectra from Bz-CNMO, displayed in Figure 2.10, also show clean NCC removal. The benzenebridge in the organosilica also gives a single peak, which is seen in other benzene-organosilicamaterials [44]. However, its large chemical shift anisotropy yields significant spinning sidebands,marked with ?*?. This prevents clean separation of the organosilica and the cellulose signals in theBz-CNMO/NCC composite spectrum, illustrated in the bottom spectrum. Higher spinning speedswere not attempted, since the main purpose was to confirm the removal of the NCC. The topspectrum shows that this has been achieved. These spectra were collected with 7000 scans, a 2 srecycle delay, and 100 hz of Gaussian line broadening.2.5.2 29Si spectra29Si CP/MAS spectra of CNMS reveals the different siloxane species. Their designations and typicalchemical shifts are in Figure 2.11. Spectra from two different CNMS samples were taken, shown inFigure 2.12. Since cross polarization is used, the intensity from 29Si spins close to proton reservoirswill be large, and the relative intensities of the peaks do not represent the relative abundance ofthe species. In fact, the intensities will depend on the cross polarization time [45]. The intense Q3peak is likely due to an abundance of this species on the surface and its close proximity to protonsfrom the OH group. These spectra used 2000 scans, a recycle delay of 2 s, and 100 hz of Gaussianline broadening.More useful is the comparison of 29Si spectra before and after cellulose removal as this showswhat impact acid hydrolysis or pyrolysis has on the material. This was done for an Et-CNMOsample: the bottom of Figure 2.13 shows the spectrum with NCC and the top shows the spectrumafter it has been removed by sulphuric acid hydrolysis. These spectra were collected with 15,000scans, a 2 s recycle delay, and 100 hz of Gaussian line broadening. Three 3 distinct silicon peaksare present: T1 at -46 ppm, T2 at -57 ppm, and T3 at -65 ppm [41]. SiO4 groups, as seen above,would give a signal between -90 and -120 ppm; its absence indicates that each silicon retains atleast one ethylene bridge group. Cleaving this bond would turn a T3 or T2 species into a Q3 or Q2species respectively. These would remain attached to the surface by the siloxane bond, so lack ofthese species proves that ethylene bond cleavage does not take place.The three peaks are fit very well by three Gaussians whose relative intensities before and afterNCC removal are given in table 2.1. The change in amplitudes of the three peaks before and afterNCC removal show that the amount of condensed CSi(OSi)3 groups (T3 peak) on the surface of132.5. NMR of CNMO films20020406080100120  (ppm)Relative IntensityCCC'C , ,CC'Et-CNMOFigure 2.9: The CP/MAS 13C spectra of an ethylene-bridged CNMO film samples prior to removalof NCC via acid hydrolysis (below) and after (above). Removal of the NCC is total as the cellulosecarbons between 60 and 120 ppm completely disappears. The peak at 5 ppm is due to the ethylenebridge carbon in the organosilica.50050100150200250300  (ppm)Relative IntensityC CC , ,CBz-CNMOBz-CNMO********Figure 2.10: The CP/MAS 13C spectra of an benzene-bridged CNMO film samples prior to re-moval of NCC via acid hydrolysis (above) and after (below). ?*? indicates spinning sidebands. Abackground from the sample rotor has been removed in both spectra. The source of the shoulderon the Bz-CNMO peak is unknown, but may be from uncondensed groups.142.6. ConclusionsFigure 2.11: The Si species seen in CNMS (left) and Et-CNMO. Approximate chemical shifts arefrom references [45, 46].the pores increases. The nature of the pore surface is important to know as it will affect the extentof adsorption and absorption properties.T1 (CSi(OSi)(OH)2) T2 (CSi(OSi)2OH) T3 (CSi(OSi)3)Before NCC removal 0.06 0.55 0.39After NCC removal 0.02 0.37 0.61Table 2.1: Relative intensities of the peaks seen in the 29Si spectra of Et-CNMO in Figure 2.13.Relative intensities errors are about ?0.03.2.6 Conclusions13C NMR spectroscopy studies enabled definitive confirmation of complete NCC template removal.29Si spectra enabled the surface groups to be identified and how they are changed by the removal ofNCC via sulphuric acid. This process is shown to promote condensation of Si-OH groups. Also, theorganosilica is unharmed by this process. These studies have helped to direct synthesis pathwaysfor the CNMS and CNMO films.152.6. Conclusions14013012011010090807060  (ppm)Relative IntensityQQQFigure 2.12: The CP/MAS 29Si spectra of two CNMS film samples. Given that cross polarizationis used, the differences in peak intensities are due to a combination of abundance and proximityto a proton reservoir. This may be affected by pore size, which are slightly different in these twosamples due to batch-to-batch variation.90807060504030  (ppm)Relative IntensityTTTFigure 2.13: The CP/MAS 29Si spectra of a Et-CNMO film samples. The spectrum prior toNCC removal is on the bottom, and after is on the top. No silica peaks are seen, indicating thechemical composition remains pure organosilica. Fits to three Gaussian peaks for the three differentcomponents are shown and explained in the text.16Chapter 3Characterization of Film Pores withNitrogen Adsorption and NMRCryoporometryThis chapter focusses on measuring pore diameters by using interfacial phenomena, which are effectscaused by interfaces between the solid, liquid, and/or vapour phases. In porous materials, the twomost useful effects for determining pore geometry are capillary condensation and the depressionof the freezing point of confined fluids. These are used by nitrogen sorption and cryoporometryrespectively to determine Pore Size Distribution (PSD) and, to some extent, the pore geometry.In this chapter, some general equations of interfacial phenomena are presented, followed by thespecific theory for these two methods and experimental results.First, some definitions are necessary. Adsorption is the when molecules from a gas or a liquid(the adsorptive) bind to the surface of a solid (the adsorbent) to become adsorbates. Desorption isthe opposite process; collectively adsorption and desorption are called sorption. Absorption is thefilling of pores by a liquid or a gas, called an absorbate.3.1 Relevant Theory of Interfacial Phenomena3.1.1 Adsorption on a surface: BET theoryAt constant temperature, a vapour will condense to a liquid phase at the condensation pressure P0.If a surface is present, then at P < P0, it will adsorb some vapour molecules due to intermolecularforces such as Van der Waals and induced dipole [6]. Recording the quantity adsorbed whilechanging P/P0 gives a sorption isotherm. Nitrogen, being ubiquitous and relatively inert, is mostcommonly used.Adsorption is a complicated process, so has many theoretical models. While the most accu-rate are based on Density Functional Theory (DFT) calculations, the most commonly used is theBrunauer, Emmett, and Teller (BET) theory [47]. It is simple and can accurately describe most ofthe features of typical adsorption isotherms. In this model, molecules in the first adsorbed layer?the monolayer?have a heat of adsorption Q0. In subsequent layers, the heat of adsorption is Q.173.1. Relevant Theory of Interfacial PhenomenaThe isotherm follows [6, 22]nnm=cx(1? x)(1 + (c? 1)x), with x =PP0and c = e(Q?Q0)/RT , (3.1)where n is the total gas adsorbed, nm is the amount of gas adsorbed in the monolayer, and R isthe gas constant.In porous materials, adsorption is used to study the pore sizes and geometry. The pressure Pis slowly increased and the weight of the sample and adsorbate measured. BET theory can givethe pore surface area and volume from the isotherm if a mean size for an adsorbed molecule isassumed [6, 22, 48]. However, after monolayer coverage, more complicated processes like capillarycondensation may occur. To understand this, the behaviour of liquids on solids must be presentedfirst.3.1.2 Wetting: the Young-Dupre? equationThe derivation below uses the definition of surface tension as a Force per Length, where the forceacts perpendicular to the length. If a circle is drawn on the surface of an interface, the surfacetension force on that circle is tangential to the interface. With this in mind, consider a drop ofliquid on a solid surface in an atmosphere of its own vapour at equilibrium. The drop makes acontact angle ? with the solid: ? = 180 ? implies the drop beads up into a sphere, ? = 0 ? is whenthe drop completely spreads out to wet the surface.Assume that 90 ? > ? > 0 ?, so that the drop spreads out (wets). By definition, the solid-vapoursurface tension ?sv encourages the spread of the drop (to reduce the interface between the solid andvapour), and the solid-liquid surface tension ?sl encourages it to bead (to increase the interfacebetween the liquid and solid). The force per length from these two surface tensions lie in the planeof the surface. The liquid-vapour interface has surface tension ?lv, and the components of theforce per length lying in the plane of the other two surface tensions?that is, in the plane of thesurface?are ?lv cos?. Equating these components leads to the Young-Dupre? equation, [6]?sv = ?sl + ?lv cos?. (3.2)In essence, this equation describes the curvature of a liquid on a solid. The next section describesthe thermodynamic effects this curvature causes.3.1.3 Pressure drop across interfaces: the Young-Laplace equationIn an isolated liquid drop, the higher the surface tension ?lv, the more the interface ?squeezes? theliquid, increasing its pressure. So there is a pressure drop ?P across the interface, described bythe Young-Laplace equation. By considering the amount of work it takes to expand the interface,183.1. Relevant Theory of Interfacial Phenomenaone derives [6, 22]?P = ?lv(1r1+1r2), (3.3)where r1 and r2 are the radii of curvature where ?P is measured. For a hemispherical interface,r = r1 = r2 and?P =2?lvr. (3.4)Due to the r?1 dependence, the pressure difference ?P is greater at small radii. These forms ofthe Young-Laplace equation above describe how the pressure inside an isolated drop of liquid ishigher than the surrounding vapour: ?P > 0. Here, the curvature is always concave.This pressure drop across a curved interface is also the source of the capillary effect. Imagine asmall glass cylindrical capillary of radius r inserted straight into a liquid. A meniscus forms, makingan angle ? with the capillary surface. If 90 ? > ? > 0 ?, the meniscus is concave and the pressureinside the fluid is higher?the same situation as an isolated drop. However, if 180 ? > ? > 90 ?, themeniscus is convex and the pressure inside the fluid is lower. Consequently, the meniscus sits abovethe bulk fluid surface in the first case, and below it in the second. The pressure drop is related tothe curvature of the meniscus (which for a cylindrical capillary is hemispherical) via [6]?P =2?lv cos?r. (3.5)3.1.4 Capillary condensation: the Kelvin equationThe Young-Laplace equation demonstrates how an interface can change the pressure inside a dropof liquid. This explains condensation and evaporation: because of the interface, the pressure Pat which liquid condenses or evaporates will be less than the vapour saturation pressure P0. TheKelvin equation gives P/P0 as a function of the liquid and gas densities ?l and ?v, surface tension?lv, meniscus contact angle, ? and mean radius of curvature r?: [6, 22]PP0= exp(?2?lv cos?r?(?l ? ?v)). (3.6)Capillary condensation is when vapour condenses in pores. In this case, the following assumptionsare made [6, 49]. First, ? is set to 0 ? (complete wetting). This is because at least a monolayer isassumed to adsorb onto the pore walls before capillary condensation, so the fluid has wet the poresurface. Also, the pores are assumed to be cylindrical, so r? = r is just the radius. Finally, thevapour density ?v is usually small compared to ?l and is omitted. So the Kelvin equation isPP0= exp(?2?lvr?l). (3.7)193.1. Relevant Theory of Interfacial PhenomenaCapillary condensation is the central idea behind calculation of Barrett, Joyner, and Holenda(BJH) PSD from sorption isotherms [50]. The isotherm adsorption branch is recorded by slowlyincreasing P/P0, and the desorption branch is recorded by slowly decreasing P/P0. There is usuallya hysteresis between the two branches.3.1.5 Freezing point depression: the Gibbs-Thomson equationThe Kelvin equation and the Gibbs-Thomson equation both describe thermodynamic effects ofconfinement, but the Kelvin equation is the constant temperature case, and the Gibbs-Thomsonequation is the constant pressure case [49]. Almost all liquids have their freezing points depressedwhen confined, as in a small droplet inside a cloud or when absorbed in a mesopore. Exceptionswhere the freezing point increase, like activated carbon, are discussed in reference [49].Cryoporometry (also called thermoporometry) uses this effect to study the pores of a material bymonitoring the freezing point depression of a liquid absorbate. NMR cryoporometry observes withan NMR spectrometer, but other techniques are used such as calorimetry, positron annihilation,and neutron scattering [51?53]. Each method can somehow distinguish between the frozen andliquid phases.There are many ways to understand freezing point depression. One microscopic point of viewis that the average cohesive energy per molecule of a solid decreases as its volume decreases. [54].Essentially, the surface effects begin to dominate. Also, smaller pores are closer to one dimensionthan larger pores, and one dimensional liquids do not freeze [6]. Equation 3.4, which described thepressure drop across a hemisphere of liquid in vapour, also applies to a solid crystal in liquid. Thiscombined with Clausius-Clapeyron relation for a curved interface determines the freezing pointdepression ?T as a function of ?P [55]. However, these derivations and others make assumptionson pore geometry (see, for example, references [56, 57]). Christenson?s derivation is more general,relying only on thermodynamical arguments [57]. Since this is a crucial equation for cryoporometry,the derivation is repeated here.Consider the difference in Gibbs free energy ?G between a solid and liquid in a pore. Therewill be a contribution from ?Gfus, which is the molar free energy of fusion (dependent on the porevolume V and the molar volume vm), and a contribution from ?Gwall, the difference in interactionenergy with the pore wall. This gives?G = ?Gwall + ?GfusV/vm = A (?l,wall ? ?s,wall) + ?GfusV/vm, (3.8)with A the area of the pore wall and ?l,wall ? ?s,wall the difference in the liquid-wall and solid-wallinterfacial energy. As for the energy of fusion, the difference in free energy between temperaturesT1 and T2 for two phases is ?G(T2) = ?G(T1)? (T2?T1)?S, if the entropy difference ?S betweenthe two phases is independent of temperature [58]. Letting T1 = T be the melting temperature ofthe liquid in the pore, ?G(T1) = 0. And by letting T2 = T0, the bulk melting/freezing temperature,203.1. Relevant Theory of Interfacial PhenomenaV?Gfus becomes?Gfus = ?(T0 ? T )V?S =?(T0 ? T )?HfusVT0vm, (3.9)with ?Hfus as the molar enthalpy of fusion. The differences in heat capacity and molar volumebetween the solid and liquid phase are ignored. In passing, it is interesting that if the differencein molar volume was important, then water, which expands when frozen, would behave differentlythan other liquids. Returning to the derivation, at equilibrium, ?G = 0, and setting ?T = T0?T ,this leads toA (?l,wall ? ?s,wall) =?T?HfusVT0vm. (3.10)Experiments show a thin film of liquid remains between the solid and the pore surface. So,(?l,wall ? ?s,wall) then describes the difference in the interfacial energy of a pore full of liquid,and a pore full of frozen liquid surrounded by a thin, unfreezing film. This difference is just thesolid-liquid surface tension, ?sl. The final form is?T =A(r)V (r)T0?slvm?Hfus. (3.11)The explicit dependence of the surface area A(r) and pore volume V (r) on the pore radius r hasbeen included. In three dimensions, A(r)/V (r) ? r?1, which links ?T to the pore diameter, wherethe proportionality constant depends on pore geometry. Equation 3.11 also shows that ?T dependson the properties of the absorbent and the absorbent-absorbate interaction energy. Lumping theseconstants together in k, the cryoporometry constant, gives?T =k2r, (3.12)k is generally calibrated for a particular absorbate and absorbent using samples with regular cylin-drical pores. This assumes that the pore shape in the test sample is cylindrical, which is usually agood approximation when they are highly interconnected.Equation 3.12 is generally modified slightly before use. Neutron scattering experiments, NMRdiffusion measurements, and NMR cryoporometry have all confirmed that a small layer of the liquidnext to the pore wall does not ever freeze [49, 51, 57]. The thickness of this layer , is typically lessthan 1 nm and does not vary greatly across absorbents. With this in mind, the final cryoporometryequation is written?T =k2r ? 2=kw ? 2, (3.13)where w = 2r.Once k and  are known for a particular absorbate and material, the change in the quantity of213.2. Hysteresis in Sorption Isotherms and Cryoporometry Isobarsfrozen absorbate as a function of temperature at constant pressure yields the PSDs. In this work,the results of these measurements are referred to as cryoporometry isobars. The isobar freezingbranch is recorded by slowly lowering the temperature (?T increasing), and the melting branchis recorded by increasing the temperature (?T decreasing). As with nitrogen adsorption anddesorption branches, there is in general a hysteresis between them, so that the average temperatureat which absorbed liquids freeze (Tf ) is different than the average temperature at which they melt(Tm).3.2 Hysteresis in Sorption Isotherms and Cryoporometry IsobarsBoth sorption isotherms and cryoporometry isobars show hysteresis between their two branches.Due to the relationship between the Kelvin equation and the Gibbs-Thomson equation mentioned inSection 3.1.5, decreasing pressure and increasing temperature are conjugate, so the isobar freezingbranch is similar to the isotherm desorption branch, and the isobar melting branch is similar tothe isotherm adsorption branch [59]. Figure 3.1 exemplifies this, where the middle metastablestates in both the sorption isotherm (left) and cryoporometry isobar (right) are the cause of thehysteresis. This behaviour is not immediately obvious from the equations describing gas sorptionor cryoporometry, but for the case of an isolated cylindrical pore, it is not hard to understand.For gas sorption, pore condensation is the primary mechanism during adsorption, but formationof a meniscus and evaporation is the primary mechanism during desorption. For a pore to condense,there is an energy barrier it must overcome [60]. The situation is less intuitive in cryoporometry.Freezing takes place from the outside in as ice fronts slowly intrude into the sample. When theice front reaches one end of a cylindrical pore, there is no energy barrier?the analogue case togas desorption. Melting occurs from the outside of the pore surface in and, as explained below,depends on the curvature of the ice crystal. This curvature increases as its side decreases, so againthere is an energy barrier to overcome [51, 61, 62].While the above example shows a simple case, interconnected materials are more complicated:here are some confounding factors. Firstly, complex network effects such as percolation (a criticalphenomenon with a phase transition) can arise. Also, ?ink bottle? pores cause bottlenecks and areanother source of hysteresis [6, 63, 64]. In gas adsorption, the presence of a mobile liquid adsorbatelayer is difficult to model [6]. Finally, in cryoporometry, a cylindrical pore with many brancheswill melt faster than a pore with fewer branches, even though they have the same diameter [65].Despite different physical causes, there is evidence that in highly interconnected materials?suchas the CNM films used in this work?the gas sorption and cryoporometry hysteresis is very similar[59, 66]. Since their interpretation is a relatively new endeavour, cryoporometry hysteresis curvesare not discussed further, and references [51, 61, 65?68] provide more details. Still, the gas sorptionhysteresis is a useful measurement.Precise modelling hysteresis in porous material for gas sorption and cryoporometry use com-puter simulations. However, despite this complexity, the gas sorption community uses an empirical223.2. Hysteresis in Sorption Isotherms and Cryoporometry IsobarsFigure 3.1: Similar causes of hysteresis in sorption isotherms (left) and cryoporometry isobars(right). Increasing pressure and decreasing temperature are conjugate, leading to a similaritybetween the adsorption isotherm and melting isobar, and between the desorption isotherm andfreezing isobar. In both cases, metastable intermediate cases are the cause of the hysteresis. Notethe presence of the unfrozen layer of thickness next to the pore wall.?hysteresis shape bank? [48, 66], giving some insight on the general geometry of the pores. Thereis also an ?isotherm shape bank? for describing sorption isotherms. Figure 3.2 shows the typicalshapes which can be understood quite well in the context of BET theory. (The ?shape bank? forcryoporometry is still under development [66].)Type I isotherms are seen in microporous materials where the adsorption is limited by the porevolume. Non-porous multilayer sorption gives Type II isotherms, where ?B? indicates monolayercoverage. Type III are quite rare and are caused by adsorbate-adsorbate interactions. Type IV areindicative of mesoporous materials, with the hysteresis between the adsorption branch (rightmostcurve) and the desorption branch (leftmost curve) caused by capillary condensation. This isothermis discussed in more detail below. Type V is caused by weak adsorbate-adsorbent interactions andis relatively rare, as is the type VI isotherm, which represents stepwise multilayer adsorption [6, 48].The type IV isotherm are of interest in this work, since they relate to mesoporous materials. Itspossible hysteresis curves (right of Figure 3.2) have empirically been shown to reveal some geometricinformation [6, 48]. Materials with H1 hysteresis curves generally have a narrow distribution ofpore sizes, such as compacted glass beads or isolated pores. In systems with a slightly broaderdistribution or greater connectivity, such as controlled pore glass or the films considered in thiswork, type H2 is often seen. Types H3 and H4 are associated with slit-like pores.233.3. Measuring Pore Size DistributionsFigure 3.2: (Left) The six different types of sorption isotherms. Point B indicates BET monolayercoverage. (Right) The four different types of hysteresis associated with isotherm IV (sorption inmesoporous materials). The hysteresis curves that can arise at very low pressures are not shown.Adapted with permission from [48]. Copyright 1985 International Union of Pure and AppliedChemistry.3.3 Measuring Pore Size DistributionsAs mentioned above in Section 3.1.1, BET theory can be used to obtain the surface area andvolume from gas sorption. By assuming cylindrical pores, the PSD can also be found. Obtainingthe pore PSD sorption isotherms and cryoporometry isobars is now discussed, with an emphasis oncryoporometry.3.3.1 Nitrogen sorptionThe most common method of obtaining the PSD from gas sorption is using the Barrett, Joyner,and Holenda (BJH) distributions [50]. BJH distributions come from applying a modified Kelvinequation to cylindrical pores to model pore condensation. Pores are assumed to have either anadsorbed multilayer t thick that depends on P/P0, or be completely condensed (full of liquid).Therefore, a modified Kelvin equation is used, [60]PP0= exp(?2?lv(r ? t)?L), (3.14)243.4. Experimental Detailsso that the presence of the multilayer simply reduces the effective pore radius by t. If the poredistribution is known, then for any given P/P0, BJH theory can calculate the total volume adsorbedfrom both condensed pores and the multilayers. The inverse calculation of determining a poredistribution from a sorption isotherm is also possible. [6].3.3.2 NMR cryoporometryThe NMR cryoporometry experiment simply measures the intensity of the signal from the liquidportion of the absorbate. The frozen phase has a much shorter T2 time than the liquid phase,allowing a spin echo (used here) or CPMG echo train to isolate the liquid phase. if A is a spin-echoamplitude, then the PSD dV/dw (where w = 2r) is given bydVdw????w = k/?T + 2=dA(T )dT(3.15)where as before T0 is the freezing temperature of the bulk liquid and ?T is the freezing pointdepression (taken as positive). Because of supercooling effects, this equation is most reliably appliedto the melting branch (melting isobar), where all the absorbate is frozen initially and slowly meltsas the temperature is raised. Figure 3.3 displays a typical melting branch for water in Et-CNMO.At low temperatures, all the absorbate is frozen and the echo intensity is zero. As the temperatureincreases, the absorbed frozen water melts in progressively larger pores (here between ?13 ?C and?7 ?C), and this is the portion used to calculate a PSD. Surface water starts melting at 0 ?C.Before this PSD can be found, the two constants,  (the thickness of the unfrozen layer) andk (the cryoporometry constant) require calibration. In this work, values of  = 0.4 ? 0.4 nm andk = 50? 3 nm K are used, which are the averages from a number of studies of water in cylindricalsilica pores tabulated by Petrov et al. [51]. Although ethane-organosilica differs from silica, the kvalue should be similarCalibration of these values for ethane-organosilica was attempted on an ethane-organosilicaMCM41-like material, described by Burleigh et al. [69]. The nitrogen sorption PSD was narrowand gave an average pore size of about 3.5 nm. Using the value of  above, k was tuned untilthe cryoporometry PSD gave the same average value, yielding k ? 75 nm K. However, due to the1/?T dependence on the pore diameter in equation 3.13, small errors in k are amplified at smaller?T (larger pores), and a significant offset was seen between the sorption and cryoporometry PSDsas the pore diameter increased. Multiple reference samples with known pore sizes spanning 3 -10 nm would be required to obtain an accurate result, and could yield a value for  specific toethane-organosilica as well.3.4 Experimental DetailsSamples E3 and E4 were flakes of Et-CNMO and sample S1 was flakes of CNMS (with NCC removedvia pyrolysis). NMR cryoporometry and nitrogen sorption measurements were performed on each253.4. Experimental Details35 30 25 20 15 10 5 0 ( )Relative echo intensity14 12 10 8 6 4 2 0Figure 3.3: A typical cryoporometry melting curve for water in Et-CNMO. The echo intensityrelates the quantity of liquid water. The water inside the pores melts in the range ?8 ?C to ?5 ?C(inset); from this, the PSD is calculated. Surface water doesn?t melt until 0 ?C, although surfacetemperature gradients spread out this transition by about 2 ?C. The decreases in intensity after themelting surface water is seen before and after corrections are applied and is likely a result of rapidprobe tuning change.sample. Kevin Shopsowitz and Andrea Terpstra supplied the nitrogen sorption measurements usingan automated Micrometrics ASAP 2020 and 2010 physisorption analyzer. About 50 - 100 mg ofthe samples was used. Prior to capturing the adsorption data, the samples were heated to 120 ?Cunder vacuum for about an hour to dry them. Immediately following adsorption, sample E3 wasput into DI water. Samples E4 and S1 were later dried again under vacuum before immersing intoDI water. Residual sulfuric acid in sample E3 and E4 caused the DI water to become slightly acidic(to about pH 4) after soaking overnight, causing ?T to decrease roughly 10 ?C. Consequently, thesamples were washed by decanting with ca. 60 mL of water five times after soaking overnight.The author performed the NMR cryoporometry experiments on the Varian spectrometer de-tailed in Section 2.5 with a double resonance probe. For each sample, about 5 mg of film flakeswere loaded into a small 1.5 cm long NMR tube and the end sealed with parafilm. The sampleswere significantly overfilled and contained 100-300 ?L total water absorbed in their pores and onthe surface. The cryoporometry experiment used 20 transients at every temperature with a recycledelay of 2 s and a 90 ? pulse width of 11 ?s. The echo time was 4 ms. The temperature wasallowed to stabilize for 2 minutes before acquisition. For samples E3 and E4, the set temperaturewas decreased from 3 ?C to -35 ?C and then increased back up in steps of 0.5 ?C in each direction.For sample S1, it was necessary to go to -45 ?C until the absorbed water in its small pores froze.Temperature calibration was achieved by repeating the experiment with a sample of neatmethanol. The average frequency difference between the CH3 and OH groups is converted into263.5. Results and Discussionan average methanol temperature using the expression by Ammann et al. [70]. Calculated temper-atures Tcalc were found for temperature settings Tset from ?35 ?C to 0 ?C and fit to a 4th-orderpolynomial (Figure 3.4). The hysteresis is from the thermal mass of the probe. Since the ex-periment was repeated in the same way each time, the error from this hysteresis is small. Thetemperature uncertainty is taken to be ?1 ?C, which is half the width of the surface water portionof a melting curve, as seen in Figure 3.3.40 35 30 25 20 15 10 5 0 ( )5.04.54.03.53.02.52.01.5()CoolingWarmingFigure 3.4: The difference between set sample temperature and calculated sample temperature forcooling and warming. Set temperature is calculated using methanol. A hysteresis is present due tothe thermal mass of the probe. The fits are 4th order polynomials.In addition to freezing and melting, the intensity of the signal was affected by temperature-dependent probe tuning and the Curie Law. Initially, the probe was retuned every 7 ?C, but inlater experiments it was only tuned once at 0 ?C and the data adjusted afterwards. A line was fitto the portion of the melting curve between the pore water and surface water melting, such as theregion between ?4 ?C and ?1 ?C in Figure 3.3. Intensities here should remain constant until thesurface water begins to melt, so any nonzero slope from the fit line was removed. The validity ofthis approach is approach is proven below: the PSDs deviate insignificantly from each other.3.5 Results and DiscussionNitrogen sorption isotherms for all samples are displayed in Figure 3.5. S1 displays the shape of atype I isotherm, which is evidence for smaller, micropore-sized materials. This is typical of CNMSafter removal of NCC via pyrolysis [35]. However, the presence of a type H2 hysteresis does indicatesome mesporosity, probably from the high level of interconnectivity. Samples E3 and E4 have type273.5. Results and DiscussionIV isotherms with H2 hysteresis, also typical of Et-CNMO [41]. Using the BJH method, theseisotherms give PSDs for the samples seen in Figure 3.6. The desorption branch is used as it yieldsa more narrow PSD. The metastability present in the adsorption branch prior to condensationbroadens the PSD, and is avoided in the desorption branch. As for cryoporometry, the meltingbranch is used to obtain the PSD.Sample E3Sample E40.0 0.2 0.4 0.6 0.8 1.0Relative pressureSample S1AdsorptionDesorptionAmount adsorbedFigure 3.5: Nitrogen adsorption and desorption isotherms. S1 demonstrates a type I isotherm,and E3 and E4 demonstrate a type IV isotherm. All have type H2 hysteresis.Using the values stated above for k and , the equation relating the freezing point depressionto the pore size for these samples is written asr =50? 3 K nm?T ? 1K+ (0.8? 0.8 nm). (3.16)To determine the error on r, ?r, the standard error propagation equation is applied.?r =?(?z?b?b)2 + (?z?a?a)2 (3.17)=?((50 K nm)(1 K)?T 2)2 + (3 K nm?T)2, (3.18)which gives ?r = 0.6 nm when ?T =10 ?C, and ?r = 1.3 nm when ?T =10 ?C. This illustrateshow errors are amplified for large pores. Since sample E3 has the smallest ?T at about -6 ?C, itwill have the largest error, with ?r ? ?1 nm. This maximum is taken as the error on r for all themeasurements.283.6. ConclusionsFigure 3.6 shows the PSDs for the three samples calculated from equation 3.16. They closelymatch the desorption PSDs, though are in general more narrow. This is likely because a smallersample amount was used leading to less variability in the pore diameter. The nitrogen sorptionis limited to pores > 2 nm, whereas cryoporometry is not; this allows a more complete PSD forsample S1 to be found. Different batches of NCC templates were used to make samples E3 and E4,and variation in the NCC crystallites is reflected in their different PSDs.Sample E3Sample E40 2 4 6 8 10 12 14Pore diameter (nm)Sample S1 Cryoporometry (melting)N  (desorption)Relative dV/dxFigure 3.6: The pore size distributions from the nitrogen desorption isotherm and the cryoporom-etry melting isobar. Error on PSDs are about ?1 nm. The cryoporometry PSD for sample S1 hasbeen smoothed.Another sample (E5) was used to test the validity of correcting for temperature-dependenttuning and Curie Law effects post-acquisition (as described in Section 3.4) and the possibility offreezing-induced damage to the pores. The PSD from the melting branch was collected three times:the first by re-tuning the probe every 7 ?C, and the next two by post-acquisition corrections. Figure3.7 shows the results: the PSD obtained by retuning the probe differs slightly but is well withinerror. The other two PSDs essentially overlap, indicating no damage to the pore structure by thecryoporometry process. CNMS (silica) films were not tested for damage, so it is possible that theirlower flexibility would make them more susceptible to damage.3.6 ConclusionsNMR cryoporometry is seen to have several advantages over nitrogen sorption, including the com-parable accuracy and improved precision in PSD determination, and the ability to use only 10%of the sample volume. Also, the freezing process does not damage the pores. However, since hys-293.6. Conclusions2 4 6 8 10 12 14 16 18 20Pore size, x (nm)dV/dx (Relative Intensity)Sample E5Run 1, tuning adjustedRun 2, data adjustedRun 3, data adjustedFigure 3.7: The results of repeated cryoporometry experiments on the same sample. No significantchange is seen, showing that freezing the absorbed water does not damage the pores for Et-CNMOfilms. Also, the validity of post-acquisition correction for temperature-dependent probe tuning andCurie Law effects is proven. There is no significant deviation between this and when the probe wasretuned every 7 ?C.teresis analysis of cryoporometry is not fully developed, these two techniques are complimentary.Et-CNMO pores were shown to be larger than CNMS pores. At present, cryoporometry calibrationhas not been for ethane-organosilica, so literature values for silica were used instead with littleadded error.30Chapter 4PFG Diffusion Studies of AbsorbedWater4.1 Theory of PFG Diffusion Measurements4.1.1 Bulk diffusionIn a bulk fluid, molecules are thermally excited and undergo Brownian motion. The moleculescollide with each other, so the result is a random walk. Analysis of a one dimensional random walkis one way to reach the well-known result, [71]?x2? = 2D0t, (4.1)where ?x2? is the mean-square displacement, D0 is the self (or bulk) diffusion coefficient, and t isthe time. For water at 20 ?C, D0 = 1.92? 10?9m2s?1 [72]. Momentum in orthogonal directions isindependent, so by considering the two or three dimensional displacements, equation 4.1 becomes?r2? = ?x2?+ ?y2? = 4D0t for 2d, and (4.2)?r2? = ?x2?+ ?y2?+ ?z2? = 6D0t for 3d. (4.3)Most PFG diffusion measurements can only detect displacement in one dimension, so equation 4.1applies here.The preceding is a microscopic viewpoint of diffusion. Macroscopically, the diffusion equationand its corresponding Green Function, or propagator (a Gaussian) applied. While not used in thiswork, it is useful for PFG diffusion in some periodic porous materials. If mean-square displacementwithin a pore is shorter than between pores, diffraction-like effects can arise. This happens becausethe echo amplitude and the average propagator are Fourier conjugates [73, 74]. These effects arenot seen in these experiments since the pores are too small.4.1.2 The PFG pulse sequenceAll NMR diffusion measurements require a spatially-dependent field gradient such that the magneticfield at r is B = B0 + r ? G. The most simple experiments involve a constant field gradient asopposed to a Pulsed Field Gradient (PFG). If the field gradient is constant, the spins develop aspatially-dependent phase following the 90 ? pulse. During the mixing period, the molecules (and314.1. Theory of PFG Diffusion Measurementstheir attached spins) diffuse from their initial position. Because of this, the 180 ? pulse is notcompletely refocussed, reducing the echo intensity. The diffusion coefficient is found by observingthe echo intensity as a function of echo time. This is the basic idea, and the actual sequence used(shown in Figure 4.1) has only three additional complications.Figure 4.1: The PFG diffusion Longitudinal Encode-Decode (LED) pulse sequence. The gradientis on the bottom. It uses a stimulated echo with the magnetization stored along the z-axis. Duringthe mixing period, two dephasing gradient pulses destroy remaining transverse magnetization.Firstly, the gradient (shown in red on the bottom) is not constant: G = G(t). Usually thegradient pulse length, called ?, is short enough that the spins are assumed to be stationary for itsduration. Hence: Pulsed Field Gradient diffusion measurements. Secondly, there are two smallergradient pulses that destroy (dephase) any transverse magnetization. These are pointless withoutthe third modification: the stimulated echo [75]. Like a spin echo, the first 90 ? pulse followed bya gradient pulse produces a fan of magnetization vectors around the z axis. The second 90 ? pulserotates this fan into a plane parallel to the z axis. Consequently, the magnetization relaxes underthe longer T1 process instead of the T2 process. The last 90 ? pulse rotates the magnetizationback into the xy plane, the final gradient pulse decodes the phases, and an echo is observed. Thissequence is a Longitudinal Encode-Decode (LED) diffusion sequence. They were first introduced byGibbs et al., but the one used here is a modification of Altieri et al.?s [76, 77]. The only differenceis that the water suppression portions of the sequence are omitted, since diffusion of water is beingobserved.With the motivation in mind, what is the expression that gives the diffusion coefficient fromthe echo amplitude? This is straightforward, once the situation is considered as diffusion of phaseinstead of particles. By solving the diffusion equation for phase, Stejskal and Tanner derived thegeneral expression for the echo amplitude [78]. The exact form depends on the shape of the gradientpulse G(t). For the rectangular pulses used here the echo amplitude A isA(?, G) = A(0) exp(?(??G)2(?? ?/3)D)(4.4)A(?, k) = A(0) exp(?k2(?? ?/3)D). (4.5)324.1. Theory of PFG Diffusion MeasurementsHere, ? is the gyromagnetic ratio, ? is the gradient pulse width, G is the gradient strength in T m?1,and D is the diffusion coefficient in m2 s?1. These values are indicated on the pulse sequence inFigure 4.1. k can be thought of as a wavenumber characterizing the phase accumulated by a spinas it travels in the direction of the gradient, and isk = ??G. (4.6)This is the origin of the ?k-space? concept discussed by Callaghan [73, 79].Figure 4.2: The RMS distance xRMS as a function of the root of diffusion time?? for bulk liquid,liquid in interconnected pores, and liquid in isolated pores. Region A is short diffusion times, wherefew particles feel the pore walls. Region B is intermediate times, where many particles feel the wallbut do not in general travel between pores in the interconnected case. Region C is long times, wheneach particle feels the pore walls many times. For details, see text.4.1.3 Restricted diffusionAn absorbed liquid in a porous material will have the diffusion of its molecules impeded by thepore walls. How this changes the echo amplitude depends on diffusion time ? and the poreinterconnectivity, so D = D(?). Stallmach et al. provide an overview of this behaviour, whichis summarized here [80]. Two length scales must be defined. The first is the average pore width, R?,and the second is the RMS distance a particle diffuses in the bulk liquid in time ?, xRMS =?2D0?.The relationship between these defines the behaviour of the diffusion coefficient D(?).Figure 4.2 summarizes the possible situations for bulk liquid, liquid in interconnected pores,and liquid in isolated pores. For bulk liquid, xRMS increases linearly with??. This is also thecase for liquid inside any pore if xRMS  R? (region A). Only a few molecules hit the pore walls,so diffusion measurements give close to the bulk diffusion coefficient, D0. In the intermediate case(region B), a significant number of molecules have hit the pore walls, but few move between pores:xRMS . R?. Now, interconnected pores and isolated pores deviate from the bulk case. In this334.2. Experimental Detailsregime, the geometry of the pores can be probed by calculating a surface area to volume ratio[81, 82]. Finally, at long ??s, when xRMS  R?, all the molecules inside isolated pores will havecollided with the walls many times, and xRMS cannot increase at all. An example is diffusion ofoil inside emulsion droplets [83, 84]. Inside interconnected pores, all the molecules have collidedwith the walls about the same amount, and the microscopic pore details are simply averaged over.However, they are free to move from pore to pore so this results in an effective diffusion coefficientD(?) < D0, and xRMS increases linearly.4.1.4 Relaxation at pore surfacesBoth the T1 and T2 relaxation rates of absorbed liquids increase inside porous media. The spinsin the layer of water close to the pore surfaces are affected the greatest. Surface intermolecularforces and the presence of paramagnetic impurities are typically implicated as the source of en-hanced relaxation [85]. T2 processes are also increased from the inhomogenous magnetic fieldsdue to susceptibility mismatches between the liquid and the porous material [86]. Yet because ofdiffusion, this layer of water next to the pore surface is constantly replaced, so the relaxation ratechanges with time and pore inteconnectivity. Brownstein and Tarr developed a method, calledNMR relaxomatery, of relating this to the surface area to volume ratio of a pore [87].In the small pores of the CNM films, at each diffusion time measured all molecules can beassumed to have spent an equal amount of time on the surface. Thus, every spin has relaxed thesame amount. With this in mind, there are two ways a PFG diffusion experiment can be run: withthe ? constant and the gradient strength G varying, or vice versa. The former approach is used forthe data here; it has the advantage that when fitting equation 4.5, each datapoint will be affectedby relaxation in the same way, since ? is the same. For this reason, relaxation processes are notconsidered in the analysis of the data.4.2 Experimental DetailsSamples E1 and E2 were films of CNMO (ethane-organosilica) and sample S1 was films of CNMS(silica). Prior to use, the films were soaked in DI water overnight to fill the pores completely. Thediffusion of absorbed water was measured in two different directions for each sample: the longitudi-nal (displacement parallel to the helical axis and surface normal) and the transverse (displacementperpendicular to the helical axis and surface normal). Figure 4.3 illustrates this. The probe gradi-ent is parallel to the B0 field, so the film flakes were mounted with their surface normals parallel(for longitudinal diffusion) and perpendicular (for transverse diffusion) to the field. Teflon mountswere built to hold the flakes in each position at the bottom of a full length 5 mm NMR tube.The probe was a Varian PFG high resolution probe (PN 01-901583-01) with a maximum gradientstrength of about 50 G cm?1.Initially, the signal from the surface water on the samples was going to be removed post-acquisition, but the effective diffusion coefficient of the absorbed water was too similar in magnitude.344.3. Results and DiscussionFigure 4.3: The transverse and longitudinal diffusion directions of absorbed liquid, as seen fromthe film surface (left), and the film pores (right; as in figure 2.8) The transverse diffusion in andout of the page will be restricted differently when a molecule is in layer A (perpendicular to poreaxis) or B (parallel to pore axis). The longitudinal diffusion does not depend on which layer themolecule is in. Photo of film flake is adapted with permission from reference [41]. Copyright 2012American Chemical SocietyInstead, following reference [88] the surface water was frozen, and the absorbed water remainedliquid due to its confinement (this is described in Chapter 3). So the signal was from the absorbedwater only. The bulk diffusion constants at temperatures below 0 ?C (supercooled water) wereobtained from Price et al. [72]. D0 decreases by roughly 0.2?10?9 m2s?1 every 5 ?C drop intemperature.4.3 Results and DiscussionThe longitudinal (Dlong??longD ) and transverse (Dtrans??transD ) diffusion coefficients were measuredfor samples E1, E2, and S1 at 25 different diffusion times (??s) arranged logarithmically from theminimum possible time of 11 ms to a maximum of 500 ms. For unrestricted diffusion or highlyinterconnected pores at long diffusion times, plotting the echo amplitude vs. k gives close to theGaussian shape described in equation 4.5.Figure 4.4 displays data from a typical experiment measuring Dlong on sample E1 at both? = 0.054 ms and ? = 0.364 ms. The fit curve for Dlong is shown, and was found using the leastsquares fitting routine from the SciPy python package. This minimized?i(A(?, ki)? yi)2 =?i(A(0) exp(?k2i (?? ?/3)Dlong)? yi)2. (4.7)The SNR of all the echos is the same so there is no error weighting factor above. From thefitting routine?s covariance matrix the standard deviation ?longD was calculated, and on the graph,Dlong ? ?longD is the region between the dotted lines. At longer diffusion times, more relaxation hasoccurred, so the data are noisier.Doing this mono-exponential fit across various diffusion times gives the results for Dlong and354.3. Results and Discussion50000 100000 150000 200000 250000=  (m )0.20.00.20.40.60.81.0Relative Echo Intensity = 0.054 ms = 0.364 msFigure 4.4: The echo intensities of water in sample E1 in the longitudinal direction as a functionof k and the fits to equation 4.5 for ? = 0.054 ms and ? = 0.364 ms. Dotted lines indicateDlong ? ?longD , which is larger at longer diffusion times due to relaxation.Dtrans and xlongRMS and xtransRMS in figures 4.5 and 4.6 for samples E1, E2, and S1. The transverseand longitudinal directions have been plotted. The most immediate pattern is that Dtrans > Dlong(or xtransRMS > xlongRMS) for all the samples. This means that on average, water molecules are lessimpeded when moving perpendicular to the helical axis than along it. Consider molecules diffusingin and out of the page (transverse direction) in Figure 4.3. At location A they move parallel to thepore axes, at location B they are moving perpendicular to them, which should be highly restrictivegiven that these pore diameters are ca. 5 nm. Since Dtrans ? D0/2, the relatively slow motionof the molecules at B must be made up by almost unimpeded motion in plane of A. This canbe confirmed by using an expression for the echo amplitude that takes into account the differentdiffusion coefficients in the transverse direction.If D? is the transverse diffusion coefficient at location A, and D? the transverse diffusioncoefficient at location B, then the echo amplitude is given by a two dimensional version of theexpression in reference [89]. This isA(?) = A(0)? pi/20exp(?k2(?? ?/3)(D? cos2(?) +D? sin2(?)))d?, (4.8)where ? is the angle of the pore axis: ? = 0 and pi/2 at locations B and A respectively. Sample E2is used to show that D? ? D0. A value is first chosen for D?: since Dlong is also perpendicularto the pores, D? = Dlong. Then, equation 4.8 is numerically evaluated at each k value, and theresulting echo amplitude fit to equation 4.5 to yield a single effective diffusion coefficient D, as was364.3. Results and Discussiondone in the experiment. At ? = 25, 50, and 100 ms, it was found that D? ? D0 in order to get theexpected value for D from Figure 4.5.Analysis of the RMS displacements in figure 4.6 shows that for most of the diffusion times,xRMS  R?, since the curves are mostly linear. This is expected, since at the smallest diffusiontime of 11 ms, unrestricted diffusion gives xRMS ? 6 ?m, which is much larger than the pore width(ca. 5 nm) or length (ca. 100 nm). The one exception is the xlongRMS for sample E1, which appearsto reach a constant value at larger diffusion times. This is due to most of the water moleculesreaching the edge of the film flake, so the RMS displacement cannot increase further.25 50 100 250 500Diffusion time,   (ms)0.00.10.20.30.40.50.6/E1 longE2 longS1 longE1 transE2 transS1 transFigure 4.5: The mono-exponential fit diffusion coefficients in the longitudinal (Dlong) and transverse(Dtrans) for samples E1, E2, and S1. The diffusion in the transverse direction is about 1.5-2x asmuch for all samples. Datapoints with ?D > 0.5 have been omitted.Figure 4.7 plots the residuals (data - fit, indicated by the colour) from the mono-exponentialfits to the echoes for all samples in the transverse and longitudinal directions. Ignoring the noise athigher diffusion times, in the transverse direction the residuals do not show any systematic error.This is consistent with the measured diffusion in the transverse direction being an average of thediffusion rate at locations A and B in Figure 4.3 and all locations in between. Individual D(?),where ? is the angle of the pore axis (0 ? at A and 90 ? at B), cannot be seen.In the longitudinal direction, slight deviations at high diffusion times can be seen. In sampleE1, the fits at lower values of k are less than the data, and higher values of k are higher thanthe data. In sample E2, the fits at high values of k are higher than the data. Again, the data374.4. Conclusions261014E1 longE1 trans26101418 E2 longE2 trans0.1 0.2 0.3 0.4 0.5 0.6 0.704812S1 longS1 transRMS displacement (m)sqrt(Diffusion time),  ( )Figure 4.6: Plots of mono-exponential fits for RMS displacement in the longitudinal (xlongRMS) andtransverse (xtransRMS) directions for samples E1, E2, and E3. The approximately linear increase showsthe microscopic details of the pores are averaged over. Datapoints with errors > 0.2 ?m have beenomitted.from sample S1 is too noisy for any patterns to be seen. The deviations in sample E1 and E2 arelikely due to the water molecules hitting the surface of the film flakes. Their thicknesses are about50 - 100 ?m, so with RMS displacements being 5 - 10 ?m at the higher diffusion times, 10% ofthe molecules will have hit the surface. This will cause the echo amplitude to become flatter andbroader from the ideal Gaussian, as is seen.4.4 ConclusionsFor both CNMO and CNMS, the effective diffusion rate in the transverse direction was about 2? asfast as in the longitudinal direction. The smaller pores of CNMS gave it an overall slower effectivediffusion. Using a simple model, it was shown that diffusion parallel to the pore axis is almostcompletely unimpeded. Finally, at long diffusion times and high gradient strengths, the presenceof the edge of the film flake can be seen.384.4. Conclusions 11 15 21 29 39 54 74102140193265364500 11 15 21 29 39 54 741021401932653645008 35 61 87 113140166192218245271 11 15 21 29 39 54 74102140193265364500 8 35 61 87 113140166192218245271Diffusion time  (s)k ?   (s m )Transverse LongitudinalSample E1Sample E2Sample E30.20 0.16 0.12 0.08 0.04 0.00 0.04 0.08 0.12 0.16 0.20Figure 4.7: The residuals (DATA ? FIT ) from the mono-exponential fit to the normalized echoamplitudes as a function of echo time ? and k (gradient strength). Sample S1?s data is the noisiestdue to its small pore size and no distinctions between longitudinal and transverse residuals areseen. However, both samples E1 and E2 show larger residuals in the longitudinal direction thanthe transverse direction, particularly at high ? and high k. This is likely due to the moleculesreaching the film surface barrier.39Chapter 5NMR of an Oriented Absorbed LiquidCrystal5.1 Overview of Liquid CrystalsA material is a liquid crystal if it has one or more intermediate phases with some degree of orderbetween its crystalline and isotropic phases. They are ubiquitous in consumer technology since mostdisplays?computer, cellphone, watches, or otherwise?are liquid crystal displays (LCDs). Otherapplications include privacy screens [90] and temperature-dependent sensors, as in this work. Interms of fundamental research, liquid crystals also provide easy access to critical phenomena. Textssuch as de Gennes [37] or Chandrasekhar [91] provide detailed discussions of this subject.Liquid crystals may be classified a number of different ways. Often, a two component sys-tem, such as surfactants in a solvent, will exhibit some ordering. As the concentration increases,the phase (such as micellular or lamellular) changes [92]. These are lyotropic, or concentration-dependent, liquid crystals. Alternatively, the phase of thermotropic liquid crystals depends ontemperature. The shape of the molecule is also important. The two most common are discotic(disc like) and calimitic (rod like) liquid crystals. Examples of these are discotic HAT6 (hexakis(n-hexyloxy)triphenylene) and the calimitic 8CB (4-cyano-4-octylbiphenyl) molecules in Figures 5.1aand 5.1b respectively. From their shape alone it is easy to see how they will pack differently. Therigid cores of the liquid crystal molecules are called mesogens.Finally, liquid crystals may be classified according to which phases they can adopt. In thebulk, 8CB has the following phase transitions: crystalline to smectic A at 21? C, smectic A tonematic at 35? C, and nematic to isotropic at 41?C; these are shown in Figure 5.2. The director nis the average direction of mesogen orientation, and n = ?n. The chiral nematic phase has alreadybeen introduced in Chapter 2 as the phase adopted by Nano-Crystalline Cellulose (NCC) when insolution. With this, the director rotates in space, tracing out a helix. The possible phases of aliquid crystal are determined by a number of factors, including shape and intermolecular forces.5.2 The Effect of External FieldsThe strong dielectric anisotropy of liquid crystals enables their use as adjustable polarizers inLCDs. Furthermore, many liquid crystal molecules, like 8CB and HAT6 in Figures 5.1b and 5.1a,have aromatic constituents, which gives causes a magnetic susceptibility anisotropy. This is weak405.2. The Effect of External Fields(a) (b)Figure 5.1: A triphenelyene-based discotic liquid crystal (hexakis(n-hexyloxy)triphenylene orHAT6) (left) and the calimitic liquid crystal used int his work (4-n-octyl-4-cyano-biphenyl or 8CB)(right).compared to the dielectric anisotropy and is of little importance in most applications, but must beconsidered in intermediate to high fields.The tensor ? describes the magnetic susceptibility?typically diamagnetic?of a bulk liquidcrystal sample. The susceptibility tensor for an individual molecule is ?molec.1 An order parameterS relates ? to ?molec, and therefore measurements of ? provide a way to determine the order of aliquid crystal. The volume dielectric tensor  is not used for this purpose since large local fieldsmake it difficult to relate this to the molecular dielectric tensor. In constrast, the local magneticfields from the diamagnetic effect are small and can usually be ignored.8CB, like many liquid crystals, shows uniaxial phases, so in the principle axis frame (with thez-axis along the director) ? is diagonal and has only two independent components:?xx = ??, ?yy = ??, and ?zz = ??. (5.1)For isotropic liquids, ? is a scalar, and for the isotropic phase of liquid crystals, ?iso is the averageof the above diagonal elements. The susceptibility anisotropy is defined as?? = ?? ? ??, (5.2)and essentially says how the director (or, if using ?molec, the molecule) will react to an external1Unfortunately, the units for ? are not standardized. In this work, volume susceptibilities in the SI system areused throughout. Therefore, the stated values in ppm are unitless, and to go to CGS units one must multiply by(2pi)?1.415.2. The Effect of External FieldsFigure 5.2: The phases of 8CB and the transition temperatures. Used from reference [93] withpermission. Copyright 2013 American Chemical Society.magnetic field. The tensor values of ? for 8CB are given below for a range of temperatures. Theyhave been converted from Frisken?s work, where ?iso and ?? were measured with a SuperconductingQuantum Interference Device (SQUID) Magnetometer [94].20 ?C 30 ?C?iso ?? ?? ?? ?? ?? ??-8.4 -1.5 -7.9 -9.4 -1.3 -7.9 -9.2Table 5.1: The 8CB magnetic susceptibility tensor values, given as ppm volume susceptibilities inthe SI system. The error is about ?0.1 ppm. They are calculated using the following values: fromreference [94], ?iso = 8.37 ? 10?9m3/kg, and 1 ? 10?9m3/kg ? ?? ? 1.6 ? 10?9m3/kg. Fromreference [95], the density of 8CB is about the same as water.Upon application of an external magnetic field H, the magnetic susceptibility tensor ? willinduce magnetization M.2 The result is a torque T described byT = M?H = ??(n ?H)n?H, (5.3)with n as the director. For 8CB, this means that in the absence of other reorientation effects, thetorques align the director with the field. On a single molecule, this effect is weak. Its strength inbulk 8CB is given by the magnetic coherence length ?. If a liquid crystal is pinned to a wall inorientation A and a magnetic field is present that induces orientation B, then the mesogens will2Since this chapter deals with magnetic fields in matter, a distinction is made between B and H, the differencebeing explained by, for example, Griffiths [96]. Again, SI units are used throughout.425.3. Behaviour of Confined Liquid Crystalstransition from A to B over one magnetic coherence length from the wall. [37]:? =??0K??B20, (5.4)where K is a coefficient that describes the elastic coupling between a mesogen and its neighbourand ?0 is the permeability of free space. For the 8CB@CNMO system described below, usingK ? 0.6 ? 10?11 N [97], B0 = 9.4 T (for the 400 MHz spectrometer), ?? ? 1 ppm, ? ? 300nm. This is much larger than the pore diameter, but given the largely unimpeded diffusion pathof water down the pore (see Chapter 4), it may be on the order of the pore length.5.3 Behaviour of Confined Liquid CrystalsMost applications of liquid crystals, like LCDs, confine them to small volumes, so the effect ofthe walls is important. The presence of a surface can be thought of as introducing a field in thebulk liquid crystal. Depending on the physical and chemical properties of the surface and theliquid crystal, the mesogens may lie perpendicular to the surface normal (the planar orientation),parallel to the surface normal (the homeotropic orientation), or at some angle in between. Figure5.3 clarifies this. Via elastic coupling, the surface orientational influence permeates into the bulksome distance. This distance is dictated by the type of surface alignment and the strength of theelastic coupling.Figure 5.3: The Planar and Homeotropic anchoring alignments. n is the local director.Even in the absence of external fields, predicting the behaviour of the liquid crystal in a confinedvolume is a complicated problem, since the energy of long-range ordering is small. A great dealof research has been done in this area, both for the purpose of improving liquid crystal-basedtechnologies and for the study of critical phenomena [90]. In general, this research is beyond thescope of this work, but a few key points are outlined below. In particular, the effect confinementhas on phase transitions and some of the orientations achieved by confined mesogens are of interest.The confining volume will affect the fluctuations of the liquid crystal, and different geometriesand surfaces will affect different fluctuations. In general, phase transitions are suppressed andspread out over a wide range. If the volume is small enough, some more highly ordered phases,435.4. The 8CB@CNMO Systemsuch as the smectic phases, cannot form. Essentially, the correlation length is severely reduced bythe confining volume. As an example, in reference [98], 8CB confined to aerogel pores of ?17 nmdiameter shows a completely continuous nematic-smectic A phase transition in a calorimetry studyinstead of the 2nd order transition seen in the bulk. However, the nematic-isotropic transition,despite being less sharp, still occurs close to the bulk transition temperature. Other studies haveshown that in even smaller pores, the nematic-isotropic transitions are spread out as well to someextent [99]. Confinement in parallel cylindrical pores of porous silicon with a diameter of 30 nmeven show evidence for a new crystalline phase [100].For the work here, understanding how the liquid crystals arrange themselves is more importantthan the behaviour of the phase transition. For 8CB, surfaces that are hydrophobic tend to havehomeotropic alignment, and those that are hydrophilic tend to have planar alignment [101]. How-ever, this will vary depending on the pore size and liquid crystal type. For example, in hydrophilic200 nm cylindrical pores (anopore membrane), 5CB and 8CB have a planar alignment, whereas12CB has a homeotropic alignment (which in a cylindrical pore is known as escaped radial, sincethere is a defect down the middle). These few examples illustrate general trends, and researchin this area is still ongoing since a great many parameters affect confined liquid crystals: elasticcoupling, surface fields, pore size, density, and applied fields [102].5.4 The 8CB@CNMO SystemThe goal of the 8CB@CNMO system was to prove that a chiral nematic phase could be imposedon a nematic liquid crystal [93]. This removes the requirement of molecular chirality and, since thehost is solid, the pitch remains constant over large environmental changes. The system, made bythe MacLachlan group at UBC, uses an Et-CNMO host and an 8CB guest. To better absorb the8CB, the CNMO was functionalized with n-Octyl (in the form of n-Octyltriethoxysilane), which isused to make the surfaces hydrophobic. The final fill factor of the liquid crystal in the films wasabout 50% of the pore volume.The composite system illustrates thermally-controlled reflective properties. The index of refrac-tion of 8CB varies between 1.51 (polarized parallel to director) and 1.68 (polarized perpendicularto the director) at 20? C (in the smectic A phase) to about 1.56 in the isotropic phase [103]. Theindex of refraction of the ethane-organosilica is about 1.5 [41]. Figure 5.4 illustrates UV-vis spectrafor the film upon melting and heating in steps of 5? C. As the 8CB enters the isotropic phase, whichoccurs near the bulk transition temperature, the refractive index becomes close to the CNMO?sand the photonic crystal property disappears. The system moves from iridescent to translucentupon heating, and this effect is reversible (with a slight hysteresis).To facilitate study with NMR, 15N-labelled 8CB was used. The shielding of the single 15Nnucleus will vary depending on whether the 8CB molecular is perpendicular or parallel to thespectrometer field B0. In this way, the position of the spectral line gives an indication of themesogen orientation. If it is oriented with Euler angles ? = (?, ?) to B0 (as in Figure A.1), then445.5. Magnetic Susceptibility EffectsFigure 5.4: The UV-vis spectrum of 8CB@CNMO upon heating (left) and cooling (right). In theisotropic phase at 330 K, no optical bandgap is seen. In the bulk, the nematic-isotropic transition isat 314 K, which is close to the behaviour of this system. These spectra are reproducible across manyheating and cooling cycles. Used with permission from reference [93]. Copyright 2013 AmericanChemical Society.via Equation A.19, its spectral line ?(?, ?) will be at?(?, ?) = ?0?iso +?0?2[3 cos2 ? ? 1 + ? sin2 ? cos2(2?)]. (5.5)(For more details, refer to Appendix A.) The 15N shielding tensor values have been approximatedby those of a CH3 para-substituted benzonitryl studied by Sardashti and Maciel [104]. They wereobtained using Magic Angle Spinning, which averages out the demagnetization effects described inthe next section [105, 106]. The tensor values are?iso = 254.7ppm, ? = ?253.9ppm, and ? = 0.11. (5.6)Note that ? is small and thermal motion causes rapid spinning around the long axis, so ? has littleeffect on the spectrum.5.5 Magnetic Susceptibility EffectsMagnetic susceptibility effects describe how the field B = ?0(H+M) felt by a nucleus is affected bythe geometry and the various magnetic susceptibilities of the surrounding environment. Typically,these susceptibilities are diamagnetic, so the field at the nucleus is less than B0. In this case, themagnetic susceptibility effects are often called demagnetization effects. In contrast, paramagneticeffects will increase the field felt by the nuclei. Ulrich et. al [106], Levitt [107] and Vander Hart[108] have comprehensive papers dealing with demagnetization effects, which are are outlined here.If the magnetic susceptibility of a macroscopic object is the same throughout, then the shape455.5. Magnetic Susceptibility EffectsFigure 5.5: The magnetic field B of an ellipsoid inside a homogeneous field. The susceptibilitydifference ?ellip ? ?0 between the ellipsoid and the surrounding medium cause the field to warpoutside the ellipsoid.of the object is the only parameter affecting demagnetization. In this work, the magnetic sus-ceptibility effects for such an object are called shape susceptibility effects. An example of this isgiven in Figure 5.5, where an ellipsoid of isotropic and homogeneous magnetic susceptibility ?ellipis immersed in a medium with isotropic and homogenous magnetic susceptibility ?0 through whicha constant magnetic field is applied. Ellipsoids have the special property of having an isotropicand homogeneous internal field B when magnetized by a constant external field. The shape sus-ceptibility effect is that the strength of this internal field will change depending on its orientationrelative to the field, expressed by the angle ? between the semimajor axis (z-axis) and the field.This will shift the frequency of the single resonance line an amount ??(?). Expressing this in ppm,referenced to the frequency of a nucleus free from susceptibility effects ?0, gives the shift as [106]??(?)?0= ?ellip[(13?Dz)+ (Dz ?Dx) sin2 ?]. (5.7)Here, Dx and Dz are the demagnetizing factors of the ellipse, and describe the demagnetizing effectsalong the x- and z-axis respectively. For an ellipse, if the lengths of each axis are ax = ay, and az,and q = az/ax, then Dx, Dy, and Dz can be found via [106]Dz(q) =q ln(q +?q2 ? 1)(q2 ? 1)3/2?1q2 ? 1(5.8)Dx = Dy = (1?Dz)/2. (5.9)Note that Dz is real for q > 1 and q < 1.For a non-ellipsoid shape with isotropic and homogeneous magnetic susceptibility, the internalfield B is no longer isotropic or homogeneous. Therefore, upon rotating this shape relative to theapplied field, the spectra will change shape and position. This is shown in Figure 5.6 for slabwith infinite length and height/width= 0.2. An ellipsoid can be inscribed that takes of most ofthe volume, the resonances from which follow approximately the sin2 ? dependence of equation465.5. Magnetic Susceptibility EffectsFigure 5.6: The changing lineshape and position of the spectrum from a slab with dimensionslength? inf, height/width= 0.2 as the angle ? between the surface normal and the the spectrometerfield B0 is changed. This is due to shape susceptibility effects. The position of the peak hasapproximately the sin2 ? dependency of equation 5.7 since the majority of the signal comes fromthe largest ellipsoid that can be inscribed into the volume. Adapted with permission from reference[106]. Copyright 2003 Elsevier.5.7. Therefore, the position of the resonance peak from the slab will also approximately follow thisdependence. The changing lineshape is due to nuclei outside the inscribed ellipse. However, for morecomplicated geometries, or for inhomogeneous fields, demagnetization from shape susceptibilityeffects must be computed numerically.Even more complicated are structure susceptibility effects. Referring again to the example ofthe ellipse, these effects would arise if ?ellip ? ?ellip is a tensor. Evidently, these effects can arise insystems like crystals, liquid crystals, or in anisotropic periodic materials such as the CNMO films.For samples with a structural susceptibility effect, the generalization of equation 5.7 is [106]??(?)?0= ?z(13?Dz)+[?x(13?Dx)? ?z(13?Dz)]sin2 ?. (5.10)Here, ?z and ?x are the susceptibilities of the object in the direction of the semimajor z and x axes.It is also possible to consider from which length scales the shape and structure susceptibilityeffects can arise. Figure 5.7 gives a diagram of this. Macroscopic sample geometry will impact onlythe shape effects. Structure effects will depend on microscopic geometry and molecular properties.It should be noted that in the literature, the magnetic susceptibility effects are often referred toas ?demagnetization effects? or ?bulk magnetic susceptibility (BMS) effects?. In this work, these475.6. Experimental DetailsFigure 5.7: The contributions to magnetic susceptibility effects from the macroscopic, microscopic,and molecular length scales.are referred to as shape or structure susceptibility effects to distinguish their origin.5.6 Experimental Details15N spectra of the samples were collected on a homebuilt 200 MHz NMR spectrometer, a BrukerAscend 850 MHz NMR spectrometer, and the 400 MHz spectrometer described in Section 2.5(referred to as ?the 200?, ?the 400?, and ?the 850?). The spectra from the 200 and 850 were takenat 20? C, and the spectra on the 400 were taken at various temperatures. A Bruker B-VT-1000variable temperature unit with an accuracy of about 1? C was used. Simple 90?-acquire sequenceswere used due to the short T2 relaxation time that made a spin-echo impossible.A homebuilt solids probe with a goniometer was used on the 200 and 400. About 300 mg ofthe sample film flakes (individually ?3x3x0.05 mm) were stacked on a small stage within the coiland pressed flat with tightly-wound Teflon tape. On the 400, an Arduino microprocessor-basedspectrometer interface was made, allowing the sample to be rotated automatically and remotely.This was realized with a servo modified for continuous rotation and homebuilt optical decoders. Onthe 850, a Bruker solids probe was used and the film flakes inserted into a 5 mm NMR tube, whichmeant they were less flat than in the goniometer probe. Solid 15NH4Cl was used as a reference(29.1 ppm on the liquid ammonia scale) [109].5.7 ResultsAs seen above in Figure 5.4, when the 8CB@CNMO system is heated the optical bandgap disap-pears. This suggests the absorbed 8CB is in an isotropic phase, and was confirmed using 15N NMRat 50? C. Figure 5.8 shows the isotropic spectra for two orientations of the film flakes, where ? isdefined as the angle between the film normal and the spectrometer field B0. Although the positionof the peak changes ?5 ppm between the two orientations, its shape remains sharp, showing thereis currently no alignment. The shift in position is due to magnetic susceptibility effects, elaboratedon below.485.7. Results200220240260280300320340 (ppm)Relative intensityFlat ( = )Side ( = )Figure 5.8: The 15N spectra of 8CB@CNMO at 50 ?C. The 8CB is in the isotropic phase, andthe shift in line position is due to magnetic susceptibility effects. Inset: ? is defined as the anglebetween the film surface normal and the spectrometer magnetic field B0.495.7. Results100150200250300350 (ppm)Relative Intensity======perpendicular edgeparallel edge200 MHz400 MHz850 MHzFigure 5.9: The 15N spectra of 8CB@CNMO at 20 ?C for multiple orientations relative to the field.At ? = 0? the mesogens are all at the same orientation due to the radial symmetry of the sample.At ? = 90? the spectrum appears to be a 2d powder, although the ? = 0? does not line up withthe perpendicular edge. This is likely due to large structural susceptibility effects not seen in theisotropic spectra of Figure 5.8. The 850 data has no line broadening, whereas the 200 and 400 have100 Hz of Gaussian broadening.505.8. DiscussionThe room temperature (anisotropic) spectra, collected on the 200, 400, and 850 at 20 ?C whenthe 8CB has some degree of order, show there is partial alignment. Both the lineshape and positionchange as the sample is rotated. The 400 and 850 spectra had ? = 0?, 18?, 36?, 54?, 72?, and 90?,whereas the 200 only had ? = 0? and 90?. The different field strengths were used in order to testthe presence of field-induced orientation effects described in equation 5.3 in the confined geometry.If the mesogens are being affected by the field, then the position and shape of the spectra wouldchange with the field strength. The results in Figure 5.9 do not show this. The broader lines seenon the 850 spectrum are due to a greater degree of misalignment of the flakes than on the 200 and400 spectra. Essentially, there is more variation in ? across the film flakes. These spectra are likelyaffected by structural susceptibility effects not seen in the isotropic spectra, as explained below.5.8 Discussion5.8.1 Isotropic spectrum magnetic susceptibility effectsDemagnetization effects affect both the isotropic and the room temperature spectra, but theisotropic spectra are easier to explain since the 8CB mesogens have no alignment. There, ther-mal motion is fast enough that only the average structural susceptibility effects of the CNMO poregeometry are seen. Alternatively, in the room temperature spectra, the 8CB mesogens do havealignment, so this contributes to the structural susceptibility effects as well.It is evident that the isotropic peak shift is due to susceptibility effects since the peak positionas a function of ? has a sin2 ? dependency, as shown in Figure 5.10. The peak position is fit to theequationPeak position = (263.7 ppm)? (4.0 ppm) sin2 ?, (5.11)showing that the observed 15N peak is close (?260-264 ppm) to the isotropic value of the 15Ntensor of ?iso = 254.7 ppm. The discrepancy is probably partially due to failing to take intoaccount magnetic anisotropy effects on the 15NH4Cl reference. The magnetic susceptibility ofNH4Cl is about -12.5 ppm [110] and the sample was in a long, thin MAS NMR sample tube. Thiscorresponds to a shift of about 2 ppm, using equation 5.7, where for a long cylinder, Dz = Dx = 0.5[106]. Therefore, all spectra may have a systematic error of +2 ppm. Another contribution isfrom using the 15N tensor for a CH3 para-substituted benzonitryl. Sardashti and Maciel look atother substitution groups which yield different ?iso values: for example, substituting a CMe3 gives?iso = 258.3 ppm for the 15N tensor. Unfortunately, bulk 15N 8CB was not available to confirmthat substituting a Bz group yields a different ?isoThe 4 ppm angular-dependent shift is explained by the shape susceptibility effect of the sample.The stack of the film flakes on the sample holder, once compressed with Teflon tape, had a heightto width ratio q of about q = 1/3. Using equation 5.9, this yields a Dz ? 0.6 and Dx ? 0.2. Withequation 5.7, a shift of about 3.2 ppm at ? = 0? and -1.6 ppm at ? = 90? is found. The magnitude515.8. Discussionof this effect is appropriate.0 20 40 60 80Film angle  (degrees)258259260261262263264265Isotropic peak position (ppm) Fit = (263.7 ppm)  (4.0 ppm)sinDataFigure 5.10: The position of the 15N spectral line for 8CB@CNMO at 50 ?C at different orientationsof ? and its fit. Shape susceptibility effects from the stack of film flakes cause a peak shift ? sin2 ?.5.8.2 Anisotropic spectrum magnetic susceptibility effectsIn the room temperature anisotropic spectra of Figure 5.9, the change in line position and shapewith ? confirms that the 8CB has some degree of alignment. This is particularly clear in the400 spectra due to its sharper lines. The single peak in the ? = 0? spectrum indicates that themesogens are all at the same angle relative to the field. This is because the film pores are radiallysymmetric (see Chapter 2). The ? = 90? spectrum appears similar to the symmetric ?2d powder?or ?transverse isotropic powder? that results from a small ? and equal distribution of ? in equation5.5 [111, 112]. The parallel and perpendicular edge for ? = 0? are indicated and correspond tomesogens aligned parallel and perpendicular to B0. So a 2d powder at ? = 90? makes sense giventhat the mesogens are distributed isotropically around the surface normal.One possibility for this distribution is that the mesogens are aligned planar (parallel to thecylindrical axis) in the pores such that at ? = 0?, ? = 90? for all the mesogens as well. However, at? = 0?, all the pores are perpendicular to B0, so in this case the peak should lie in the same positionas the perpendicular edge on the ? = 90? spectrum, which it does not. Another possibility is thatat ? = 0?, 90? > ? > 0? for all the mesogens such that they effectively lie in a cone at angle ? to thesurface normal. The spectra can be fit using this model for the ? = 0? and ? = 90? orientations,but deviates significantly in between, since the perpendicular edge appears much sooner than inthe data.Finally, the possibility that the magnetic field affects the alignment of the mesogens somehowwas tested using three different magnetic field strengths. No difference in the peak position of shape525.8. Discussionwas found. Evidently, the magnetic coherence length, which is on the order of 100 nm, is less thanthe pore diameter or mean free path down the pore. The large surface area to volume ratio of thepores and the irregular surface may also inhibit field-induced ordering.The remaining explanation is a simple planar alignment (? = 90? when ? = 0?) with addedline shifting due to structural magnetic susceptibility effects not seen in the isotropic spectra. Thisorientation-dependent shift (?40 ppm) is much larger than shape magnetic susceptibility effects inthe isotropic spectra. There are a few possibilities for the origin of this shift. Its magnitude is thesame in the 400 and 850 spectra, and since the sample was packed differently between the two,it must be due to a structural susceptibility effect. However, contributions from the aligned 8CBmesogens cannot account for this. Depending on the temperature, 8CB in the nematic phase has?? ? 1.6 ppm, which is too small. Since the magnetic susceptibility of the glass is about -12 ppm[106], and the octyl chains about -8 ppm [113], the susceptibility differences are not a likely source.It is interesting that the direction of this shift (downfield when ? = 90?) is in the opposite directionof the shift in the isotropic spectra (upfield when ? = 90?).There are a few precedents for this large shift in the literature. Many NMR studies have beendone using stacks of thin glass plates to orient lipid molecules. Rainey et al. took 31P spectra usingthis method with POPC lipid molecules. They observed a 22 ppm shift between the horizontal andvertical orientations, with the vertical orientation more negative, like the data here [114]. Forse et al.observed a 9 ppm shift in 11B and 19F spectra of electrolyte-infused microporous carbon electrodes,which also have a thin, flat geometry [115]. On similar studies using Li-based electrolytes, Treaseet al. also observed a 8.5 ppm shift in 7Li [116]. With these, the shift was also more negative in thevertical orientation. While these shifts are appreciable, they do not approach the magnitude seenin the data here. Trease et al. also study shifts seen from nuclei near paramagnetic ions, whichresults in a much larger 30 ppm shift. However, there should not be any paramagnetic species inthe 8CB@CNMO sample. Currently, the exact source of this shift is unknown. Still, as shown inthe next section, the shift does have the sin2 ? dependence typical of magnetic susceptibility effects.5.8.3 Lineshape simulationIn order to simulate the spectra, equation 5.5 is modified in three ways. First, ? is scaled by afactor ? to account for the rapid thermal averaging. According to the values for the chemical shifttensor given above, a full 2d powder pattern should span ?390 ppm, whereas at room temperaturefor the 8CB@CNMO it is only about ?70 ppm due to thermal averaging. Secondly, ?iso is shiftedby a term C0. This probably results from both the shift seen in the isotropic spectra (see section5.8.1) and incomplete thermal averaging. The final modification is the addition of a term C1sin2?to account for the unexplained magnetic susceptibility effects, where C1 is the magnitude of theeffect. With these, equation 5.5 becomes?(?, ?)?0= ?iso + C0 + C1 sin2 ? +?2?[(3 cos2 ? ? 1) + (? sin2 ? cos2(2?))]. (5.12)535.8. DiscussionThe mesogens have a thermal average orientation ?0 to the surface normal. It is assumed they arelying parallel to the cylindrical pore axis, so on average ?0 = 90?. This is shown in figure 5.11:essentially, the 8CB now lies in the same orientation as the NCC crystallite templates. As theFigure 5.11: The assumed average alignment of the 8CB mesogens is down the pore, in the sameorientation as the NCC crystallite templates. ?0 is the average angle the mesogens make with thesurface normal, and is shown here as 90?.sample is rotated, ?, the angle the liquid crystal molecules makes to B0, will vary depending onthe pore. For example, when ? = 90?, 0 ? ? ? 90?. Figure 5.12 shows this for an exaggerated ?0such that the mesogens (red lines) are clearly lying in a cone. To find ? for a mesogen lying along(r, ?0, ?) in spherical coordinates when the sample is rotated by ? around the x-axis, a rotationmatrix is applied to this vector in Cartesian coordinates. Converting the result back to sphericalcoordinates yields? = arccos(sin ?0 sin ? sin? + cos? cos ?0). (5.13)Lastly, all of the parameters are given some Gaussian distribution via a mean x? and standarddeviation sx:? ? ?? ? s? (5.14)?0 ? ??0 ? s?0 (5.15)?? ??? s?. (5.16)For ? and ?0, this is motivated by the inability to pack the film flakes completely flat on the sampleholder. For ?, some mesogens, such as those in the middle of a pore, will undergo more thermallyaveraging than others, such as those close to a pore wall.The lineshape simulation was written in Python using Numpy and SciPy. Equation 5.12 was545.8. DiscussionFigure 5.12: If ?0 6= 90?, then the 8CB mesogens (red lines) will effectively form a cone aroundthe surface normal (z-axis). Here, the deviation in ?0 is greatly exaggerated. When the sample isrotated an angle ? around the x-axis, ? will depend on the azimuthal angle ?.integrated over all the pores and molecular orientations at each value of ?. The resulting histogramwas converted to an FID with T2 parameter and then the real part of the Fourier transform usedto get the fits seen in Figures 5.13a and 5.13b. The parameters used for the spectrum are given intable 5.2.T2 ??0 s?0 s? ?? s? C0 C1400 1.5 ms 90? 7? 10? 5.0 1.0 40 ppm -13 ppm850 1.5 ms 90? 25? 20? 5.0 1.0 40 ppm -13 ppmTable 5.2: The parameters for the lineshape simulation of the 15N spectra of 8CB@CNMO collectedon a 400 and 850 MHz spectrometer. Parameters are defined by equation 5.12. The results areshown in Figures 5.13a and 5.13bThe parameters that should remain constant despite different sample packing are T2 , ??, ands?. These were set equal between the 400 and 850 spectra whereas the other parameters could varyindependently. The differences in shape between the two spectra are adequately explained by thepoor sample packing on the 850, leading to a large variation in ? and, due the cylindrical natureof the NMR tube, ?0.555.9. Conclusions5.9 ConclusionsUsing 15N labelled 8CB, the alignment of the mesogens inside an octyl-functionalized CNMO filmswas studied. At room temperature, the mesogens are on average lying parallel to the pore axis,in the same orientation as the removed NCC crystallite templates. A large, unexplained ?40 ppmshift is seen when the film flakes are on their side relative to the B0 field. This is probably due tostructure magnetic susceptibility effects. After the 8CB is melted, the mesogens are confirmed tobe in an isotropic state with no average alignment.565.9. Conclusions100150200250300350 (ppm)Relative Intensity======400 MHzSpectrometer(a)100150200250300350 (ppm)Relative Intensity======850 MHzSpectrometer(b)Figure 5.13: The lineshape simulation for the 400 and 850 ?N spectra of 8CB@CNMO using theparameters in table 5.2. Data is shown in thin blue lines and the fit is in thick red lines.57Chapter 6Conclusions and Future WorkPorous materials in general and mesoporous materials in particular can be designed to have inter-esting chemical, physical, and optical properties. The CNMS and CNMO films analyzed in thiswork are novel mesoporous, free-standing materials with a chiral nematic pore structure, made bytemplation with NCC. They are strong polarizers and appear iridescent, properties that reversiblydisappear upon introduction of refractive index-matching fluids into the pore.NMR spectroscopy was used to study the bulk and surface chemistry of these films. First, 13CNMR allowed the removal of the NCC template to be confirmed. This was used to identify suitablesynthesis pathways for different organosilica precursors, including Bz-organosilica. Furthermore,the Si surface groups could be observed via 29Si NMR. In CNMO, removal of the NCC with sulphuricacid was seen to increase the number of T2 groups and decrease the number of T1 groups, showingthe condensation of Si-OH groups. Also, it was proven that the acid doesn?t cleave organosilicabonds, since no silica species were detected.Next, the pore sizes were studied with nitrogen adsorption and NMR cryoporometry. NMRcryoporometry offers the same accuracy and improved precision in obtaining PSDs over nitrogensorption. Also, only about 10% of the adsorption sample size was used, and even smaller sampleamounts would be possible. Freezing of the absorbed water was proven to be non-destructive.Using post-acquisition corrections for probe tuning and the Curie Law effects, the entire processcan be completely automated. Gas sorption and cryoporometry are complimentary techniques.Cryoporometry can probe smaller pore volumes than gas sorption and with a higher resolution, butanalysis of the sorption hysteresis can yield some geometric information. Analysis of the hysteresisin cryoporometry is currently under development by other researchers. The pores of Et-CNMO wereshown to be larger than CNMS, with both methods in good agreement. Future work will endeavourto find the cryoporometry constants k and  for Et-organosilica using multiple calibration samples.Due to the ? 1/r dependence of the freezing point depression, a single calibration sample did nothave the required accuracy.For the PFG diffusion measurements, the main finding was the difference in the transverse andlongitudinal diffusion rates through CNMO and CNMS films, with the former roughly 2? the latter.For the same diffusion time, molecules displace on average further in the direction perpendicular tothe helical axis than parallel to it. Motion parallel to the pore axis is therefore almost unimpeded,with a diffusion simulation showing that this must be close the bulk diffusion constant D0. CNMSfilms, which have a smaller pore diameter, have an overall slower diffusion rate, indicating that theirsmall pores are more tortuous as well. The diffusion times are long enough that each measurement58Chapter 6. Conclusions and Future Workcan be fit to a Gaussian with a single diffusion coefficient. Deviations from this in the longitudinaldirections, as seen in the residuals, are due to the molecules hitting the barrier of the film surface.The last portion of work was concerned with determining the alignment of absorbed 8CBliquid crystals in an octyl-functionalized CNMO film. This host-guest system shows the usualiridescent and polarization characteristics until the melting point of the absorbed 8CB is reached,at which point it becomes clear. NMR spectra of 15N-labelled 8CB showed definitive lack of mesogenalignment when the sample is clear. Shape magnetic susceptibility effects give a small dependenceon the sample orientation in this case. 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A rotatable flat coil for static solid-state nuclearmagnetic resonance spectroscopy. Review of Scientific Instruments 76, 086102?086102 (2005).[115] Forse, A. C. et al. Nuclear magnetic resonance study of ion adsorption on microporouscarbide-derived carbon. Phys Chem Chem Phys 15, 7722?7730 (2013).[116] Trease, N. M., Zhou, L., Chang, H. J., Zhu, B. Y. & Grey, C. P. In situ NMR of lithiumion batteries: Bulk susceptibility effects and practical considerations. Solid State NuclearMagnetic Resonance 42, 62?70 (2012).67Appendix AInteractions and the Chemical ShiftTensorA.1 Representing InteractionsTo determine how an interaction affects a spin, the local magnetic field on the nuclei Bloc (andelectric field Eloc in the case of the quadrupolar interaction) is calculated. Once these local fieldsare known for a spin I, the interaction Hamiltonian is simplyH?A = ??I I? ?Bloc (A.1)= ??I(I?xBlocx + I?yBlocy + I?zBlocz ). (A.2)With this, the problem is essentially solved. The Zeeman Hamiltonian,H?Zeeman = ??I I?zB0= ?0I?z,(A.3)and RF Hamiltonian,H?RF = ~? cos(?RF t)I?x, (A.4)are straightforward cases of equation A.2. With these, ?I is the gyromagnetic ratio of the nuclei, I?zand I?x the spin operators (with eigenvalues in units of ~) parallel and perpendicular to spectrometerfield B0 respectively, ?0 is the Larmor precession frequency, and ?RF is the RF frequency.The Zeeman and RF Hamiltonians are simple because the application of an external fielddirectly yields the form (A.2). Internal interactions are more difficult: as discussed in the sectionsbelow, each nuclei?s electrical and magnetic local environment?produced by nearby electrons andspins?is unique. Even though the strength of these local fields is ? 10?6 smaller than the B0 andB1 fields, an NMR spectrometer?s can detect the resulting subtle changes in precession frequency,revealing the local environment. This is what makes NMR so powerful. The general form of theseHamiltonians is exemplified below using the chemical shift Hamiltonian.68A.2. The Chemical Shift Hamiltonian and Tensorial RepresentationA.2 The Chemical Shift Hamiltonian and TensorialRepresentationParamagnetic and diamagnetic effects result from the interaction of atomic and molecular electronsto an external magnetic field, which in turn produce their own field. Since no molecular electronclouds perfectly spherical, the chemical shift interaction depends on molecular orientation relativeto the external field. Consequently, the rank two chemical shielding tensor ? is introduced. In theNMR spectrometer, the B0 field is the external field in the lab frame, so the local field from theparamagnetic and diamagnetic effects isBloccs = ? ?B0. (A.5)The relative difference in Bloccs fields is called the chemical shift. This is the most common mea-surement made in NMR experiments.With the local field given by equation (A.5), equation (A.1) yields the chemical shift Hamilto-nian:H?cs = ??I I? ? ? ?B0. (A.6)This is the general form of any interaction Hamiltonian. Given some external magnetic or electricfield J, a spin I will interact with it via a coupling tensor A, contributing a local field Blocint = A ?J.The most general form of equation (A.1) is thenH?int = ??I I? ?A ? J. (A.7)In general, H?int will have a time-independent secular part that is a first order perturbation onthe Zeeman Hamiltonian, and a time-dependent nonsecular part has has higher order corrections.The eigenstates of the secular part are still those of the Zeeman Hamiltonian, whereas the nonsecularpart has different eigenstates. Except in extreme cases, only the influence of the secular part willbe of concern.A.2.1 Tensor representationThe chemical shielding tensor ? is frame-dependent. As a second rank tensor describing a threedimensional system, ? can be written as a 3? 3 matrix in any frame, like? =????xx ?xy ?xz?yx ?yy ?yz?zx ?zy ?zz??? . (A.8)In the Principle Axis Frame, or PAF, ? = ?PAF is diagonal. The three nonzero values on thediagonal are ?PAFxx , ?PAFyy , and ?PAFzz . Considering these are perpendicular vectors, they define69A.2. The Chemical Shift Hamiltonian and Tensorial Representationan ellipsoid, and 2nd rank tensors are often visualized this way. Knowledge of these three valuesgive a complete description of the tensor; from the PAF, it can be rotated into any frame. FigureA.1 shows an ellipsoidal representation of ?PAF for a 15N nucleus on the end of the liquid crystalmolecule studied in Chapter 5. To find its representation in the lab frame, it is rotated throughEuler angles ? = (?, ?, ?).Figure A.1: The PAF ellipsoid for the 15N shielding tensor on the liquid crystal studied in Chapter5. It can be expressed in the lab frame (blue lines) by rotating it through Euler angles ? = (?, ?, ?).Note that ? has no effect on the resulting frequency.Although the definition above is straightforward, a different set of three values is often usedto describe the tensor. These are the isotropic value ?iso, the anisotropy ?, and the asymmetry ?,defined as?iso =13(?PAFxx + ?PAFyy + ?PAFzz ) = Tr(?), (A.9)? = ?PAFzz ? ?iso, (A.10)? =?PAFxx + ?PAFyy?. (A.11)These three values come about naturally once the tensor is rotated, as will be seen in the nextsection.A.2.2 Tensor rotations: the spherical tensor representationAs mentioned before, once a tensor is known in one frame, its can be rotated into any frame. Asimple way to do this is with the standard rotation matrix R. For example, to rotate ? by anEuler angle ? = ?(?, ?, ?), it is simply ?? = R(?) ? R?1(?). This method is straightforward, andindeed the analysis of many NMR spectra can be accommodated by this formalism. For example,Mehring illustrates how it can be used to determine the spectra of a single crystal as it is rotatedrelative to B0. [3] However, a more general representation of rotations is advantageous and is often70A.2. The Chemical Shift Hamiltonian and Tensorial Representationneeded to ease theoretical analysis of many problems. This is done by representing the interactiontensors as spherical tensors, instead of 3x3 matrices.Spherical tensors come about naturally once rotations are considered. One property of second-rank tensors, such as ?, is that rotating them always affects three parts of the tensor separately.These are the isotropic, traceless antisymmetric, and traceless symmetric parts. Each of these are aseparate subspace closed under rotations?that is, a rotated isotropic part yields another isotropicpart, and similarly for the other two. Since ? as represented by the matrix in equation A.8 canbe broken down like this, it is called a reducible representation. Separating it out into parts thatdon?t change under rotations is called the irreducible representation. While possible to write threeseparate 3x3 matrices for the reducible representation, a less cumbersome way is to use sphericaltensors, which are functions, not matrices. Moreover, they are functions of spherical harmonics.Therefore, not only is cumbersome matrix algebra avoided, but the advantages of working withspherical harmonics?such as their orthogonality?is gained.With the motivation in place, the reducible representation of ? in spherical tensors is? =2?l=0+l?m=?1Rl,m (A.12)= R0,0 ++1?m=?1R1,m ++2?m=?2R1,m. (A.13)The first term is a scalar and corresponds to the isotropic part, the second and third are tensorsand correspond to the traceless antisymmetric and traceless symmetric parts, respectively. Inspectroscopy, only the isotropic and traceless symmetric portions are measurable; therefore, theisotropic part is ignored from now on (see for example Chapter 3 in Haeberlen [2]). A sphericaltensor operator like for ? above can also be found for the I?z operator in equation A.6. Sinceoperators are inherently tied to the laboratory frame, this is invariant under rotations. If thePAF is rotated relative to the lab frame by Euler angles ? = (?, ?, ?), then the chemical shiftHamiltonian in spherical tensor representation isH?cs = ??I?l=0,2l?m=?1(?1)mRl,?m(?)T?l,m. (A.14)The T?l,m?s are the components of the reducible representation of I?z, known as the operator part,and the Rl,?m(?)?s are the spatial part which alone contains rotations. Using Wigner rotationmatrices D (which are just functions of spherical harmonics), the Rl,?m(?)?s can be expanded asRl,?m(?) =?lDlk,m(?)?lk, (A.15)71A.2. The Chemical Shift Hamiltonian and Tensorial Representationwhere the relevant ?lk?s (ignoring the antisymmetric part and ?l,?2?s since T?2,?2 = 0) are given by?00 = ?iso (A.16)?20 =?3/2? (A.17)?2?2 = ?? (A.18)Now finally the motivation behind the definitions for the isotropic value ?iso, the anisotropy ?,and the asymmetry ? defined above becomes clear! They naturally fall out of the spherical tensorrotations.Putting all of this together, the chemical shielding Hamiltonian of equation A.6 gains the familiarform. Using the Wigner rotation matrices and their spherical harmonic representation, equationA.14 is expanded. Taking the secular part givesH?cs ? ~?0?iso ?~?0?2[(3 cos2 ? ? 1) + (? sin2 ? cos2(2?))]. (A.19)This equation gives the expression to find the resonant frequency as a function of orientation, aslong as ?PAF (or ?iso, ?, and ?) are known. This is used in Chapter 5 to find the orientation ofliquid crystals using a 15N nuclei (as in Figure A.1) as a probe.72

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