GAS DIFFUSION IN STRONGLY CONFINING AND INTERACTING MICROPOROUS SORBENTS BY NMR By AKSHITA RANJAN DUTTA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMEN TS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2017
2017 Akshita Ranjan Dutta
To my Mom, Dad, and younger b rother for their sacrifices and e ncouragement throughout my educ ation
4 ACK NOWLEDGMENTS I would like to acknowledge the guidance and mentorship of my doctoral co advisors, Dr. Sergey Vase n kov and Dr. Kirk J. Ziegler for advancing my skills as an engineer, a researcher, and a contributor t o the scientific community. The constant a nd unwavering support, and the time they have devoted to my graduate studies has been invaluab le throughout this journey. I would like to thank Poorvajan Sekar for his assistance in the completion of this research during his graduate studies. I would also like to thank my lab mates, Dr. Robert Mueller, Evan Forman, and Matthias Trujillo for their support. Being a part of the Energy Frontiers Research Center has been a priceless experience. I would like to thank all the graduate student s and post doctoral r esearchers of this center for th eir companionship and insights. The cooperation of the following collaborators, who are also members of the center is especially appreciated: Dr. David Sholl for his advice and guidance with respect to the single file diffus ion and hydrogen sulfide work; Dr. Ryan Lively Dr. J R Schmidt and Guanghui Zhu for all work in relation to hydrogen sulfide. I would like to thank Dr. James Collins at the Advanced Magnetic Resonance Imaging and Spectroscopy (AMRIS) facility for his tech nical assistance and expert input offered during the development of the new methodology. I am indebted to my mother Mini Dutta, and father Ranjan Dutta, for their emotional support and their unaltered confidence in me. I would also like to thank my brot her for his constant encouragement. The support of my family and friends throughout my doctoral studies is unparalleled and I would like to acknowledge their contribution in helping me achieve my goals: Pushpa Mohan, Yash Pal Dutta, Kanta
5 Dutta, Himani Nad gauda, Lauren Miller, Nicole Taylor, Subit Chakrabarti, Shreyas Honrao, Abhinav Bha ndari, Yash Mehta, Arati Sakhalkar, Sharvari Bhagwat. I am grateful to acknowledge the financial support of the Donors of the American Chemical Society Petroleum Research F und (PRF#53140 ND10), NSF grant NO. CHE 0957641 and the NSF CAREE R award No. CBET 0957641 for this work. A portion of this work was performed in the McKnight Brain Institute at the National High Magnetic d by National Science Foundation Cooperative Agreement No. DMR 1157490 and the State of Florida.
6 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ .......... 9 LIST OF SYMBOLS ................................ ................................ ................................ ...... 11 LIST OF ABBREVIATIONS ................................ ................................ ........................... 13 ABSTRACT ................................ ................................ ................................ ................... 14 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 16 2 NORMAL AND ANOMALOUS DIFFUSION ................................ ............................ 18 elf Diffusion ................................ ................................ ............ 18 2.2 Random Walk Model and Single File Diffusion ................................ ................. 21 2.3 Deviations from Single File Diffusion in Finite Systems ................................ .... 23 3 NMR AND PULSED FIELD GRADIENT DIFFUSOMETRY THEORY .................... 25 3.1 Fundamentals of Nuclear Magnetic Resonance ................................ ............... 25 3.1.1 Spin, Precession, and Magnetization ................................ ...................... 25 3.1.2 Longitudinal and Transverse Relaxation ................................ ................. 28 3.1.3 NMR Spectroscopy ................................ ................................ .................. 31 3.2 Pulsed Field Gradient NMR ................................ ................................ .............. 31 3.2.1 Field Gradient ................................ ................................ .......................... 32 3.2.2 Spin Echo ................................ ................................ ................................ 32 3.2.3 PFG NMR Spin Echo Sequences ................................ ............................ 35 3.3 Experimental NMR Details ................................ ................................ ................ 37 4 SINGLE FILE DIFFUSION IN MIXTURES ................................ .............................. 40 4.1 Motivation ................................ ................................ ................................ ......... 40 4.2 Review of Published Single Fil e Diffusion Studies ................................ ............ 40 4.3 Sorbent and Sorbates Studied ................................ ................................ .......... 42 4.4 Experimental Details ................................ ................................ ......................... 43 4.5 Observation of Single File Diffusion in Mixtures ................................ ............... 45 4.6 Summary ................................ ................................ ................................ .......... 54 5 ABSENCE OF TRANSITION TO CENTER OF MA SS DIFFUSION ....................... 55
7 5.1 Single File Diffusion in Finite Systems ................................ .............................. 55 5.2 Relationship between Single File Mobilities ................................ ...................... 58 5.3 Summary ................................ ................................ ................................ .......... 60 6 METHODOLOGY DEVELOPMENT CALIBRATION OF MAGIC ANGLE SPINNING PROBE FOR NMR DIFFUSOMETRY ................................ .................. 62 6.1 Motivation ................................ ................................ ................................ ......... 62 6.2 Gradient Calibration ................................ ................................ .......................... 63 6.3 Reference Measurement in Microporous Material ................................ ............ 67 6.4 Summary ................................ ................................ ................................ .......... 76 7 WORK IN PROGRESS: DIFFUSION IN THE PRESENCE OF HYDROGEN SULFIDE ................................ ................................ ................................ ................. 78 7.1 Motivation ................................ ................................ ................................ ......... 78 7.2 Diffusion of Hydrogen Sulfide ................................ ................................ ........... 79 7.2.1 Experimental Details ................................ ................................ ................ 79 7.2.2 Results and Discussion ................................ ................................ ........... 80 7.3 Influence of Hydrogen Sulfide Exposure on Transport Properties .................... 82 7.3.1 Experimental Details ................................ ................................ ................ 83 7.3.2 Results and Discussion ................................ ................................ ........... 85 7.4 Summary ................................ ................................ ................................ .......... 87 8 CONCLUSIONS ................................ ................................ ................................ ..... 94 LIST OF REFERENCES ................................ ................................ ............................... 97 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 102
8 LIST OF TABLES Table page 4 1 Gas concentration (in mol/kg of AV) obtained for the studied samples using NMR signal analysis. ................................ ................................ .......................... 44 4 2 Single F ile mobilities (F helix ) estimated from attenuation curves in Figure 4 1 and 4 2 using Eq 4 6. ................................ ................................ ......................... 51 5 1 E stimates of fractional loadings ( ), the values of MSD at which a transition from SFD to CO M diffusion is predicted by Eq 5 1 ( ), and the minimum cluster size required by Eq 5 1 for not observing this transition under the reported experimental conditions of this study. ................................ .. 57 6 1 V alues of maximum gradient strength, g I (T/m), calculated using Eq 6 1 for two spacers of varying lengths. ................................ ................................ .......... 65 6 2 Ensemble fractions and the corresponding diffusivities obtained by fitting Eq 6 1 to the bi exponential attenuation curves in Figure 6 8. ................................ 74 7 1 Concentration (mmol/g) of C 2 H 4 molecules adsorbed in pure and exposed ZIF 8 crystals as determined using NMR signal analysis. ................................ .. 90 7 2 The parameters corresponding to the bi exponential curve fit to the T 1 relaxation data reported in Figure 7 4. ................................ ................................ 91 7 3 Parame ters corresponding to the bi exponential (Eq 6 1) and mono exponential fits used to describe attenuation curves for both pure and exposed ZIF 8 at selected diffusion times. ................................ ......................... 92
9 LIST OF FIGURES Figure page 2 1 Illustration of measurement o f diffusion ................................ ............................. 19 3 1 Precession of ma gnetic dipole moment in response to the application of a magnetic field in the z direction is a function of the initial direction of the spin. .. 27 3 2 Schematic of a collection of nuclear spins at thermal equilibrium after being subjected to a magnetic field. The result is in a net macroscopic magnetization aligned along the direction of the magnetic field. ........................ 29 3 3 The transfer of the net magnetization from the z axis to the y axis is possible with the application of a r.f. pulse. The transverse magnetization then continues to precess in the xy plane. ................................ ................................ 30 3 4 Schematic illus tration of the combined effect of a static, homogeneous magnetic field and a spatial dependent field gradient that results in a spatially dependent precession. ................................ ................................ ....................... 33 3 5 Schematic illustration of the three steps that describe all NMR diffusometry experiments. ................................ ................................ ................................ ....... 33 3 6 Schematic presentation of the sp in echo based PFG NMR sequences ............. 39 4 1 Schematic illustration of molecules restricted to move in a one dimensional channel that has a radius smaller than the diameter of the molecules. Mutual passage is forbidden under these conditions. ................................ ..................... 41 4 2 13 C PFG NMR attenuation plots measured at 298 K for CO, CH 4 and CO 2 diffusion in AV ch annels loaded with a single gas ................................ .............. 49 4 3 13 C PFG NMR attenuation plots measur ed for CO, CH 4 and CO 2 diffusion as mixtures in AV channels at 298 K ................................ ................................ ....... 50 4 4 MSD dependence on times p lotted on a double log scale ................................ .. 52 5 1 Illustration of difference in molecular motion in SFD and COM diffusion ............ 56 6 1 A schematic of the MAS experimental setup combined with the pulsed field gradient coil, g(z), in a high resolution MAS probe ................................ ............. 63 6 2 The image profile acquired along the rotor axis for a mixture of H 2 O/ D 2 O at two gradient strengths ................................ ................................ ........................ 68 6 3 A plot of the NMR signal obtained at the smallest gradient strength normalized to the signal recorded at the largest gradient strength ..................... 68
10 6 4 The temperature dependence of sample temperature on the spin rate plotted as a function of the system temperature determined using neat methano l ........ 69 6 5 Attenuation plots acquired using the MAS PFG NMR setup at various spin rates and diffusion times for a sample of water at 298 K ................................ .... 70 6 6 The 1 H NMR spectrum of non activa ted ZIF 8 bed obtained using the high resolution MAS probe ................................ ................................ ......................... 72 6 7 1 H spectrum obtained of H 2 O molecules in a saturated ZIF 8 bed under c onditions of MAS and static NMR ................................ ................................ ..... 72 6 8 Overlay of the attenuating signal of H 2 O saturated in the ZIF 8 bed resulting from increasing applied gradient. ................................ ................................ ........ 73 6 9 Measured PFG NMR attenuat ion curves for H 2 O diffusion in a saturated ZIF 8 bed under MAS conditions with a spin rate of 5 kHz. ................................ ...... 75 6 10 Attenuation curve obtained using deconvolution of overlapping peaks observed in th e PFG NMR sp ectra of H 2 O diffusion in ZIF 8 .............................. 75 7 1 Schematic of transport on different length scales in a cross section of a membrane ................................ ................................ ................................ ......... 78 7 2 The evolution of the peak identified as H 2 S in ZIF 8 acquired using the 13 interval PFG NMR sequence ................................ ................................ .............. 88 7 3 Example of 1 H spectra acquired of H 2 S in ZIF 8 using the 14 T spectro meter and diff30 gradient probe ................................ ................................ ................... 89 7 4 T 1 Relaxation curves obtained using 13 C NMR show a bi exponential distribution ................................ ................................ ................................ .......... 90 7 5 Examples of measured attenuation curves obtained using 13 C (black symbols) and 1 H (red symbols) PFG NMR of E thylene diffusion at 298 K .......... 91 7 6 SEM images provided by collaborators in Dr. Ryan pure and exposed ZIF 8 ................................ ................................ ................................ ... 93
11 LIST OF SYMBOLS B 0 static magnetic field c,c* molar concentration/of labeled particles d diameter of single file AV channel D, D* transport diffusivity/of labeled particles D D 0 D COM self diffusivity/of isolated particles/center of mass diffusivity F, F helix single file mobility/along the helical AV channel g,g i ,g a G radient J, J* molar transport flux/of labeled particles l,l i ,l j elementary step of a particle/i/j in the diffusion model L length of AV crystals M s spin azimuthal quantum number n number of steps in the diffusion manual N* t otal number of labeled molecules p geometrical factor relating true path length and helical path length p i ensemble factions P probability distribution mean squared displacement angular spin momentum S spin quantum number S(g) signal intensity as a function of magnetic field t T ime T 1 longitudinal NMR relaxation constant
12 T 2 transverse NMR relaxation constant one dimensional mean squared displacement length of spacer gyromagnetic ratio site occupancy length of elementary step in single file diffusion manual attenuation elementary time between each step in the diffusion model/si ngle file diffusion model magnetic dipole moment angular frequency
13 LIST OF ABBREVIATIONS MSD mean square displacement MOF metal organic framework NMR nuclear magnetic resonance PFG NMR pulsed field gradient nuclear magnetic resonance PGS TE pulsed gradient stimulated echo PGSTEBP LED pulsed gradient stimulated echo using bipolar gradients with longitudinal eddy current delay r.f. R adiofrequency SEM scanning electron microscope ZIF zeolitic imidazolate framework
14 Abstract of Disserta tion Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy GAS DIFFUSION IN STRONGLY CONFINING AND INTERACTING MICROPOROUS SORBENTS BY NMR By Akshita Ranjan Dutta August 2017 Chair: Sergey Vasenkov Cochair: Kirk. J. Ziegler Major: Chemical Engineering Gas separations is a process relevant to a variety of chemical industries ranging from petrochemicals and biofuels to pharmaceuticals. The importance of this process has insp ired the development of novel materials and technology tailored for efficient membrane based technolog ies The main long term goal of the work presented in this dissertation is to develop the fundamental science for a conceptually n ew type of gas separations which is based on induction of single file diffusion (SFD) for a selected component of a gas mixture inside a microporous membrane. SFD can be exhibited by molecules confined in narrow unidimensional channels that prevent mutual molecular passages The unique properties associated with this anomalous diffusion mechanism offer the possibility of highly selective separations. In this work a diffusion NMR technique at high magnetic field and high magnetic field gradient has enabled the first experimental observation of SFD of molecular mixtures and provided insight i nto the relationsh ip between SFD rates of pure sorbates and their respective mixtures. The studied gases include methane and carbon dioxide, which are relevant for natura l gas separations. Molecular clustering in the investigated
15 model single file channels is proposed to explain the observed relationship between the mixture and one component mobilities and to reconcile r eported experimental data with predictions of a rando m walk model. In addition to carbon dioxide, hydrogen sulfide represents another hazardous impurity in natural gas Zeolitic imidazolate frameworks (ZIFs) are promising candidates for natural gas separation and processing. Diffusion NMR was applied to stu dy dynamics of hydrogen sulfide in ZIF 8. Preliminary results reveal an extremely low mobility of hydrogen sulfide in this material. Investigat ion of the influence of hydrogen sulfide exposure on the transport properties of ZIF 8 is currently underway. NMR diffusometry is a powerful technique for studying diffusion phenomena in porous materials. However, short NMR transverse relaxation times under conditions of extreme confinement in microporous materials limit applicability of standard diffusion NMR. In an effort to overcome this challenge, a NMR pro b e capable of performing diffusion measurements under magic angle spinning (MAS) was calibrated and applied for diffusion studies in ZIFs. These studies demonstrate that MAS can drastically improve conditions of NMR diffusion measurements.
16 CHAPTER 1 INTRODUCTION Separation processes account or 10 due to the high energy cost associated with thermally driven processes such as distillation, drying, and evaporation. 1 2 Low energy separation processes like membrane based separations are both economically and environment ally friendly alternatives. Universal implementation of membrane tech nology is currently hindered by barriers that limit their potential. 3 Progress on this front requires advancements in process innovation as well as development o f novel materials. In this dissertation, both these approaches are acknowledged. Highl y selective membrane based separations are challenging because they demand stringent control and uniform distribution of pore sizes. 4 These porous materials must th en be tailored and customized for a given separation. 5 6 A conceptuall y new type of separation based on induction of single file diffusion can relieve some of these requirements. Single file channels forbid molecules inside the channels from passing each other during a diffusion process by geometrically constraining them an d, hence, conserving their rela tive order in the channels. This correlated movement manifests as unique properties that can be exploited to achieve effective separations. Incorporating this diffusion mechanism in applications requires a fundamental underst anding of single file diffusion of mixtures. While there have been numerous theoretical and simulation studies on mixtures confined to single file channels, there have been no previous experimental reports on it. In Chapter 4 the first experimental observ ation of single file diffusion of mixtures is presented. CO, CO 2 and CH 4 gas molecules confined under single file conditions in
17 L Ala L Val nanochannels as both pure and binary mixtures is investigated using pulsed field gradient (PFG) NMR at the high fie ld of 17.6 T The unique relationship between single file mobilities of the sorbates in pure and mixture form is also discussed. Chapter 5 prese n ts a comparison of theoretical prediction s and the reported experimental trends of single file diffusion of mix tures in finite systems. This comparison drives the discussion of the presence of molecular clusters in the studied pure and mixture samples. Complementary to the investigation of the fundamental science of a unique transport phenomenon, microscopic diffu sion studies in existing sorbents is the focus of Chapters 6 and 7. Novel, microporous materials are often introduced and reported in literature as promising materials for challenging and industrially important membrane based separations. To identify and c haracterize the potential of such materials, a thorough understanding of diffusion on a microscopic scale is essential. The development of a new m ethodology that can expand the range of microporous systems that can be studied in this manner is presented in Chapter 6. Chapter 7 reports the study of microscopic diffusion in the presence of a tox ic and hazardous gas, hydrogen s ulfide. The possible presence of strong interactions between the fr amework of the porous material and hydrogen sulfide is revealed here In addition, the influence of exposure to this corrosive gas on the transport properties of the sorbent is being investigated. In the next two chapters, Chapters 2 and 3, the theory of diffusion, NMR spectroscopy, and NMR diffusometry is presented.
18 C HAPTER 2 NORMAL AND ANOMALOUS DIFFUSION Diffusion is the process of migration of molecules tending towards elimination of spatial variations and is driven by non zero internal energy of systems at temperat ures above absolute zero. 7 8 It is a manifestation of the tendency of m atter to approach maximum entropy. 9 This is a fundamental transport phenomenon that has been studied since the 19 th century. Adolf Fick introduced two laws that describe macroscopic diffusion by studying the rate of interdiffusion of two gases, at constant pressure. is given by: (2 1) where J is the molar flux, D is the diffusivity, c is the conce ntration, and z is the spatial coordinate 8 The molar flux is defined as the net number of molecules that flow in the z direction per un it area normal to this direction used to define the diffusiv e flux of a species, and according to Eq 2 1, it opposes the concentration gradient of the species. The diffusivity, D, generally implies the diffusivity of a species in a given medium such as another solvent, a biological membrane, or microporous material s, which are studied in this work For diffusion in one dimension, as described in Eq 2 1, provided that diffusivity is aw is: (2 2) and is a consequence of the application of the conservation of mass principle.
19 Figure 2 1. Measurement of self diffusion in an (a) equilibrated system is illustrated here half of the molecules and tracking their flux or (c) displacements of each molecule. Brownian motion is the m olecular movement under conditions of macroscopic equilibrium. It is driven by thermal fluctuations, resulting in random displacements of molecules. In the absence of a macroscopic concentration gradient, the diffusivity is classified as self diffusivity ( D ) established by Einstein. 10 Figure 2 1 illustrates this connection by invoking a concentration gradient in an initially equilibrated of exactly half the molecules. 11 The system is then allowed to come to equilibrium as shown in Figure 2 1 (c). In analogy t o Eq 2 1, the self diffusivity can be defined in terms of the flux, J*, and concentration, c*, of labeled molecules by: (2 3) under the constraint that total concentration of molecules is constant. Since the flux of labeled molecu les is equal in magnitude but opposite in direction to unlabeled molecules, and their concentrations are equal, the diffusivities of both labeled and
20 unlabeled are identical and defined as the self diffusion coefficient, D Hence, it follows, econd law for this system of labeled molecules in a one dimensional coordinate system is: (2 4) where the total concentration of molecules remains constant. For the initial condition that the labeled species are confined at z=0, t he solution to Eq 2 4 has the form: (2 5) wher e A is a constant. The total number of molecules, N is preserved and can be determined by integrating over the infinite length in the z direction by: (2 6) The resulting solution for is otropic diffusion from a point source is derived by combining Eq 2 5 and 2 6 : (2 7) The mean square displacement (MSD) of the labeled molecules after time t can be calculated by integration: (2 8) Eq uation 2 s equation and provides a direct correlation between the self 4 a nd an experimentally observable relation, the MSD dependence on time.
21 Synonymously, the propagator for a molecule initially located at the origin and allowed to occupy three dimensional space can be derived and the corresponding MSD relation to time is: (2 9) where r is the final position of the molecule. 2.2 Random Walk Model and Single File Diffusion An alternative approach to understand self diffusion, both qualitatively and quantitatively, is the random walk model. The random walk is a sequence of steps a particle can take in any direction with equal probability and at constant frequency given that each step is of equal length. 12 This section outlines the random walk model for a particle allowed to move in one dimension. It also extends the model to more complex systems that give rise to non random correlations between successive steps. A single particle in a one dimensional system takes steps of l with equal probability, allowing for time, between each step. In this scenario, the mean square distance traveled by the particle after n steps, is given by: (2 10) A ssuming the direction of a given step is uncorrelated to the previous step, the term in Eq 2 10 can be rewritten as whe re Hence, Eq 2 10 simplifies to: 6 (2 11) substituted in Eq 2 11 gives:
22 (2 12) where D =l 2 /2 The corresponding probability distribution of displacements for a particle subjected to a one dimensional random walk is: 12 (2 13) It should be noted that t his probability distribution is equivalent to the diffusion 7. Both these expressions (Eq 2 7 and Eq 2 13) describe the normal self diffusion process. The fo undation for the mathematical description of normal diffusion is the concept of uncorrelated movements of molecules. A system, in which particles are confined to move in one dimensional channels so narrow that th eir mutual passage is forbidden, deviates fr om the features of normal diffusion. Here the order of molecules is preserved. This specific anomalous diffusion mechanism is known as single file diffusion (SFD). The correlated motion of particles in single file system can be examined in the context of the MSD expression describing a random wal ker in one dimension (Eq 2 10), where the term is negative. This is because u nder single file confinement, a particle can attempt to step in either direction with equal probability but can only move successfully if directed to a vacant site. 6 The corresponding MSD dependence on time is given by: (2 14) where F is the single file mobility factor. The growth of MSD with the square root of time is a unique signature of SFD. The s ingle file mobility factor, F, is a function of site
23 described by the following equation: 12 (2 15) The diffusion propagator for single file diffusion, as for normal diffusion, is a Gaussi an function. The relationship between the mobility factor, F, of a particle in a single file system and it s self diffusivity D 0 as a sole particle in this system is: (2 16) It should be noted that, Eq uation 2 14 is only v alid for sufficiently large observation 12 2.3 Deviations from Single File Diffusion in Finite Systems In finite, single file channels, with fast molecular exchange at boundaries a transition to the time scaling of MSD for normal diffu sion is predicted by theoretical and mathematical models This behavior is understood by considering a selected time interval in which a particle enters the channel at one end while another exits at the opposite end. The coincidence of these events results in a shift in the chain of molecules that occupy the channel by one position. Ho wever, this shift in the chain of molecules at a given moment is unco rrelated wit h the previous one which is a distinct feature of normal diffusion. This form of molecular pr opagation is known as center of mass (COM) diffusion. 13 14 The MSD dependence on time in this diffusion regime can be 8), where the effective diffusivity D COM can be described as follows:
24 (2 17) i n terms of the diffusivity of an isolated particle in the single file channel, the site occupancy, the channel length, and the elementary step of a particle, 15 Over sufficiently long observation times, in a finite single boundaries that allow for molecular exchange, center of mass diffusion becomes the domin ating mode of transport The MSD at which a transition from SFD to COM diffusion is expected is described by the parameters listed in Eq 2 17 : 15 (2 18) The case of reflecting boundaries would also result in deviation s but towards smaller displacements Since motion in finite, single file channels is sensitive to boundary conditions observation of SFD can be a challenge. 12
25 CHAPTER 3 NMR AND PULSED FIELD GRADIENT DIFFUSOMETRY THEORY Nuclear magnetic resonance (NMR) is an experimental technique that exploits the interaction between certain atomic nuclei and resonant radio frequency (RF) radiation in the presence of an external magnetic field to gain molecular information. The at omic nuclei that are sensitive to this analytical technique must have a non zero magnetic moment. This non invasive technique has wide applications ranging from imaging of biological subjects to determination of complex molecular structures. The work prese nted here is based on an NMR diffusometry method called P ulsed F ield Gradient (PFG) NMR, which is used to study both normal and anomalous diffusion in microporous materials. This chapter provides a brief overview of the fundamentals of NMR and PFG NMR. 3. 1 Fundamentals of Nuclear Magnetic Resonance 3.1.1 Spin, Precession, and Magnetization Spin is an intrinsic quantum mechanical property of elementary par ticles. 16 Each of these particles has a spin quantum number, S, associated with it. It is described to be a form of angular momentum that a particle possesses. In quantum mechanic s, the angular momentum is quantized and the total angular momentum , of a particle with spin takes a value of [S(S+1)] 1/2 2S+1 spin sublevels exist, that are described by the azimuthal quantum nu mber, M s 17 This quantum number can take on specific values between S and S. In the absence of a magnetic field, all 2S+1 states for a given value of S are degenerate. The introduction of a magnetic field can break this degeneracy.
26 Since nuclei are made up of elementary particles that have spin, consequently, they too have a spin (I) and azimuthal (m) quantum number associated with them. The relationship between these numbers and the information provided by them is analogous to that for particle spin quantum numbers described above. While the quantum mechanical perspective of spins is described thus far, for the sake of clarity the f ollowing discussion will be based on semi classical approach. Fundamental particles and, hence, atomic nculei also have an intrinsic magnetic dipole moment , that is closely related to the spin angular momentum, ,by (3 1) where is a proportionality constant known as the gyromagnetic ratio 17 The direction of spin angular momentum, known as polarization spin axis, can point in any possible direction, in complete analogy to the angular momentum of a rotating object. In the presence of an external magnetic field, the response of the polarization is to move around the magnetic field This motion is physically described to result from a torque, that is applied to the atomic spin as given by: 18 (3 2) Torque is the time derivative of angular momentum, leading to the following relationship: (3 3) Traditionally in NMR the applied magnetic field is aligned along the z direction such that =(0,0,B 0 ). Combining this with the cross product shown in Eq 3 3 reveals the time evolution of the magnetic dipole momen t in each Cartesian component as a function of
27 Figure 3 1. Precession of magnetic dipole moment in response to the application of a magnetic field in the z direction is a function of the initial direction of the spin. its direction before application of the magnetic field. If visualized, the magnetic moment moves on a cone, maintaining a constant angle between itself and the field as shown in Figure 3 1. This motion is called precession and a quantum treatment of it shows that the frequency of precess ion, is given by: (3 4) where is the gyromagnetic ratio and is the magnetic field applied Eq 3 4 also applies to nuclear spins for which the angular frequency is termed Larmor frequency. The energy of a magnetic dipole, E, in the presence of a magnetic field is given by: (3 5) This energy depends on the relative direction of the magnetic dipole and magnetic field vectors. As mentioned earlier, a magnetic field. B 0 applied in the z direction, simplifies
28 Eq 3 5 to show that each spin state will have a distinct energy level an effect known as Zeeman splitting : (3 6) Eq uation 3 5 indicates that the magnetic energy is lowest if the magnetic moment is parallel to the magnetic f ield. The Boltzmann distribution describes the population distribution of the spin states to a good approximation. 9 Accordingly, the population of nuclear spins in the lowest energy state is higher than in the higher energy state. In NMR spectroscopy, a macroscopic ensemble of molecules is generally observed. The application of a magnetic field to a large sample o f nuclear spins results in a magnetization vector, that is the sum of all nuclear dipole moments: (3 7) The net magnetization is in the direction parallel to and arises from the relative difference in population of spins at different energy states. Since the energy is not dependent on the x and y components, the vector sum of the nuclear dipole moments along these coordinates is zero. Hence, at thermal equilibrium, the magnetization is M eq aligned along the z direction as visualized in Figure 3 2 3.1.2 Longitudinal and Transverse Relaxation is a term widely used to refer to the re establishment of thermal equilibrium after application of a perturbation. In NMR studies, before the introduction of a magneti c fi eld, the system is in equilibrium in a state where all spins orientations are equilibrium state shown in Figure 3 2. The time it takes for magnetization to reach this latter state is dictated by the time constant, T 1 also known as the longitudinal relaxation
29 Figure 3 2. Schematic of a collection of nuclear spins at thermal equilibrium after being subjected to a magnetic field. The result is in a net macroscopic magnetiza tion aligned along the direction of the magnetic field. time constant. The build up of this magnetization is described by the following equation: (3 8) The relaxation constant T 1 is a function of the sample and the n uclear isotope. It can range from seco n ds to even days in exceptional cases. The magnitude of the magnetization M eq is almost undetectable, making experimental study of this magnetization impractical. There is no n et magnetization in the x y plan e as disc ussed in the previous section. To measure the magnetization that does exist along the z radians around the x axis by applying the appropria te radiofrequency (r.f.) pulse. Accordingly, the ne t polarization along the z axis is transferred to align with the y axis. Transverse magnetization refers to this net magnetic moment perpendicular to the magnetic field direction. Once the r.f. pulse is turned off, every single spin begins to
30 precess, and hence the net magnetic moment also precesses. Figure 3 3 is a schematic demonstrating this motion. The microscopic field experienced by these spins fluctuates over time, causing their precession frequency to vary. This causes the precessing nuclear spins that make up the net magnetic moment to get out of phase with each other. The result is an irreversible decay of the transverse magnetization. The time constant, T 2 dictates the time it takes for this magnetization to decay. It is referred to as the tran sverse relaxation time. The time evolution of the components of the net magnetization after the r.f. pulse is removed is described by: (3 9) (3 10) The value of T 2 also depends on the system being studied. For small molecules in liquids, it is on the same order of magnitude as the T 1 relaxation time. For ga s molecules in porous materials, it can be as short as a few milliseconds. Figure 3 3. The transfer of the net magnetization from the z axis to the y axis is possible with the application of a r.f. pulse. The transverse magnetization then continues to precess in the xy plane.
31 3.1. 3 NMR Spectroscopy The transverse magnetization is detectable as it oscillates at a well defined frequency (Eq 3 9 and Eq 3 10). The rotation of the magnetic moment produces a ons, a changing magnetic field is related to an electric field which can generate an electric current if a coil is introduced perpendicular to the transverse plane. The resulting oscillating current is called the free induction decay (FID). To briefly sum marize, the NMR spectrometer is capable of: (i) applying a large magnetic field to produce a net magnetization; (ii) applying r.f. pulses to rotate the spin polarization into the transverse plane; (iii) detecting the electric current generated by the prece ssing transverse magnetization. The frequency and duration of the r.f. pulse influences the degree of rotation of the net magnetization. Hence, the application of a resonant r.f. pulse can rotate the net magnetization between the longitudinal and transvers e plane. 3.2 Pulsed Field Gradient NMR Pulsed Field Gradient (PFG) NMR, an application of NMR, is able to trace molecular displacements on the order of 0.1 m to hundreds of micrometers by determining the mean propagator for displacements. It is a powerfu l technique to investigate diffusion in nanoporous and microporous systems as it can distinguish between various features of mass transfer such as intra crystalline diffusion, permeation of surface resistances, and long range diffusion. 12 PFG NMR measurements are performed under conditions where there is no significant convective flow and there is a uniform macroscopic chemical potential throughout the sample. Hence, the
32 measurements conducted are of self diffusion of a given sorbate. The following sections elaborate on the fundamentals of PFG NMR as w ell as the type of measurements. 3.2.1 Field Gradient PFG NMR is a technique that is based on creating an inhomogeneous magnetic field B eff that varies with position. This is achieved by the superposition of a magnetic field, gz, that depends on position on to the constant field, B 0 : (3 11) where the field gradient, g, is applied along the z direction. The consequence of the application of the position dependent field gradient is the spatial dependence of the Larmor frequency of nuclei. Th e resulting effective Larmor frequency, is given by: (3 12) Figure 3 4 is a schematic illustration of the effect of the effective B eff on the effective Larmor frequency, The phase angle accumulated by the nuclear spins as a result of their proces sion motion during the application of the field gradient pulse applied for a time, is described by Eq 3 13: (3 13) The average of positions of nuclear spin s as recorded by the angular phase, becomes acces sible through the NMR signal. 3.2.2 Spin Echo The measurements of diffusion based on PFG NMR technique are based on the application of two field gradients applied over short time intervals. Hence, this yields the difference in location of molecules during these gradient pulses. Figure 3 5 illustrates
33 Figure 3 4. Schematic illustration of the combined effect of a static, homogeneous magnetic field and a spatial dependent field gradient that results in a spatially dependent precession. Figure 3 5. Schematic illustration of the three steps that describe all NMR diffusometry experiments that involve the application of two gradients of equal magnitude and duration but opposite sign and separated by a certain time delay allowed for diffusion.
34 the thr ee general components of all NMR based diffusometry experiments: (i) labeling the initial position during the application of the first field gradient; (ii) a time delay that allows molecules to diffuse to a new position; (iii) reading the final position by means of the second field gradient that is equal in magnitude and duration but opposite in sign to the that employed in (i). In practice, PFG NMR sequences are based on the spin echo experiment. As r.f. pulse is initially appli ed to rotate net magnetization arising from the presence of homogeneous magnetic field, B 0 into the transverse plane. The application of a gradient field then causes a distribution of the precession frequencies which translates to a reduction in the obser ved magnetization vector, which following the first gradient inverts the accumulated phase. For example, if a phase A is acquired during time, in which the field g radient was present, it is inverted to A with after the second r.f. pulse, the spins continue to precess and accumulate an additional phase B in the presence of the gradient field. In the ab sence of translational diffusion and transverse relaxation, there is no phase difference acquired between the two intervals, resulting in the acquisition of maximum magnetization. In reality, however, the phase accumulation in these two intervals is dist inct due to self diffusion of molecules as well as fluctuations in the microscopic field. Subsequently, the maximum amplitude of the spin echo decreases with time. While the measurement of the magnitude of transverse magnetization after a spin ec h o can re veal information about self diffusion and transverse relaxation, it is
35 limited in its utility for studying more complex systems such as porous materials. However, the concept of spin echo is employed in other PFG NMR sequences that are traditionally used f or such experiments, as outlined in the next section. 3.2.3 PFG NMR Spin Echo Sequences Stejskal and Tanner introduced the first PFG NMR pulse sequence known as the pulsed gradient spin echo (PGSE) pulse sequence. 19 In this sequence, the gradient field is pulsed for short intervals, during the time interval, between r.f. pulses. Figure 3 6 (a) presents a schematic of this s equence. As presented in the discuss ion of spin echoes, the application of the gradient pulses results in accumulation of spatially dependent phases which can be related to measure d transverse magnetization, by the following expression: 19 (3 14) where M eq is magnetization at thermal equilibrium, , and are time intervals as labeled in Figure 3 6 (a). The self diffusion process can be studied by altering the experimental parameters g, and while keeping constant and measuring the transverse magnetization. Keeping consistent eliminates the additional varying decay caused by T 2 r elaxation effects with changing time. More sophisticated pulse sequences have appeared in the study of self diffusion using PFG NMR since the introduction of PGSE. Tanner published a new sequence known as the pulsed gradient stimulated echo (PGSTE) sequen ce that has greater utility in the field of transport in porous materials. 20 The PGSTE sequence is useful for cases where T 2 is much shorter than T 1 as is often true for molecules in microporous
36 mate rials. A schematic presentation of the pulse sequence is shown in Figure 3 6 (b). pulses. These new consecutive pulses share the same r.f. phase and are separated by a time interval equal to During this time interval, the magnetization is stored in the longitudinal direction. Along this axis, the nuclei relax as per the T 1 relaxation time constant allowing for much longer diffusion times as compared to in the PGSE sequence The expected transverse magnetization based on the pulse seq uence parameters presented in Figure 3 6 (b) are as follows: (3 15) Any additional unwanted magnetization in the transverse plane between the application of t he 2 nd and 3 rd gradient pulse known as the spoiler gradient. H eterogeneous materials, suc h as microporous materials can create internal gradients which arise from interfaces created by s urfaces of varying magnetic susceptibilities. The presence of internal gradients has unintended effects on the magnetic field and can manifest as distortions leading to misleading measured attenuation. To minimize these effects, Cotts et al. proposed the 1 3 interval sequence also known as the pulsed field gradient stimulated echo bipolar sequence (PGSTEBP). 21 Figure 3 6 (c) is schematic of the 13 interval sequence which employs gradients of opposing pola rity. The effects of internal gradients are hence refocused after the second interval. The relationship between the transverse magnetization and the experimental parameters is predicted by Eq 3 16:
37 (3 16) 3.3 Ex perimental NMR Details The NMR experiments presented in this dissertation were performed at the Advanced Magnetic Resonance Imaging and Spectroscopy Facility of the National High Magnetic Field Laboratory, located at the University of Florida. Tw o Bruker B iospin spectrometers, 17. 6 T and 14 T, were used for all measurements discussed. Magnetic field gradients were generated by d iff60 and d iff30 diffusion probe s that are capable of producing gradients of amplitude as large as 30 T/m and 18 T/m, respectively 13 C and 1 H nuclei were probed based on the nature of the sorbate and sorbent being studied. The use of high magnetic field afford ed enhanced NMR sensitivity. In particular, in the reported 13 C NMR studies using high field offered vast improvements in s ignal to noise ratio of measured NMR signals. The focus of this work is on sorbents that have a strong influence on the sorbate dynamics being measured, either through confinement or interaction. Hence, the use of high field gradients offered the possibili ty to measure very small displacements, in the ran ge of 100s nm to 10 m, and perform measurements of very slowly diffusing molecules. Application of large gradients also allows performing measurements under conditions of short T 2 relaxation times because, in this case, even short gradients can result in large signal attenuations Diffusion measurements were carried out using the 13 interval or PFGSTEBP pulse sequence due to its resilience towards potential artefacts produced by the samples of interest in this work. In all cases, unless otherwise specified, the pulse sequence included a longitudinal eddy current delay (LED). During this additional time interval, magnetization is stored along the longitudinal axis as any additional eddy
38 currents induced in t he gradient coil decay. In addition, spoiler gradients and phase cycling were employed to minimize any distortions in the NMR signal and in turn increasing the reliability of the results. Measured attenuation curves were used to acquire the self diffusion data for all samples. Attenuation ( ) is defined as the PFG NMR signal obtained with the application of gradient strength, g(z), normalized to the signal obtained with the application of the smallest gradient possible, g(z) 0. All other experimental parame ters of the pulse sequence are kept constant as the amplitude of the gradient is altered. The NMR signal recorded is obtained by integrating the area under the peak belonging to the sorbate of interest. While the relationship between the attenuation of si gnal and the experimental parameters depends on the diffusion mechanism, certain parameters are consistent. Under the conditions that allow normal self diffusion of multiple molecular ensembles that exhibit distinct effective diffusivities the diffusion induced signal attenuation can b e predicted by the following attenuation equation: (3 17) where p i and D i are the population facti on and self diffusivity, respectively, of ensemble i. The factors q and t are calculated by the following equations: (3 18) (3 19)
39 A B C Figure 3 6. Schematic presentation of the spin echo based PFG NMR sequences: ( A) PGSE, (B) PGSTE, and (C) PGSTEBP or 13 interval
40 CHAPTER 4 SINGLE FILE DIFFUSION IN MIXTURES 4.1 Motivation Molecules restricted to diffuse through unidimensional channels so narrow that they cannot pass each other may exhibit single file diffusion (SFD ) Figure 4 1 visualizes this type of anomalous diffusion, where the sequential order of d iffusing molecules is preserved. This results in a deviation from the usual time dependence of the mean square displacement (MSD) observed for normal diffusion Whil e MSD increases proportionally with time for normal diffusion, for SFD in sufficiently long channels when all boundary effects at the channel margins can be neglected it grows with the square root of time (4 1) where F is the single file mobility factor and t is t he diffusion time. 12 Single file systems exist in various fields, from living systems to catalysis. 22 25 However, experimental evidence and understanding of diffusion in single file systems is sparse. The work o utlined in this c hapter is aimed towards investigating the fundamentals of SFD of mixtures. 4.2 Review of Published Single File Diffusion Studies An important consequence of SFD is a significant slowdown in the growth of MSD with time in comparison to that for normal diffu sion. This property opens a This chapter i s based on previously published work. Reprinted with permission from: Dutta, A. R.; Sekar, P.; Dvoyashkin, M.; Bowers, C. R.; Ziegler, K. J.;Vasenkov, S., Relationship between Single File Diffusion of mixed and pure gases in Dipeptide Nanochannels by High Field Diffusion NMR. Chem. Commun 201 5, 51 (69), 13346 13349 Dutta, A. R.; Sekar, P.; Dvoyashkin, M.; Bowers, C. R.; Ziegler, K. J.;Vasenkov, S., Relationship between Single File Diffusion of Gas Mixtures in Nanochannels of the Dipeptide Ala Val: High Field Diffusion NMR Study J. Phys. Chem. C 201 6, 120 (18 ), 9914 9919
41 Figure 4 1. Schematic illustration of molecules restricted to move in a one dimensional channel that has a radius smaller than the diameter of the molecules. Mutual passage is forbidden under these conditions. p ossibility fo r highly selective separations if one co mponent in the mixture exhibits normal diffusion and another SFD. 12, 26 29 The potential use of this property in controlled catalysis has also been proposed. 30 33 The integration of SFD in these applications has motivated studies of SFD in mixtures of different types of molecules using molecular level simulations. In particular, kinetic Monte Carlo simulations were performed to study counter diffusion, tracer exchange and reaction under s ingle file conditions. 34 35 Molecular simulations have al so demonstrated that binary mixtures of small gas molecules diffusing in AlPO 4 5 molecular sieve can exhibit different diffusion regimes. 26 In one of these regimes both species in the mixture diffused according to the mechanism of SFD, and in another regime only one type of molecules exhibited SFD while the other exhibited normal diffusion. The potential application of this latter diffusion regi me in molecular separations became the focus of Adhangale and Keffer, who investigated using molecular dynamics (MD) simulations, the diffusion of a methane/ethane mixture in AlPO 4 5. 27 A recently published analytical model based on the Maxwell Stefan equations proposed usi ng pure component mobilities and the corresponding adsorption isotherm data to estimate their mixture behavior in single file systems. 36 More recently, an MD simulation study explored single file mobilities as a function of loadings of binary gas mixtures in single walled carbon nanotubes. 37
42 While diffusion of mixtures of different types of molecules in single file channels has been the focus of many theor e tical and simulations st u dies, there have been no previously published reports on experimental investigations of transport of sorbate mixtures under the SFD conditions. In section 4. 5 .1 the first experimental observation of SFD of molecular mixtures is presented. A thorough analysis of the relationship between the mobilities of pure sorbates and their corresponding mixtures is discussed in section 4. 5.2 4 .3 Sorbent and Sorbates Studi ed Self assembled L Ala L Val (AV) hydrophobic dipeptide crystalline nanotubes were used in these diffusion studies. AV is a microporous material containing helical channels of mean diameter ( d ) of ~0.51nm. 38 It was se lected for this work because of recent studies of diffusion and tracer exchange of Xe atoms in AV nanochannels published by our group These previous studies were performed using a combined application of 129 Xe PFG NMR and hyper polarized 129 Xe spin tracer excha nge NMR. These studies reve a l ed that Xe in AV nanochannels is a highly ideal single file system. AV nanotubes were purchased from Bachem and used as received. The average length of the AV crystal ( L ) is around 50 micrometer s, as determined from the images recorded by a JEOL 6400 scanning electron microscope. 39 Diffusion of CO, CO 2 and CH 4 (Sigma Aldrich, 99 at om% 13 C) in AV nanochannels was investigated by 13 C PFG NMR. These spe cies were chosen because they simultaneously satisfy the spatial criteria for SFD observation and are industrially relevant gases. The spatial criteria require that the diameters of molecules are smaller than the diameter of AV channels, but greater than t he radius of the channels. The latter requirement prevents mutual passage of the molecules. The kinetic diameters of CO,
43 CO 2 and CH 4 molecules are, respectively, a round 0.34, 0.33, and 0.41 nm. 40 Clearly, all three speci es satisfy the SFD criteria for diffusion in AV nanochannels. 4.4 Experimental Details PFG NMR can be used to directly measure the dependence of MSD on diffusion time in porous materials. Therefore, this technique is ideally suited for distinguishing SFD, characterized by the MSD scaling given by Eq 4 1. Five millimeter diameter, medium wall NMR tubes (Wilmad Glass) were loaded with ~55 mg of AV nanochannels. Each tube was then separately, connected to a custom made vacuum system and the nanochannels wer e activated at an elevated temperature of 373 K overnight. The activation was performed under high vacuum conditions at pressures below 0.001 mbar. This degassing of AV is intended to remove any initially present sorbates The activation temperature is w ell below the AV melting point of 511 K. 38 Upon activation, each NMR sample was loaded with either a pure gas or a two component gas mixture by cryogenic transfer of the corresponding gas or g as mixture from the calibrated volume of the vacuum system. The gas loaded samples were flame sealed and separated from the vacuum system. Three NMR samples loaded with one component gas (CO, CO 2 and CH 4 ) and two samples loaded with two component mixtu res (CO/CH 4 and CO/CO 2 ) were prepared. The total gas pressure in the gas phase of each sample at 298 K was around 10 bar. In mixture samples, the partial pressure of each sorbate was 5 bar. For each sample the concentration of each gas in AV was determined by comparing the NMR signal of adsorbed gas(es) in these samples with the NMR signal in an NMR tube containing only pure gas at a known pressure, in the same way as d iscussed in previous work published by our group 41
44 Table 4 1. Gas concentration (in mol/kg of AV) obtained for the studied samples using NMR signal analysis. Single Sorbate CO/CH 4 CO/CO 2 CO 0.56 0.06 0.34 0.03 0.24 0.02 CH 4 0.48 0.05 0.20 0.02 CO 2 0.52 0. 05 0.34 0.03 Total gas concentration for mixture samples 0.54 0.02 0.58 0.03 The gas loadings for all samples are shown in Table 4 1. This table indicates that the total gas concentration in each sample was the same, within the limits of exp erimental uncertainty. It also shows that the concentration o f each gas in the mixture was comparable. This is significant as it allows for direct comparison of the rate of the diffusion of the molecules in single sorbate and mixture samples. Longitudinal (T 1 ) 13 C NMR relaxation times of the studied gases inside AV nanochannels were estimated using the 13 interval PFG NMR sequence by changing sequence and keeping all other time intervals constant. Similarly, transverse (T 2 ) 13 C NMR relaxation times were estimated using the same sequence by changing the time and keeping all other time interval s constant. Under the conditions of these measurements, it was confirmed that there was no attenuation of the signal due to diffusion of adsorbed gas molecules. In all of the studied samples, the measured NMR relaxation data showed no evidence for distrib ution of T 1 and T 2 relaxation times for adsorbed gases. The T 1 NMR relaxation times for CO, CO 2 and CH 4 molecules adsorbed in the channels were estimated to be around 0.6, 2.7, and 1.5 s, respectively. The corresponding T 2 NMR
45 relaxation times for CO, CO 2 and CH 4 were found to be around 16, 18, and 22 ms, respectively. All diffusion and relaxation NMR measurements were performed at 298 K. PFG NMR diffusion measurements were performed using a 17.6 T Bruker BioSpin NMR spectrometer operating at 13 C resona nce frequency of 188.6 MHz. The in Gainesville, FL. Sine shaped, bipolar magnetic field gradients with the effective duration of 1500 2200 s were generated using a di ff60 or diff30 diffusion probe (Bruker BioSpin) and Great60 gradient amplifier (Bruker BioSpin). The range of the gradient amplitudes used was between 0.3 and 23 T/m. 4. 5 Observation of Single File D iffusion in Mixtures The 13 interval bipolar PFG NMR puls e sequence with longitudinal eddy current delay was used for all diffusion measurements The longitudinal eddy current delay was between 3.5 and 5 ms. Diffusion data were obtained by measuring the PFG NMR signal intensity (S (g) ) as a function of the magnet ic field gradient ( g ) while keeping all other sequence parameters constant. For each type of the studied gas molecules S was equal to the area under the single 13 C NMR line of these molecules. In the mixture samples, the different species were identified b y their distinct 13 C chemical shifts resulting in non overlapping 13 C NMR lines of these species. Under the measurement conditions, the 13 C NMR chemical shifts CO, CO 2 and CH 4 were equal to 190.2, 135.6, and 5.4 ppm, respectively. 4.5 .1 Attenuation P lots A ttenuation plots record the decrease in PFG NMR signal with increasing gradient strength, g, normalized to the signal obtained at the smallest gradient, as a
46 function of q 2 where q is equal to 2 g is the gyromagnetic ratio, and is the effective gr adient duration. The attenuation ( ) is given by the following equation: (4 2) where S(g) is the signal obtained with the application of gradient g and S(g 0) is the signal corresponding to the smallest gradient that can be accurately p roduced by the hardware. The PFG NMR echo attenuation due to one dimensional diffusion in straight randomly oriented, channels is expected to follow: (4 3) where is the MSD along the channel axis at diffusion time t. Eq uation 4 3 simplifies to the following expression after evaluation of the integral: 12, 39, 42 (4 4) It is importan t to note that due to the helical topology of the AV channels, the actual MSD inside the AV channels along the diffusion path ( ) is larger than the MSD along the channel direction ( ) in Eq 4 4. The values of can be est imated as where p is the geometrical factor equal to 1.3. The value of this factor was determined using the values of the pitch of the helix (~1 nm) in AV channels and the maximum Xe atom capacity per unit pitch 42 Hence, for AV channels, Eq uation 4 4 can be re written as: (4 5)
47 For SFD in AV channels, Eq 4 1 can be substituted into Eq 4 5: (4 6) where F helix denotes the single file mobility factor inside the AV channels, viz. mobility factor along the helical path. Figures 4 2 and 4 3 show exampl es of the measure 13 C PFG NMR attenuation curves as a function of q 2 t. In the presentation of these figures, the attenuation curves measured for each sample and species at different times, but otherwise consistent conditions, are expected to collapse into a single curve if the MSD grows with the square root of diffusion time, as prescribed by Eq 4 1. It is seen in Fig ures 4 2 and 4 3 that for each studied sample and type of molecules the attenuation curves measured for a broad range of diffusion times coin cide, within experimental uncertainty, in this presentation of the figures. Hence, it can be concluded that in each case the diffusion process obeys the SFD mechanism. This is an expected result because the channel radius is smaller than the collision diam eter of the studied molecules, leading to the SFD conditions for these molecules. 4.5.2 Mobilities of Sorbates in Pure and Mixture S amples Figure 4 2 shows PFG NMR attenuation curves for the diffusion in AV channels loaded with a single gas. In Figure 4 2 (a), such curves are shown for CO and CH 4 i.e. the gases used in the CO/CH 4 mixture. In Figure 4 2 (b) the attenuation curves are shown for CO and CO 2 i.e. the gases used in the CO/CO 2 mixture. Attenuation curves measured for each single gas coincide wi thin experimental error for the broad range of diffusion times reported. It is seen in Figure 4 2 that for each gas pair used in the mixtures the attenuation curves in the samples loaded with a single gas are different for
48 different gases. Least squares fi t of the attenuation curves in Figure 4 2 for the samples loaded with the single gas (CO, CH 4 and CO 2 ) to Eq 4 6 resulted in the single file mobility factors for these gases (Table 4 2). The corresponding best fit curves are shown in Figure 4 2 S mall devi ations between the experimental data and t he best fit curves in Figure 4 2 for the single component samples can result from some small differences between the actual distributions over the channel directions in the samples and the fully random distribution assumed in the derivation of Eq 4 6. Figure 4 3 ( a, b ) shows PFG NMR attenuation curves for the diffusion of CO/CH 4 and CO/CO 2 mixtures in AV channels. It is seen in the figure that for each mixture the attenuation curves measured for different types of m olecules in a mixture coincide within the experimental uncertainty. According to Eq 4 6 this coincidence of the attenuation data indicates that both types of molecules in a mixture diffuse with an identical single file mobility factor F helix Identical tr ansport rates for different types of molecules in a mixture are expected under the SFD conditions because under such conditions molecules cannot pass one another in the channels Hence, the observation of identical rates of transport for different types o f molecules in the studied mixtures provides additional evidence for the SFD diffusion mechanism in the mixtures. Least squares fit of all the attenuation curves measured for the same mixture type to Eq 4 6 resulted in the values of F helix for the studied two mixtures that are reported in Table 4 2. The best fit curves are shown in Figure 4 3 It is seen that there are some small systematic deviations between these curves and the measured data. In complete analogy with single gas samples, t hese deviations can be attributed to some small differences between the distributions over the channel
49 Figure 4 2. 13 C PFG NMR attenuation plots measured at 298 K for CO, CH 4 and CO 2 diffusion in AV channels loaded with a single gas. The attenuation plots are shown for the following gases used to prepare the gas mixtures: (a) CO and CH 4 and (b) CO and CO 2
50 Figure 4 3 13 C PFG NMR attenuation plots measured for CO, CH 4 and CO 2 diffusion in AV channels at 298 K. The attenuation plots are shown for the samp les containing the following gas mixtures: (a) CO/CH 4 and (b) CO/CO 2
51 Table 4 2. Single File mobilities (F helix ) estimated from attenuation curves in Figure 4 1 and 4 2 using Eq 4 6. Mixtures Single Sorbate Samples CO + CH 4 CO + CO 2 CO CH 4 CO 2 F helix x 10 13 (m 2 s 1/2 ) 31 2 25 2 46 4 8.0 0.6 15.0 1 directions in the PFG NMR samples and a truly random distribution, which was assumed in the derivation of Eq 4 6. 12, 39, 42 Figure 4 4 presents the values of as a function of diffusion time for each of the two mixture samples together with the corresponding data for the one component samples The MSD values were obtained from least square fitting of individual PFG NMR attenuation curves meas ured for each sorbate and sample at each diffusion time to Eq 4 4. These values were then scaled to account for the helical topology as previously discussed. The proportionality between the values of MSD and square root of time in Figure 4 4 was verified b y least square fitting of the data for each sample to Eq 4 1, where and F are substituted by and F helix respectively. In the double logarithmic presentation of Figure 4 4 the best fit lines are solid lines, which in all cases sho w a good agreement with the measured data. For comparison, the dashed line shows the linear growth of MSD with diffusion time which is expected for normal, Fickian diffusion. It was verified that the values of F helix obtained from the fitting of the data i n Figure 4 4 are in agreement, within experimental error, with the corresponding data in Table 4 2 which resulted from direct fitting of the attenuation curves. Table 4 1 shows that the intrachannel gas concentration in each of the studied single componen t samples was the same, within the experimental uncertainty. The
52 Figure 4 4. MSD dependence on times plotted on a double log scale for (a) CO/CH 4 mixture and (b) CO/CO 2 mixture along with the corresponding single component samples. The MSD is obtained by fitting the attenuation curves shown in Figures 4 2 and 4 3 with Eq 4 4. The MSD reported here accounts for the helical topology of the channels. Solid lines are best fit lines of the measured time dependences of MSD to Eq 4 1. Dashed line shows the gro wth of MSD with time expected for normal diffusion.
53 concentrations in the single component samples were also nearly identical with the total gas concentration in the mixture samples, where the fractions of the individual gas components were comparable t o each other (Table 4 1). All these considerations allow for a direct comparison of the single file mobility factors in the single component samples and their relationship with the single file mobilities in the c orresponding mixture samples. The data in Ta ble 4 2 indicates that in the single component samples the fastest diffusing sorbate is CO, followed by CO 2 and CH 4 This trend in decreasing single file mobilities of CO, CO 2 and CH 4 is consistent with the combination of larger molecular mass and size of CO 2 and CH 4 relative to CO. Table 4 2 shows that the single file mobilities in the mixture samples is intermediate in comparison with the corresponding single sorbate mobilities. This observation suggests that under the SFD conditions the replacement of th e slower diffusing molecules (CH 4 or CO 2 ) by an approximately the same number of faster diffusing molecules (CO) in AV channels leads to a noticeable increase in the single file mobility of the slower diffusing sorbates. In contrast, under the conditions of normal diffusion in microporous solids such a replacement of a fraction of a slower diffusing species by faster diffusing species often does not lead to any significant change in the diffusivity of the slower diffusing molecules. 12 This difference between the SFD and normal diffusion of mixtures c an be rationalized by noticing that under the conditions of SFD the displacement of any molecule strongly depends on the displacements of its next neighbors. Such dependence arises in SFD from the fact that movements of next neighbors determine the availab ility of space for displacements of any particular molecule.
54 In the studied mixture samples, fast local mobility of CO molecules surrounding CH 4 or CO 2 molecules can lead to faster diffusion of these CH 4 or CO 2 molecules in comparison to the single compon ent CH 4 or CO 2 samples. This consideration provides a tentative explanation for the difference between the single file mobilities of CH 4 or CO 2 in the mixture samples in comparison to those in the correspon ding single component samples. However, further an alysis and discussion presented in the next chapter indicate that the relationship between the single file mobilities of the studied pure and mixed sorbates is likely related to molecular clustering. 4.6 Summary Single file diffusion of single component ga ses and their binary mixtures in AV nanochannels was investigated by 13 C PFG NMR at a high magnetic field using large field gradients for a broad range of diffusion times. The SFD mechanism was confirmed by verifying the proportionality of the MSD with the square root of time for each sorbate in both single component and mixture samples. This represents the first experimental evidence of SFD in a mixture of different types of molecules. The SFD mobilities of both gases in the two component mixture samples were found to be identical, which is consistent with the exclusion of mutual passages of molecules under the single file conditions. At comparable total molecular concentrations in the channels, the mixture SFD mobilities were observed to be intermediate b etween observed for normal diffusion where the replacement of a fraction of a slower diffusing component by a faster diffusing one usually does not significantly influen ce the diffusivity of the former.
55 CHAPTER 5 ABSENCE OF TRANSITION TO CENTER OF MASS DIFFUSION 5.1 Single File Diffusion in Finite Systems Theoretical treatment and computer modeling of diffusion in channels of finite length under the SFD conditions predic t a transition from the single file regime to the center of mass (COM) regime at sufficiently large values of root MSD 14 15, 35 The COM diffusion is characterized by fully correlated displacements of all the molec ules in any particular channel. Figure 5 5 illustrates the difference in molecular motion between SFD and COM diffusion. Consecutive movements of such molecular chains are not correlated with each other resulting in applicability of the laws of normal dif fusion. In particular, a linear relationship between the MSD and diffusion time is expected in the COM regime. Hence, it is possible to define COM diffusivity in the same way as for normal diffusion by using the Einstein relation. COM diffusivity is great ly reduced in comparison to the diffusion coefficient under the conditions of normal diffusion, where an exchange of the relative positions of molecules in channels is allowed. For the single file diffusion process modeled as a random walk of particles wi th hard core interactions a transition from the SFD to COM regime is predicted to occur at the MSD given by the following relation: 15, 35 (5 1) This chapter is based on previously published work. Reprinted with permission from: Dutta, A. R.; Sekar, P.; Dvoyashkin, M.; Bowers, C.R.; Ziegler, K.J.; Vasenkov, S., Possible Role of Molecular Clustering in Single file Diffusion of mixed and pure gases in Dipeptide Nanochannels. Microporous and Mesoporous Mater 2017
56 Figure 5 1. Illustration of difference in molecular motion in SFD and COM diffusion. After sufficient time in finite channels with fast exchange at boundaries, motion of molecules becomes extremely correlated and is independent of i ts previous movement, exhibiting Fickian motion. However, such a transition is yet to be ob served experimentally. For all mixture and one component systems reported in Chapter 4 the crosso ver MSD was estimated using Eq 5 1 The The average channel length l along the diffusion path in Eq 5 1 was determined using the known average length of the studied AV crystals (L = 50 m). This length corresponds to the channel length along the helix axis. The value of l was obtained by multiplying L with the geometrical factor p = 1.3, which was discu ssed earlier. to the corresponding maximum theoretical concentration. Although the maximum theoretical concentrations for CO, CH 4 and CO 2 in AV channels have not been l SFD COM
57 reported, these lo adings can be estimated using the known maximum loading for Xe in these channels. 38 For each studied gas the maximum loading was roughly estimated as the maximum loading of Xe atoms in AV mult iplied by the ratio of the kinetic diameters of Xe and the studied gas. Using the reported kinetic diameters, which are equal to 0.44, 0.34, 0.33, and 0.41 nm for Xe, CO, CO 2 and CH 4 respectively, the frac tional loadings obtained for each sample are give n in Table 5 1 40, 43 models as the distance between two adjacent sorbate molecules at a maximum loading in channels. Previous studies of Xe in AV channels have reported that the most efficient packin g is achieved when Xe atoms are stacked abreast in a staggered arrangement. 38 The distance between two adjacent Xe atoms at the maximum loading was estimated to be , of the kinetic diameter of a Xe atom. Owing to the similarity between the kinetic molecules as of the kinetic diameter. Table 5 1 Estimates of fractional loadings ( ), the values of MSD at wh ich a transition from SFD to COM diffusion is predicted by Eq 5 1 ( ), and the minim um cluster size required by Eq 5 1 for not observing this transition under the reported experimental conditions of this study. Mixtures Single Sorbate Samples CO + CH 4 CO + CO 2 CO CH 4 CO 2 0.22 0.02 0.16 0.02 0.20 0.01 0.21 0.01 0.18 0.02 (m 2 ) 14 14 14 14 14 Minimum cluster size (number of molecules) 140 70 110 30 50 Table 5 1 lists the crossover MSDs, ob tained using Eq 5 1 with the parameters estimated as discussed above. It is seen that the values of in Table
58 5 1 are in all cases smaller than the smallest M SD measured by PFG NMR (Figure 4 4 ). Henc e, a linear dependence of MSD on time, a sig nature of COM diffu sion, should have been observed for all samples under the reported experimental conditions. However, contrary to this, the experimental data in Figure 4 4 show the MSD dependence on diffusion time as given by Eq 4 1 for SFD. 5.2 Re latio nship between Single File M obilities The theoretical model (Eq 5 1 ) and the experimental data presented in Chapter 4 can be reconciled if presence of long range interactions between molecules is considered, which could induce molecular clustering. Formatio n of molecular clusters was previously demonstrated under the SFD conditions using MD simulations. 44 46 These studies demonstrate that single file diffusion of small gas molecules can be dominated by correlated mov ements of many molecules, viz. molecular clusters, rather than by movements of individual molecules. In the scenario where individual molecules form large clusters that perform SFD, the actual elem 1 is determined by the si ze of clusters, which is much larger than the collision diameter of the molecules used to estimate the c rossover values shown in Table 5 1 larger values for all samples studied. Hence, by assuming that the mi nimum crossover MSD is as large as the greatest e xperimentally measured MSD, Equation 5 1 can be used to estimate the smallest cluster size for each system. This is done by using the inherent proportionality between the number of molecules in a clu ster and manipulating Eq 5 1 system. The projected values of the minimum cluster sizes are reported in Table 5 1 They vary between 30 to 140 molecules, depending on the system.
59 The presence of molecular clustering in the context of single file diffusion can also explain the relationship observed between the SFD mobilities of single component sorbates and their mixtures. In accordance with the random walk model, the SFD mobility varies linearl y with the produc D 0 of a single molecule/cluster in the infinite dilution limit. 12 D 0 is expected to be inversely be expected that the single component sorbate that exhibits the largest SFD mobility (CO) forms larger clusters than its slower counterparts (CH 4 and CO 2 ). One reason for CO molecules to form larger clusters than CH 4 and CO 2 can be related to dipole dipole interactions between CO molecules. The average cluster in mixture samples would likely be intermediate in size with respect to their corresponding single sorbate samples. This would translate into intermediate SFD mobilities of mixtures relative to their respective one component sorbates, which is consistent with the experimental observations reported here. More studies are needed to investigate the cluster ing of molecules. In particular, measurements of adsorption isotherms can be useful. It is discussed in R ef 4 7 that the shape of the adsorption isotherm can reveal the extent of interaction between molecules, especially if it exceeds their interaction with the host. 47 Such a strong interaction could result in clustering. While such an analysis can be insightful in general, the framework of AV nanochannels i s flexible and can alter with loading resulting in adsorption isotherms that become difficult to interpret conclusively. In addition to molecular clustering discussed above, the existence of strong transport barriers at the ends of AV channels can be res ponsible for the absence of a
60 transition to COM diffusion at the MSDs reported in Table 5 1 15, 35 Such transport barriers, howev er, are not expected to appreciably extend the length scale of the SFD regime, which remains unperturbed by boundary effects at the channel ends. Since t he MSD time scaling of SFD (Eq 4 1) is observed for all sorbates, both in pure and mixture form, at len gth scales beyond those of COM transition, it is unlikely that transport barriers are singularly responsible for preventing this transition. However, additional theoretical studies are needed to clarify the contribution of transport barriers at channel bou ndaries to the transition from SFD to COM diffusion. The existence of clusters can be experimentally investigated through techniques such as Fourier Transform Infrared Spectroscopy. The exact size of these clusters, however, would be difficult to corrobora te given the resolution limitations of FTIR. Distinguishing between clusters consisting of molecules in the wide range between 50 to 150 molecules is challenging under the conditions of this experimental technique. A possible combination of experimental an d computational study is recommended to provide further insight into the role of clusters in this study. 5.3 Summary Simulation studies and theoretical considerations predict a transition from the SFD to COM regime at sufficiently large diffusion times. The crossover is characterized by a shift from the MSD dependence on the square root of time to a linear dependence on time. However, this transition to COM regime was not experimentally observed for any of the systems studied. Molecular clustering is pro posed to explain the absence of this transition. Eq uation 5 1 was used to estimate the minimum cluster size needed to sustain SFD under our experimental conditions. The clustering hypothesis was also used to qualitatively explain the unique relationship be tween the SFD mobilities of
61 components in the single sorbate and mixture samples. This data suggest s that molecular clustering in single file channels can be one of the main factors determining the rate of single file diffusion of mixed and pure sorbates.
62 CHAPTER 6 METHODOLOGY DEVELOPMENT CALIBRATION OF MAGIC ANGLE SPINNING PROBE FOR NMR DIFFUSOMETRY 6.1 Mot i vation Pulsed Fi e l d Gradient NMR is a powerful technique to investigate diffusion in porous materials. Our group, in conjunction with the Nationa l High Magnetic Field Labor atory, has developed an experimental setup that combines high magnetic field and large field gradients to study diffusion properties on different length scales in nanoporous and microporous materials. However, there are som e diff iculties associated with such materials, e specially when studying mixture sorbates. 48 Firstly, the restricted mobility of molecules in pores can result in line broadening. Secondly differences in local magnetic fields originating from sample heterogeneities can contribute further to large broadening of the signal. This eff ect of line broadening limits the applicability of PFG NMR and the corresponding reduction in resolution can inhibit identification of components in mixtures principally for 1 H NMR Thirdly, the line broadening also manifests the reduction in T 2 NMR relax ation times, which restrict the duration of gradients that can be applied and hence restricts the range of displacements that can be probed. The combination of Magic Angle Spinning (MAS) with PFG NMR presents a possible solution to the listed challenges a ssoci ated with conventional PFG NMR. 49 50 In field, as portrayed in Figure 6 1, and rotated about this axis at a desired frequency. 17 Influence of a nisotropic interactions such as dipolar coupling and susceptibility effects on the line width and related T 2 NMR relaxation time can be reduced or suppressed
63 Figure 6 1. A schematic of the MAS expe rimental setup combined with the pulsed field gradient coil, g(z), in a high resolution MAS probe. The spinning axis along which the sample is oriented is at an angle of 54.7 with respect to the external magnetic field, B 0 completely by adopting the MAS experimental design. The extent of suppression depends on the interaction strength and the achievable rotation frequency. The combination of MAS and PFG NMR to study diffu sion of single and multi component sorbates in various host materials, including zeolites has been successfully demonstrated by others. 49, 51 53 To enhance the capability of PFG NMR at the National High Magnetic Fi eld Laboratory (NHMFL) this methodol ogy of MAS PFG NMR calibration and testing was introduced at its facility at UF. This chapter is devoted to the development of this experimental setup on the 600 MHz Bruker NMR spectrometer located at the AMRIS facility of the NHMFL 6.2 Gradient Calibration The high resolution MAS probe available at the NHMFL at UF supports a 4 mm rotor and is equipped with gradient coils. The steps involved in making this probe usable for PFG NMR measurements are outlined as follows : determine strength of gradient evaluate gradient uniformity p erform temperature calibrations test accuracy of technique using diffusion measurements
64 The absolute value of the gradient strength was determined by recording the image ( 1 H NMR) of a spacer of known precise thickness, immersed in a solution of H 2 O/D 2 O, in the presence of an applied field gradient From the resulting spectra, the maximum gradient strength, g l of the coil can be calculated by: 54 (6 1) w here is the width of the image of a spacer of known length, o btained at a controlled gradient strength which is indicated by a dimensionle ss parameter, expressed as the percentage of the maximum possible gradient strength and is the gyrogmagnetic constant of protons. Two spacers of different lengths, 2.9 mm and 4.0 mm, were used for measuring g I The images of each of these sp acers were recorded for different values of g a to improve both the accuracy and precision of the meas urement. For these experiments the samples were oriented along the magic angle axis but remained static. Table 6 1 shows the calculated values of the maximum g radient strength estimated using both spacers and the method described above. Based on this data, the gradient strength of the high resolution MAS probe is determined to be 0. 56 0.01 T/ m. It should be noted that the value of g I measured using the longer spacer of 4 mm can generate a larger error associated to it. This is because there is expected to be greater variation in the magnetic field and the gradient strength at the edges of the rotor as compared to its middle. Hence, the longer sample is suscepti ble to these deviations, resulting in decreased confidence in the calculated value of g I Evaluating the uniformity of the gradient along the rotor axis is essential to assessing the potential accuracy of PFG NMR measur ements. Large variations in the
65 Tab le 6 1. Values of maximum gradient strength, g I (T/m), calculated using Eq 6 1 for two spacers of varying lengths. g a 10 % 0.55 0.51 20 % 0.57 0.57 30 % 0.57 gradient profile along the length of the rotor can translate to inaccurat e measurements of diffusivities. To assess the uniformity, a pulse sequence proposed by Hurd et al was employed. 55 It is composed of a Hahn echo with two gradient pulses g, before and after another gradient (g r ). Spectrum acquisition under the conditions of a large gradient, g, results in a heavily diffusion weighted profile. Comparison of such spectra with application of a weak and strong gradient can provide insight into the uniformity of the gradient strength along a chosen axis. This sequence was used on a sample of H 2 O that filled the length of the rotor. Under static condi tions, th e spectrum of the sample was obtained at two different gradient strengths, 6% and 50%. Figure 6 2 show s the gradient profile obtained at the mentioned gradient strengths. By normalizing the intensity of the NMR signal obtained at smallest g radient strength ( 0.03 T/ m) to the largest gradient strength ( 0. 28 T/ m) applied, the variation in the gradient profile along the magic angle axis is captured. This normalized intensity along the length of the rotor is shown in Figure 6 3 While there is large variation al ong the edges of the rotor caused by experimental artefacts the gradient profile is essentially flat al ong the bulk of the rotor axis. 54 Hence, there is sufficient uniformi ty of the gradient to proceed with further calibration.
66 Accurate diffusion measurements require precise control and knowledge of the sample temperature. Under MAS conditions, where the sample is rotating at rates as large as 8 kHz, there is a potential for the temperature of the sample to vary with the application of this mechanical force While this change in temperature is also a function of the thermal conductivity of the sample itself, this contributi on cannot be generalized However, an assessment of temperature variation as a function of spinning rate can be performed as a preliminary correction for accurate sample temperature control. The actual sample temperature was systematically estimated using neat methanol under various spinning conditions. Neat methanol can be utilized as an NMR thermometer as the difference in frequency of its two proton peaks is a strong function of temperature it experiences 56 This dependence on temperature has been well established in literature and the corresponding empirical formula was used here to estimate the temperature of the sample (neat methanol) at different spinning rates and system temperatures. The system temperature corresponds to the temperature that is set by the user for the NMR spectrometer to operate at. Figure 6 4 plots the sample temperature of methanol as a function of the system temperature for s pin ning rates in the range of 4 kHz to 8.5 kHz. As anticipated, the increase in sample temperature from the system temperature is larger for high er spin ning rates. The deviation of the sample temperature can be as large as 8 K. The data presented in Figure 6 4 is fit with the least squares line that predicts sample temperature (y) as a function of system temperature (x). These formulas can be used to determine the system temperature that should be used for a desired sample temperature at a given spin rate d uring diffusion measurements.
67 The final assessment of the accuracy of gradient and temperate calibration in this MAS PFG NMR setup culminates in the measurement of self diffusivity of water. Figure 6 5 compares the attenuation plots obtained using the 13 i nterval sequence under various conditions of a sample of water in a 4 mm rotor. There is very strong agreement between the attenuation curves acquired under static conditions and two different spinning conditions (4 kHz and 5 kHz). The system temperatures were set based on the temperature calibration presented in Figure 6 4 such that the sample temperature corresponded to 298 K. In addition, two different diffusion times (50 ms and 200 ms) were observed for each spin ning rate to evaluate the gradient calibr ation. Given the consistency in the attenuation curves for all these conditions, a single least square fit was obtained as shown in Figure 6 5. The corresponding self diffusivity of H 2 O at 298 K as evaluated from this fit is (2.2 0.1) x 10 9 m 2 /s. This v alue is consistent with the published value of self diffusivity of water at 298 K, obtained used conventional PFG NMR. 6.3 Reference Measurement in M icroporous M aterial After establishing this new methodology through the preliminary measurements of free w ater, a progression was made towards performing measurements on porous materials. As discussed in the beginning of this chapter, the re are limitations to PFG NMR related to studying diffusion in nanoporous and microporous materials. Hence, the MAS PFG NMR setup was used to demonstrate its potential in investigating diffusion in porous structures, specifically zeolitic imidazolate framework (ZIF). ZIF 8 is a MOF that consists of zinc ions coordinated with four imidazolate rings. ZIF 8 was saturated with H 2 O and the self diffusivity of adsorbed H 2 O was measured by employing MAS PFG NMR.
68 Figure 6 2. The image profile acqu ired along the rotor axis for a mixture of H 2 O/ D 2 O at two gradient strengths. The sample was oriented along the 57.4 but kept static. Fi gure 6 3 A plot of the NMR signal obtained at the smallest gradient strength normalize d to the signal recorded at the largest gradient strength. The presented gradient profile shows that the gradient is uniform along most of the magic angle rotor axis. -2000 18000 38000 58000 78000 98000 118000 -40000 -20000 0 20000 40000 Signal Itensity Frequency (Hz) g=3.3 G/cm g=28 G/cm 0 5 10 15 20 -40000 -20000 0 20000 40000 Normalized Intensity Frequency (Hz)
69 Figure 6 4. The temperature dependence of sample temperature on the spin rate plotted as a function of the system temperature determined using neat methanol. The formulas listed are obtained by the least squares fit and the relate the system temperature ( x) to the sample temperature for spin rates between 4 kHz and 8.5 kHz. y (4 kHz) = 1.03x 6.3 y (5 kHz) = 0.95x + 18 y (6 kHz) = 0.96x + 15 y (8.5 kHz) = 0.87x + 49 292 294 296 298 300 302 304 306 308 291 293 295 297 299 Sample Temperature (K) System Temperature (K) 4 kHz 5 KHz 6 kHz 8.5 kHz
70 Figure 6 5. Attenuation plots acquired using the MAS PFG NMR setup at various spin rates and diffusion times for a sample of water at 298 K. Due to the strong agreement between th e various attenuation curves a single least squares line was fit to describe them.
71 Before saturating ZIF 8 with water, a conventional 1 H NMR of ZIF 8 (non activated) was acquired under conditions of MAS at a spin rate of 7 kHz. Figure 6 6 illustr ates how effective MAS can be at improving spectral resolution. The two large peaks with line widths between 900 Hz and 1200 Hz belong to the protons in the ZIF 8 framework. The peaks marked with an asterisk are spinning side bands. The much narrower peak with a width of 40 Hz is assigned to the water present in ZIF 8. This material was not activated and hence was expected to contain some water adsorbed under ambient conditions. For comparison, the line width of adsorbed water measured using this spectromet er with a conventional PFG NMR probe is around 1 2 00 Hz Figure 6 7 compares the spectrum obtained of H 2 O in a saturated bed of ZIF 8 as obtained under MAS conditions and static conditions. A vast improvement in the line widths is observed under MAS conditi ons. The ZIF 8 bed wa s then saturated with water and after equilibration was introduced into the calibrated high resolution MAS probe. Figure 6 8 shows the overlay of 1 H spectra at different gradient strengths obtaine d using the 13 interval PFG NMR sequen ce with eddy c urrent delays at a spin rate of 5 kHz. For the spectra shown, a diffusion time of 200 ms was allowed and sine shaped gradient pulses of duration 2 ms were applied. Diffusion times in the range of 40 ms to 200 ms were investigated to provide a comprehensive understanding of self diffusivity of water in ZIF 8. The two overlapp ing peaks observed in Figure 6 8 can be assigned to water molecules in two different environments i.e., confined and unconfined (free) water. The slower attenuating, narrow er peak might belong to the molecules inside the pores of the ZIF while the faster diffusing, slightly broader peak might be assigned to free water that
72 Figure 6 6. The 1 H NMR spectrum of non activated ZIF 8 bed obtained using t he high resolution MAS probe. The sample was rotated at 7 kHz. The asterisk mark the spinning side bands and the remainder of the peaks are assigned to the protons of the ZIF 8 framework and residual, adsorbed H 2 O. Figure 6 7. 1 H spectrum obtained of H 2 O molecules in a saturated ZIF 8 bed under conditions of MAS and static NMR. There is a vast improvement in line widths under MAS conditions
73 Figure 6 8 Overlay of the attenuating signal of H 2 O saturated in the ZIF 8 bed resu lting from increasing applied gradient. The two overlapping peaks belong two water molecules in different environments of the ZIF 8 bed. is either between the crystals of the material or in a thin layer that coats the surface of the crystals. The peaks fr om the framework essentially disappear in these experiments due to their significantly shorter T 2 relaxation times in comparison to that of water. Under standard, static NMR conditions, overlapping peaks are difficult to resolve into individual ones due to large deviations from Lorentzian line shapes. However, under the stated MAS conditions, each overlapping peak can be deconvoluted using software. As a thorough analysis, two different approaches are demonstrated here to evaluate the diffusion properties o f H 2 O in ZIF 8. In the first approach, the two peaks were treated as one and the attenuation of the signal was calculated by normalizing the integrated area o f the combined peaks. Figure 6 9 shows the resulting attenuation curves corresponding to each repo rted diffusion time It is clear from this figure that a bi exponential curve is required to
74 characterize the two different diffusi vities that pertain to each peak. The following equation was used to fit each attenuation curve individually: (6 2) where p i corresponds to the fraction of molecules exhibiting diffusivity D i Table 6 2 lists the fitted values for each diffusion time obtained using E q 6 1. The diffusivity of the slower ensemble of mo lecules is approximately (1.1 0.3) x 10 10 m 2 /s and is consistent with the diffusivity of adsorbed water molecules in ZIF 8, ~1 x 10 10 m 2 /s, measured using conventional PFG NMR. For consistency, the comparison of fitted values for the attenuation curve s acquired under the same gradient duration is appropriate. From Table 6 2, the diffusivity of the faster ensemble of molecules is (4.4 0.4) x 10 9 m 2 /s. The faster diffusivity, which is notably larger than that of bulk water, can be tentatively assigned to molecules exchanging between ZIF crystals and the volume between the crystals occupied by water in the liquid and/or vapor phase during the diffusion time used. Also, the trend of increasing fraction with diffusion time for the faster ensemble of molec ules is a signature of long range diffusion, i.e. diffusion under conditions of fast exchange between intra ZIF volume and the liquid or vapor phase between ZIF crystals. Table 6 2. Ensemble fractions and the corresponding diffusivities obtained by fitti ng Eq 6 1 to the bi exponential attenuation curves in Figure 6 8. Diffusion Time (ms) Gradient Duration (ms) p 1 D 1 x 10 9 (m 2 /s) p 2 D 2 x 10 10 (m 2 /s) 1 00 2.0 0.74 4.5 0.4 0.26 1.2 0. 2 150 2.0 0.78 4.4 0.4 0.22 0.96 0.2 200 2.0 0.82 4.6 0.4 0.1 8 1.1 0.3
75 Figure 6 9 Measured PFG NMR attenuation curves for H 2 O diffusion in a saturated ZIF 8 bed under MAS conditions with a spin rate of 5 kHz. The measurements were performed using the 13 interval sequence with eddy curr ent delay. All measurements were performed at a sample temperature of 298 K. Figure 6 10. Attenuation curve obtained using deconvolution of overlapping peaks observed in the PFG NMR spectra of H 2 O diffusion in ZIF 8. These atten uation curves correspond to the narrower, slower diffusing peak.
76 In the second approach, software was use d to deconvolute the overlapping peaks into two distinct Lorentzian peaks and the corresponding attenuation curves obtained at different diffusion time s are shown in Figure 6 10. To keep the comparison brief, the attenuation curves for just one of the peaks is discussed here. The attenuation curves in Figure 6 10 correspond to the narrower, slower diffusing peak as it is assigned to the adsorbed molecule s in ZIF 8 pores. Since there is an exchange of water molecules inside the pores and the environment outside, there is some deviation from the mono exponential behavior expected after deconvolution. Hence, a since bi exponential fit is used to describe all attenuation curves in Figure 6 10. The diffusivity of 0.93 fraction of molecules is estimated to be ( 1.2 0. 3 ) x 10 10 m 2 /s and agrees very closely with the diffusivity of adsorbed water molecules as measured by the first approach. Th e corroboration of t he diffusivities as calculated by the two approaches provides further confidence in the interpretation of the results reported. 6.4 Summary Combining magic angle spinning with pulsed field gradient NMR expands the range of porous systems that can be explor ed due to the remarkable gains in resolution. To exploit the se advantages, a MAS probe was calibrated for diffusion measurements in collaboration with the National High Magnetic Field Laboratory. The gradients of a high resolution MAS probe were calibrated and tested to perform diffusion measurements. This experimental setup was also used to demonstrate the accuracy of PFG NMR measurements in porous materials under MAS conditions. While MAS PFG NMR offers a promising alternative to conventional, static PFG NMR, it is limited by the maximum strength of the gradient. The state of the art gradient probes available can produce gradients as large as 30 T/ m. Whereas, most high
77 resolution MAS probes are equipped to apply gradients up to 0. 50 0.60 T/ m. Depending on the sorbate and host molecules of interest, a suitable experimental setup should be chosen.
78 CHAPTER 7 WORK IN PROGRESS: DIFFUSION IN THE PRESENCE OF HYDROGEN SULFIDE 7 .1 Motivation Zeolitic Imidazolate Frameworks (ZIFs) are composed of imidazolate lin kers and metal ions and are a subclass of metal organic frameworks (MOFs). Recently there has been an explosion in research studies devoted toward s synthesizing ZIFs and integrating them in industrial applications. 57 58 This is because ZIFs possess properties such as large surface area, exceptional chemical and thermal stability, and high crystallinity which make them attractive candidates for membrane based separations applications. 59 62 Their vast functionality and tunable pore size can be e xploited to tackle challenging gas separations. The Energy Frontiers Research Center (EFRC) in Atlanta is devoted to studying these mate rials for acid gas separations and the results pr esented here are in an effort to further this goal. Figure 7 1. Performance of membrane separation systems depends on the transport properties of the material. A fundamental understanding on these properties at various length scales is essential when inve stigating novel materials for membrane based separations.
79 Hydrogen Sulfide (H 2 S) is a toxic and corrosive gas that plagues a vast number of industrial processes ranging from natural gas processing to wastewater treatment. 63 64 Long term exposure to H 2 S can result in adverse health effects, that vary based on it s c on centration. 65 It is responsible for catalyst poisoning, and corrosion of pipelines and chemical storage containers. H 2 S can also form flammable mixtures that in turn produce harmful vapors such as sulfur dioxide upon ignition. It is evident that separation of H 2 S is critical from both an economic and safety perspective. Membra nes have complex structures and can exhibit different transport properties at varying length scales (Figure 7 1) Advancements in membrane technology require a fundamental understanding of these distinct transport properties. This chapter presents the firs t microscopic diffusion study of H 2 S in ZIFs as well as its influence on their transport properties due to prolonged exposure. 7 .2 Diffusion of Hydrogen Sulfide ZIF 8 is a MOF that consists of zinc ions coordinated with fou r imidazolate rings. 57 The large spherical cavities of ZIF 8 are accessible through small apertures that are 3.4 in diameter. 66 It is extensively studied due to its vast functionality and ease of assembly into various forms 60, 67 Given the abundant interest in ZIF 8, it was the primary focus of diffusion studies presented here. 7 .2.1 Experimental Details PFG NMR was employed to measure self diffusivity of H 2 S in ZIF 8 crystals. ZIF 8 crystals of average dia meter 5.3 the group of Ryan Lively at Georgia Institute of Technology. equipped to store and use H 2 S for experimental studies. Hence, all NMR samples containing H 2 S were prepared by Dr.
80 NMR samples were prepared in a fashion similar to that described in section 4.4. A pproximately 100 mg of ZIF 8 w as loaded into two separate 5 mm, medium wall, NMR tubes (Wilmad Glass) each The tubes were then activated at an elevated temperature and evacuated under vacuum, overnight. Each tube was then attached to a manifold connected to a source of H 2 S gas mixture. The gas mixture contains 5% H 2 S with the remainder being N 2 The bed of ZIF 8 crystals was equilibrated wi th 0.9 atm of this mixture for 11 hours and subsequently immersed in liquid nitroge n to cryogenically condense H 2 S into the tube. The NMR tube was then flame sealed and separated from the manifold. 1 H NMR was used to perform PFG NMR mea surements of these samples. Primarily, a 600 MHz Bruker spectrometer in conjunction with a diff30 gradient probe was e mployed. Some experiments were also attempted on a 750 MHz Bruker sp ectrometer equipped with a diff5 0 gradient probe in an effort to reproduce results and el iminate influence of artefacts. All measurements were performed at 298 K. 7.2.2 Results and Discussion The combination of low sorbate concentration and large susceptibility effects associated with 1 H NMR measurements in porous materials has produced many challenges in diffusion measurements of H 2 S. The first obstacle was the accurate identific ation of H 2 S signal. Acquisition of a 1 H NMR spectrum of ZIF 8 loaded with H 2 S showed large baseline distortions which obscured the visibility of any distinct peak. T he 13 in t erval PFG NMR sequence can be used to minimize the susceptibility effects and provide a more discernable spectrum S hortening the time interval between the first and second and second and thi applying the smallest gradient possible reveals various features Using this procedure, the spe ctra shown in
81 Figure 7 2 were acquired and they show a distinct peak centered at 3 .4 ppm. The time evolution of this peak recorded here occurs by decreasing the time available for T 2 radiofrequency pulses. The gradients applied were sine shaped of amplitude 0.30 T/ m. The response of this peak to the systematic manipulation of the 13 interval sequen ce agrees with the predicted behavior of a real peak, uninfluenced by susceptibility effects. However, this is not enough evidence to confirm the identification of this fe ature as H 2 S. The broad linewidth of this peak encompasses the chemical shift of bulk H 2 O (4.8 ppm). While the ZIF 8 crystals were activated prior to sorbate loading, the presence of strongly, adsorbed residual water is possible. Measuring the self diffusivity of the species that belongs to this peak can al so elucidate its identity s ince the diffusivity of adsorbed H 2 O in ZIF 8 is known (Chapter 6). Using a 600 MHz spectrometer, the 13 interval sequence (with eddy current delay) with bipolar gradients of duration 4.5 ms was employed with gradients as large as 10 T/ m. A large range of diffu sion times from 10 ms to 500 ms were used but with no visible attenuation of the NMR peak (Figure 7 3) This result implied a very low diffusivity of the species, several orders lower that the reported value of H 2 O. Hence, the peak can be assigned to H 2 S i n ZIF 8. Any further increase in experimental parameters such as gradient duration and diffusion time was limited by T 1 and T 2 relaxation processes that result in sev ere loss of signal. However, larger gradient strengths can be achieved using the diff5 0 p robe that has been calibrated by our group to perform diffusion measurements in conjunction with a 750 MHz spectrometer a t the AMRIS facility. This experimental setup was used to
82 repeat the PFG NMR measurements mentioned above. While the maximum gradient s tren gth produced was as high as 19 T/ m, the poor signal to no ise observed at this higher spectrometer frequency restricted the magnitude of diffusion times. The largest diffusion time used at this setup was 40 ms. Under these conditions, neither the area n or intensity of the peak change d with increasing gradient strength. This confirmed the reproducibility of results at two different spectrometer frequencies, providing additional confidence in the results. The PFG NMR measurements reported here suggest tha t the self diffusivity of H 2 S in ZIF 8 is extremely small. It is estimated to be less than 5 x 10 15 m 2 /s. Strong interactions between the ZIF 8 framework and H 2 S molecules can explain this very small diffusivity. To provide further insight on a molecular scale of the strength of these interactions, this work will be continued in collaboration with Dr. J. R. Schmidt and Dr. David Sholl, fellow members of EFRC. They are currently working on MD simluations to estimate the energy associated with H 2 S molecules moving from a given cage in the ZIF 8 framework to another and using this to estimate the magnitude of its self di ffusivity to compare with the experimental results discussed. 7 3 Influence of Hydrogen Sulfide Exposure on Transport Properties Degradation of ZIF 8 crystals is a potential concern due to the corrosive nature of H 2 S. Changes to the surface or bulk properties of ZIF 8 could alter the transport properties of the material. To address this issue, a comprehensive study comparing self dif fusion of v arious sorbates (C 2 H 4 and CO 2 ) in pure ZIF 8 with ZIF 8 exposed to dry H 2 S conditions is underway
83 7 3 .1 Experimental Details Following H 2 S diffusion measurements described in section 7 .2, and three weeks of enclosed exposure of ZIF 8 to dry H 2 S condi tions, the ZIF 8 crystals were retrieved returned to our laboratory at University of Florida for comprehensive analysis of possible influence of H 2 S exposure on transport prope rties of ZIF 8. To understand the effect of H 2 S ex posure, pure ZIF 8 crystals are compared with exposed ZIF 8, where both samples of crystals were obtained from the same synthesized batch. This was done to limit effects of heterogeneity in crystals on the observed transport prop erties. PF G NMR is employed to inve stigate self diffusivity of C 2 H 4 and CO 2 in the pre and post H 2 S expos ed samples. The NMR samples are prepared as follows. Two NMR tubes, identical to those used to prepare H 2 S loaded samples as d escribed above, were separately filled with about 95 mg of pure and 70 mg of exposed ZIF 8 respectively The NMR tubes containing the porous material were connected to a custom made vacuum system and any residual sorbates were removed by application of hi gh vacuum at an elevated temperature (383 K), for 10 hours. A desired mass of each chosen sorbate was cryogenically condensed into the sample tubes using liquid nitrogen. Following sorbate loading, each tube was flame sealed and separated from the vacuum s ystem. Single 13 C enriched, high purity, CO 2 and C 2 H 4 gases are used as sorbates for this exposure study. The pure and exposed samples were loaded under equilibrium with a certain pressure in the gas phase. The self diffusivity of certain sorbates has bee n shown to depend on its concentration. Hence, achieving equivalent adsorbed concentration, within experimental error, is essential for direct comparison of results.
84 The results reported here thus far pertain to C 2 H 4 sorbate studies. Ana lysis and interpret ation of CO 2 sorbate study are in progress. T he concentration of each gas in all ZIF 8 samples i s determined by comparing th e NMR signal of adsorbed gas in these samples with the NMR signal in an NMR tube containing only pure gas at a known pressure, in t he same way as d iscussed in our previous work. These loadings are reported in Table 7 1 and are noted to be comparable, within experimental error. Longitudinal (T 1 ) 13 C NMR relaxation time s of the adsorbed gas molecules in ZIF 8 were estimated using the 1 3 interval PFG NMR sequence by changing the time and keeping all other time intervals constant. Similarly, transverse (T 2 ) 13 C NMR relaxation times were estimated using the same sequence by changing the time interval keeping all other time intervals constant. Under the conditions of these measurements, it was confirmed that there was no attenuatio n of the signal due to diffusion of adsorbed gas molecules. Figure 7 4 plots the observed magnetization, normalized to the intensity of the magnetization at the smallest T 1 interval as a function of these intervals on a log linear scale. I n all of the stud ied samples, the measured NMR relaxation data showed a bi exponential distribution for distribution of T 1 relaxation times for adsorbed gases. A bi exponential curve was used to fit this data and the corresponding parameters are recorded in Table 7 2. To s ummarize, the larger faction of molecules exhibit a smaller T 1 relaxation time while the smaller faction of molecules has a larger T 1 relaxation time. The presence of two separate factions of molecules could be due to the simultaneous
85 presence of crystal a ggregates and single crystals. There is no apparent distribution in the T 2 relaxation values The transverse relaxation time in pure and exposed ZIF 8 crystals was 13 ms and 1 4 ms, respectively. All measurements were performed at 298 K. PFG NMR diffusion measur ements were performed using a 14 0 T Bruker BioSpin NMR spectrometer operating at 13 C resonance frequ ency of 150 MHz. Selected experiments were also performed at the 1 H resonance frequency of 600 MHz. The spectrometer is located at in Gainesville, FL. Sine shaped, bipolar magnetic field gradients with the effective duration of 1.2 2.3 m s were generated using a diff30 d iffusion probe (Bruker BioSpin). The range of the gradi ent amplit udes used was between 0. 30 and 11 T /m. 7.3.2 Results and Discussion Figure 7 5 presents examples of the measured attenuation curves for ethylene diffusion in both pure and exposed ZIF 8 obtained using 1 H and 13 C PFG NMR at selected diffusion time s The coincidence of the diffusion data obtained using the two n uclei rules out measurement arte facts. At the smaller diffusion time, a bi exponential curve corresponding to Eq 6 1 is shown to describe the data well. A t the larger diffusion time, a singl e exponential fit provide s reasonable description of the data. Table 7 3 lists the parameters of each fit at various diffusion times The respective fractions of each diffusing e nsembles are consistent with those obtained using the T 1 relaxation distributi on, within the limits of experimental uncertainty. By comparison, the faster diffusing molecules exhibit a longer longitudinal relaxat ion ti me
86 The diffusi vity of the faster component at shor t diffusion times is much larger than previously reported estima tes of self diffusivity of C 2 H 4 in ZIF 8. 41 The slower diffusivity agrees with these published values of intra crystalline diffusion of C 2 H 4 in defect free ZIF 8, within experimental unc ertainty. Furthermore, w ith increasing observation time, the diffusivity of the faster molecules is shown to decrease. These observations can be rationalized by assigning the faster diffusing molec ules to those diffusing a fraction of the diffusion time in macro and mesopores (i.e. defects) between single crystals in densely packed ZIF 8 crystal aggregates. Such aggregates account for the majority of the sample as seen in the recorded SEM images in Figure 7 6. The remaining fraction of time these molecules diffuse inside ZIF 8 crystals where the diffusivity is lower. With increasing total diffusion time the fraction of time spent in macro and mesopores by the molecules exhibiting the largest diffusivity would decrease. Hence, their diffusivity would also d ecrease and approach the average diffusivity inside an aggregate. The latter, averaged, diffus i vity can be similar to the intracrystalline diffusivity of ethylene in ZIF 8. Figure 7 5 presents the attenuation curves measured at longer diffusion times for both pure and exposed crystals Due to the coincidence of all curves shown, a single mono exponential fit was used to describe them. The corresponding self diffusivity of ethylene molecules moving inside the crystal aggregates is (1.1 0. 8 ) x 10 10 m 2 /s f or both samples Based on PFG NMR measurements of ethylene in ZIF 8, there are no significant alterations in the intracrystalline structure that manifest themselves in bulk transport properties. The self diffusivities of ethylene in ZIF 8, along with other diffusion time dependent trends, are in agreement between the pure and H 2 S exposed crystals. To
87 reveal any alterations to surface properties, a faster diffusing sorbate such as CO 2 is appropriate. 7.4 Summary High field and high gradient PFG NMR was appl ied to investigate microscopic self diffusivity of H 2 S in ZIF 8. While H 2 S signal in ZIF 8 was successfully identified using 1 H NMR, its diffusivity was too low to measure with gradients as large as 19 T/ m. The reported experimental data suggests the prese nce of strong interactions between H 2 S and the ZIF 8 framework which result in a self diffusivity that is estimated to be lower than 5 x 10 15 m 2 /s. Molecular simulation results expected to be provided by collaborators can aid in better understanding the n ature of these interactions. Additionally, effects of H 2 S exposure on ZIF 8 do not alter the bulk transport properties of ZIF 8 as measured using ethylene. This suggests ZIF 8 to be a stable sorbent, immune to corrosion upon exposure to small H 2 S concentra tions.
88 Figure 7 2 The evolution of the peak identified as H 2 S in ZIF 8 acquired using the 13 interval PFG NMR sequence so that strong susceptibility effects are minimized. The time interval, was systematically decreased such that various features present can be visualized.
89 Figure 7 3. Example of 1 H spectra acquired of H 2 S in ZIF 8 using the 14 T spectrometer and diff30 gradient probe. Here, the 13 interval sequence with eddy current delay was employed with the application of gradients in the range of 0.30 and 11 T/ m for a diffusion time of 500 ms. No visible attenuation was observed under these and any other conditions.
90 Table 7 1. Concentration (mmol/g) of C 2 H 4 molecules adsorbed in pure and exposed ZIF 8 crystals as determined using NMR si gnal analysis. Pure ZIF 8 Exposed ZIF 8 Ethylene (C 2 H4) 2.5 0.2 2.3 0.2 A B Figure 7 4. T 1 Relaxation curves obtained using 13 C NMR show a bi exponential distribution in both (A) pure ZIF 8 and (B) exposed ZIF 8 crystals.
9 1 Table 7 2. The pa rameters corresponding to the bi exponential curve fit to the T 1 relaxation data reported in Figure 7 4. Pure ZIF 8 Exposed ZIF 8 A 0.64 0.63 B 0.36 0.37 T 1 (A) 42 45 T 1 (B) 320 363 Figure 7 5. Examples of measured a ttenuation curves obtained using 13 C (black symbols) and 1 H (red symbols) PFG NMR of E thylene diffusion at 298 K in pure and exposed ZIF 8 samples. A single m ono exponential curve is fit to the data shown due to the coincidence of all attenuation curves, w ithin experimental uncertainty
92 Table 7 3. Parameters corresponding to the bi exponential (Eq 6 1) and mono exponential fits used to describe attenuation curves for both pure and exposed ZIF 8 at selected diffusion times. Parameters Diff usion Time (ms) 10 40 200 600 Pure ZIF 8 A 0.79 0.2 0.72 0.2 1 1 B 0.21 0.1 0.28 0.1 0 0 D A x10 11 ( m 2 /s ) 8.9 1.5 8.3 1.2 11 2 10 2 D B x10 11 ( m 2 /s ) 53 8 24 3 0 0 Exposed ZIF 8 A 0.68 0.1 0.65 0.1 0.72 0.2 1 B 0.32 0.1 0.35 0.1 0.28 0.1 0 D A x10 11 (m 2 /s) 8.7 1.5 8.4 1.5 9.4 2.0 12 2 D B x10 11 (m 2 /s) 51 7 37 4 24 4 0
93 A B Figure 7 6. (A) pure and (B) exposed ZIF 8. These images show that both samples consist largely of aggregates of crystals. No significant differences are observed between the pure and exposed samples.
94 CHAPTER 8 CONCLUSIONS Gas separation processes can be an energetically expensive process when conducted using the rmally driven technologies. The main focus of this dissertation was to investigate the fundamentals of a conceptually novel process of gas separations that offers an energetically favorable alternative. It is based on the induction of si ngle file diffusion of a chosen sorbate in a mixture which can result in highly selective separations Single file diffusion of sorbate mixtures was observed and reported for the first time using diffusion NMR that combined high magnetic field and large fi eld gradients. The following gases were studied in an ideal system of microporous channels as single sorbate and mixtures: CO, CO 2 and CH 4 SFD of all components of single sorbate and mixture samples of CO/CO 2 and CO/CH 4 was confirmed by the growth of th eir mean squared displacement with the square root of diffusion time, a unique signature of single file motion. Additionally, the rates of SFD of both components in a given mixture are identical, within experimental uncertainty, as expected under single fi le confinement where molecules cannot pass each other. As sing le sorbate samples, each gas exhibited a distinc t rate of single file diffusion. The collective single file mobility of the mixture components is shown to be intermediate of its respective compo nents in pure form This trend is qualitatively different from the case of normal diffusion in microporous materials where an addition of a faster diffusing component to a slower diffusing component does not change si gnificantly the diffusivity of the slow er diffusing component. The predictions of a random walk model describe a transition from SFD to center of mass diffusion in finite channels under the SFD conditions. However, this
95 transition is not experimentally observed for the systems studied. Molecula r clustering is proposed to explain the absence of such a transition. The minimum cluster size needed to maintain the SFD regime observed under the experimental conditions reported here was estimated to be in the range of 30 and 140 molecules. The presence of molecular cluster s is also shown to offer a plausible explanation for the relationship experimentally observed between the one component and mixture mobilities. Hydrogen sulfide, a hazardous impurity, often needs to be isolated and removed for both eco nomic and safety reasons ZIFs are promising candidates for such a separation and the microscopic diffusion of H 2 S in ZIF 8 was investigated here. Preliminary measurements reveal the self diffusivity of H 2 S to be lower than 5 x10 15 m 2 /s. This extremely lo w diffusivity could be a result of strong interactions between the sorbate and the ZIF 8 fram ework. The potential influence of exposure to H 2 S on the transport properties of ZIF 8 was also studied. Comparison of the trends of self diffusion of ethylene in pure and exposed ZIF 8 did not reveal any alteration of the bulk transport properties of the microporous sorbent. Further analysis of changes in the external surface properties is currently underway by comparison of self diffusivity of carbon dioxide in pu re and exposed ZIF 8 crystals. In an effort to overcome some of the limitation s of standard diffusion NMR associated with studying sorbates in porous materials, an experimental setup that combines magic angle spinning with PFG NMR was introduced at the Na tional High Magnetic Field Laboratory. A high resolution MAS probe equipped with gradients was calibrated and tested to perform diffusion measurements. The maximum gradient strength of the probe was determined to be 0.56 0.01 T/m and the gradient field w as
96 shown to be uniform along the bulk of the rotor length. Changes in the temperature of the sample due to the mechanical heating that results from high spinning rates was also evaluated. Finally, the accuracy of the reported calibration pr ocedure was asse ssed by demonstrating the agreement between self diffusivity of H 2 O in ZIF 8 between PFG NMR under static and magic angle spinning conditions.
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102 BIOGRAPHICAL SKETCH Akshita Ranjan Dutta was born in Aligarh, India to Ranjan and Mini Dutta. She spent the first eight years of her life in India before relocating to Hong Kong with her family. She was raised there along with her younger brother, Mudit Dutta. As she was about to enter high school, she once again relocated this time to Florida. After completing her high school education at Coral Glades High school in Coral Springs in 2009, she was accepted to purse a B.S. in c hemical e ngineering at the University of California, Berkeley. While at Berkeley, Akshita had the opportunity to engage in various undergraduate research opportunities in various fields related to alternative energy. Some of her research accomplishments include participation in the REU program at the National High Magnetic Field Laboratory and an inter national research internship at the Indian Institute of Technology in Kharagpur, India. She was an active member of the Berkeley chapter of American Institute of Chemical Engineering and served as an officer for two years. She graduated f rom UC Berkeley wi th a B.S. in chemical e ngineering in May 2013. Following the completion of her undergraduate degree, she joined the University September of 2013, Akshita joined the labs of Dr Sergey Vasenkov and Dr. Kirk J. Ziegler to study transport of molecules subjected to extreme conditions in porous materials. She has shared her research with colleagues through numerous poster and oral presentations at various conferences. She was awarde d 3 rd place in AIChE TED Separations competition in 2016.
103 During the completion of her doctoral studies, Akshita mentored several as a teaching assistant and was awarded Outstanding Teaching Assistant of the Year for the 2016 Spring Semester. Akshita has been involved in several outreach programs that promote interest in science of elementary, middle, and high school students.