SELF-DIFFUSION OF CARBON DIOXIDE IN SAMARIA/ALUMINA AEROGEL CATALYST USING HIGH FIELD NMR DIFFUSOMETRY

Transcription

1 SELF-DIFFUSION OF CARBON DIOXIDE IN SAMARIA/ALUMINA AEROGEL CATALYST USING HIGH FIELD NMR DIFFUSOMETRY By SUIHUA ZHANG A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA

2 2013 Suihua Zhang 2

3 To my Mom, my Dad and my Grandparents 3

4 ACKNOWLEDGMENTS Being a part of the Gators for the last two and a half years will be a precious and wonderful memory in my life. I would like to sincerely and gratefully thank Professor Sergey Vasenkov for his patient guidance and constant encouragement. His commitment to work is so inspiring that I have learned a lot from him. I would like to thank my colleague Mr. Robert Mueller for his immense support and guidance. His enthusiasm towards research and his concentration on details are so impressing that I will keep this as my creed. I would like to thank my lab mates Ms. Aiping Wang, Mr. Han Wang, Mr. Eric Hazelbaker and Mr. Muslim Dvoyashkin for their constant support. I am grateful to the staff at the Advanced Magnetic Resonance Imaging and Spectroscopy (AMRIS), especially Dr. Daniel Plant for their guidance and support. I will also thank Dr. Peng Jiang for being a committee member for my master s defense. My family are always the source of encouragement for me. I would like to sincerely thank my parents for their love and care. I would like to express my best wishes to my grandma who suffered from medical problem during the completion of this thesis. I am also grateful to Ms. Deborah Sandoval for praying for my grandma. Lastly, I would like to thank all my teachers and friends for their constant support and guidance. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS... 4 LIST OF TABLES... 6 LIST OF FIGURES... 7 LIST OF ABBREVIATIONS... 8 ABSTRACT CHAPTER 1 INTRODUCTION Mass Transport in Catalytic Materials Catalytic Aerogels Basics of Diffusion Basics of NMR Longitudinal & Transverse Magnetization NMR Relaxation Signal Detection Pulsed Field Gradient (PFG) NMR Pulsed Gradient Stimulated Echo Pulse Sequence PFG NMR Stimulated Echo Longitudinal Encode-Decode Pulse Sequence Attenuation Equation Applied in PFG NMR SELF-DIFFUSION OF CO2 IN SAMARIA/ALUMINA AEROGEL CATALYST Experimental Details Samaria Aerogel Catalyst Synthesis and Structural Characterization Results and Discussion Conclusion LIST OF REFERENCES BIOGRAPHICAL SKETCH

6 LIST OF TABLES Table page 1-1 Self-diffusivities of CO 2 as a Function of Equilibrium Loading Pressure

7 LIST OF FIGURES Figure page 1-1 Schematic of spin precession in an external magnetic field Emergence of longitudinal net magnetization in external magnetic field B Graph of free-induction decay (FID) in a PFG NMR diffusion measurement Schematic of PFG NMR stimulated echo pulse sequence Schematic of the PFG NMR Stimulated echo longitudinal encode decode pulse sequence SEM image of the studied alumina stabilized samaria aerogel particles bed Pore size distributions for the studied alumina stabilized samaria aerogel CO2 adsorption isotherm measured for the particles bed sample at 297 K PFG NMR attenuation curves for CO 2 diffusion in monoliths and particles bed samples at pressure of 0.3 atm CO2 self-diffusivities in the studied samaria aerogel catalyst samples at T = 297 K as a function of sample equilibrium loading pressure Interparticle tortousity factors for different equilibrium loading pressures of CO2 in the catalyst powder bed samples

8 LIST OF ABBREVIATIONS FID MSD NMR PFG NMR PGSTE PGSTE LED r. f. pulse B 0 B 1 D g J c J c z M r 2 μ γ W 0 M z Free Induction Delay Mean Square Displacement Nuclear Magnetic Resonance Pulsed Field Gradient Nuclear Magnetic Resonance PFG NMR Stimulated echo pulse sequence PFG NMR Stimulated echo longitudinal encode decode pulse sequence Radio Frequency Pulse Amplitude of the External Static Magnetic Field Amplitude of the Oscillating Microscope Magnetic Field due to r.f. Pulse Diffusion Coefficient Amplitude of the Magnetic Field Gradient Flux of Molecules Concentration of Molecules Flux of Labeled Molecules Concentration of Labeled Molecules Position of Molecule or Spin is at z Coordinate The Total Number of Diffusing Molecules Mean Square Displacement Magnetic Moment of Nuclear Spin Gyro magnetic Ratio Larmor Frequency Net Total Longitudinal Magnetization 8

9 M 0 P t τ 1 τ 2 δ T 1 T 2 T LED Ψ Net Total Longitudinal Magnetization at Equilibrium State Distribution of Spin Phase Time Duration between the First and Second r.f. pulses in PFG NMR Echo Longitudinal Encode Decode Pulse Sequence Duration between the Second and Third r.f. pulses in PFG NMR Echo Longitudinal Encode Decode Pulse Sequence Diffusion Time Duration of the Magnetic Field Gradient Pulse Spin-Lattice NMR Relaxation Time Spin-Spin NMR Relaxation Time Duration between the Fourth and Fifth r.f. pulses in PFG NMR Echo Longitudinal Encode Decode Pulse Sequence for Dissipating the Eddy Current Attenuation of the Amplitude of Signal in PFG NMR Experiment 9

10 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science SELF-DIFFUSION OF CARBON DIOXIDE IN SAMARIA/ALUMINA AEROGEL CATALYST USING HIGH FIELD NMR DIFFUSOMETRY Chair: Sergey Vasenkov Major: Chemical Engineering By Suihua Zhang December 2013 Pulsed field gradient (PFG) NMR was used to investigate the self-diffusion of carbon dioxide in alumina stabilized samaria aerogel catalyst, a promising porous catalyst for gas-phase reactions featuring high porosity and high surface area. For diffusion studies the catalyst was prepared in two sample packing types, macroscopic monoliths and powder beds with particle sizes around 200 µm that are considered for catalytic applications. Studies of diffusion in these samples revealed how macroscopic packing influences the catalyst transport properties. Application of a high magnetic field of 17.6 T in the reported PFG NMR studies enabled diffusion measurements for relatively low carbon dioxide densities in the catalyst samples corresponding to a gas loading pressure of around 0.1 atm. As a result, it was possible to perform diffusion measurements for a large range of the carbon dioxide loading pressures between 0.1 and 10 atm. The measured carbon dioxide diffusivities in the beds of catalyst particles are interpreted in the context of a simple diffusion-mediated exchange model previously used for zeolites and other porous materials. 10

11 CHAPTER 1 INTRODUCTION Mass Transport in Catalytic Materials Porous amorphous catalysts often have a fractal internal surface down to molecular scales. In the chemical processes occurring inside porous catalysts, the reactant(s) and product(s) molecules have to diffuse through the extensive porous network in the catalyst in order to reach and react on the active sites of the catalyst. According to the research on these processes, diffusion often serves as the rate determining factor of the total reaction rate, influencing not only the reaction rate but also the selectivity of the reaction(s). On the other hand, the optimization of existing catalysts is one of the most important issues of catalytic research today despite the development of new catalysts consisting of abundant and cheap materials. Although many important and successful porous catalysts have been developed on the basis of empirical knowledge and experience, the elucidation of the diffusion mechanism(s) occurring inside catalysts is required to make progress in the development of porous catalysts. 1-3, 8 Mass transport of reactant and product molecules can play an important role in the overall performance of catalysts and is a multi-length scale process: the rate of diffusion inside individual catalyst particles can be quite different from that at macroscopic length scales comparable with the size of a particle bed. For many decades, the scientific community has intensely researched catalyst designs optimized for mass transfer, and it continues to be a very active area of research. Novel tailored porous materials have been developed, investigated and applied throughout the catalytic community

12 In this work, the relationship between the structural and transport properties of aerogel catalysts is investigated through studies of the self-diffusion of carbon dioxide in two different packing of alumina stabilized samaria aerogel catalyst using pulsed field gradient NMR. Catalytic Aerogels Aerogels are transparent, highly porous, open cell and low density foams. The microstructure including nano-sized pores and linked primary particles can be tailored and controlled to a large extent by solution chemistry and the conditions of the mild preparation procedure. These light-weight materials exhibit many unusual and interesting properties due to their unique microstructure. Such unique properties comprised of large surface areas, ultra-low densities and high-degree of porosity have made aerogel an attractive candidate as a catalyst support. Historically, Steven Kistler firstly recognized the potential catalytic applications of aerogels. Catalytic aerogels, such as carbon aerogels can be obtained in the form of beads, powders, monoliths or thin films. They have a three-dimensional network structure of interconnected nano-sized primary particles, which leads to fast mass transport within an aerogel particle and this transport has been determined to be virtually identical in velocity to gas-phase diffusion. Concerning the pore structures, micropores (pore diameters< 2 nm) are related to the intraparticle structure, while mesopores (2 nm< pore diameters< 50 nm) and macropores (pore diameters > 50 nm) are produced by the interparticle structure. Hence it is possible to control the concentration of micropores and mesopores independently, which is one of the advantages of catalytic aerogel as a porous material. As a result, aerogels, such as 12

13 carbon aerogels and silicon aerogels, have been recognized as excellent catalysts and catalyst supports for many years. There are numerous references of this application for various aerogels and doped aerogels Basics of Diffusion Development of transport-optimized catalysts requires a fundamental understanding of diffusion in catalytic materials. Diffusion is defined as a random, thermal motion of molecules at temperature above absolute zero. Two different types of diffusion can be distinguished: transport diffusion and self-diffusion. Transport diffusion involves the movement of molecules which tends to eliminate macroscopic gradients of chemical potential. Self-diffusion, on the other hand, is the random motion of matter which occurs in the absence of chemical potential gradients or macroscopic concentration gradients. The relationship between the flux of molecules and a concentration gradient has been described by the transport diffusion coefficient. Dating back to the 1850s, Adolf Fick came up with Fick s First Law of Diffusion 14, 15 for one dimensional diffusion describing the mass flux J in the z direction, J = D c z, (1-1) in which D is corresponding transport diffusion coefficient and c(z) is the concentration. At low molecular concentration, the diffusion coefficient is often independent of the concentration. While at high molecular concentration, the diffusion coefficient can be a function of the concentration. Combining Equation 1-1 with the mass balance leads to Fick s second law of diffusion 13

14 c = D 2 c, (1-2) t 2 z where D is assumed to be independent of concentration. In the case where diffusion coefficient depends on the concentration, the Equation 1-2 could be written as c = [D(c) c z ] t z. (1-3) Self-diffusion is the random translational motion of molecules driven by thermal energy which occurs in the absence of a macroscopic gradient of chemical potential. Fick s Laws (Equation 1-1 through Equantion 1-3) can also describe self-diffusion by arbitrarily dividing the identical molecules into labeled and unlabeled molecules. If an uneven distribution of unlabeled and labeled molecules is considered under the situation that the overall molecular concentration remains constant at every point of the region under consideration, the self-diffusion coefficient can be derived from the Fick s First Law. The diffusive flux J of the labeled molecules can be derived as J = D c s z c=const, (1-4) in which D s is the self-diffusivity, c is the concentration of the labeled molecules, and c = const stands for the constancy of total concentration of labeled and unlabeled molecules. Under the assumption of a parallel sided container of unit cross-sectional area and infinite length in z direction and the constancy of diffusivity, the total number of diffusing molecules (M) can be written as 13 in which M = + c dz, (1-5) 14

15 c = A t e z2 /4Dt, (1-6) in which A is an arbitrary constant. For the case of isotropic diffusion (where the self-diffusion coefficient remains the same in all directions), the concentration distribution can be written as c = e r2/4dt M (4πDt) in which r stands for the position vector from the origin. 3/2, (1-7) This is a Gaussian function and can be termed as the Propagator 13, which leads to the mean square displacement in three-dimensional diffusion system by integration: r 2 (t) = r 2 e r2 /4Dt (4πDt) 3/2 dr = 6Dt. (1.8) Similarly, the mean square displacement for one-dimensional and twodimensional diffusion systems can be derived respectively as: r 2 (t) = 2Dt (one dimensional) (1-9) r 2 (t) = 4Dt (two dimensional). (1-10) The equations 1-8 through 1-10 are called Einstein Relations 16. Basics of NMR Spin or spin angular momentum is a quantum-mechanical property of all atomic nuclei. Instead of arising from any rotational motion as in classical mechanics, the spin angular momentum is actually an intrinsic property of elementary particles, including that of protons and neutrons making up atomic nuclei. Because of the possession of a non-zero intrinsic magnetic moment, all atomic nuclei with non-zero spin angular momentum are called magnetically active. With the interrelation between the magnetic 15

16 moment and the spin angular momentum, a constant known as the gyromagnetic ratio (γ) is defined, which can be positive or negative. In the absence of any external magnetic field the distribution of spin-polarization axes and corresponding magnetic moments of individual nuclei in a macroscopic sample is completely isotropic, in which the net nuclear magnetization in the sample is zero. Under this situation, every NMR active nucleus can be thought as a microscopic magnetic top and its magnetic dipole moment can be written as μ = γι, (1-11) in which Ι is the nuclear spin angular momentum vector and γ is the nuclear gyromagnetic ratio. However, when the NMR active nucleus is placed in an external magnetic field B 0, its magnetic dipole moment μ couples with the magnetic field with an energy of interaction given by E = μ B 0. (1-12) This coupling is called Zeeman coupling. As a result of applying an external magnetic field, a torque is exerted on the nuclear magnetic moment of individual nuclei and causes the spin polarization to move on a cone always keeping the same angle between the directions of magnetic moment and the field (Figure 1-1) in the absence of relaxation processes. Such type of motion is known as precession. The frequency of precession ( w 0 ) for a magnetically-active nuclei, which is referred to as the Larmor frequency, is dependent on the amplitude of the external magnetic field B 0 and is given by 16

17 w 0 = γb 0, (1-13) in which γ is the gyromagnetic ration of the nucleus under observation. Figure 1-1. (A) Schematic motion of spin precession in external magnetic field B 0 (B) The angle of the precession depends on initial spin polarization axis. Longitudinal & Transverse Magnetization In the absence of an external magnetic field, the distribution of the spin polarizations of the sample is isotropic, which gives rise to zero net magnetization in the sample. If an external magnetic field is applied, the isotropy in distribution of spin polarization directions breaks down, which results in the formation of a net macroscopic nuclear magnetization in the sample along the direction of the applied magnetic field. (Figure 1-2). This direction is referred to as the longitudinal direction and by convention denoted as the z direction. Detection of nuclear magnetization along the longitudinal direction is impractical because the nuclear magnetism is comparatively small to the strong applied magnetic field needed for NMR. In NMR, the nuclear magnetization is detected in a plane perpendicular to longitudinal direction, which is achieved by the application of an additional oscillating magnetic field B 1 whose direction is normal to the direction of static magnetic field B 0. The effect of the alternating magnetic field on the magnetization can 17

18 be understood as producing an applied torque, which tends rotates magnetization about the axis of B 1. The frequency of the B 1 field is chosen to be sufficiently close to the Larmor frequency, which are typically in the range of radio waves. Hence the application of B 1 is usually referred to as application of a radiofrequency (r.f.) pulse in NMR jargon. The duration of r.f. pulse can be chosen in such a way that it rotates the net magnetization by 90 from the original z direction into the transverse plane, in which the magnetization precesses in the X-Y plane with the Larmor frequency. After rotation into the X-Y plane, the direction of the net magnetization gradually returns to the direction of the static external magnetic field B 0 due to a relaxation process. Figure 1-2. Thermal equilibrium and emergence of longitudinal net magnetization in presence of external magnetic field B0. NMR Relaxation The process of returning of the net nuclear magnetization to its equilibrium state is referred to as NMR relaxation. Two types of NMR relaxation can be distinguished: 1) Spin-Spin/ Transverse relaxation or T 2 relaxation and 2) Spin-Lattice/ Longitudinal relaxation or T 1 relaxation. T 2 relaxation leads to a gradual decay of transverse magnetization (e.g. the X-Y plane magnetization) to zero. This decay occurs because of a de-phasing of the precessing nuclei in other words, the precessions frequencies 18

19 vary in the sample and the spin precessions become out of sync - leading to destructive interference. This interference is caused by perturbations of the local magnetic field by neighboring nuclear spins and electronic clouds as well as the fluctuating microscopic magnetic field. The net rate of transverse relaxation can be characterized by a time constant T 2, as shown in the following equation t ( M X Y (t) = M X Y (0) e T2 ), (1-14) in which M X Y (t) is the net transverse magnetization at time t. T 1 relaxation is a process that describes a gradual growth of the net nuclear magnetization to its equilibrium value along the +z direction in the presence of a static magnetic field. After the application of 90 r.f. pulse the net magnetization grows along the z-direction and the rate of growth of which can be by characterized by t ( M Z (t) = M 0 (1 e T1 ) ), (1-15) in which M Z (t) is the net longitudinal magnetization at time t and M 0 is the net equilibrium magnetization pointing along +z direction. The Equation 1-15 holds for the cases when the net longitudinal magnetization is equal to zero at t=0, in presence of an external magnetic field B 0. Signal Detection After the application of an r.f. pulse, the precessing transverse magnetization oscillates at a very well-defined frequency. An electric field generated by this rotating magnetic moment gives rise to an oscillating electric current flow in the signal detection coil placed near the sample. Hence the oscillating electric current induced by the precessing nuclear transverse magnetization is called the NMR signal or free-induction decay (FID, Figure 1-3). 19

20 Figure 1-3. Graph of free-induction decay (FID) in a PFG NMR diffusion measurement. Pulsed Field Gradient (PFG) NMR Pulsed-field gradient NMR spectroscopy allows for the measurements of the diffusion propagator and the related diffusivity as well as the mean square displacement by exploiting the magnetic field dependence of the Larmor frequency of the precessing nuclear spins. In PFG NMR, a gradient of magnetic field along the z direction is applied when the net nuclear magnetization is in the transverse plane, under which condition the Larmor frequency of precession is given by ω = γ(b 0 + gz), (1-16) where ω is the Larmor frequency, γ is the gyromagnetic ratio, B 0 denotes the amplitude of the static external magnetic field along the z direction and g is the linear gradient of 20

21 the magnetic field superimposed on the B 0 field and z is the spatial coordinate along the z axis. Equation 1-16 shows that application of the magnetic field gradients thus makes the Larmor frequency to be position dependent, effectively labeling the nuclear spins based on their spatial coordinate along the z-direction. Pulsed Gradient Stimulated Echo Pulse Sequence Figure 1-4. Schematic of PFG NMR stimulated echo pulse sequence. Figure 1-4 shows the standard PFG NMR stimulated echo pulse sequence (PGSTE) used in PFG NMR. After the application of the first π r.f. pulse, the longitudinal 2 magnetization is brought to the transverse plane with nuclear spins precessing at the same Larmor frequency. Meanwhile the application of the first field gradient pulse makes the Larmor frequency of spins to be dependent on the spins positions along the z direction. After the second π r.f. pulse, the net magnetization is reoriented from the 2 transverse plane to the z axis. After time interval τ 2, the third π r.f. pulse is applied and 2 the net magnetization is tilted from the z axis back to the transverse plane, and an identical field gradient pulse as the first one is applied. The time interval between the first and the second π r.f. pulse and that between the third π r.f. pulse and the beginning 2 2 of the signal acquisition are defined as the de-phasing intervals and re-phasing 21

22 intervals, respectively. The process during the de-phasing intervals τ 1 is longitudinal T 1 relaxation and that during the re-phasing intervals is transverse T 1 relaxation. Due to the molecular diffusion between the applications of the two field gradient pulses, the rephrasing of the spins is incomplete giving rise to the decrease of the intensity. Hence the rate of diffusion and the mean square displacement can be measured using the signal decay. This pulse sequence provides a good approach to the measurement for the systems required for large diffusion times with short T 2 relaxation time, as well as systems in which T 2 relaxation time is much shorter than T 1 relaxation time. PFG NMR Stimulated Echo Longitudinal Encode-Decode Pulse Sequence Figure 1-5. Schematic of the PFG NMR Stimulated echo longitudinal encode decode pulse sequence. Figure 1-5 shows the schematic of the process of the PFG NMR stimulated echo longitudinal encode- decode or longitudinal eddy current delay pulse sequence (PGSTE-LED). During the switch-on and switch-off of the field gradient pulses (especially high-amplitude pulses), the introduced eddy currents by the changing magnetic field in the gradients coils give rise to the inhomogeneity in the magnetic field. Such field inhomogeneity is addressed in PGSTE-LED by two more π r.f. pulse between 2 the second field gradient pulse and the beginning of the signal acquisition, which is 22

23 known as the LED delay (T LED ) and allows for the dissipation of the eddy current. The PGSTE-LED sequence offers a clear advantage over the PGSTE sequence under the measurement conditions with short T 2 relaxation times and large field gradients which are common conditions in measuring diffusion in porous materials. Attenuation Equation Applied in PFG NMR is given by Under the field gradient pulse, the sum of the magnetization phase angle of spins t 0 φ(t) = γ (B 0 + gz)dt. (1-17) After the de-phasing and re-phasing processes, the difference in the magnetization phase angle is accumulated and can be written as τ φ z1 = γ { (B 0 + gz 1 )dt 0 2τ τ (B 0 + gz 2 )dt} = γgδ z, (1-18) in which z 1 and z 2 represent the position of the spins when the first and second gradient are applied. If the positions of the spins change during the process ( z 0), the value of φ z will be non-zero so that we can acquire the attenuation of the intensity of the NMR signal, which can be written as in which ( φ z1 ) is the distribution of the spin phase at z 1. as Ψ = cos ( φ z1 )P( φ z1 )d φ z1, (1-19) In terms of mean square displacement z 2, the equation above can be rewritten φ z1 2 = (γδg) 2 z 2, (1-20) in which φ z1 2 denotes the average change of phase accumulation between the first and second field gradient pulses. 23

24 Consequently the equation of attenuation curve for all spins can be given by Ψ = exp ( (γδg) 2 t eff D). (1-21) Typically, the effective diffusion time t eff in PGSTE and PGSTE-LED experiment is given by t eff = δ/3. (1-22) In order to find the experimental diffusion coefficient, the attenuation is usually plotted as the function of either g or t eff, when the value of the unchosen parameter is fixed. 24

25 CHAPTER 2 SELF-DIFFUSION OF CO2 IN SAMARIA/ALUMINA AEROGEL CATALYST In this project, we investigate gas self-diffusion in alumina stabilized samaria aerogel catalyst using pulsed field gradient (PFG) NMR. For this catalyst, the oxidative coupling of methane (OCM) is an interesting reaction. 21 Apart from the desired reaction products, ethane and ethylene, total oxidation to CO2 is an important side reaction. Based on this consideration CO2 was chosen as a probe molecule for the reported PFG NMR diffusion studies. Application of a high magnetic field of 17.6 T enabled studies of CO2 self-diffusion for a broad range of loading pressures between 0.1 and 10 atm. Since alumina stabilized samaria aerogel can be easily formed into monoliths and powder beds, which are considered for catalytic applications, the reported diffusion studies were performed for both sample types. PFG NMR data obtained for powder beds are interpreted in the framework of a diffusion-mediated exchange model. 22 Experimental Details Samaria Aerogel Catalyst Synthesis and Structural Characterization Alumina stabilized samaria aerogels were prepared using the epoxide addition sol-gel method established by Gash et al. 23, 24 The aerogels samples were obtained from our collaborators Dr. Marcus Bäumer and Dr. Björn Neumann in Germany and were prepared as monoliths and as particle beds. The aerogel monoliths used in this study were cylindrical in shape with a diameter of about 2-3 mm and mm in length. The particle beds were obtained by mechanically grinding the material, followed by sieving using sieves with mesh sizes between 280 and 100 µm. These mesh sizes determine the range of particle sizes in the bed. Figure 2-1 shows a representative SEM 25

26 image of the bed obtained by our collaborators in Germany. This image indicates a relatively low degree of a particle shape asymmetry. Nitrogen adsorption isotherms were measured at 77 K using a QuadraSorb sorption analyzer (Quantachrome Instrument Corp.) by our collaborators Dr. Marcus Bäumer and Dr. Björn Neumann in Germany. All samples were degassed for 120 hours at 523 K prior to sorption experiments. Pore size distributions were obtained from the measured N2 adsorption isotherms by applying density functional theory (DFT) and the Barrett-Joyner-Halenda (BJH) methods for micro-/meso-porous and meso-/macroporous regions, respectively. Figure 2-2 presents the estimated pore size distribution from about 1 nm to 250 nm. The first moment of the pore size distribution indicates an average pore size around 75 nm. Figure 2-1. SEM image of a bed of alumina stabilized samaria aerogel particles used in this study. The image was acquired using a JOEL JSM microscope with a thermal cathode at 5 kv acceleration voltage. To reduce charging effects, the particles were coated with a gold layer (2-5 monolayers) using a commercial gold sputter coater. 26

27 0.8 DFT BJH 0.08 MFP in Catalyst dv, cm 3 g -1 nm dv, cm 3 g -1 nm d, nm 0.00 Figure 2-2. Pore size distributions for the studied alumina stabilized samaria aerogel. The distributions were obtained from N2 adsorption isotherms measured at 77 K by applying density functional theory (open circles, left-axis) and Barrett- Joyner-Halenda (filled squares, right-axis) methods. The pore size distributions have been scaled such that the areas under the distributions are proportional to the calculated total volume by their corresponding adsorption model. Also shown (bar above the distribution) is the range of the values of the mean free path of CO2 molecules (MFP) estimated from kinetic theory of gases for the CO2 intrapore densities corresponding to the loading pressures in the range between 0.1 and 0.9 atm. NMR Measurements To prepare the samples for PFG NMR studies, about mg of aerogel powder or monoliths was placed into a 5 mm NMR tube. The tube was connected to a custom-made vacuum system where the aerogel samples were degassed under high vacuum (< 10-3 Pa) at an elevated temperature (around 473 K) for at least 24 hours. The initial evacuation of samples from atmospheric pressure to high vacuum was conducted slowly over a period of at least 12 hours at room temperature to prevent any potential alterations in the sample microstructure caused by rapid desorption of water or other sorbates. Samples were prepared at the following equilibrium pressures of CO2: 0.1, 0.3, 0.9, 3.1 ± 0.5 and 10 ± 1.5 atm. For sub-atmospheric to near-atmospheric loading pressures, CO2 was loaded into the degassed aerogel samples by exposing the 27

28 samples to the gas at the desired gas pressure for at least 4 hours at ~298 K to allow for sorption equilibrium. Greater than atmospheric loading pressures were obtained by cryogenically freezing the desired mass of sorbate into the sample tubes (wall thickness of 0.78 mm) using liquid nitrogen. Upon loading, all NMR sample tubes were flame sealed and separated from the vacuum system. In all cases carbon dioxide with 99% 13 C isotopic enrichment (Sigma-Aldrich) was used to increase the signal-to-noise ratio of NMR measurements. The sorption loadings of the catalyst samples were determined by comparing the NMR signal of CO2 in these samples with the signal originating from the NMR tubes containing only a known quantity of CO2 gas (no porous materials present), in the same way as discussed in S. Vasenkov and J. Kärger. 29 Figure 2-3 shows these loadings as a function of CO2 pressure in the surrounding gas phase at 297 K. It is important to note that if any strongly-bound, i.e. chemisorbed, CO2 molecules are present in the catalysts samples, the NMR adsorption isotherm in Figure 2-3 is not expected to show any contributions from such molecules because the NMR line width of such molecules is expected to be large. The experimental uncertainties of the loadings in Figure 2-3 are expected to be within about 15%. The data in Figure 2-3 can be well described by a Langmuir isotherm (solid line in the figure), q = C bp 1+bp, (2-1) where q is the concentration of sorbate in the catalyst, p is the equilibrium loading pressure in the gas phase, C is the sorption capacity and b is the affinity constant. Least-squares fit of Equation 2-1 to the data in Figure 2-3 resulted in the following parameters: C = 5.11 ± 0.5 mmol g -1 and b = 0.11 ± 0.02 atm

29 1 q, mmol / g p, atm Figure 2-3. CO2 adsorption isotherm measured for the bed of the particles of alumina stabilized samaria aerogel catalyst by 13 C NMR at 297 K. The solid line shows the best fit of the sorption data to a Langmuir isotherm (Equation 2-1). The inset presents the isotherm for the 0 to 1 atm pressure region using linear scales. The fit of the inset data to Henry s law is shown with a dashed line and corresponds to a Henry s Law constant of 0.54 mmol atm -1 g -1. Diffusion measurements of carbon dioxide in the alumina/samaria aerogel catalyst samples were performed using a wide-bore 17.6 T NMR spectrometer (Bruker Biospin) operating at a resonance frequency of MHz for 13 C. The high field of 17.6 T provided high sensitivity and, as a result, enabled direct measurement of diffusion of CO2 molecules in the adsorbed phase as well as in the bulk gas phase at relatively low sorbate densities corresponding to around 0.1 atm in the gas phase. A Diff60 diffusion probe and Great60 gradient amplifier (Bruker BioSpin) were used to generate magnetic field gradients. In this study, the pulsed gradient stimulated echo (PGSTE) and pulsed gradient stimulated echo with longitudinal eddy-current delay (PGSTE LED) pulse sequences were used. The gradient pulses were kept short ( µs effective duration) with maximum gradient amplitudes ranging between about T m -1 ; spoiler gradients of 0.3 T m -1 were applied; and the LED delay time was 1.5 ms. The absence of disturbing magnetic susceptibility effects and other measurement artifacts 29

30 was confirmed by verifying that a variation of the time interval between the first and second π/2 radiofrequency pulses of sequences in the range between ms (while keeping the effective diffusion time constant) does not change the measured diffusivities. Self-diffusivities were obtained by measuring the attenuation of the area under the 13 C NMR line of CO2 with increasing magnetic field gradient amplitude, g, and keeping all other PFG NMR pulse sequence parameters constant. In the case of normal (i.e., Fickian) self-diffusion with a single diffusion coefficient (D) PFG NMR attenuation curves can be described using the following relation for the pulse sequences used in this study: 22, 25, 26 Ψ S(g) S(g = 0) = exp ( Dq2 t), (2-2) in which Ψ is the NMR signal attenuation, S is the area under the NMR signal, t is the time of observation of diffusion process (i.e. diffusion time) and q is defined to be γgδ where γ is the gyromagnetic ratio, δ is the effective gradient pulse length. In the case of normal self-diffusion in three dimensions, the mean square displacement (MSD) can be obtained using the Einstein relation: 22 r 2 (t) = 6Dt. (2-3) The square root of Equation 2-3 yields the root mean squared displacement (root MSD), a characteristic displacement of a diffusion process. Under our experimental conditions for CO2 in the aerogel samples, the longitudinal (T 1 ) 13 C NMR relaxation time varied between about 30 ms at an equilibrium sorbate loading pressure of 0.1 atm and 250 ms at 10 atm. Likewise, the transverse (T 2 ) relaxation time varied between about 2 ms at a loading pressure of 0.1 atm and 30

31 about 70 ms at 10 atm. The increasing relaxation times with gas density is a well-known NMR property of gases. 27 Longitudinal and transverse relaxation times were measured using the inversion-recovery and Carr-Purcell-Meiboom-Gill pulse sequences, respectively. At all loading pressures, the obtained relaxation curves were consistent with only one relaxation ensemble. Results and Discussion Figure 2-4 shows a representative set of PFG NMR attenuation curves for CO2 diffusion in both powder bed and monolith aerogel samples for a range of diffusion times between 3.2 and 23.2 ms. In Figure 2-4, the attenuation contribution of CO2 molecules that diffuse only in the gas phase of the samples over the duration of diffusion observation time has been subtracted away for clarity of the presentation in the same way as discussed in ref. 29. It is seen in Figure 2-4 that the attenuation behavior of CO2 inside the monolith is mono-exponential (e.g. a straight line in the semi-log presentation of Figure 2-4) and time-invariant as seen by the coincidence of the attenuation curves for different diffusion times. Such attenuation behavior is in agreement with Equation 2-2 and corresponds to normal, isotropic diffusion in the monolith with a single, time-independent diffusivity. 31

32 1 3.2 ms 7.2 ms 15.2 ms 23.2 ms x x x x10 6 q 2 t, s 1 m -2 Figure 2-4. PFG NMR attenuation curves for carbon dioxide diffusion in alumina stabilized samaria aerogel catalyst prepared as a monolith (filled symbols) and as a powder bed of about 200 μm sized particles (open symbols) at 297 K with an equilibrium pressure of CO2 in the gas phase of 0.3 atm. Symbol shape denotes the effective diffusion time in the PFG NMR measurement. These curves were obtained from the measured PFG NMR attenuation curves by subtracting away the contributions from CO2 molecules that diffuse only in the gas phase of the samples during the diffusion time used in the measurement. Solid lines show the best fit results using Equation 2-2 for the monolith sample and Equation 2-4 for the powder bed sample. For the powder bed sample, the PFG NMR attenuation is not mono-exponential. It also depends upon the effective diffusion time in the presentation of Figure 2-4. This behavior is consistent with the existence of an exchange of CO2 molecules between the catalyst particles and the gas phase between the particles in the bed during the diffusion time. In this case the attenuation curves can be described by the weighted sum of the contributions from the molecules that do not leave the particles during the diffusion time (intraparticle ensemble) and those that exchange between the particles and the surrounding gas phase (long-range ensemble): 22, 26 Ψ(t) x intra (t)exp ( D intra q 2 t) + x lr (t)exp ( D lr q 2 t), (2-4) in which x is the ensemble fraction of diffusing molecules, intra subscripts denote the intraparticle ensemble and lr subscripts denote the long-range ensemble. The diffusivity 32

33 of the intraparticle ensemble can be expected to be smaller than that of the long-range ensemble. It is expected that as the diffusion time (and thus the root MSD) increases, the intraparticle ensemble fraction decreases in favor of the long-range ensemble fraction. This behavior is apparent in Figure 2-4; as the diffusion time increases, the curves become more mono-exponential (linear in Figure 2-4) indicating an increasingly larger fraction of the long-range ensemble with the largest diffusivity. Sets of attenuation curves similar to those shown in Figure 2-4 were also obtained for other CO2 loading pressures between 0.1 and 10 atm used in this work. Least squares regression of the PFG NMR attenuation curves to Equation 2-2 for the monolith samples and to Equation 2-4 for the powder samples yielded intramonolith, intraparticle and long-range diffusivities. Figure 2-5 and Table I present these diffusivities as a function of the CO2 loading pressure. The values of the root MSD obtained by Equation 2-3 for the measured intraparticle diffusivities were found to be comparable with the particle sizes in the powder bed which confirms the existence of the exchange under the PFG NMR measurement conditions. Figure 2-5 also shows for comparison the corresponding diffusivities of the bulk gas phase. The latter diffusivities were measured in separate PFG NMR experiments using samples that contain only CO2 without the catalyst. These measured diffusivities were in satisfactory agreement with the predictions of the kinetic theory of gases. The experimental uncertainty shown in Figure 2-5 and Table I takes into account contributions from the signal-to-noise ratio in any particular PFG NMR measurement, the reproducibility of the results of several identical PFG NMR measurements of the 29, 30 33

34 same sample and the reproducibility of the results of identical PFG NMR measurements performed for different preparations of the same type of sample. It is seen in Figure 2-5 and Table I that for any loading pressure used in this work the intramonolith and the intraparticle diffusivities are the same within the experimental uncertainty. The monolith and powder samples are derived from the same material and are expected to have the same microstructure as well as the related transport properties. However, it is important to note that the coincidence of these two diffusivities suggests that there are no significant transport barriers at the external surface of the particles in the powder samples. The presence of such barriers would decrease the measured intraparticle diffusivities under our experimental conditions when a large fraction of molecules reaches the particle boundaries during the diffusion time. 22 Table 1-1. Self-diffusivities of CO2 in the studied alumina stabilized samaria aerogel catalysts at 297 K as a function of equilibrium loading pressure. These selfdiffusivities were determined by the regression of PFG NMR attenuation curves to Equation 2-1 for intramonolith diffusivities (D mono ) and to Equation 2-2 for the intraparticle diffusivities (D intra ) and long-range diffusivities (D lr ) of the powder bed. p, atm D mono, 10 7 m 2 s 1 D intra, 10 7 m 2 s 1 D lr, 10 7 m 2 s ± ± 3 15 ± 3 55 ± ± 2 12 ± 2 26 ± ± ± 2 8 ± 2 12 ± 3 10 ± ± ± ±

35 10-4 Gas Monolith Powder, Long-Range Powder, Intraparticle Long-Range Model D, m 2 s p, atm 10-8 Figure 2-5. CO2 self-diffusivities in the studied alumina stabilized samaria aerogel catalyst samples at T = 297 K as a function of sample equilibrium loading pressure. The data are shown for the following diffusion types: diffusion inside the monolith samples ( ), intraparticle diffusion in the particle bed samples ( ), long-range diffusion in the particle bed samples ( ). Also shown for comparison are the gas-phase self-diffusivities of CO2 ( ). The diffusivities were obtained from the measured PFG NMR attenuation curves using Equation 2-2 or Equation 2-4. The solid line shows the long-range diffusivity obtained from the two-region exchange model (Equation 2-7) and using an interparticle tortuosity factor of 1.7. The data in Figure 2-5 and Table I shows that the dependence of the CO2 diffusivity inside the catalyst particles on the sorbate loading pressure is very weak or even nonexistent in the loading pressure range between 0.1 and 0.9 atm. This observation can be a consequence of the domination of the diffusion process in the catalyst mesopores, which represent the most abundant pore type of the studied material, by Knudsen diffusion. Figure 2-2 shows that indeed in this loading pressure range the mean free path (MFP) of CO2 molecules inside intraparticle mesopores is comparable with the mesopore sizes. The values of MFP, λ, shown in Figure 2-2 were estimated using the following relation from the kinetic theory of gases: 22, 29 λ = 1 2πnσ 2 (2.5) in which n is the sorbate molar concentration inside the catalyst and σ = 3.3 Å is the 35

36 kinetic diameter of the sorbate molecule. 31 Here we assumed that CO2 molecules are randomly distributed inside the catalyst mesopores and that the presence of small (by volume) fraction of micropores can be neglected for the purpose of the estimate. Such assumptions are in agreement with the observation that for the considered range of the loading pressures the dependence of the intraparticle concentration of CO2 on the loading pressure can be well described by a linear dependence in agreement with Henry's law (Figure 2-3). For CO2 loading pressures around 1 atm or larger a transition from the Knudsen to the molecular diffusion regime is expected with increasing loading pressure. This explains an observation of the decreasing intraparticle diffusivity with the increasing pressure in Figure 5 for p 1 atm. The long-range self-diffusivity of CO2 in the powder samples depends on the sorbate loading pressure as apparent in Figure 2-5. It is seen that the long-range diffusivity is greater than the intraparticle diffusivity at low loading pressures (e.g. factor of 4 for 0.3 atm loading pressure) and asymptotically approaches the intraparticle diffusivity as the loading pressure increases. To describe the measured long-range diffusivities quantitatively the following well-known relation for the two domain exchange model can be used: 22, 32, D lr = p intra D intra + p inter D inter, (2-6) in which p intra and p inter are the fractions of molecules inside of the porous particles and in the space between the particles, respectively. D inter is the interparticle diffusivity, i.e. self-diffusivity for diffusion in the interconnected voids between the particles for displacements much larger than the particle size. In Equation 2-6, p intra and p inter can be written in terms of the volumetric void fraction of the bed, ϕ and molar densities of 36

37 sorbate, ρ. The void fraction was determined by comparing the macroscopic mass density of the powder bed to that of the monolith sample, yielding a value of ϕ = 0.38 ±0.05. D inter is taken to be the bulk gas phase diffusivity (D gas ) divided by an interparticle tortuosity factor, η. 29, 36 In terms of these variables, Equation 2-6 can be rewritten as: D lr = 1 (1 φ)ρ intra +φρ gas [(1 φ)p intra D intra + φp gas D gas η ]. (2-7) Equation 2-7 was used to calculate the values of interparticle tortuosity factor η, which is the only unknown parameter in the equation. The calculated values are presented in Figure 2-6 for each of the loading pressures used in the measurements. It is seen that all the values of the tortuosity factor agree within the experimental uncertainty, with an average value of (1.7 ± 0.2). This tortuosity factor appears to be reasonable as it is within the experimental uncertainty of the theoretical value (η = 1.61) for a bed of randomly packed, non-overlapping spheres 37 and also the experimentally determined tortuosity factor for water self-diffusion in a bed of glass spheres (η = 1.51) with the same void fraction (φ = 0.38) and similar particle sizes (e.g µm) as the studied aerogel catalyst bed. 38 Hence, Equation 2-7 provides a satisfactory description of the measured long-range diffusivities. This is also demonstrated in Figure 2-5 that shows a good agreement between the values of the long-range diffusivities calculated using Equation 2-7 with η = 1.7 and the corresponding measured diffusivities. 37

38 3 2 inter p, atm Figure 2-6. Interparticle tortousity factors for different equilibrium loading pressures of CO2 in the catalyst powder bed samples. The tortuosity factors were calculated using Equation 2-7 where all the parameters characterizing diffusion, porosity and sorption properties were obtained experimentally. Conclusion Self-diffusion studies using high field PFG NMR were performed for carbon dioxide in alumina stabilized samaria aerogel catalysts for a broad range of equilibrium loading pressures. The studied catalyst samples were prepared in the form of macroscopic monoliths and powder beds. Under our measurement conditions the values of the root MSD of CO2 molecules in the monolith samples were much smaller than the monolith sizes. In this case, no influence of the external monolith surface on the measured intramonolith diffusivities is expected. For each CO2 loading pressure a single intramonolith diffusivity independent of the root MSD value was observed, thus indicating that the transport properties of monoliths are uniform. For the powder beds, the values of the root MSD of CO2 molecules were comparable with the sizes of catalyst particles in the beds. In this case PFG NMR diffusion studies revealed the presence of two distinct diffusivities for two ensembles of molecules: an ensemble of molecules inside the catalyst particles and an ensemble of molecules which rapidly exchange 38

39 between the intraparticle and interparticle space. The observed coincidence of the corresponding intraparticle and intramonolith diffusivities suggests that there are no significant transport barriers at the external particle surface in the beds. It was found that the measured long-range diffusivities, i.e. diffusivities in the beds of catalyst particles for displacements much larger than the sizes of individual particles, can be described quantitatively using a simple two-region exchange model. This model allowed estimating the tortuosity factor for diffusion in the voids of the bed. The value of this tortuosity factor was found to be in agreement with the expectations based on the previously published data, thus confirming the applicability of the model. The long-range diffusivities, which characterize the rate of self-diffusion in the particle beds on the macroscopic length scales of displacements approaching the size of monoliths, were compared with the corresponding intramonolith diffusivities. Such comparison gives direct information on the relationship between the mean residence times of sorbate molecules in the beds and monoliths having the same overall size. It was observed that at low loading pressures of CO2 the long-range diffusivity is much larger (e.g., by about a factor of 4 at a 0.3 atm loading pressure) than the corresponding intramonolith diffusivity. With increasing loading pressure this difference was found to decrease and at a loading pressure of about 10 atm both diffusivities become indistinguishable within the experimental uncertainty. 39

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