A Simulation Study of Diffusion in Microporous Materials

by

Mahmoud Kamal Forrest Abouelnasr

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

In

Chemical Engineering

In the

Graduate Division

Of the

University of California

Committee in charge:

Professor Berend Smit, Chair

Professor Jhih-Wei Chu

Professor Phillip Giessler

Spring 2013

A Simulation Study of Diffusion in Microporous Materials

Copyright 2013

By

Mahmoud Kamal Forrest Abouelnasr

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Abstract A simulation study of diffusion in microporous materials

By Mahmoud Kamal Forrest Abouelnasr

Doctor of Philosophy in Chemical Engineering University of California

Professor Berend Smit, Chair

The goal of this work is to develop molecular simulation techniques to characterize the diffusion properties of crystalline microporous materials for separation applications. The conventional simulation type used to study the diffusion behavior of adsorbates in a microporous material is Molecular Dynamics. However, for slowly diffusing systems, these simulations become intractably long. In such situations, the diffusion process can be considered as a series of rare cage-to-cage hops, with the majority of the time (and computational effort) spent on unimportant movements within a cage. Recent work in the field has focused on the application of transition state theory (TST) to this process, allowing an estimation of the diffusion properties with a Monte Carlo simulation. In some cases, for example the diffusion of methane in zeolite LTA at low loading, TST gives a good approximation of the true (MD) diffusion. For the general case, the TST result requires a correction factor, which is calculated with a Bennett-Chandler simulation. The correction factor is the conditional probability the system will undergo a transition given that it is at the transition state; this correction factor is influenced by the number of particles in each cage. For the system of methane in zeolite LTA, there are between zero and fifteen particles in either cage at any time, meaning that 120 different correction factors must be calculated. We developed a mixing rule that relates the correction factor between two cages of unequal loading (a and b) to the correction factors between two cages of equal loading (a and a; b and b). This reduced by an order of magnitude the number of Bennett-Chandler simulations required, from 120 to 16. Next, we investigated a fundamental change in the packing of methane adsorbed in zeolite LTA that occurs at high loadings, where a sub-lattice develops within each supercage leading to increased blocking a divergence between the self- and collective-diffusion coefficients. This qualitative change was replicated in a model kMC system that accounted for this topological shift. As particles move within a cage, their speed fluctuates until at one moment, they happen to be travelling quickly enough to hop out of their potential energy well and through the window. As they fall into the potential energy well of the next cage, they speed up for some time until they re-equilibrate. During this time, they are more likely to hop again. Hopping is no longer a Markovian process, without memory of past events. This has a tremendous impact the diffusion behavior. This behavior was observed for methane adsorbed in zeolites ASV, LTA, and CGS, where various different departures from an expected random walk of Poisson-distributed hops were investigated. Because of the immense number of frameworks available for study, we

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developed a high-throughput computational screening method, applying TST to a vast database of >80,000 hypothetical zeolite structures in order to asses their suitability for carbon capture. From this large set of structures, several materials were identified with higher predicted performance for carbon capture by a factor of four or more. These high-performing structures were observed to exhibit certain structural similarities. The materials in this large database did not exhibit a Robseon upper bound.

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Table of Contents

Table of Contents ....................................................................................................................... i

Acknowledgements ................................................................................................................ iii

Chapter 1. A review of diffusion simulations for carbon capture............................ 1 Carbon dioxide and global climate change ................................................................................ 1 Carbon Capture .................................................................................................................................... 2

Carbon Capture methods ............................................................................................................................ 2 Post-Combustion capture with sorbents .............................................................................................. 3 Membrane Separations ................................................................................................................................ 4 Zeolites and other Microporous Materials .......................................................................................... 6

Diffusion ................................................................................................................................................. 6 Diffusion Coefficients. ................................................................................................................................... 6 Diffusion in the adsorbed phase and the Reed-Ehrlich Model .................................................... 7 Diffusion of mixtures and the Maxwell-Stefan formulation ......................................................... 9

Diffusion Simulations ..................................................................................................................... 11 Molecular Dynamics ................................................................................................................................... 11 Monte Carlo simulations .......................................................................................................................... 13 Transition State Theory ............................................................................................................................ 14 Bennett-Chandler simulations ............................................................................................................... 15 Kinetic Monte Carlo Simulations .......................................................................................................... 17

Chapter 2. Diffusion in Confinement: Kinetic Simulations of Self- and Collective Diffusion Behavior of Adsorbed Gases ........................................................................... 19

Abstract ............................................................................................................................................... 19 Introduction ...................................................................................................................................... 20 Theory.................................................................................................................................................. 21

Re-Crossing Coefficient for asymmetrical loadings ...................................................................... 21 Calculation of Hop Rates with Explicit Contribution From Concentration Fluctuations............................................................................................................................................................................. 23 Kinetic Monte Carlo .................................................................................................................................... 23

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Methods ............................................................................................................................................... 23 Simulation System ...................................................................................................................................... 24 Molecular Dynamics ................................................................................................................................... 24 Canonical Monte Carlo .............................................................................................................................. 24 The Re-Crossing Coefficient .................................................................................................................... 25 Kinetic Monte Carlo .................................................................................................................................... 25 Calculation of Diffusion Coefficients ................................................................................................... 25

Results and Discussion .................................................................................................................. 26 Diffusion of Methane in LTA ................................................................................................................... 26 Zeolites SAS and AFI: Asymmetrical Re-Crossing Coefficients ................................................ 32 Diffusion of Isobutane in Zeolite MFI .................................................................................................. 33

Conclusions ........................................................................................................................................ 35

Chapter 3: A simulation study of rare diffusive hop events of methane adsorbed in zeolites .............................................................................................................. 37

Abstract ............................................................................................................................................... 37 Introduction ...................................................................................................................................... 37 Results and Discussion .................................................................................................................. 38 Conclusion .......................................................................................................................................... 45

Chapter 4. Large-scale Screening of Zeolite Structures for CO2 Membrane Spearations .............................................................................................................................. 46

Abstract ............................................................................................................................................... 46 Introduction ...................................................................................................................................... 46 Methods ............................................................................................................................................... 48

Molecular Dynamics Simulations ......................................................................................................... 49 Efficient Diffusion Coefficient Calculations ...................................................................................... 49 Grand Canonical Monte Carlo Simulations ....................................................................................... 50

Results and Discussion .................................................................................................................. 51 Conclusion .......................................................................................................................................... 59

Appendix A: Supplementary information for “Chapter 2. Diffusion in Confinement: Kinetic Simulations of Self- and Collective-Diffusion Behavior of Adsorbed Gases” .................................................................................................................... 60

Appendix B: Supplementary information for “Chapter 4. Large-scale Screening of Zeolite Structures for CO2 Membrane Spearations” ............................................. 68

Illustration of a CH4/CO2 membrane separation process ................................................. 69 Detailed derivation of an ideal membrane system for binary separation application. ........................................................................................................................................ 70 Relationship between CO2 Henry coefficient vs self diffusion coefficient for large database of zeolite materials ...................................................................................................... 72

References ................................................................................................................................ 73

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Acknowledgements

I would like to acknowledge the support provided to me by the U.S. Department of

Energy’s EFRC for Gas Separations Relevant to Clean Energy Technologies, and the

Advanced Research Projects Agency – Energy (ARPA-E) for funding. Since both of

these are funded with U.S. federal government money, I would like to thank all American

citizens for supporting this work; I will strive to be a good investment.

I would like to thank my cohort of Chemical Engineering graduate students at Cal who

arrived in 2008. Thank you John Alper, Melissa Bartel, Anthony Conway, Matt Dodd,

Kris Enslow, Tony Ferrese, Joe Gomes, Ben Hsia, Victor Ho, Eddy Kim, Shannon Klaus,

Joe Lee, Anton Mlinar, Miguel Modestino, Katie Pfieffer, Chris Shymansky, Peter Soler,

Alex Teran, and Nick Young. Knowing there were others like me suffering through the

same ordeal inspired me not to give up. At the moment I write this, none of us have given

up. I would also like to acknowledge all the other people whose friendships have given

me strength: thank you Matt Pavlovich, Annie Sauthoff, Jon King, Josh Howe, Coffee

Lab Guy, Janice at Yali’s, and the entire Clark lab.

During my five years here I have had the privilege of working with a group of friendly

and intelligent co-workers, and I would like to thank you for your help. Thank you Joe,

Frederick, Kevin, Josh, Ayelet, Li-Chiang, Shachi, Johanna, Corey, Tae, Teresa, Kristin,

Drew, Rocio, Bei, Manju, Fangyong, Jihan, Nils, An, Jesper, Roberta, Sergey, Rupert,

Wendy, Maciek, and Richard.

Thank you Prof. Smit for your mentorship, trust, and especially patience.

Finally, I would like to thank my family. Mom, Dad, Miriam, and Reese, thanks for the

years of encouragement and emotional support. Tahina, Hallawa, and Gryphy, thank you

for always helping cheer me up. And a special thank you to Kierston; without you I

wouldn’t have had the strength to finish this thing.

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Chapter 1. A review of diffusion simulations for carbon

capture

Carbon dioxide and global climate change

Greenhouse gases like carbon dioxide are a major cause of global climate change. Carbon dioxide concentrations in the atmosphere are at their highest levels in at least 400,000 years,1,2 with the latest spike clearly correlated to the eighteenth-century industrial revolution that caused continuously increasing anthropogenic CO2 emissions. Global temperatures have historically tracked closely with atmospheric CO2 concentrations, leading to a recent global warming trend.3 Consequences of this warming include4 increased ocean level, global humidity, ocean acidity, and local changes in climate, all of which can have negative environmental, economic, and social consequences. It is therefore crucial to find methods to mitigate these effects. Carbon capture and sequestration is a promising method to reduce CO2 emissions, as it can be retrofitted to existing point sources of CO2, such as hydrocarbon-combustion power plants, or decentralized to remove CO2 from the ambient atmosphere. Carbon dioxide is present in a variety of streams from which separation would be desired. The flue gas of a hydrocarbon-combustion power plant is primarily nitrogen gas with a small but significant quantity of CO2. A separation of oxygen from air would allow for the gas exhaust from so-called oxy-fuel combustion to be relatively pure CO2 (after the relatively easy removal of water via condensation). Reacting a hydrocarbon anaerobically with water (“steam reformation”) creates H2 and CO2, a process called precombustion capture. Efficient removal of CO2 from natural gas (i.e. methane) deposits allows for greater use of natural gas, ideally replacing the use of other carbon fuels that produce more CO2 per unit energy when burned, like coal. And the removal of

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CO2 from ambient atmosphere, while less efficient because of lower CO2 concentrations than at CO2 sources, has advantages because it does not require the cooperation or consent of those responsible for CO2 emissions, and can be decentralized and distributed anywhere on the planet’s surface.

Table 1.1. Typical gas compositions (volume %) for CO2-containing effluent streams for separation.2,5,6

stream CO2 N2 H2 O2 H2O CO Coal power plant flue gas 15-16% 70-75% 3-4% 5-7% 0.002% Natural Gas 0-40% or more Air 0.035% 78% 21% 0-3% Oxy-Fuel effluent 80-98% 0-20%

Precombustion effluent

35.5% 0.25% 61.5% 0.2% 1.1%

Carbon Capture

Carbon Capture methods

Several mechanisms exist to capture CO2. Most of the processes discussed here deal with capture from point sources, since the cost and difficulty of CO2 separation increases for more dilute streams, as they contain more entropy which must be removed per unit of CO2 captured. Nevertheless, some methods to remove CO2 from the ambient atmosphere are being investigated7,8 because of the decentralizability of such schemes. For example, two problems with alternative energy sources are intermittence and distance from electricity consumers; using this energy instead for carbon capture would only require access to air. Another advantage to this method is that it does not require the cooperation or consent of carbon producers, whose intransigence should be expected.9 For carbon capture, separating O2 from air would suffice to effect complete capture, since combustion of fuel in the presence of pure O2 creates a stream of just CO2 and water, which can be separated in a very straightforward manner. This process is called “oxy-fuel” combustion.10 The absence of N2 in the combustion unit would raise the temperature of the resulting exhaust, allowing an increase in the efficiency of the power plant. This is because the maximum possible amount of work extracted from a transfer of heat, known as the Carnot efficiency, is limited from the second law of thermodynamics by the ratio of the temperatures (in an absolute scale) of the two streams:11 However, current research in this field has focused on recycling a portion of the captured CO2 through the system to erase this temperature gain, because the materials of modern power plants are not designed to handle the higher temperatures possible with a pure O2 feed. Another method of carbon capture is pre-combustion separation.12 First, the hydrocarbon is reacted with limited O2 to form a

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mixture of CO and H2 (“gasification”), and then with H2O to form a mixture of CO2 and H2 (“steam reformation”), which can be separated relatively easily. The resulting H2 can be burned with air for power, forming only water. Both of these methods require significantly different process equipment from that currently used in power plants, and a source of O2, whose separation from air is itself problematic. For these reasons, post-combustion carbon capture is an area of extensive research, regardless of the inherent disadvantages.

Post-Combustion capture with sorbents

Currently, the most mature method of post-combustion carbon capture is absorption in a solution of monoethanol amine (MEA) and water.13–15 This solution is sprayed down through a large tank, while the flue gas, a mixture of CO2 and N2, flows upward through the tank. The CO2 is absorbed by the liquid while the N2 passes through and is emitted harmlessly into the atmosphere. The CO2-rich liquid solution is then collected at the bottom of the absorber and heated in a stripper, forcing the dissolved CO2 back into the gas phase. The CO2 gas is then compressed and sequestered, while the regenerated MEA solution can be used again. Notably, this separation process requires heat input in the form of heat for the stripper, which significantly increases the overall cost of electricity from such a power plant by 50-90%.16 Because the MEA solution is comprised mostly of water, the cycle of heating and cooling water is seen as one of the primary areas of possible improvement. Recent work has focused on identifying different absorbents with high absorption selectivity for CO2 but don’t require as much heat input for regeneration.17 Ionic liquids in particular show significant promise because of their stability and extremely high degree of customizability.18,19 An attractive alternative to the absorption process involves a solid sorbent in a packed (or fluidized) bed. This sorbent would ideally adsorb CO2 selectively from N2, and release the CO2 with minimal heat or pressure applied. Such a process would involve several steps. First, the CO2/N2 mixture flows through the bed of solid sorbent, where the CO2 is selectively adsorbed while the N2 passes through and is released to the atmosphere. Once the solid sorbent is saturated with CO2 (“breakthrough”), the inlet is halted and the outlet it directed to a CO2 storage space, while the bed is heated and/or depressurized to remove CO2. Finally, the bed is purged (“swept”) of CO2 with a portion of the previously separated N2, and the bed is regenerated and ready for another cycle. A material’s adsorption selectivity is defined as the ratio of the amounts of either species adsorbed under given pressure and temperature conditions. For Langmuir or other classical adsorption types, the adsorption selectivity of e.g. CO2 over N2 is enhanced when the heat of adsorption for CO2 in that material has a larger magnitude (more negative) than the heat of adsorption for N2 in that material. This is interpreted to mean that CO2 finds the adsorbed phase more energetically favorable than N2 does. However, a higher heat of adsorption means more heat must be applied to remove CO2 from the sorbent, leading to a trade-off between selectivity and thermodynamic efficiency. These systems are usually not diffusion-limited because the timescale for

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concentration in a particle to equilibrate is on the order of , where D is the diffusion coefficient and r is the radius of the material chunk. For slowly diffusing systems the adsorbent material can be ground into smaller chunks. However it should be noted that adsorption and diffusion properties are poorly understood at the material surface; this area could benefit from further study.

Membrane Separations

Removing an energy-intensive regeneration cycle is not possible for sorption-based processes, but a membrane separation provides attractive efficiency, especially for flue gas separation.20 The appropriate selectivity to consider for a membrane separation is

the permeation selectivity,

, which will vary over the thickness of the

membrane (KH,i is the Henry coefficient for species i and DC,i is the collective-diffusion coefficient of species i; both are explained in more detail in a later section). The process efficiency is often a strong function of the total permeation of the desired component. Generally there is a weak inverse relationship between the Henry coefficient and the diffusion coefficient (see Figure B.3 in Appendix B). This is because favorable adsorption properties (i.e. a high heat of adsorption) usually indicate several extremely favorable sites, which must then be separated by a very large barrier, causing slow diffusion. However, this inverse relationship is not strictly observed, since some materials exhibit an adsorption landscape closer to the ideal for maximum diffusion and adsorption: a uniformly attractive channel, spanning the entire structure. In this case both factors contributing to the permeation will be enhanced in the material bulk, if not at the surface. Processes designed to separate CO2 from a coal power plant flue gas stream have recently converged onto one setup for membranes.16 In this setup, the flue gas is first split via membrane separation into two streams, and the N2-rich stream is further treated with an air sweep across a membrane, extracting even more CO2. This air stream is fed into the combustion unit so that any CO2 captured this way does not leave with the N2, but instead is forced to recycle through the system again. This setup allows a more efficient separation because air, which currently has negligible CO2 partial pressure,2 provides a large driving force for the separation at no cost. Side effects of this integrated (i.e. not simply tail-end) process have yet to be thoroughly investigated, but could include O2 depletion in the incoming air stream.

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Figure 1.1. Simplified process flow diagram of a membrane separation system for carbon capture.

Historically, there has been a trade-off in membranes between high permeation and high permeation selectivity as demonstrated in the classic example of polymer membranes.21,22 This is because polymers tend to have a distribution of window sizes. Perfectly sized windows will effectively exclude one species while minimally impeding the movements of another, while too-small windows will exclude both and too-large windows will permit both. Tightening up the distribution of window sizes in a polymer allows an increase in both permeation and permeation selectivity (breaking through the “Robeson upper bound”). The extreme case, with uniform window sizes, can be achieved using crystalline materials. Diffusion in a crystalline microporous material is characterized in part by the size of the largest possible sphere that can traverse the structure.23 Assuming a certain potential energy of the adsorbate is off-limits, soft-core potentials (Lennard-Jones and Coulomb) can be combined to determine the largest possible free sphere. For a given adsorbate, the infinite-dilution diffusion behavior decreases asymptotically as that characteristic diameter decreases, down to the “kinetic diameter” of the adsorbate, below which the molecule cannot diffuse (Figure 1.2).

Figure 1.2. Infinite-dilution self-diffusion coefficient of methane at 300K for some structures provided by

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the International Zeolite Association (IZA), determined from Molecular Dynamics. X-axis is the maximum free-sphere diameter.

Zeolites and other Microporous Materials

Zeolites are a class of crystalline microporous material consisting of oxygen and silicon, with two oxygen atoms for every silicon atom. In some cases a proportion of silicon atoms is replaced with aluminium atoms, necessitating the presence of charge-balancing cations as well, like hydrogen or sodium. Currently, more than 200 zeolite structures have been synthesized,24 but millions of hypothetical zeolite structures have been postulated.25,26 Zeolites could be exceptionally useful in many applications because the pore diameters are about the same size as many common molecules, including CO2. The considerable database of hypothetical zeolites has been analyzed using a variety of high-throughput simulation techniques27,28 (see also chapter 4) for their utility. This huge population of structures is likely to contain materials suited to a variety of applications, among them carbon capture. The interested reader is referred to several excellent reviews on the simulation of adsorbed molecules in zeolites.29–31 Metal-Organic Frameworks (MOFs) consist of metal atoms connected by organic linkers. Because they can be constructed with a variety of metal clusters, organic linkers, and topologies, they are immensely customizable. The rate of production of new MOFs has been increasing exponentially since their invention.32–34 This huge variety and customizability suggests that a MOF could be designed to efficiently capture CO2.35–38 In addition to zeolites and MOFs, more classes of microporous materials are being developed at an accelerating pace.39–42 The process of synthesizing a new structure and testing its adsorption, diffusion, and stability properties is costly and time-consuming. Computer simulations are an ideal choice to fill this gap, since they can be pursued much more cheaply and easily scaled up.

Diffusion

Diffusion Coefficients.

Diffusion behavior can be characterized by a number of different fundamental diffusion coefficients. At infinite dilution, all diffusion coefficients for a given system are the same. The macroscopically observed “transport”-diffusion coefficient is defined by Fick’s law,43 the relationship between net molar flux J and the spatial gradient of concentration, c . (1.1) Recognizing that the driving force for diffusion is actually the spatial gradient of the chemical potential,44,45 µ, one obtains a Fick-like relationship involving the “collective”-diffusion (or “corrected”-diffusion) coefficient DC: . (1.2)

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These two diffusion coefficients can be equated through the relationship between concentration and fugacity, which is known as the thermodynamic factor

. (1.3)

The chemical potential is related to the fugacity by ,

where is the chemical potential of some reference state. In general, the thermodynamic factor in a pure-component system is greater than or equal to one since the concentration of adsorbates always increases with increasing fugacity. This means the transport-diffusion coefficient is generally greater than or equal to the collective-diffusion coefficient. Disregarding the motions of the aggregate population of a given chemical species, but instead focusing on the movements of each molecule individually, one obtains the “self”-diffusion coefficient DS, also known as the “tracer” diffusion. Another way to describe these diffusion coefficients is through Einstein’s relationship,46 allowing equilibrium measurements of an inherently transient property. The self-diffusion coefficient is related to the mean-squared displacement (MSD) of each particle in the system, or equivalently the velocity autocorrelation-function:

(1.4)

(1.5)

where N is the number of particles in the system and d is the number of dimensions considered. Similarly, the collective-diffusion coefficient is related to the mean-squared displacement of the center-of-mass of all particles in the system (or equivalently the velocity correlation function):

(1.6)

(1.7)

This means that macroscopically-observed molar net flux cannot be directly calculated from the self-diffusion coefficient, but for most systems they are within the same order of magnitude29. One way to interpret the difference between self- and collective-diffusion is that collective-diffusion accounts for velocity correlations between particles at different times by including velocity correlations between two different particles at two separate times, thus keeping track of momentum transfer between particles. For the adsorbed phase, the ordering of the diffusion coefficients is generally DT>DC>DS.

Diffusion in the adsorbed phase and the Reed-Ehrlich Model

Molecules in the adsorbed phase exhibit fundamentally different properties than in the gas or liquid phases29,47 because interactions between an adsorbate and the host

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material are usually the principal energetic considerations, over interactions between adsorbates. Adsorption properties thus play a central and inseparable role in the diffusion phenomenon. The host material often imposes a characteristic hop length because the diffusing adsorbate often spends enough of its time in specific attractive adsorption sites that each hop between such sites is de-correlated from the previous one. For sufficiently shallow barriers this decorrelation is not complete, meaning the characteristic hop distance is larger than the material’s site-to-site distance. These energetically favorable adsorption sites can be modeled48 as a lattice of equivalent sites, and adsorbates’ movements between these sites are often uncorrelated, resulting in a random walk. If we assume that the only interactions between adsorbates is site exclusion, we obtain the Reed-Ehrlich model.49 Without further interactions between adsorbed particles, the probability for a successful hop is equal to the fraction of sites that are vacant, leading to an equation for collective-diffusion , (1.8) where is the relative adsorbate loading (i.e. the adsorbate loading divided by the maximum possible “saturation” loading for that chemical species, ; for convenience adsorption loading q is considered rather than concentration c). This is a significant departure from the Darken approximation, in which DC is assumed to be constant with respect to loading (the Darken approximation is thought to apply in “weakly confined” adsorption systems50). In the limit of highly-connected sites, DS and DC converge at all loadings; for poorly-connected topologies, DS<DC at nonzero loadings.51 (In the low-connection topological limit, “single-file diffusion” occurs, which

is not diffusion as defined here. In that case the MSD is proportional to where the proportionality constant is analogous to the diffusion coefficient.52,30) Identical adsorption sites and a lack of interaction between adsorbed particles beyond site exclusion also describes a system which will experience ideal Langmuir adsorption,48 for which

, (1.9)

where is a constant and p is the partial pressure of the adsorbate in the gas phase. Assuming that f=p (valid for an ideal gas), the thermodynamic factor for a Langmuir system is then

(1.10)

(For Langmuir systems, or indeed most adsorption systems, converges to unity in the limit of low loading because Henry’s law,53 which requires that the loading is linearly proportional to the pressure, governs the system. Therefore , resulting in at infinite dilution). The transport-diffusion coefficient for a simple Reed-Ehrlich system is therefore constant with respect to loading,

. (1.11)

Relaxing the assumption that all sites are equivalent, and that particles do not interact beyond site exclusion, leads to a more general Reed-Ehrlich formulation for diffusion

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that considers the number of connections per site (“connectivity”) z, and an interaction energy between particles in adjacent sites, , to give (following Krishna et al54)

, where (1.12)

and

.

It can be seen that at (no interactions beyond site exclusion), this converges to the previous case, i.e. . For positive , which implies particle repulsion, we see that ; for sufficiently large that , the collective-diffusion coefficient at some nonzero loading is higher than at infinite dilution. For negative , which implies particle attraction, we see that at all nonzero loadings the collective-diffusion coefficient is lower than it would be if . The Reed-Ehrlich lattice model described above, with equivalent adsorption sites that can each accommodate only one particle, is not always accurate. Often more than one type of adsorption site exists, multiple adsorbates can fit into one site, or adsorbed molecules affect the favorability of sites (even removing or creating sites). Adsorption topologies often fall into three broad categories: open framework, intersecting channels, or cage-and-window. An open framework has no diffusion-limiting windows that are small enough to strongly impair the movement of an adsorbed molecule. Intersecting channel topologies include systems that can often be described well by the Reed-Ehrlich model. However, cage-and-window topologies usually consist of very large cages that can accommodate more than one adsorbate, separated by small windows that allow only one adsorbate to pass at a time. Since the effects of intermolecular forces will be much more pronounced between particles sharing a cage, these systems will exhibit fundamentally different diffusion behavior. In the limit of larger cages and smaller windows, adsorbates in a cage will become fully mixed between hop events, leading to the condition . This is because particles pass each other in this lattice easily, so a particle’s motion is a random walk from cage to cage as described above, without the possibility of being blocked by other adsorbates present. These systems demonstrate a divergence of DC from DS at high enough loadings that the time-scale of mixing within a cage slows down to the same magnitude as the timescale of cage-to-cage hops, allowing particles to block each other. This blocking behavior is the core cause of the separation between DC and DS in such systems. When two particles block each other, the self-diffusion coefficient is lower than if the two particles had switched positions. However, the center-of-mass of all particles in the system, which defines the collective-diffusion behavior, is unchanged in both situations. This also explains why the approximation that applies better to well-connected intersecting channel topologies.51

Diffusion of mixtures and the Maxwell-Stefan formulation

The Maxwell-Stefan formulation provides a theoretical perspective to view the diffusion phenomenon in mixtures. This formulation considers the spatial gradient of the

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chemical potential as the driving force for diffusion, and accounts theoretically for momentum-transferring interactions between chemical species, leading to a matrix of cross-exchange coefficients. Once constructed, such a matrix can be used to predict the “Maxwell-Stefan” diffusion coefficient at any mixture concentration. Following the work of Kapteijn et al55 the gradient of the chemical potential is now influenced by other molecules present,

(1.13)

where ui is the average velocity of component (chemical species) i, Ði is the Maxwell-Stefan diffusion coefficient (the same as the collective-diffusion coefficient), and the Maxwell-Stefan Diffusion interchange coefficients Ðij describe the momentum interactions between components i and j. These Maxwell-Stefan interchange diffusion coefficients can be calculated from the Onsager coefficients,56–58

(1.14)

where i and j are molecule types and g and h are specific molecules (momentum is conserved, so ). Ðij can be then be written (for e.g. a binary mixture):

(1.15)

The self-diffusion coefficient is related to the Onsager coefficient by

(1.16)

(1.17)

where xi is the mole fraction of component i among all components present. Assuming that these velocity cross-correlations (between molecules) are small compared to the autocorrelations (same molecule), it can then be shown that for an n-component mixture, the Maxwell-Stefan interchange diffusion coefficient is approximated by

(1.18)

These interchange diffusion coefficients have also been estimated from various empirical relationships like the Vignes relation,59 which is the concentration-weighted geometric mean of the collective-diffusion coefficients,

. (1.19)

The molar flux of any species in the mixture now depends on the concentration gradients of all n species present, since they can be thought to push and pull each other: [Kapteijn eqs 14 and 12]

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(1.20) where parenthesis denotes an n-length vector, e.g. (J)= (J1 J2 … Jn) and θi is the loading of component i divided by the maximum loading of that component. The matrix of thermodynamic factors [ ] is defined, e.g. for a binary mixture, as

(1.21)

where pi is the partial pressure of component i in the system. The matrix [B] contains the Maxwell-Stefan diffusivities:

Alternatively, the cross-exchange coefficients can be interpreted from the

equation:60

(1.22)

This means that in the limit of large , interchange events do not have any influence

and DS=DC. But for smaller , interchange events have a larger impact and DS<DC. The

interchange diffusion coefficients thus define the relationship between self and

collective behavior.

Diffusion Simulations

Molecular Dynamics

For simulating diffusion in microporous materials, van der Waals interactions are modeled with the Lennard-Jones potential:61

, (1.23)

where and are constants. These interactions are truncated to a certain cutoff radius on the order of 3 , and shifted so that the potential is continuous at the cutoff radius. Coulombic interactions are calculated using the Ewald summation method.62 Common features of atomistic simulations are the assumption of a rigid adsorbent framework and the implementation of periodic boundary condition (PBC) with a simulation space large enough that a particle will never experience Lennard-Jones interactions with any of its own periodic images. A Molecular Dynamics63–66 (MD) simulation evaluates Newton’s second law of motion67 F=ma for a given configuration, calculating the force on each atom. The system then steps forward in time by a time step, dt, updates all positions and velocities, and repeats the force calculation. For a sufficiently small time step, energy within the system is conserved. A common method to integrate Newton’s laws of motion is the Verlet algorithm,68 defining the next vector position

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from the current and previous positions and , the force F, and the mass m:

The three terms on the right-hand-side of the equation correspond to the first three terms of a Taylor expansion of the position over time, i.e. position, velocity, and acceleration. An alternative scheme is the velocity-Verlet algorithm, which records the velocity explicitly:

(recording the velocity explicitly allows the use of the velocity correlation in equations 1.5 and 1.7 to calculate the diffusion properties). Selection of the time step dt is important: too large, and unphysical energy drift results; too small, and simulations of meaningful time-scales become impractical. Further discussion of the construction of an MD simulation can be found elsewhere.69 A typical MD simulation keeps a constant volume, number of particles, and energy (NVE), which is known as a Micro-Canonical ensemble. From a statistical mechanics perspective a temperature corresponds to an ensemble of energies, and a system must be sufficiently large so that each single particle can sample a “natural” distribution of energies from its interactions with all the other particles. However, for smaller systems this exchange of energy between particles is usually not sufficient to reproduce a natural energy fluctuation. For a molecule adsorbed inside a microporous material and thus surrounded by atoms with which to exchange energy, this would not normally be an issue; however, to improve computational efficiency, the material’s atoms are often assumed to be entirely rigid70, in which case an adsorbate cannot exchange energy with them. In the extreme case a particle is stuck forever in a potential energy well without the energy to escape. Two solutions to this are to allow the framework to be flexible (computationally very expensive), or to implement a thermostat, of which good candidates are Nosé-Hoover71,72 and Andersen73 (the methodical simulation designer will select parameters for their thermostat to replicate the equilibration behavior, e.g. from a step change in temperature, of a particle in a flexible framework74). Another consideration in the use of these thermostats is that they do not appropriately simulate particle dynamics unless a sufficient number of particles are present in the system75. If infinite-dilution behavior is desired, then 20 or more particles can be inserted into the system, and interactions between these adsorbates can be set to zero, thus allowing important implicit interactions through the thermostat. The resulting particle trajectory from an MD simulation of adsorbed molecules can be analyzed to yield the diffusion coefficient. Analysis is enhanced by considering a plot of the mean-squared-displacement (MSD) versus time (Figure 1.3). The first region, at t < 1ps, corresponds to ballistic motion, where a particle’s movements are small enough

13

that the forces acting on it are relatively constant (hence “ballistic”) and the slope in log-log space is two. The next region, at 1ps < t < 100ps, corresponds to the typical amount of time a particle in this system spends trapped in the host framework’s adsorption sites, which are in this case cages, and the slope in log-log space approaches zero. At higher loadings, or in bulk fluid simulations, the same portion of the MSD curve corresponds to the particle trapped within a meta-stable cage comprised of other particles. The final feature in this curve is the diffusive regime, in which the particle has hopped more than a few times from one cage (or meta-stable cage) to another, and the slope in log-log space converges to unity, which implies a linear relationship between time and MSD as suggested by equations 1.4-7. This is the reason for the limit of large times in that equation.

Figure 1.3. Log-log plot of mean-squared-displacement of methane in zeolite LTA at 300K at infinite

dilution, measured from MD simulations. The dashed line is a linear fit to the MSD data in the diffusive regime.

Monte Carlo simulations

Monte Carlo simulations76,77 use random displacements in the system to enhance importance sampling. They can use the same potential energy definitions as described above. However, instead of stepping through time, a new configuration for the system is proposed, which is not required to be connected to the current configuration by any realistic means (e.g. teleportation of particles), and the proposed configuration is accepted or rejected based on various acceptance rules. One popular choice is the Metropolis Algorithm, which takes into account the potential energy U of the system: , (1.24) where

and .

This acceptance rule incorporates temperature so that the ensemble of configurations sampled by the system reflects the imposed temperature. Proposed moves must obey detailed balance,78 specifically ,

14

where P is the probability to be in a configuration and K is the rate to move from one configuration to another. Another way to interpret the detailed balance requirement is that at equilibrium, there is no net flow around a closed cycle of configurations,

,

where is the total movement from configuration A to configuration B (i.e. ). Detailed balance ensures that at equilibrium each process the system might undergo is balanced by the reverse process; the system’s dynamics are reversible and memoryless (“Markovian”79). Beyond the detailed balance requirement, configuration proposals for MC simulations are limited only by creativity. Different ensembles are possible in an MC system than were discussed for an MD system. One might constrain the simulation volume, the number of particles, and the temperature to obtain the Canonical ensemble. Alternatively, one might constrain the energy80 of an MC system by introducing a “demon” with a sack full of energy; if a proposed move involves a decrease of energy in the system, that energy goes into the sack, but if it involves an increase, the energy must come from the sack, and if there isn’t enough energy, the move is rejected. This is the Micro-Canonical ensemble, and the sack represents forms of energy not directly simulated in the MC simulation (i.e. kinetic energy). In the limit of many demons, this converges to the Canonical ensemble. Keeping temperature and volume constant, but imposing a constant chemical potential (µVT) rather than a constant number of particles results in a Grand-Canonical ensemble, achieved through MC moves that allow particles to spontaneously appear and disappear, with somewhat modified acceptance rules that impose a constant chemical potential or equivalently a constant fugacity.81 This type of simulation is very useful for obtaining adsorption properties (e.g. average loading at a given fugacity and temperature) that can be used to calculate the thermodynamic factor .

Transition State Theory

Transition State Theory (TST) has been applied to adsorbate diffusion in zeolites successfully since the 1970s.82–85 Canonical MC simulation can be used to calculate the probability density of the system at the transition state, i.e. of particles at the free-energy barrier surface between cages. For the case of methane adsorbed in zeolite LTA, this corresponds to the probability density of methane pseudoatoms at the plane x=0, shown in Figure 1.4.

15

Figure 1.4. Probability density of methane in zeolite LTA at infinite dilution, determined from Canonical

Monte Carlo simulations. The free energy F(q) at reaction coordinate q (in this case position) is defined from the probability P:

(1.25)

which can be calculated by constructing a histogram of particle occupancy. The TST then predicts the hop rate KTST from a cage to be

(1.26)

(at infinite dilution the denominator is related to the Henry coefficient of the adsorbate in the material). At this point the self-diffusion coefficient can be calculated directly with these rates and knowledge of the topology, assuming a particle performs a random walk on this lattice. For a cubic lattice of equivalent adsorption sites,

, (1.27)

where λ is the cage-to-cage distance. Transition-state theory has been applied to a large number of systems to estimate the diffusion properties very quickly27 (see also chapter 4).

Bennett-Chandler simulations

For an application of transition state theory, it is important that the probability density of the system identifies the correct transition state (in a phase space which includes all degrees of freedom). For the case of a single point particle diffusing in a microporous material, the transition state is a literal two-dimensional surface at the window, usually assumed to be a plane. However, for more complex systems, this transition surface might exist within a higher-dimension phase space, e.g. a multi-atomic molecule that

16

has to be in the right location, orientation, and shape in order to pass through a window. Or, in the case of higher concentrations, the diffusing molecule must wait for a moment when no other molecules are blocking the window. In some cases, even selecting the exact transition state will not guarantee that the reaction rate obeys transition state theory; if the window between two cages is instead a long tunnel, then the particle exhibits a “diffusive” barrier crossing86,87 in which it performs a slow random walk along the top of the barrier until finally falling to one side or the other. In either the case of complex transition surfaces or of diffusive barrier crossings, the rate calculated from transition state theory can still be applied to the system correctly by applying a correction factor . This correction factor is simply the conditional probability that the system undergoes a transition, given that the system is at the supposed transition surface, and can be calculated explicitly with a Bennett-Chandler simulation.88,89 For this method the system is first simulated with Monte Carlo dynamics and constrained at the supposed transition surface. Dubbeldam and co-workers75,90–92 extended this method for application to particle movement in a microporous framework. In the example75,93 of single pseudoatoms of methane hopping between cages of zeolite LTA, this means constraining one particle to the barrier plane between two cages. Snapshots of this simulation (with sufficient decorrelation time between snapshots) are used as the starting points for short micro-Canonical MD simulations, with starting velocities sampled from the Boltzmann distribution. These short MD simulations are run forward and backward in time, allowing for a measurement of the net flux across the transition surface, relative to the assigned initial velocity of the particle. In other words, assigning a random velocity allows the designation of “expected donor” and “expected receiver” (i.e. TST expects that a particle on the barrier moving toward one cage will always end up in that cage, and always came from the other cage), and so the normalized net flux from expected donor cage to expected receiver cage defines the correction factor. Once the particle reaches deep enough into either cage, that MD simulation is ended. Realizing that a molecule moving at an angle θ from the vector orthogonal to the transition surface contributes only a weight of cosθ to the flux can enhance efficiency of the measurement of this flux. For the case of multi-pseudoatom molecules (e.g. propane), this method can be exploited by calculating the probability that one particular pseudoatom (e.g. the CH2 group) is on the barrier, and performing a Bennett-Chandler simulation from that constraint. In fact any arbitrary surface in phase space can be treated as the transition state, and a Bennett-Chandler simulation will find the accurate correction factor for that surface, so for any choices of

transition surface i or j in phase space, the product

; in other words

an inaccurate choice of the transition state, which includes some configurations that won’t react, will be exactly balanced by a correction factor that determines the fraction of configurations at the transition state that do react. A more appropriate choice of transition state will allow for a faster calculation of the correction factor because the correction factor will be larger, meaning that fewer of the short MD runs need to be executed in order to obtain an accurate answer (in the limit of low correction factor, only a small fraction of these MD simulations will be successful. But for accuracy one requires a certain minimum number of successful runs, so the total number of required

17

MD runs will be inversely proportional to the correction factor). A more thorough discussion of this method can be found in the textbook by Frenkel and Smit.69 The correct transition surface, if it exists, can be efficiently located using the string method.94,95 More complicated systems will have multiple parallel saddle points in the probability density of phase space between two stable points, meaning that a single transition state may not be sufficient to characterize the movement between them. This requires a new treatment to determine the kinetics of transition between two states: transition path sampling.96,97 For this simulation method an MD (or other type of) trajectory of the system is considered as only one member of an ensemble of trajectories. The simulation starts with a given time-reversible trajectory that starts at the reactant state and ends in the product state. Neighboring trajectories are visited with “shooting moves”: choosing a time slice of the current trajectory, modifying the velocities by a small random amount, and simulating the system forward and backward again. The move is rejected if the trajectory no longer traverses the distance between two states. Sampled trajectories are analyzed to determine the overall transition properties between the two states.

Kinetic Monte Carlo Simulations

The collective-diffusion coefficient cannot be calculated analytically from random-walk theory. A kinetic Monte Carlo (kMC) method98–100 provides a relatively efficient simulation method to generate particle trajectories that can be analyzed to calculate the diffusion coefficients. In contrast to a standard Monte Carlo simulation, a kMC simulation connects actual events in time, rather than disconnected states. First, the rates ri of all possible events that can occur to the system in the current configuration are collected,

(1.28)

Second, the simulation time is increased by

(1.29)

because the average amount of time the system spends in a given state is inversely proportional to the rate at which a system in that state will change to another state. Third, one event is selected with probability proportional to its rate, meaning the fastest events are the most likely to be selected. This can be accomplished by constructing a cumulative rate table where , and executing the event corresponding to . This event selection can be accelerated using various methods, primary among them the binary search algorithm. Applying this type of simulation to a diffusion process brings the advantage that, regardless of how slow the hop rate is, one “hop event” will occur during each computational cycle, rather than taking an arbitrary large number of cycles for an MD simulation. However, ensuring that the kMc model accurately reproduces the more

18

detailed (e.g. MD) system entails simplifying the material framework into a collection of lattice sites, collecting information about the movement rates between these sites, and finally accounting for more complicated adsorbate-adsorbate interactions that affect these rates. These three steps are not as simple as they may seem. One must first ensure that the barrier between two sites in a proposed topological simplification is indeed large enough to guarantee that once a molecule moves from one site to the other the molecule’s next move is not correlated with its previous one. For barriers smaller than kBT, the particle passes through the site without stopping, but for barriers above about 4 kBT, the particle’s direction of exit from the site is completely decorrelated from the direction of entry. The difficulty arises for moderately sized barriers, where the particle’s motion is neither completely correlated nor decorrelated. Once a simple topology is decided upon, the rate of movement between two sites can be calculated TST as described above if applicable, with correction factors as necessary. Accurately modeling the adsorbate-adsorbate interactions (blocking, attraction, repulsion, and coordinated movement) is the key to accurately simulating the separation between collective-diffusion behavior and self-diffusion behavior. One strength of a kMC simulation is to the ability to use any arbitrarily complex (repeating) lattice topology and quickly calculate the self-diffusion coefficient at infinite dilution, especially because accurate knowledge of adsorbate-adsorbate interactions is not required for that case. This should be substantially easier than the derivation of an analytical expression relating a collection of hop rates to the diffusion coefficient. It should be noted that stiffness is still a possible problem in this situation, wherein the hop rates within a group of two (or more) sites are several orders of magnitude higher than the hop rate to exit that group in any direction. The solution in that case is to consider that group as one super-site. In that case some information is lost, as the super-site can now accommodate more than one particle, so the rates of entry into such a super-site must now depend on the probability that a particle within the super-site is blocking the entrance, rather than checking to see if any particle is there. This probability can be determined with a Bennett-Chandler simulation, as described above. Various notable algorithms have been developed to improve the speed and applicability of kMC simulations.101–103

19

Chapter 2. Diffusion in Confinement: Kinetic Simulations

of Self- and Collective Diffusion Behavior of Adsorbed

Gases

Material in this chapter is based on Abouelnasr and Smit, Phys. Chem. Chem. Phys., 14, 11600 (2012).

Abstract

The self- and collective-diffusion behaviors of adsorbed methane, helium, and isobutane in zeolite frameworks LTA, MFI, AFI, and SAS were examined at various concentrations using a range of molecular simulation techniques including Molecular Dynamics (MD), Monte Carlo (MC), Bennett-Chandler (BC), and kinetic Monte Carlo (kMC). This chapter has three main results. (1) A novel model for the process of adsorbate movement between two large cages was created, allowing the formulation of a mixing rule for the re-crossing coefficient between two cages of unequal loading. The predictions from this mixing rule were found to agree quantitatively with explicit simulations. (2) A new approach to the dynamically corrected Transition State Theory method to analytically calculate self-diffusion properties was developed, explicitly accounting for nanoscale fluctuations in concentration. This approach was demonstrated to quantitatively agree with previous methods, but is uniquely suited to be adapted to a kMC simulation that can simulate the collective-diffusion behavior. (3) While at low and moderate loadings the self- and collective-diffusion behaviors in LTA are observed to coincide, at higher concentrations they diverge. A change in the adsorbate packing scheme was shown to cause this divergence, a trait which is replicated in a kMC simulation that explicitly models this behavior. These phenomena were further investigated for isobutane in

20

zeolite MFI, where MD results showed a separation in self- and collective- diffusion behavior that was reproduced with kMC simulations.

Introduction

Microporous materials like zeolites and, more recently, Metal-Organic Frameworks (MOFs) have been the focus of much attention in recent years because of their industrially useful properties. Of particular interest are these materials in the context of gas separations.6,104–106 Microporous materials have the potential to exhibit extremely favorable separation properties due in part to the capability of closely matching the diameter of a rate-limiting diffusion pathway in the material to that of an adsorbed molecule, resulting in a “molecular sieve” action.29 The measurement of diffusion properties poses a problem for someone wishing to quickly scan the millions of hypothetically possible materials25,27 for a candidate material to use in a particular separation process. Molecular simulations provide a promising route; however the benchmark technique for obtaining the diffusion coefficients of gases in these materials, Molecular Dynamics (MD), is often too slow to efficiently generate particle trajectories over the time-scales necessary to characterize the diffusion properties.29 In particular, for systems that display the potentially useful “molecular sieve” trait, MD can be especially time consuming. For these systems, the diffusion phenomenon can often be viewed as a hopping process between large cages separated by narrow windows. June et al107 proposed an application of the Transition State Theory (TST)108 with the Bennett-Chandler method88,89 to model diffusion in confinement, considering adsorbate movement as uncorrelated hops on a lattice of adsorption sites and calculating the self-diffusion coefficient DS. Beerdsen et al90 used this method to successfully replicate the self-diffusion coefficient values found from a Molecular Dynamics simulation. In that article, the Canonical Monte Carlo and Bennett-Chandler simulations were executed in a system where the microscopic concentration was allowed to vary, but the global concentration was fixed. This resulted in an “implicit” treatment of inter-particle interactions, where an adsorbate is assumed to be subject to an effective force field comprised of all other particles in all possible configurations, weighted by the probability of each configuration. While this approach was shown to accurately predict the self-diffusion coefficient at a range of concentrations, the collective-diffusion coefficient cannot be obtained using those means. Krishna et al54 formed a model of the collective-diffusion behavior. Their application of the Reed-Ehrlich model49 of diffusion on a lattice results in a reasonable fit to the collective-diffusion coefficients found from MD, using an adjustable parameter. Although this method gives important insights, it should be noted that, since it requires

21

prior knowledge of the desired property, that method is not aimed to predict the collective-diffusion coefficient. The approach of this chapter is to use a kinetic Monte Carlo (kMC) simulation,98 which, like an MD simulation, generates a set of particle trajectories that can be analyzed to calculate the desired collective-diffusion coefficient, but which can have an advantage over MD in computational speed. Starting with the formulation laid out by Beerdsen et a,l90 we calculated the TST hop rates and re-crossing coefficients. However, where Beerdsen et al computed the average hopping rates of a single particle as a function of global concentration, we computed these hop rates in the case where the exact loading of the participating cages is controlled. This is similar to the approach of Jee and Sholl,109 who investigated the effect on hop rate of local cage occupancy, rather than global concentration. Calculation of these loading-specific hop rates required the development of a model to predict the re-crossing coefficient in the case of a hop between two cages of different loadings. Next, this approach was used to analytically calculate the self-diffusion coefficient while explicitly accounting for fluctuations in cage loading, allowing a comparison to previous methods. These values were then used in a subsequent kMC simulation to calculate the self- and collective-diffusion coefficients. Building on these insights, some of the theoretical approaches developed in the methane/LTA model were tested in a variety of other systems, including methane in zeolite SAS, helium in zeolite AFI, and isobutane in zeolite MFI. The rest of this chapter is divided into three parts. First, the theoretical basis for this work is laid out, including a mixing rule for the re-crossing coefficient and a method to explicitly account for microscopic density fluctuations. Next, the simulation details are enumerated to allow reproduction of the data presented in this work. Finally, results of those simulations are presented.

Theory

The corrected hop rates provided by the dynamically corrected transition state theory (dcTST) formulation described in chapter 1 cannot be applied to a kinetic Monte Carlo simulation to measure the collective-diffusion coefficient, since the interactions between particles are accounted as an average effect; the effects of fluctuations are averaged. Before detailing our solution to this problem, it is first necessary to discuss the re-crossing coefficient for asymmetrical loadings.

Re-Crossing Coefficient for asymmetrical loadings

Here we will derive an expression to estimate the re-crossing coefficient between two cages of unequal loading from the re-crossing coefficients calculated at equal loadings. Let us first consider a particle observed on the barrier between two cages A and B,

22

which have respective loadings i and j, and the particle is currently moving toward cage B. We assume that the probability Pb(j) that a particle, upon entering a cage, is bounced back out the way it came, depends only on the loading j of that cage. The probability that this particle will end up in cage B is therefore

(2.1)

(2.2)

(2.3)

by simplifying the geometric series. Similarly, the probability that the particle ends up in cage A is

. (2.4)

Because the model system behaves the same whether the system is moving forward or backward in time, the probability that the particle started in cage A can be calculated similarly as

, (2.5)

and likewise the probability that the particle started in cage B can be calculated as

. (2.6)

The re-crossing coefficient is therefore the normalized net flux across the barrier,

(2.7)

Substituting Equations 2.3-6 into Equation 2.7 and simplifying yields

, (2.8)

Or, for the case where the two cages contain the same number of particles i

. (2.9)

Solving quadratically for Pb(i), and noting that Pb(i) must be positive, yields

(2.10)

which, when substituted into Equation 2.8, gives the final mixing rule

. (2.11)

Equation 2.11 may be recognized as the harmonic mean. This function satisfies the necessary symmetry κij =κji, which is required for detailed balance.

23

Calculation of Hop Rates with Explicit Contribution From Concentration

Fluctuations

Next will be presented a method to separate the contribution that each specific cage-loading case makes toward the mean value and re-mix them properly. For this purpose, the Canonical MC simulation results are analyzed to give the TST hop rates for the case of exactly i, rather than an average of <n>, particles in a cage, through Equation 1. Next, the normalized probability Pocc,<n>(j) of observing a cage with exactly i particles in a system with on average <n> particles per cage is recorded. The related normalized probability of observing a particle in a cage with occupancy i in a system with average occupancy <n> per cage is defined as

.

(2.12)

The resulting corrected hop rate is calculated as the summation over all possible hops from a cage with i particles to a cage with j particles, weighted by the probability of observing that situation in a system with <n> average particles per cage,

. (2.13)

Combined with Equations (2.2), (2.11), and (2.12), this allows an explicit calculation of the self-diffusion coefficient as

. (2.14)

It should be noted that the re-crossing coefficients used in Equations (2.13) and (2.14) should account for the fact that one particle, the one executing the hop, is no longer in its original cage at the time of the calculation.

Kinetic Monte Carlo

Now the groundwork is laid for a kinetic Monte Carlo simulation. The only necessary inputs for such a simulation are the lattice topology and the rates at which hop events occur. The hop rate of a particle from its current cage A into a neighbor cage is given by

, (2.15)

where i is the loading of the origin cage (not including the hopping particle), j is the loading of the destination cage, and statistics for the free energies are collected from a simulation with exactly i particles in the cage. Once these values are tabulated, a kinetic Monte Carlo simulation can be run which faithfully recreates some aspects of the diffusion behavior.

Methods

24

Simulation System

Here we describe the simulations systems used for MD, MC, and Bennett-Chandler simulations. Methane and isobutane pseudo-atoms were simulated using the Lennard-Jones (LJ) parameters described by Dubbeldam et al,110,111 and helium atoms were simulated with the LJ parameters reported by Talu and Myers.112 Simulation details of the zeolite frameworks are enumerated in Table 2.1. The all-silicon zeolite frameworks91,113 (coordinates available in Appendix A) were assumed to be rigid, so to enhance computational efficiency the truncated and shifted LJ potential energy between an adsorbate pseudo-atom and all oxygen atoms of the framework was tabulated for a grid of resolution 0.15Å in the simulation box. During the simulation the LJ interactions between an adsorbate and the framework were found by cubic spline from this lookup table. Adsorbate-adsorbate interactions were calculated by directly evaluating the truncated and shifted Lennard-Jones potential during the simulation. Periodic Boundary Condition (PBC) was used and inaccessible pores in the zeolite were blocked by prohibiting adsorbate entry into specified inaccessible pockets.114 (Supplementary Information available in Appendix A: structure files for frameworks LTA, MFI, SAS, and AFI, as well as blocking information for framework LTA.)

Table 2.1. Simulation parameters for zeolite framework systems.

Zeolite name

Simulation box size (x) [Å]

Simulation box size (y) [Å]

Simulation box size (z) [Å]

Cut-off radius [Å]

Number of super-sites

LTA 24.555 24.555 24.555 12.0 8 SAS 28.644 28.644 20.848 10.0 16 AFI 23.774 27.452 25.452 10.0 96 MFI 40.044 39.798 26.766 12.0 32

Molecular Dynamics

To directly calculate the diffusion coefficients, the Canonical Ensemble (NVT) was simulated, using the Nosé-Hoover method to regulate temperature.71,72 For part of the Bennett-Chandler simulations, the Micro-Canonical Ensemble (NVE) was simulated. Step sizes for all MD simulations were 0.5 fs. Except for the MD simulations used as part of the Bennett-Chandler simulations, MD simulations were equilibrated for 1 ns or more, and total simulation time was 20 ns or more. The Verlet Algorithm68was used.

Canonical Monte Carlo

The MC simulations used the same system and potentials as described above. The Metropolis acceptance algorithm77 was used to impose the temperature, with the maximum displacement tuned to achieve an acceptance ratio of 20%. Each MC simulation was run for 1.5 x 108 steps or more. The probability density of observing a particle at a given position can be calculated from a Canonical MC simulation; the coordinates of each particle are recorded at regular intervals, and sorted into planar

25

bins that are orthogonal to the straight path between cage centers. Dividing the number of hits in each bin by the size of the bin yields the probability density for that bin P(x), allowing calculation of the Helmholtz Free Energy through the relationship . (2.16)

The Re-Crossing Coefficient

Bennett-Chandler simulations88,89 were executed by first running a Monte Carlo simulation of the Canonical Ensemble as described above, where one particle is constrained to the Free Energy barrier plane between two adsorption sites. Uncorrelated snapshots of this system 1000 MC steps apart were used as starting points for short Micro-Canonical (NVE) MD simulations. In these MD simulations, each particle was given a random velocity sampled from the Boltzmann distribution at the given temperature. This allowed a designation of “expected donor” cage and “expected receiver” cage. Each short MD simulation was run until either the particle in question moved past the center of either cage, or 20 ps of simulation time elapsed. The re-crossing coefficient was then calculated as the normalized net flux of particles across the barrier over many such simulations (more details can be found in Frenkel and Smit69 or Dubbeldam et al75).

Kinetic Monte Carlo

In principle it is possible to run a kMC simulation using an unvarying hop rate Khop

<n>×κ<n>,but in that case a simulation is not necessary as the result converges to that of a random walk on a lattice. Furthermore, in that case there is no explicit interaction between particles, meaning that DC=DS, which does not reflect reality. Instead, the hop rates used were those for the exact loading case, Khop

i×κij, explicitly accounting for microscopic fluctuations. If correctly executed, the simulated system should have the same Pocc,<n> cage-loading distribution values as in the all-atomistic simulations, and therefore should yield the same DS as the analytical solution above. All kMC simulations advanced forward in time using the Gillespie Algorithm,115 with a binary search algorithm to efficiently determine the selected event. Each kMC simulation was run for at least 100 ns, after at least a 10 ns equilibration period.

Calculation of Diffusion Coefficients

Each kMC or MD simulation yields an ensemble of particle trajectories. The self-diffusion coefficient can be calculated29 as

, (2.17)

where r is the position of a given particle. This self-diffusion coefficient characterizes the movement of a given adsorbate. The collective-diffusion coefficient can be calculated as

, (2.18)

26

which characterizes the movement of the center of mass of all N adsorbate particles in the system. The collective-diffusion coefficient can then be used to find the molar flux as a function of the chemical potential, µ: ; (2.19)

If the adsorption isotherm is known, the collective-diffusion coefficient can be converted into the Fick diffusion coefficient, which is the diffusion coefficient required in mass transfer problems in the design of industrial processes using these materials.

Results and Discussion

Diffusion of Methane in LTA

Transition State Theory

In our method we rely on the assumption that adsorbate movement can be treated as uncorrelated hops between adsorption sites. To test this, we must first collect the hop rates of these particles between cages, which are heavily influenced by the number of particles loaded into each cage. This information could be measured in either of two systems: first, one where the global concentration is fixed, but the instantaneous loading of each super-cage is allowed to fluctuate, faithfully reproducing the system’s natural fluctuations; and second, where the number of particles in each cage is constrained. Figure 2.1 shows the free energy in LTA for both situations. Systematic differences can be seen between the two cases.

Figure 2.1. Free Energy of methane as a function of the position in the cage in LTA at 300K. Lines are the result when the local

concentration is allowed to fluctuate, and dots are the result when the number of particles in a cage remains fixed. The legend shows

the number of particles in each cage.

In our scheme we assume that movements within a cage occur on a much faster time-scale than hops between two separate cages. This requires that the free energy barriers within a cage are relatively small compared to the barriers between cages. The derivative of the Free Energy of methane in LTA at 300K with respect to position is shown for loadings of seven, eight, nine, and ten particles per cage (Figure 2.2). At a position of approximately 2 Å from the barrier, the function is observed to cross the x-

27

axis at densities of nine particles per cage and higher. This indicates the emergence of local maxima in the probability density at those locations, marking the beginning of a new packing scheme at high densities. The shaded areas represent the depth of the local minimum in the Free Energy. Whereas at lower densities the particles wander around the cage relatively freely, at higher densities they are stuck, for short time scales, in a sub-cage formed in part by other pseudo-stationary adsorbates.

Figure 2.2. The derivative of the Free Energy with respect to position for methane in LTA at 300K. Loading

(molecules per super-cage) is indicated in the legend.

Re-crossing Coefficients

The re-crossing coefficient of methane in LTA was calculated for an average loading, and compared to the values obtained by Beerdsen et al,91 showing quantitative agreement (Figure 2.3). Lower values of the re-crossing coefficient correspond to a higher likelihood that the entering particle will be blocked by another adsorbate.

Figure 2.3. Re-crossing coefficient of methane in zeolite LTA at 300K at various average loadings. The non-participating particles are

allowed to move between cages during the Canonical MC simulation, but the total number of particles in all eight cages was constant

throughout each simulation.

Increasing LoadingdF

/dx (

J/m

ol/Å

)

−8000

−6000

−4000

−2000

0

2000

position (Å)

0 0.5 1.0 1.5 2.0

7

8

9

10

28

The re-crossing coefficient was then calculated for various exact asymmetrical loadings, and compared to the new model presented in this chapter (Figure 2.4). The model is observed to be accurate in LTA as long as both cages have fewer than nine particles per cage, with a correlation coefficient of 0.99. The breakdown of the model, starting at nine particles per cage, corresponds to the concentration where additional local maxima in the probability density start appearing (Fig 2). This leads to adsorbate behavior that violates the core assumption of the model: that the probability of a cage to reject an entering particle is constant. If particles spend a longer amount of time stationary near the entrance of a cage than the hopping particle spends bouncing between two cages, then the probability of a rejection during the current attempt is correlated with the probability of a rejection during the last attempt; this is not accounted for in the model.

Figure 2.4. Re-crossing coefficients of methane in zeolite LTA at 300K with constrained cage loadings. The x-axis denotes the

loading of one cage, while the legend denotes the loading of the other cage. Symbols are the observed values, while dotted lines show

values predicted by the model presented in this work.

Analytical Calculation of Self-diffusion Coefficient

For LTA, the probability Pocc,<n> observing a cage with i particles in a system with an average <n> particles per cage was recorded from Canonical MC simulations (Figure 2.5). This probability distribution was used in the novel DS calculation that explicitly accounts for fluctuations in concentration (presented in the Theory section). The distributions are narrower than would be predicted from non-interacting particles (not shown), indicating a free-energy penalty for higher cage loadings. The cage occupancy probability was also recorded from simulations of the first kMC model. It can be seen that the cage occupancy distributions offer precise quantitative agreement at lower average loadings, but begin to disagree at an average loading of eight particles per cage.

29

Figure 2.5. Cage occupancy probability for different average loadings (molecules per cage, see legend) of methane in LTA at 300K.

Points indicate data from Canonical Monte Carlo simulations, while lines indicate data from simulations of the first kMC model.

Figure 2.6 compares the self-diffusion coefficients of methane in LTA at 300K as obtained using MD, the analytical expressions which account for concentration fluctuations implicitly and explicitly (see Theory section), and the values obtained by Beerdsen et al91 using the implicit treatment of concentration fluctuations. Quantitative agreement is observed at most loadings, in particular at loadings below the key concentration of nine molecules per cage.

Figure 2.6. Self-diffusion coefficient of methane as a function of loading in LTA at 300K from MD, random walk theory using data from cages with an imposed average loading, random walk theory with explicit

concentration fluctuation contributions as presented in this chapter, and the values found by Beerdsen et al.

91

Kinetic Monte Carlo

30

The self- and collective-diffusion coefficients of methane in LTA at 300K were obtained using MD and kMC simulations (Figure 2.8). In the first type of kMC simulation, in which a lattice site represents an entire super-cage, no separation is seen between Ds and Dc. This replicates the behavior observed from MD at low loadings. However, at higher loadings, a second type of kMC simulation becomes necessary, since more local maxima are observed in the probability density (Figure 2.2). This second kMC model consisted of a 3×3×3 cube of super-cages, each subdivided into fifteen lattice sites90 (Figure 2.7): one “A” site in the center, eight “B” sites arranged as vertices of a cube around that, and six “C” sites arranges as faces of the cube around those. The central “A” site is connected to the eight adjacent “B” sites, which are each connected to the nearest three “C” sites. Each “C” site is also connected to the nearest “C” site of the adjacent super-cage. Each of these lattice sites can accommodate at most one particle. The hop rates between these sites were defined as

,

where

.

The re-crossing coefficient is accounted for in the definition of Khop above, where the particle is effectively barred from entering an occupied site. This second model qualitatively replicates the separation between DS and DC observed in the MD simulation, but quantitatively is not in perfect agreement.

31

Figure 2.7. (a) Free energy surfaces of methane in zeolite LTA at 300K. Three types of local minima are observed, as described in the text. (b) Schematic of sub-cage adsorption lattice sites comprising one super-cage of LTA. Each “C” site is connected to the nearby “C” sites of a neighboring super-cage (for clarity, the two “C” sub-cage sites in line with the central “A” site along the z-direction are omitted).

32

Figure 2.8. Self- and collective-diffusion coefficients of methane as a function of loading in LTA at 300K. Points are from MD

simulations, lines are from the kMC models. For loadings up to nine molecules per cage, the results of the first kMC model (large

super-cages with perfect mixing) are shown. For loadings at nine and above, results from the second kMC model are shown (fifteen

discrete lattice sites within a super-cage, as shown in Figure 2.7).

Zeolites SAS and AFI: Asymmetrical Re-Crossing Coefficients

In zeolite SAS, the re-crossing coefficient was calculated for methane at 300K (Figure 2.9). Modeling the asymmetrical re-crossing coefficients as the harmonic mean of the two symmetrical re-crossing coefficients gives a correlation coefficient of 0.995.

Figure 2.9. Re-crossing coefficients of methane in SAS at 300K for various loadings in each participating cage. The loading of one

cage is indicated on the x-axis, and the loading of the other cage is indicated in the legend. Symbols are data from explicit Bennett-

Chandler simulations, while lines are the predictions of the model presented in this work.

33

In zeolite AFI, the re-crossing coefficients of helium at 300K were computed (Figure 2.10). The predicted value from the model (Equation 2.11) gives quantitative agreement as long as there are three or fewer particles in each cage.

Figure 2.10. Re-crossing coefficients of helium in AFI at 300K for various loadings in each participating cage. The loading of one

cage is indicated on the x-axis, and the loading of the other cage is indicated in the legend. Points are data from explicit Bennett-

Chandler simulations, while lines are the predictions of the model presented in this work.

Diffusion of Isobutane in Zeolite MFI

In zeolite MFI, isobutane was analyzed in detail at 600K. From Canonical MC simulations, the probability density distribution was calculated (Figure 2.11) in both the x and y directions at infinite dilution. This information was used to calculate TST hop rates of the molecules between sites. Two different models were considered for this system. In the first model, the molecules were thought to move from intersection to intersection, spending a negligible amount of time between intersection sites. In the second model, two additional sites were considered in each channel, corresponding to the local free energy minima observed there.

34

Figure 2.11. Free energy profiles of isobutane (central pseudo-atom) in zeolite MFI at 600K; (a) shows the profile in the y-direction,

and (b) shows the profile in the x-direction.

Several Bennett-Chandler simulations were executed, constraining the central pseudo-atom of isobutane to the central barrier of either channel type, and allowing the simulation to run until that pseudo-atom reached either adjacent intersection. This characterizes the hop rate between intersection sites. However, for the more finely detailed model, four more Bennett-Chandler simulations were executed, one for each of the barrier types: intersection to x channel site, intersection to y channel site, x channel site to x channel site, and y channel site to y channel site. In this more detailed model, we did not allow both channel-type sites within the same channel to be occupied at the same time, in order to reflect the dynamics of the true system. KMC simulations were run for either model type based only on the hop rates obtained from CMC and BC simulations run at infinite dilution. The resulting self- and collective-diffusion coefficients are shown in figure 2.12, along with results from direct MD simulations. While the central pseudo-atom of the adsorbate was observed to spend only 0.2% of the time in the channels, the associated “passing” behavior is seen to be an important aspect of the overall diffusion activity, as the kMC simulation which disallows it has an exaggerated DC/DS separation compared to MD, but one which accounts for it has a closer DC/DS separation, corresponding more closely to the MD simulation. Quantitatively, the kMC with channel sites offers a correlation coefficient of 0.89 to the MD for the self-diffusion coefficient. These diffusion coefficient values are near the

35

reported values for this system,116–118 although the system is known to be sensitive to force-field parameters119 and the rigidity of the framework.120

Figure 2.12. Self- and collective-diffusion coefficients of isobutane in zeolite MFI at 600K. Solid lines represent self-diffusion

coefficients and dotted lines represent collective-diffusion coefficients.

Conclusions

The mixing rule for asymmetric re-crossing coefficients is seen to be quantitatively accurate in systems where the assumptions of that model are upheld. Some systematic deviations between the observed results and the model’s predictions occurred in each system at a high enough adsorbate density to nullify the model’s assumptions. The DS results from the analytical expression that explicitly accounts for fluctuations in cage loading agree in the LTA system quantitatively with the results of MD, and with the results of the analytical expression that implicitly accounts for such fluctuations. The hop rates obtained with this method are necessary to run a kMC simulation that includes particle interactions, so that the microscopic fluctuations are properly accounted for. In the LTA system, the separation between DS and DC at higher loadings was shown to be a result of a change in adsorbate packing behavior, demonstrated by the local minima that developed in the free energy at those loadings. Until then the particles in a super-cage were reasonably well mixed, with no correlation between a particle’s previous cage and next cage. But at higher loadings, a particle is detained at various distinct adsorption sites within a cage, leading to these correlations and thus the separation between the self- and collective-diffusion behaviors. This change in behaviors was demonstrated in the two kMC simulations, one which assumes each cage

36

is well-mixed, finding that DC=DS, and the other which subdivides the cage into smaller sites to allow a stronger interaction between particles, finding that that DC>DS. This phenomenon is further demonstrated with isobutane in MFI, where the kMC simulations replicate the DC/DS separation observed with MD. Even more encouraging is the fact that, for isobutane in MFI, quantitatively close predictions of both the self- and collective-diffusion properties at high loadings are obtained from kMC simulations that use only the hop rates calculated at infinite dilution.

37

Chapter 3: A simulation study of rare diffusive hop events

of methane adsorbed in zeolites

Abstract

We investigated the rare hop events of methane adsorbed in various microporous zeolites at infinite dilution, examining the breakdown, in three ways, of the assumption of a random walk among adsorption sites. In zeolite ASV, the diffusion rate observed in Molecular Dynamics (MD) simulations exceeded that predicted by transition state theory, a discrepancy which was explained by correlations in hop direction, implying a “multi-jump” process. When these correlations were explicitly accounted for, the MD results were replicated. In zeolites ASV, CGS, and LTA, the hop rate was observed to be higher immediately after a previous hop, and to gradually decrease to the rate predicted from transition state theory. This is because the molecule’s briefly elevated kinetic energy required some nonzero time to equilibrate. Finally, in zeolite CGS, backward hops were seen to be more likely because of the zig-zag nature of the channel system, in which there is a significant possibility to bounce backward immediately.

Introduction

There are several different types of diffusion coefficient. This work investigates the diffusion phenomenon at infinite dilution, at which all the diffusion coefficients that describe single-component diffusion all have the same value. Molecular dynamics (MD) methods calculate the force on each particle at each time, and then integrate Newton’s laws over small time steps to generate particle trajectories that can be analyzed to calculate the diffusion coefficient. However, MD can often require impractical amounts of wall time, especially for the most interesting systems. Transition state theory (TST)

38

allows a much faster calculation of diffusion properties by calculating a movement rate between adsorption sites. By assuming that a particle executes a random walk of Poisson-distributed hops between adsorption sites, TST can provide a prediction of the diffusion coefficient which often matches the result from MD very closely, while requiring a much shorter simulation time.93 However, when the particle’s hops between cages are not random but instead correlated, that prediction is no longer accurate. This work examines several situations where the particle does not execute a random walk of Poisson-distributed hops, but deviates from this idealized process in several key ways. First, methane in zeolite ASV was observed to undergo “multi-jump” events, where a molecule quickly hops several times in the same direction before its kinetic energy is resdistributed isotropically. Second, methane in zeolite LTA was observed to have a residence time distribution different from what would be predicted from a Poisson distribution. Third, methane in zeolite CGS was observed to have a significantly higher probability to hop backward initially, an effect explained at least partially by the shape of the cage. A more complete discussion of diffusion coefficients, molecular dynamics, and transition state theory can be found in chapter 1. Simulations in this work implemented the Lennard-Jones potential to model particle-particle interactions, imposed a periodic boundary condition, and assumed a rigid framework except where noted. Energy exchange with the rigid framework was thus impossible, so this effect was modeled with a Nose-Hoover thermostat. Flexible frameworks were simulated using the method of Nicholas et al121 as adapted by Vlugt and Schenk122. Unless otherwise noted, simulation procedures and parameters are the same as in previous chapter.

Results and Discussion

For methane in zeolite ASV, MD simulations were run at various temperatures and infinite dilution. To avoid adverse thermostat effects,75 20 methane particles were present in a given MD simulation, but Lennard-Jones interactions between them were disabled (epsilon = 0), allowing them to interact only indirectly through the thermostat. The resulting diffusion coefficients are shown on an Arrhenius plot in Figure 3.1. These results are compared to the transition state theory prediction at that temperature. For temperatures lower than 50 K, the results were seen to match with the TST predictions. But at higher temperatures, the MD results were observed to exceed the TST predictions. This was shown to be because of correlations between hop directions at those temperatures, where a molecule is more likely to hop forward, in the direction it had previously hopped, than backward. Two kinetic Monte Carlo models were created to account for these correlations: one with just a simple correlation in hop directions, and another with multi-jump events, in the same direction, consistent with the observed correlations. Both kMC models used the hop rate taken from TST. The MD result is usually somewhere between the two kMC

39

results. Figure 3.2 shows a histogram of residence times for methane molecules. A significant asymmetry can be seen between forward and backward hops at times shorter than 200 ps. Many more forward hops are seen at shorter times than backward hops, indicating “multi-jump” events where a particle continues moving forward for more than one cage before its elevated kinetic energy equilibrates, de-correlating its hop direction once again.

Figure 3.1. Arrhenius plot of the diffusion coefficient of methane in zeolite ASV, at infinite dilution. Open symbols represent the prediction from transition state theory. Black symbols represent the result from

explicit MD simulations. Red symbols represent the result from transition state theory, explicitly accounting for the correlations in jump direction. Blue symbols represent the result from transition state theory which assumes virtually instantaneous “multi-jump” events forward, but only single-jump events

backward.

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Figure 3.2. Histogram of cage residence times for methane in zeolite ASV at 150K and infinite dilution,

with logarithmically-spaced time bins. The blue line and shaded area represent forward hops. The red line and shaded area represent backward hops. The dotted black line represents a hypothetical Poisson

process with the same mean.

For methane in zeolite LTA, MD simulations were run at infinite dilution and the cage residence times recorded for about 30,000 hop events. The resulting histogram is shown

in Figure 3.2. The rates of forward and backward hops were approximately equal to each other, and to about one-fourth of the turn hop rate, as would be predicted for a

random walk on a cubic lattice. This overall rate is fit relatively well with the sum of two Poisson processes, one for thermal equilibrium and another at an elevated event rate. The transition between these two processes, around 300,000 fs after entry to the cage,

coincides with the thermal equilibration of the molecule as shown from the average instantaneous molecule speed.

41

Figure 3.3. Black symbols represent the proportion of total hops observed during each (log-spaced) time bin after the molecule enters the cage. The red and blue shaded areas indicate two Poisson processes,

optimized such that their sum (blue line) fits the MD data. The red line is the mean instantaneous speed as a function of residence time.

The average speed as a function of residence time for methane adsorbed in zeolite LTA was calculated from MD simulations with rigid or flexible framework atoms, and the results are compared in Figure 3.3. It can be seen that the adsorbate in a flexible framework equilibrates much more quickly after hopping than with a rigid framework.

42

Figure 3.3. Average speed as a function of residence time for methane adsorbed in zeolite LTA at 300K,

simulated with either a rigid framework (black), or a flexible one (red).

TST predictions of the hop rate at various temperatures allowed the construction of an Arrhenius fit of hop rate as a function of temperature. Using the average instantaneous molecule speed upon entry, an equivalent “temperature” of the particles is calculated as a function of residence time. Then, using this Arrhenius relationship, an instantaneous hop rate is predicted. This prediction provides quantitative agreement with the hop rate observed in MD simulations, as a function of residence time, as shown in Figure 3.4.

43

Figure 3.4. Predicted and observed hop rate for methane in zeolite LTA at 300K. The green line represents the prediction of the hop rate from an Arrhenius fit of the rates predicted by transition state theory, as

applied to the equivalent temperature corresponding to the average instantaneous molecule speed. The black symbols represent the actual observed hop rate as a function of residence time in the cage.

Methane adsorbed in zeolite CGS was analyzed similarly, and the hop rate was found to vary with molecule residence time (Figure 3.5). At short times, the molecule is observed to be more likely to hop backward than forward, even after excluding (“spurious”) hop events that did not pass center of the cage before hopping backward, which would account for a sub-optimal choice of the transition surface. This backward hop correlation can be explained from the nature of the cage shape, as shown in Figure 3.6. Upon entering, the molecule hits a wall and is bounced backward, where it has a chance to return to its previous cage. In order to hop forward, it must remain in that cage long enough for its kinetic energy to be redistributed isotropically, which seems to require about 5000 fs.

44

Figure 3.5. Observed hop rate (blue and red = forward and backward, respectively) as a function of residence time for methane in zeolite CGS at 1000K. The black curve represents the instantaneous equivalent “temperature”, calculated from the mean instantaneous speed after entry into a cage.

45

Figure 3.6. Two-dimensional density plot of methane in zeolite CGS at 1000K; data from 356,000,000 snapshots of MD simulations. A square’s color represents the number of times a particle is observed in that bin [area 0.0067 Ångstrom

2], as indicated in the legend. The numbers denoting each bin color vary

exponentially. All non-white squares had at least one observation.

Conclusion

Diffusion in micro-porous materials was observed to display a non-Poisson distribution of hop events, because the thermal equilibration of a particle’s kinetic energy is not instantaneous. This will be true whether the energy transfer between the adsorbate and the framework is modeled with a thermostat and a rigid framework, or explicitly with a flexible framework. This non-Poisson behavior occurs even with large free-energy barriers between cages, as in the case of methane in LTA at 300K, where there are no correlations between hop directions. For barriers with less of an entropic component, multi-hop events are observed, with hop correlations either forward or backward depending on the cage shape. Accounting explicitly for these departures from the random walk assumption allowed for the recovery of observed diffusion behavior. These departures from a random walk of Poisson-distributed jumps have a significant effect on the diffusion behavior at infinite dilution, and should also have a significant effect on the diffusion properties at nonzero loading.

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Chapter 4. Large-scale Screening of Zeolite Structures for

CO2 Membrane Spearations

Material in this chapter was submitted for publication by Kim, J.; Abouelnasr, M.; Lin, L.-C.; and Smit, B. on March 25, 2013.

Abstract

This chapter presents a large-scale screening of zeolite materials for CO2/CH4 and CO2/N2 membrane separation applications using the free energy landscape of the guest molecules inside these porous materials. We show how advanced molecular simulations can be integrated with the design of a simple separation process to arrive at a metric to rank performance of over 87,000 different zeolite structures, including the known IZA zeolite structures. Our novel, efficient algorithm using the graphics processing units can accurately characterize both the adsorption and diffusion properties of a given structure in just a few seconds and accordingly find a set of optimal structures for different desired purity of separated gases from a large database of porous materials in reasonable wall time. Our analysis reveals that the optimal structures for separations usually consist of channels with adsorption sites spread relatively uniformly across the entire channel such that they feature well-balanced CO2 adsorption and diffusion properties. Our screening also shows that the top structures in the predicted zeolite database outperform the best known zeolite by a factor of 4 to 7. Finally, we have identified a completely different optimal set of zeolite structures that are suitable for inverse process in which the CO2 is retained while CH4 or N2 is passed through a membrane.

Introduction

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The elevated CO2 concentrations in the atmosphere are considered to be the primary cause of global warming.4 Because of the ever-increasing amount of CO2 emissions and our continuing reliance on fossil fuels, it remains imperative to search for various methods to mitigate the emission process. Amongst many suggested solutions, carbon capture and sequestration (CCS) is emerging as a viable technique:123 CCS consists of utilizing materials to capture CO2 emissions from point sources such as electric power plants, cement and steel plants, or natural gas field and to inject the adsorbed CO2 molecules to geological reservoirs. Some of the main barriers for the large scale implementation of CCS are the energy requirements and cost of the capture process. The currently available technology uses amines to selectively absorb CO2. These amines are very efficient in absorption of CO2 but the regeneration of the amine solution is relatively energy intensive. Alternative technologies, such as adsorption by adsorbents28 or separations using membranes16 have the potential of significantly reducing the energy costs. Both technologies de- pend on the development of novel materials that have optimal properties for a given separation with important classes of materials being nano-porous solids, such as zeolites and metal-organic frameworks.6,28,33,105,124 By changing the pore topology and chemical composition, one could in princi- ple synthesize millions of different materials, making it difficult to experimentally characterize and test all these materials. This provides a great opportunity for molecular simulations to identify the optimal materials, in silico, and guide the direction of the experimental research. To this end, we present a novel computational approach that lets us efficiently predict the permeation of a material for membrane separation appications. Recently, there have been several articles that pertain to computational screening of a large database of porous materials in search for optimal materials for CCS.28,125–130 However, most of these calculations focus on adsorption processes and in contrast, far less attention has been given to screening for membrane processes.27,131 The main reason being that the screening of membranes requires in addition to the adsorption properties, also information about the diffusion coefficients. However, most diffusion coefficient calculations requires expensive molecular dynamics (MD) simulations, and as such, much work in the past has only been focused on analyzing 10-20 structures.125 To avoid conducting MD simulations for thousands of structures, one can apply a geometric criterion to select those materials for which one component can enter but not the other.27,132,133 This is a very efficient method to screen materials with very high diffusive selectivity, but not necessarily for high permeability. Moreover, the geometric approach ignores the energy interactions between the guest particle and the host framework atoms, which leads to predicted diffusion properties that are independent of the specific chemistry used to functionalize a material. In this work, we demonstrate that a reliable estimate of the diffusion coefficient can be obtained from a free energy calculation. In this approach, we take full advantage of the information contained within the free energy landscape throughout the entire unit cell of the crystal structure and

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apply the transition state theory (TST) to calculate the diffusion properties.75,93 Mapping this algorithm to the high-throughput processing power of the graphics processing units (GPUs), we have accurately characterized the adsorption and the diffusion properties of over 87,000 experimental International Zeolite Association (IZA) and predicted pure-silica zeolite structures from Deem’s database.26,134 For the predicted zeolite database, PCOD (predicted crystallography open database), a set of 330,000 structures within +30 kJ/mol Si of α-quartz was further reduced down to 139,397 removing structures with largest free sphere diameter below 3.25 Å.23 From the reduced set, we select over 87,000 zeolite structures that have orthogonal unit cells to facilitate calculations. Similar work has been conducted by Haldoupis et al. but their approach has been limited to computing diffusion coefficients of spherical molecules such as CH4.35 Our method can compute diffusion coefficients of both spherical and non-spherical molecules within a single structure in just a few seconds, providing the speedup required to screen thousands of different structures. Moreover, our algorithm explores the entire channel profile and identifies multiple channels and free-energy barrier locations, which can provide a more accurate picture of diffusion in porous materials. At this point, it is important to note that zeolitic membranes have been syn- thesized.135,136 However, these studies have been limited to a few pore topologies, so an important practical question we would like to address is whether these materials are close to the optimal performance, or whether significant gains can be expected if one would try to synthesize a membrane using a different zeolite topology. To illustrate how our screening can be used to find the optimal material, we use as an example, the separation of CO2 from CH4 in natural gas reservoirs. Natural gas reservoirs may contain up to 70% CO2, and the exploitation of these reservoirs would require the separation of the CO2 from the natural gas and injection back to the reservoir. As the CH4/CO2 has high pressure, membranes are ideal to carry out this separation efficiently. At this point, it is important to emphasize that the ideal material of a separation depends on the actual process requirements. We use our screening approach to illustrate this point by a simple model that mimics the separation of CH4 from CO2 at typical conditions of a natural gas reservoir. The increase in efficiency of our method allows us to screen many materials and identify the optimal structures for an entire class of materials. Establishing such a theoretical limit provides important guidance for future material synthesis. Our study identifies the general characteristics of the best performing structures. It can be expected that in other classes of materials, structures with similar characteristics will also perform very well.

Methods

Although we focus mostly on zeolite structures with CO2, N2, and CH4 as resident gas molecules, the techniques developed to compute the diffusion coefficients can readily extend to other gas molecules and to other materials. In our calculations, we assume that the zeolites are perfect, infinitely large crystals such that the periodic boundary

49

conditions can be used. The number of unit cells is chosen such that the simulation box extends at least twice the cutoff radius of 12 Å in all three perpendicular directions. The host framework atoms are assumed to be rigid, and the pair-wise gas-gas and gas-host interactions consist of Lennard-Jones forces and electrostatic interactions. The force fields developed by Garcia-Perez et al. are used in all of our work as they have been shown to be transferrable for variety of zeolite structures137. The temperature is set to be 300 K in all of the work. The MD simulations were conducted utilizing the CPU cores in our own cluster while the efficient diffusion coefficient calculations and the grand canonical Monte Carlo simulations were conducted utilizing the NVIDIA Tesla C2050 GPU cards from the Dirac Cluster at the National Energy Research Supercomputing Center (NERSC).

Molecular Dynamics Simulations

For a molecular dynamics simulation, the gradient of the potential energy with respect to position is calculated for each adsorbate particle, including the van der Waals forces and Coulombic forces. This energy gradient manifests as a force which, constrained by intra-molecular considerations, results in an acceleration according to Newton’s second law of motion. In an MD simulation, the force on each particle is sampled periodically, allowing an update to each particle’s position and velocity. With sufficiently small time-steps (0.5 fs) and sufficiently long simulations (> 1 ns), a collection of trajectories produced from MD simulations can be analyzed to calculate the self-diffusion coefficient.29 In this study MD simulations were carried out in the canonical ensemble, using a Nose-Hoover thermostat.

Efficient Diffusion Coefficient Calculations

At the start of the simulation, an energy grid that contains detailed information about the free energy profile of the gas molecules inside the porous material is constructed and subsequently analyzed to obtain both the adsorption and the diffusion properties of the system. A sufficiently fine mesh size of 0.1Å is chosen for all structures and the resulting grid is superimposed on top of a single unit cell, where each of the grid points represents the total pair-wise free-energy summation between the gas molecule and all of the framework atoms. For gas molecules such as CO2 and N2, which cannot be represented as a point particle, 250 randomized center-of-mass rotations of the molecule are conducted on the grid point to obtain an average Boltzmann-weighted free energy of the gas molecule at that point. The expression for free energy, Fi, at a specific grid point is expressed as follows,

.

where Ntot = 250 and Ej is the potential energy of CO2 (or N2) molecule of a given randomized j configuration. For zeolite structures, the number of energy grid points is typically on the order of 106 and 107 and the calculations only take a few seconds using our efficient GPU code.

50

From the constructed energy grid, points where Fi < 15kBT are considered accessible, while the rest are inaccessible. The choice of the 15kBT cutoff was made such that energy values higher would be considered inaccessible during a typical experimental timescale.128 The binary infor- mation (i.e. accessible/inaccessible) stored in a separate grid can be used to determine both the number of the channels and the channel direction. For example, in determining the number of channels along a given spatial direction (e.g. x-direction), a two-dimensional flood fill algorithm at x = 0 along the y,z plane is used to combine the adjacent accessible points together. The flood fill algorithm implemented here is similar to the one we utilized to determine blocked regions in porous materials.128 After identfiying the distinct number of accessible regions at x = 0, each of these sections are analyzed separately in subsequent analysis. To analyze the entire channel, we compute the sum of Boltzmann weights along the y,z plane slice at a given x value for all grid points that are connected to the initial accessible region at x = 0. The algorithm continues from x = 0 to x = lx where lx is the unit cell size along the x-direction. If at any point, we encounter a dead-end (i.e. y,z plane where sum of Boltzmann weights is zero) we conclude that the channel does not exist along this region and proceed to the next possible candidate either along the same x direction or along y or z directions. Upon successful traversal to the end of the unit cell, the sum of Boltzmann weights for each value of x can be utilized to identfiy the peak/trough of the free energy profiles for that specific channel along the x-direction. The entire free-energy profile is utilized to compute the diffusion coefficient given that there can be multiple lattice sites along the same channel. The diffusion coefficient value of an individual channel can be obtained from the transition state theory (TST)83,107 taking into account multiple hop-rates generated from the analysis assuming a random walk along the lattice sites. The total diffusion coefficient value along a given direction (e.g Dx) consists of linear combination of the channel diffusion coefficients weighted by its local Henry coefficient values. Finally, the total self diffusion coefficient is calculated as DS =(Dx +Dy +Dz)/3. Throughout this work, the effects of adsorbate concentration on the diffusion behavior were neglected in order to allow for extremely fast diffusion characterization. The sensitivity of the diffusion coefficients for different loading values will vary based on the structure topology, but is not expected to impact these results significantly. In fact, the assumption of constant diffusion coefficient is commonly used in applications.29

Grand Canonical Monte Carlo Simulations

Grand canonical Monte Carlo (GCMC) simulations were utilized to obtain the gas uptake value as a function of fugacity. In GCMC, the chemical potential, volume, and temperature are kept constant throughout the simulation and random insertion, deletion, and translation moves were used to propagate the MC system from one cycle to the next. We have utilized various efficient techniques such as density biased sampling, energy grid usage, and parallelization of energy calculations to reduce the average overall wall time of a single GCMC simulation to under a minute.138 The number

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of equilibration cycles and production cycles were set respectively at 250,000 and 100,000. The mixture isotherms were obtained from the computed pure isotherm data using the ideal adsorbed solution theory (IAST) as IAST has been demonstrated to be generally applicable to make good predictions about mixture behaviors for various porous materials.47 It is important, however, to note that some particular cases may need some variant theories of IAST,139 which is regarded to be out of the scope for this current work.

Results and Discussion

The transport of molecules through a membrane can be characterized by its permeability. The permeability is defined as the product of the solubility and the diffusion coefficient of the gas molecules. As such, permeability is a crucial component for evaluating membrane performance,16 and it requires computation of both adsorption and diffusion properties of the system. For the adsorption part, we use existing computational methods based on GPUs.128 The methodology used to compute the diffusion coefficients described in the Methods section has been implemented for this work. We have selected a set of representative experimental zeolite structures from the IZA database to test our method. Figure 4.1 compares the self-diffusion coefficient (DS) of CO2, N2, and CH4 gas molecules at infinite dilution and T = 300 K for the two methods, and it shows that our method provides a reasonably accurate description of the diffusion. The discrepancies between the two methods result from a variety of different reasons such as correlated hops for large diffusion values75 and the presence of complicated channel profiles that makes very accurate TST analysis difficult. In general, the MD simulation wall time scales with the inverse of DS, becoming intractable in slowly-diffusing systems as hops across a large barrier becomes increasingly rare. Accordingly, given that our model based on the TST uses an algorithm where the wall time remains independent of the diffusive coefficient values, an enormous speedup (few seconds versus several weeks) can be gained compared to MD simulations in structures with small DS. Figure 4.1 also illustrates that the zeolite data points for CO2 and N2 are concentrated in the highly diffusive region (i.e. DS > 10−9 m2/sec) whereas CH4 data points are scattered across a wider range of DS values. Because the kinetic diameter of CH4 is larger than both CO2 and N2, there exists more structures with relatively smaller diffusion coefficients for CH4. Also, due to the long-range electrostatic interactions present for the polar CO2 and N2 molecules, the likelihood of finding structures with a relatively high energy barrier remains small as the contributions of the non-local interactions spread across the entire channel. On the other hand, the lack of electrostatic interactions in CH4 molecules translates into free energy landscapes that are determined solely by close-range interactions from the neighboring framework atoms, which enhances the likelihood of finding channels with a narrow, pinched region (i.e. large energy barrier) separating the adsorption sites for certain topologies.

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Figure 4.1. Comparison of two different methods (MD and GPU-implemented TST) to compute the self-

diffusion coefficients of CO2 (red; circle), N2 (blue; triangle), and CH4 (green; upside-down triangle) molecules for IZA zeolite structures. The dashed line indicates the region of perfect agreement between the two methods. The error bars from the MD simulations are provided for only a few selected zeolite

structures.

Next, we explore the different characteristics between CO2/CH4 and CO2/N2 separations for the zeolite structures. For analysis, we included over 87,000 predicted zeolite structures in order to detect possible patterns that can emerge for the entire class of materials and for selected separations that might not be obvious in analyzing just a few structures. In most membrane research, the relative performance of a material is estimated from a Robeson plot, which gives the relationship between permeability and the permeation selectivity.21 In all cases, the permeation selectivity value less than one indicates that the membrane is selective but the CO2 ends up in the retentate. Figure 4.2(a) and (b) shows the zeolite Robeson plots for CO2/N2 and (c) and (d) for CO2/CH4 separations. We considered two different methods to compute the adsorption component of the permeability and permeation selectivity: (1) using the Henry coefficient, KH, which gives an accurate description of the adsorption at low pressures and (2) using the grand canonical Monte Carlo (GCMC) simulations for obtaining pure component isotherm and applying the ideal adsorbed solution theory (IAST) for estimating mixture adsorption at given condition.47 In general, using KH values overestimates the permeability as shown in the Robeson plots in Figure 4.2(a) and (c) (red data points). Since most gas separation occurs at higher pressures, the uptake values at the actual separation pressure provides a better measure of permeability

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53

compared to KH. For CO2/N2 separations, the flue gas operating condition of total fugacity equal to 1 bar (14% CO2 and 86% N2) was used while for CO2/CH4, the total fugacity of 10 bar (50% CO2 and 50% CH4) was used. Upon increasing the pressure, the gas uptake begins to saturate and thus, the adjusted permeability value based upon GCMC-IAST becomes smaller at pressures outside of the linear Henry regime. Because CO2/N2 separation occurs at a lower pressure compared to CO2/CH4 separation, the overall shift in the data points in the Robeson plots (indicating decrease in CO2 permeability) in Figure 4.2(c) becomes more apparent for CO2/CH4.

Figure 4.2. Permeation selectivity as a function of CO2 permeability for (a) CO2/N2 separations (KH - red,

GCMC-IAST - blue), (b) CO2/N2 separations for GCMC-IAST in predicted (blue) and IZA (orange) zeolites, (c) CO2/CH4 separations (KH - red, GCMC-IAST - blue), (d) CO2/CH4 separations for GCMC-IAST in predicted

(blue) and IZA (orange) zeolites.

It is instructive to compare our results with the well-known Robeson plots for polymer materials. For these materials, one typically observes a limiting behavior, the Robeson upper bound, as materials that have high selectivity have low permeability and materials with high permeability, a low selectivity.21,140 In the literature, deviations from this Robeson upper bound have been reported for nanoporous materials.141 Figure 4.2 shows, however, that nano-porous materials have a qualitatively very different behavior, in which the concept of a Robeson upper limit has little value. The reason is that the difference in solubility of guest molecules in these materials can vary orders of

54

magnitudes, while in the Robeson upper limit, it is assumed that all materials have a similar solubility.140 In Figure 4.2, most of the data points in the CO2/N2 Robeson plot are concentrated in a narrow band of permeation selectivity values and indicates a general positive correlation between CO2 permeability and CO2/N2 permeation selectivity. On the other hand, the data points for the CO2/CH4 Robeson plots are spread across a wider range of selectivity values with less meaningful correlations being found between permeability and selectivity. Because both CO2 and N2 molecules are polar, linear, and have comparable kinetic diameters, the diffusion properties of the two gas molecules are similar. Moreover, the values of KH

N2 of the entire database are in a much narrower range as the corresponding values of KH

CO2. Accordingly, given this narrow range of KHN2

values and similar diffusive properties between CO2 and N2, the shape of the Robeson plot is dictated largely by KH

CO2, which is positively correlated in both the permeability and the permeation selectivity. The CO2/N2 points located along a narrow band can be also seen in the CO2/CH4 Robeson plot as well. However, additional, more scattered data points exist in this Robeson plot, which is caused by the dissimilarity between the CO2 and the CH4 molecules. Most of these outliers correspond to zeolite structures that possess very narrow channels, leading to relatively low CH4 diffusion coefficient values or from different CO2/CH4 adsorption sites. To identify the structures most promising for CO2/N2 and CO2/CH4 separations, a suitable metric that can quantify the material performance needs to be constructed. A diagram that illustrates the CO2/CH4 separation process is shown in Figure B.1 (Appendix B). The target for the CO2/CH4 separation is to obtain a high purity CH4 stream in the retentate side. The conventional approach in identifying the top performing structures for such a membrane separation is to select those materials that have the highest permeability and selectivity. We use a simple membrane design to illustrate that from a practical point of view, this criterion does not provide us the optimal material. The argument that one needs to be in the upper right corner in the Robeson plot (i.e. high permeability and selectivity region) assumes that selectivity is equally important as permeability. Our analysis shows that for a given separation, one needs a minimum selectivity; the best material is the one with highest permeability out of all materials that satisfy this minimum selectivity criterion. Selectivity is the dominant factor only for separations that require an extremely high purity. Baker et al. reached a similar conclusion for the N2/CO2 separation.16 In an ideal membrane system, the area of the membrane is assumed to be a measurement of the cost of the entire process, and this area is shown to be mainly dominated by and inversely proportional to the CO2 permeability (For more detailed derivation, see Appendix B). Hence, we can rank those materials that satisfy the minimum selectivity criterion based on the membrane area that is required for the separation.

55

Figure 4.3. (a) Membrane area as a function of CH4 feed loss for zeolite structures that satisfy the

minimum purity requirement of 95% (red) and 99% (purple). Amean,95% is defined as the logarithm averaged area with the given 95% purity. The segments within each box from bottom to top represent the 5%, 25%, 50%, 75%, and 95% points for each bin with the circle indicating the average value for the whole bin. (b)

Heat of adsorption as a function of CO2 KH for all zeolite data sets (black), selected candidate sets that satisfy the 95% purity requirement (red), and the top 1% performing structures from the set of all

candidate structures (blue).

With a working performance metric at hand, we plot the membrane area as a function of CH4 feed loss (i.e. 1 - methane recovery) for materials that satisfy the methane purity criteria of 95% and 99% purities in Figure 4.3(a). The feed loss gives us the amount of methane that we inject with the CO2 in the reservoir. As we are screening many materials, we find a clear trend but also some structures that have nearly ideal properties and have therefore, an exceptionally high performance. However, from a synthetic point of view, it might be very difficult to synthesize exactly these materials. In Figure 4.3(a), the box representation is used to indicate the exceptional materials and show the general trend: the lines above and below the boxes show the structures with good and poor performance, and the boxes show the trends as represented by a large number of structures that have the same properties. In the following, we focus on these general trends. As expected, the membrane area tends to increase for smaller membrane feed loss, indicating that if we require a higher selectivity, we will have less materials to choose from.

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56

For a specified purity requirement, we can isolate the top structures and identify common features that separate these structures from the others. For example, we plot the CO2 heat of adsorption as a function of the Henry coefficient, KH

CO2 , for the top 1% structures that satisfy the product purity requirement of 95%. The best structures possess Henry coefficient that is in the intermediate range (10−5 < KH

CO2 < 10−4 mol/kg/Pa). In structures that have very small KH

CO2 , the overall CO2 permeability and the permeation selectivity is too small, making them sub-optimal for membrane separations. In structures that have too large of KH

CO2 (KHCO2 > 10−3 mol/kg/Pa), one key

factor come into play that degrade performance; there is an inverse relationship between KH

CO2 and DS as can be seen from Figure B.3 in Appendix B. Structures with large KH

CO2 possess strong adsorption sites that cause the diffusion rate to decrease as the effective barrier separating one adsorption site to another becomes large. Amongst structures that are inside the optimal Henry coefficient range, the optimal structures tend to have low heat of adsorption values (blue data points in Figure 4.3(b)). A high heat of adsorption often corresponds to strong adsorption sites, which lead to low diffusion coefficients.

Figure 4.4. CO2 permeability as a function of largest free sphere diameter for (a) 95% CH4 purity and (b) 99.9% CH4 purity for all the zeolites in the dataset (solid-fill, black), the candidate subsets that fulfill the purity requirement (solid-fill, red), and the top 1% structures (i.e. top 1% in area with recovery higher

than 90%) CO2 (no-fill, blue).

At this point, it is instructive to compare our results with those from geometrical screening.27 Geometrical screening identifies the best materials as the ones that possess pore diameter values similar to the kinetic diameter of the molecules that have to be separated. Intuitively, this makes sense as one can imagine that the separation process will be optimized when one guest molecule species is just small enough to diffuse across the channel, while the larger one is blocked. To gain insights in the differences between the two methods, we plot the CO2 permeability as a function of the largest free sphere diameter (Df)23 for all structures shown in Figure 4.4. The largest free sphere diameter is a measure of the size of the molecule that can enter a particular structure. A simple geometric criterion is to select those structures that have a Df large enough for CO2 to enter but too small for CH4. In our analysis, two important points emerge that cannot be deduced from a pure geometrical analysis. First, few high performing structures possess

57

very large Df values, which deviates from what is predicted from geometrical analysis. Because most geometrical analysis focuses on selectivities, only the structures that consists of pore diameters that are close to the size of the guest molecules are deemed interesting and worthwhile for investigation. Our energy-based method reveals that for 95% purity requirement, one can identify structures that lie outside of this region (blue data points for Df > 4 Å in Figure 4.4(a)). Once the desired purity is set to be very large at 99.9% (Figure 4.4(b)), the best performing structures tend to be concentrated in region between Df = 3-4 Å. For such a high purity, it is essential to have high selectivity and the geometric criterion ensures this. The second important point is that our method can readily differentiate large and small CO2 permeability values in structures that are located within the optimal Df (3-4 Å) values as our method can compute accurate DS values. Accordingly, the set of potential candidates for membrane separation can be further refined using the energy-based analysis.

Figure 4.5. Permeation selectivity as a function of N2 permeability for (a) CO2/N2 inverse sep- arations (KH - red, GCMC-IAST - blue), (b) CO2/N2 inverse separations for GCMC-IAST in predicted (blue) and IZA (orange)

zeolites and permeation selectivity as a function of CH4 permeability for (c) CO2/CH4 inverse separations (KH - red, GCMC-IAST - blue), (d) CO2/CH4 inverse separations for GCMC-IAST in predicted (blue) and IZA

(orange) zeolites.

Until now, we have assumed that CO2 is the component with the highest permeation. As this is the case for most known porous materials, one normally does not consider a separation where CH4 or N2 is assumed to be the component with the highest

58

permeability. Such a material, how- ever, would be of interest as it would allow for separation in which the CO2 is retained while preferring permeability of CH4 or N2. This process can be attractive since in the conventional process, the CO2 rich stream is the low pressure permeate and hence needs to be repressurized for transportation and storage, while in the inverse process, the CO2 rich stream is the retentate and re-pressurizing is not required. Such process will be particularly attractive for those natural gas fields in which the CO2 concentration exceeds 50%. Figure 4.5 shows the Robeson plots corre- sponding to this separation where the same data points from Figure 4.2 were plotted with inverted permeation selectivity values. In general, the permeability as well as the selectivity for the top performing structures are predicted to be lower as the number of structures that possess larger CH4 and N2 uptake values compared to the CO2 uptake values at the interested separation process is very small. We did not identify any of the known experimental (IZA) zeolite structures that possessed CH4/CO2 or N2/CO2 permeation selectivity values higher than 10. On the other hand, screening of the predicted zeolite structures does reveal a large number of structures that would allow for such a separation, suggesting such a separation is possible. It is interesting to make comparisons between the best predicted zeolite structure and the best IZA structure we identified from our screening analysis. In our evaluation, the membrane area, which is dominated by and inversely proportional to the CO2 permeability is used as a metric that needs to be minimized to have the best membrane separation performance. The ratio between the smallest membrane area found in the IZA and the PCOD structures are defined as the performance gain.26 For CO2/CH4, this gain ranges from 4 to 7× for differing minimum selectivity requirements that range from 85 to 98% purity with the recovery set at 0.90. Amongst the IZA structures, ABW and GIS zeolite structures were identified to have the largest CO2 permeability as both of these structures possess relatively strong CO2 adsorption sites spread throughout the entire main channels. The KH

CO2 values computed at T = 300 K are 5.42×10−5 and 1.45×10−4 mol/kg/Pa, while the DS values are 8.01×10−9 and 3.51×10−9 m2/sec respectively for ABW and GIS. These values indicate that the best IZA structures do not possess exceptional adsorption or diffusive properties but are well-balanced in both. Figure 4.6(a) shows the CO2 free energy landscape within the GIS structure where blue regions indicate relatively strong adsorption regions. The best PCOD structures for CO2/CH4 is identified to be COD8186909. Similar to ABW and GIS, the best predicted structures also show well-balanced adsorption and diffusion properties. Finally, for the structures optimal for the inverse process shown in Figure 4.5, the best structures tend to have strong CO2 binding sites that reduce its diffusion coefficient to lower the CO2 permeability. Concurrently, these structures possess strong enough CH4 binding sites with fast enough diffusive properties for CH4 to make them optimal for the inverse process. A sample structure (i.e. PCOD8198030) is shown in Figure 4.6(b) where a significant energy barrier exists in the main channel with the largest adsorption sites.

59

(a) (b)

Figure 4.6. (a) CO2 free energy landscape within a unit cell of GIS structure, which is predicted to be one of the best structures for both CO2/N2 and CO2/CH4 separations. (b) CO2 free energy landscape within a unit

cell of PCOD8198030 structure, which is predicted to be one of the best structures for CH4/CO2 inverse separation. The blue regions indicate low-energy adsorption sites (minima value of -4493 and -6775 Kelvin

for GIS and PCOD8198030, respectively), the red indicates relatively high energy regions while the rest represents inaccessible regions.

Conclusion

In summary, we have implemented an efficient method to determine diffusion coefficients based on the application of the transition state theory to the energy landscape of a large collection of structures. These diffusion properties, when combined with adsorption properties, can characterize the membrane performance for a given nano-porous material. Applying this method to a database of over 87,000 predicted zeolite structures, we found that for CO2/CH4 separations, the best-performing predicted structure can improve the performance as measured by the required area of a membrane by a factor of 4 to 7 compared to the best known zeolite structure. Robeson plots for the CO2/CH4 separation reveals two distinct sub-classes of structures: (1) a group with a relatively low permeation selectivity which varies in proportion to the permeability of CO2, and (2) a group with much higher permeation selectivities, which is presumed to arise mainly from diffusion selectivity, i.e. molecular sieving. These two different groups are not as clearly observed for the CO2/N2 separation, since the two particles have more similar dimensions, making molecular sieving less possible. The best material achieves a high selectivity without creating adsorption sites that slow down the diffusion. A simple experimental signature to recognize such a material for CO2/CH4 separation is a material that has intermediate range of Henry coefficient (i.e. 10−5 < KH

CO2 < 10−4 mol/kg/Pa) and a relatively low heat of adsorption (i.e. -30 to -20 kJ/mol).

60

Appendix A: Supplementary information for “Chapter 2.

Diffusion in Confinement: Kinetic Simulations of Self- and

Collective-Diffusion Behavior of Adsorbed Gases”

This appendix contains coordinate files used for the simulations executed for Chapter 2. Please note all atomic or blocking coordinates listed are on a relative scale. Files in Appendix A include: LTA.cssr LTA.block AFI.cssr MFI.cssr SAS.cssr

61

LTA.cssr: 24.5550 24.5550 24.5550 90.000 90.000 90.000 SPGR = 1 P 1 OPT = 1 576 0 Created by Raspa-1.0 0 LTA_SI : LTA_SI 1 Si 0.00000 0.09316 0.18499 0 0 0 0 0 0 0 0 0.000 2 Si 0.00000 0.18715 0.09042 0 0 0 0 0 0 0 0 0.000 3 O 0.00000 0.11367 0.24663 0 0 0 0 0 0 0 0 0.000 4 O 0.00000 0.14459 0.14591 0 0 0 0 0 0 0 0 0.000 5 O 0.05379 0.05865 0.17152 0 0 0 0 0 0 0 0 0.000 6 Si 0.00000 0.90684 0.18499 0 0 0 0 0 0 0 0 0.000 7 Si 0.00000 0.81285 0.09042 0 0 0 0 0 0 0 0 0.000 8 O 0.00000 0.88633 0.24663 0 0 0 0 0 0 0 0 0.000 9 O 0.00000 0.85541 0.14591 0 0 0 0 0 0 0 0 0.000 10 O 0.94621 0.94135 0.17152 0 0 0 0 0 0 0 0 0.000 11 Si 0.00000 0.09316 0.81501 0 0 0 0 0 0 0 0 0.000 12 Si 0.00000 0.18715 0.90958 0 0 0 0 0 0 0 0 0.000 13 O 0.00000 0.11367 0.75337 0 0 0 0 0 0 0 0 0.000 14 O 0.00000 0.14459 0.85409 0 0 0 0 0 0 0 0 0.000 15 O 0.94621 0.05865 0.82848 0 0 0 0 0 0 0 0 0.000 16 Si 0.00000 0.90684 0.81501 0 0 0 0 0 0 0 0 0.000 17 Si 0.00000 0.81285 0.90958 0 0 0 0 0 0 0 0 0.000 18 O 0.00000 0.88633 0.75337 0 0 0 0 0 0 0 0 0.000 19 O 0.00000 0.85541 0.85409 0 0 0 0 0 0 0 0 0.000 20 O 0.05379 0.94135 0.82848 0 0 0 0 0 0 0 0 0.000 21 Si 0.18499 0.00000 0.09316 0 0 0 0 0 0 0 0 0.000 22 Si 0.09042 0.00000 0.18715 0 0 0 0 0 0 0 0 0.000 23 O 0.24663 0.00000 0.11367 0 0 0 0 0 0 0 0 0.000 24 O 0.14591 0.00000 0.14459 0 0 0 0 0 0 0 0 0.000 25 O 0.17152 0.05379 0.05865 0 0 0 0 0 0 0 0 0.000 26 Si 0.18499 0.00000 0.90684 0 0 0 0 0 0 0 0 0.000 27 Si 0.09042 0.00000 0.81285 0 0 0 0 0 0 0 0 0.000 28 O 0.24663 0.00000 0.88633 0 0 0 0 0 0 0 0 0.000 29 O 0.14591 0.00000 0.85541 0 0 0 0 0 0 0 0 0.000 30 O 0.17152 0.94621 0.94135 0 0 0 0 0 0 0 0 0.000 31 Si 0.81501 0.00000 0.09316 0 0 0 0 0 0 0 0 0.000 32 Si 0.90958 0.00000 0.18715 0 0 0 0 0 0 0 0 0.000 33 O 0.75337 0.00000 0.11367 0 0 0 0 0 0 0 0 0.000 34 O 0.85409 0.00000 0.14459 0 0 0 0 0 0 0 0 0.000 35 O 0.82848 0.94621 0.05865 0 0 0 0 0 0 0 0 0.000 36 Si 0.81501 0.00000 0.90684 0 0 0 0 0 0 0 0 0.000 37 Si 0.90958 0.00000 0.81285 0 0 0 0 0 0 0 0 0.000 38 O 0.75337 0.00000 0.88633 0 0 0 0 0 0 0 0 0.000 39 O 0.85409 0.00000 0.85541 0 0 0 0 0 0 0 0 0.000 40 O 0.82848 0.05379 0.94135 0 0 0 0 0 0 0 0 0.000 41 Si 0.09316 0.18499 0.00000 0 0 0 0 0 0 0 0 0.000 42 Si 0.18715 0.09042 0.00000 0 0 0 0 0 0 0 0 0.000 43 O 0.11367 0.24663 0.00000 0 0 0 0 0 0 0 0 0.000 44 O 0.14459 0.14591 0.00000 0 0 0 0 0 0 0 0 0.000 45 O 0.05865 0.17152 0.05379 0 0 0 0 0 0 0 0 0.000 46 Si 0.90684 0.18499 0.00000 0 0 0 0 0 0 0 0 0.000 47 Si 0.81285 0.09042 0.00000 0 0 0 0 0 0 0 0 0.000 48 O 0.88633 0.24663 0.00000 0 0 0 0 0 0 0 0 0.000 49 O 0.85541 0.14591 0.00000 0 0 0 0 0 0 0 0 0.000 50 O 0.94135 0.17152 0.94621 0 0 0 0 0 0 0 0 0.000 51 Si 0.09316 0.81501 0.00000 0 0 0 0 0 0 0 0 0.000 52 Si 0.18715 0.90958 0.00000 0 0 0 0 0 0 0 0 0.000 53 O 0.11367 0.75337 0.00000 0 0 0 0 0 0 0 0 0.000 54 O 0.14459 0.85409 0.00000 0 0 0 0 0 0 0 0 0.000 55 O 0.05865 0.82848 0.94621 0 0 0 0 0 0 0 0 0.000 56 Si 0.90684 0.81501 0.00000 0 0 0 0 0 0 0 0 0.000 57 Si 0.81285 0.90958 0.00000 0 0 0 0 0 0 0 0 0.000 58 O 0.88633 0.75337 0.00000 0 0 0 0 0 0 0 0 0.000 59 O 0.85541 0.85409 0.00000 0 0 0 0 0 0 0 0 0.000 60 O 0.94135 0.82848 0.05379 0 0 0 0 0 0 0 0 0.000 61 Si 0.59316 0.50000 0.31501 0 0 0 0 0 0 0 0 0.000 62 Si 0.68715 0.50000 0.40958 0 0 0 0 0 0 0 0 0.000 63 O 0.61367 0.50000 0.25337 0 0 0 0 0 0 0 0 0.000 64 O 0.64459 0.50000 0.35409 0 0 0 0 0 0 0 0 0.000 65 O 0.55865 0.55379 0.32848 0 0 0 0 0 0 0 0 0.000 66 Si 0.40684 0.50000 0.31501 0 0 0 0 0 0 0 0 0.000 67 Si 0.31285 0.50000 0.40958 0 0 0 0 0 0 0 0 0.000 68 O 0.38633 0.50000 0.25337 0 0 0 0 0 0 0 0 0.000 69 O 0.35541 0.50000 0.35409 0 0 0 0 0 0 0 0 0.000 70 O 0.44135 0.44621 0.32848 0 0 0 0 0 0 0 0 0.000 71 Si 0.59316 0.50000 0.68499 0 0 0 0 0 0 0 0 0.000 72 Si 0.68715 0.50000 0.59042 0 0 0 0 0 0 0 0 0.000 73 O 0.61367 0.50000 0.74663 0 0 0 0 0 0 0 0 0.000 74 O 0.64459 0.50000 0.64591 0 0 0 0 0 0 0 0 0.000 75 O 0.55865 0.44621 0.67152 0 0 0 0 0 0 0 0 0.000 76 Si 0.40684 0.50000 0.68499 0 0 0 0 0 0 0 0 0.000 77 Si 0.31285 0.50000 0.59042 0 0 0 0 0 0 0 0 0.000 78 O 0.38633 0.50000 0.74663 0 0 0 0 0 0 0 0 0.000 79 O 0.35541 0.50000 0.64591 0 0 0 0 0 0 0 0 0.000 80 O 0.44135 0.55379 0.67152 0 0 0 0 0 0 0 0 0.000 81 Si 0.50000 0.68499 0.40684 0 0 0 0 0 0 0 0 0.000 82 Si 0.50000 0.59042 0.31285 0 0 0 0 0 0 0 0 0.000

83 O 0.50000 0.74663 0.38633 0 0 0 0 0 0 0 0 0.000 84 O 0.50000 0.64591 0.35541 0 0 0 0 0 0 0 0 0.000 85 O 0.55379 0.67152 0.44135 0 0 0 0 0 0 0 0 0.000 86 Si 0.50000 0.68499 0.59316 0 0 0 0 0 0 0 0 0.000 87 Si 0.50000 0.59042 0.68715 0 0 0 0 0 0 0 0 0.000 88 O 0.50000 0.74663 0.61367 0 0 0 0 0 0 0 0 0.000 89 O 0.50000 0.64591 0.64459 0 0 0 0 0 0 0 0 0.000 90 O 0.44621 0.67152 0.55865 0 0 0 0 0 0 0 0 0.000 91 Si 0.50000 0.31501 0.40684 0 0 0 0 0 0 0 0 0.000 92 Si 0.50000 0.40958 0.31285 0 0 0 0 0 0 0 0 0.000 93 O 0.50000 0.25337 0.38633 0 0 0 0 0 0 0 0 0.000 94 O 0.50000 0.35409 0.35541 0 0 0 0 0 0 0 0 0.000 95 O 0.44621 0.32848 0.44135 0 0 0 0 0 0 0 0 0.000 96 Si 0.50000 0.31501 0.59316 0 0 0 0 0 0 0 0 0.000 97 Si 0.50000 0.40958 0.68715 0 0 0 0 0 0 0 0 0.000 98 O 0.50000 0.25337 0.61367 0 0 0 0 0 0 0 0 0.000 99 O 0.50000 0.35409 0.64459 0 0 0 0 0 0 0 0 0.000 100 O 0.55379 0.32848 0.55865 0 0 0 0 0 0 0 0 0.000 101 Si 0.68499 0.59316 0.50000 0 0 0 0 0 0 0 0 0.000 102 Si 0.59042 0.68715 0.50000 0 0 0 0 0 0 0 0 0.000 103 O 0.74663 0.61367 0.50000 0 0 0 0 0 0 0 0 0.000 104 O 0.64591 0.64459 0.50000 0 0 0 0 0 0 0 0 0.000 105 O 0.67152 0.55865 0.44621 0 0 0 0 0 0 0 0 0.000 106 Si 0.68499 0.40684 0.50000 0 0 0 0 0 0 0 0 0.000 107 Si 0.59042 0.31285 0.50000 0 0 0 0 0 0 0 0 0.000 108 O 0.74663 0.38633 0.50000 0 0 0 0 0 0 0 0 0.000 109 O 0.64591 0.35541 0.50000 0 0 0 0 0 0 0 0 0.000 110 O 0.67152 0.44135 0.55379 0 0 0 0 0 0 0 0 0.000 111 Si 0.31501 0.59316 0.50000 0 0 0 0 0 0 0 0 0.000 112 Si 0.40958 0.68715 0.50000 0 0 0 0 0 0 0 0 0.000 113 O 0.25337 0.61367 0.50000 0 0 0 0 0 0 0 0 0.000 114 O 0.35409 0.64459 0.50000 0 0 0 0 0 0 0 0 0.000 115 O 0.32848 0.55865 0.55379 0 0 0 0 0 0 0 0 0.000 116 Si 0.31501 0.40684 0.50000 0 0 0 0 0 0 0 0 0.000 117 Si 0.40958 0.31285 0.50000 0 0 0 0 0 0 0 0 0.000 118 O 0.25337 0.38633 0.50000 0 0 0 0 0 0 0 0 0.000 119 O 0.35409 0.35541 0.50000 0 0 0 0 0 0 0 0 0.000 120 O 0.32848 0.44135 0.44621 0 0 0 0 0 0 0 0 0.000 121 O 0.94621 0.94135 0.82848 0 0 0 0 0 0 0 0 0.000 122 O 0.05379 0.05865 0.82848 0 0 0 0 0 0 0 0 0.000 123 O 0.05379 0.94135 0.17152 0 0 0 0 0 0 0 0 0.000 124 O 0.94621 0.05865 0.17152 0 0 0 0 0 0 0 0 0.000 125 O 0.82848 0.94621 0.94135 0 0 0 0 0 0 0 0 0.000 126 O 0.82848 0.05379 0.05865 0 0 0 0 0 0 0 0 0.000 127 O 0.17152 0.05379 0.94135 0 0 0 0 0 0 0 0 0.000 128 O 0.17152 0.94621 0.05865 0 0 0 0 0 0 0 0 0.000 129 O 0.94135 0.82848 0.94621 0 0 0 0 0 0 0 0 0.000 130 O 0.05865 0.82848 0.05379 0 0 0 0 0 0 0 0 0.000 131 O 0.94135 0.17152 0.05379 0 0 0 0 0 0 0 0 0.000 132 O 0.05865 0.17152 0.94621 0 0 0 0 0 0 0 0 0.000 133 O 0.44135 0.44621 0.67152 0 0 0 0 0 0 0 0 0.000 134 O 0.55865 0.55379 0.67152 0 0 0 0 0 0 0 0 0.000 135 O 0.44135 0.55379 0.32848 0 0 0 0 0 0 0 0 0.000 136 O 0.55865 0.44621 0.32848 0 0 0 0 0 0 0 0 0.000 137 O 0.44621 0.32848 0.55865 0 0 0 0 0 0 0 0 0.000 138 O 0.55379 0.32848 0.44135 0 0 0 0 0 0 0 0 0.000 139 O 0.55379 0.67152 0.55865 0 0 0 0 0 0 0 0 0.000 140 O 0.44621 0.67152 0.44135 0 0 0 0 0 0 0 0 0.000 141 O 0.32848 0.44135 0.55379 0 0 0 0 0 0 0 0 0.000 142 O 0.32848 0.55865 0.44621 0 0 0 0 0 0 0 0 0.000 143 O 0.67152 0.44135 0.44621 0 0 0 0 0 0 0 0 0.000 144 O 0.67152 0.55865 0.55379 0 0 0 0 0 0 0 0 0.000 145 Si 0.00000 0.59316 0.68499 0 0 0 0 0 0 0 0 0.000 146 Si 0.00000 0.68715 0.59042 0 0 0 0 0 0 0 0 0.000 147 O 0.00000 0.61367 0.74663 0 0 0 0 0 0 0 0 0.000 148 O 0.00000 0.64459 0.64591 0 0 0 0 0 0 0 0 0.000 149 O 0.05379 0.55865 0.67152 0 0 0 0 0 0 0 0 0.000 150 Si 0.00000 0.40684 0.68499 0 0 0 0 0 0 0 0 0.000 151 Si 0.00000 0.31285 0.59042 0 0 0 0 0 0 0 0 0.000 152 O 0.00000 0.38633 0.74663 0 0 0 0 0 0 0 0 0.000 153 O 0.00000 0.35541 0.64591 0 0 0 0 0 0 0 0 0.000 154 O 0.94621 0.44135 0.67152 0 0 0 0 0 0 0 0 0.000 155 Si 0.00000 0.59316 0.31501 0 0 0 0 0 0 0 0 0.000 156 Si 0.00000 0.68715 0.40958 0 0 0 0 0 0 0 0 0.000 157 O 0.00000 0.61367 0.25337 0 0 0 0 0 0 0 0 0.000 158 O 0.00000 0.64459 0.35409 0 0 0 0 0 0 0 0 0.000 159 O 0.94621 0.55865 0.32848 0 0 0 0 0 0 0 0 0.000 160 Si 0.00000 0.40684 0.31501 0 0 0 0 0 0 0 0 0.000 161 Si 0.00000 0.31285 0.40958 0 0 0 0 0 0 0 0 0.000 162 O 0.00000 0.38633 0.25337 0 0 0 0 0 0 0 0 0.000 163 O 0.00000 0.35541 0.35409 0 0 0 0 0 0 0 0 0.000 164 O 0.05379 0.44135 0.32848 0 0 0 0 0 0 0 0 0.000 165 Si 0.18499 0.50000 0.59316 0 0 0 0 0 0 0 0 0.000 166 Si 0.09042 0.50000 0.68715 0 0 0 0 0 0 0 0 0.000 167 O 0.24663 0.50000 0.61367 0 0 0 0 0 0 0 0 0.000 168 O 0.14591 0.50000 0.64459 0 0 0 0 0 0 0 0 0.000 169 O 0.17152 0.55379 0.55865 0 0 0 0 0 0 0 0 0.000 170 Si 0.18499 0.50000 0.40684 0 0 0 0 0 0 0 0 0.000

62

171 Si 0.09042 0.50000 0.31285 0 0 0 0 0 0 0 0 0.000 172 O 0.24663 0.50000 0.38633 0 0 0 0 0 0 0 0 0.000 173 O 0.14591 0.50000 0.35541 0 0 0 0 0 0 0 0 0.000 174 O 0.17152 0.44621 0.44135 0 0 0 0 0 0 0 0 0.000 175 Si 0.81501 0.50000 0.59316 0 0 0 0 0 0 0 0 0.000 176 Si 0.90958 0.50000 0.68715 0 0 0 0 0 0 0 0 0.000 177 O 0.75337 0.50000 0.61367 0 0 0 0 0 0 0 0 0.000 178 O 0.85409 0.50000 0.64459 0 0 0 0 0 0 0 0 0.000 179 O 0.82848 0.44621 0.55865 0 0 0 0 0 0 0 0 0.000 180 Si 0.81501 0.50000 0.40684 0 0 0 0 0 0 0 0 0.000 181 Si 0.90958 0.50000 0.31285 0 0 0 0 0 0 0 0 0.000 182 O 0.75337 0.50000 0.38633 0 0 0 0 0 0 0 0 0.000 183 O 0.85409 0.50000 0.35541 0 0 0 0 0 0 0 0 0.000 184 O 0.82848 0.55379 0.44135 0 0 0 0 0 0 0 0 0.000 185 Si 0.09316 0.68499 0.50000 0 0 0 0 0 0 0 0 0.000 186 Si 0.18715 0.59042 0.50000 0 0 0 0 0 0 0 0 0.000 187 O 0.11367 0.74663 0.50000 0 0 0 0 0 0 0 0 0.000 188 O 0.14459 0.64591 0.50000 0 0 0 0 0 0 0 0 0.000 189 O 0.05865 0.67152 0.55379 0 0 0 0 0 0 0 0 0.000 190 Si 0.90684 0.68499 0.50000 0 0 0 0 0 0 0 0 0.000 191 Si 0.81285 0.59042 0.50000 0 0 0 0 0 0 0 0 0.000 192 O 0.88633 0.74663 0.50000 0 0 0 0 0 0 0 0 0.000 193 O 0.85541 0.64591 0.50000 0 0 0 0 0 0 0 0 0.000 194 O 0.94135 0.67152 0.44621 0 0 0 0 0 0 0 0 0.000 195 Si 0.09316 0.31501 0.50000 0 0 0 0 0 0 0 0 0.000 196 Si 0.18715 0.40958 0.50000 0 0 0 0 0 0 0 0 0.000 197 O 0.11367 0.25337 0.50000 0 0 0 0 0 0 0 0 0.000 198 O 0.14459 0.35409 0.50000 0 0 0 0 0 0 0 0 0.000 199 O 0.05865 0.32848 0.44621 0 0 0 0 0 0 0 0 0.000 200 Si 0.90684 0.31501 0.50000 0 0 0 0 0 0 0 0 0.000 201 Si 0.81285 0.40958 0.50000 0 0 0 0 0 0 0 0 0.000 202 O 0.88633 0.25337 0.50000 0 0 0 0 0 0 0 0 0.000 203 O 0.85541 0.35409 0.50000 0 0 0 0 0 0 0 0 0.000 204 O 0.94135 0.32848 0.55379 0 0 0 0 0 0 0 0 0.000 205 Si 0.59316 0.00000 0.81501 0 0 0 0 0 0 0 0 0.000 206 Si 0.68715 0.00000 0.90958 0 0 0 0 0 0 0 0 0.000 207 O 0.61367 0.00000 0.75337 0 0 0 0 0 0 0 0 0.000 208 O 0.64459 0.00000 0.85409 0 0 0 0 0 0 0 0 0.000 209 O 0.55865 0.05379 0.82848 0 0 0 0 0 0 0 0 0.000 210 Si 0.40684 0.00000 0.81501 0 0 0 0 0 0 0 0 0.000 211 Si 0.31285 0.00000 0.90958 0 0 0 0 0 0 0 0 0.000 212 O 0.38633 0.00000 0.75337 0 0 0 0 0 0 0 0 0.000 213 O 0.35541 0.00000 0.85409 0 0 0 0 0 0 0 0 0.000 214 O 0.44135 0.94621 0.82848 0 0 0 0 0 0 0 0 0.000 215 Si 0.59316 0.00000 0.18499 0 0 0 0 0 0 0 0 0.000 216 Si 0.68715 0.00000 0.09042 0 0 0 0 0 0 0 0 0.000 217 O 0.61367 0.00000 0.24663 0 0 0 0 0 0 0 0 0.000 218 O 0.64459 0.00000 0.14591 0 0 0 0 0 0 0 0 0.000 219 O 0.55865 0.94621 0.17152 0 0 0 0 0 0 0 0 0.000 220 Si 0.40684 0.00000 0.18499 0 0 0 0 0 0 0 0 0.000 221 Si 0.31285 0.00000 0.09042 0 0 0 0 0 0 0 0 0.000 222 O 0.38633 0.00000 0.24663 0 0 0 0 0 0 0 0 0.000 223 O 0.35541 0.00000 0.14591 0 0 0 0 0 0 0 0 0.000 224 O 0.44135 0.05379 0.17152 0 0 0 0 0 0 0 0 0.000 225 Si 0.50000 0.18499 0.90684 0 0 0 0 0 0 0 0 0.000 226 Si 0.50000 0.09042 0.81285 0 0 0 0 0 0 0 0 0.000 227 O 0.50000 0.24663 0.88633 0 0 0 0 0 0 0 0 0.000 228 O 0.50000 0.14591 0.85541 0 0 0 0 0 0 0 0 0.000 229 O 0.55379 0.17152 0.94135 0 0 0 0 0 0 0 0 0.000 230 Si 0.50000 0.18499 0.09316 0 0 0 0 0 0 0 0 0.000 231 Si 0.50000 0.09042 0.18715 0 0 0 0 0 0 0 0 0.000 232 O 0.50000 0.24663 0.11367 0 0 0 0 0 0 0 0 0.000 233 O 0.50000 0.14591 0.14459 0 0 0 0 0 0 0 0 0.000 234 O 0.44621 0.17152 0.05865 0 0 0 0 0 0 0 0 0.000 235 Si 0.50000 0.81501 0.90684 0 0 0 0 0 0 0 0 0.000 236 Si 0.50000 0.90958 0.81285 0 0 0 0 0 0 0 0 0.000 237 O 0.50000 0.75337 0.88633 0 0 0 0 0 0 0 0 0.000 238 O 0.50000 0.85409 0.85541 0 0 0 0 0 0 0 0 0.000 239 O 0.44621 0.82848 0.94135 0 0 0 0 0 0 0 0 0.000 240 Si 0.50000 0.81501 0.09316 0 0 0 0 0 0 0 0 0.000 241 Si 0.50000 0.90958 0.18715 0 0 0 0 0 0 0 0 0.000 242 O 0.50000 0.75337 0.11367 0 0 0 0 0 0 0 0 0.000 243 O 0.50000 0.85409 0.14459 0 0 0 0 0 0 0 0 0.000 244 O 0.55379 0.82848 0.05865 0 0 0 0 0 0 0 0 0.000 245 Si 0.68499 0.09316 0.00000 0 0 0 0 0 0 0 0 0.000 246 Si 0.59042 0.18715 0.00000 0 0 0 0 0 0 0 0 0.000 247 O 0.74663 0.11367 0.00000 0 0 0 0 0 0 0 0 0.000 248 O 0.64591 0.14459 0.00000 0 0 0 0 0 0 0 0 0.000 249 O 0.67152 0.05865 0.94621 0 0 0 0 0 0 0 0 0.000 250 Si 0.68499 0.90684 0.00000 0 0 0 0 0 0 0 0 0.000 251 Si 0.59042 0.81285 0.00000 0 0 0 0 0 0 0 0 0.000 252 O 0.74663 0.88633 0.00000 0 0 0 0 0 0 0 0 0.000 253 O 0.64591 0.85541 0.00000 0 0 0 0 0 0 0 0 0.000 254 O 0.67152 0.94135 0.05379 0 0 0 0 0 0 0 0 0.000 255 Si 0.31501 0.09316 0.00000 0 0 0 0 0 0 0 0 0.000 256 Si 0.40958 0.18715 0.00000 0 0 0 0 0 0 0 0 0.000 257 O 0.25337 0.11367 0.00000 0 0 0 0 0 0 0 0 0.000 258 O 0.35409 0.14459 0.00000 0 0 0 0 0 0 0 0 0.000

259 O 0.32848 0.05865 0.05379 0 0 0 0 0 0 0 0 0.000 260 Si 0.31501 0.90684 0.00000 0 0 0 0 0 0 0 0 0.000 261 Si 0.40958 0.81285 0.00000 0 0 0 0 0 0 0 0 0.000 262 O 0.25337 0.88633 0.00000 0 0 0 0 0 0 0 0 0.000 263 O 0.35409 0.85541 0.00000 0 0 0 0 0 0 0 0 0.000 264 O 0.32848 0.94135 0.94621 0 0 0 0 0 0 0 0 0.000 265 O 0.94621 0.44135 0.32848 0 0 0 0 0 0 0 0 0.000 266 O 0.05379 0.55865 0.32848 0 0 0 0 0 0 0 0 0.000 267 O 0.05379 0.44135 0.67152 0 0 0 0 0 0 0 0 0.000 268 O 0.94621 0.55865 0.67152 0 0 0 0 0 0 0 0 0.000 269 O 0.82848 0.44621 0.44135 0 0 0 0 0 0 0 0 0.000 270 O 0.82848 0.55379 0.55865 0 0 0 0 0 0 0 0 0.000 271 O 0.17152 0.55379 0.44135 0 0 0 0 0 0 0 0 0.000 272 O 0.17152 0.44621 0.55865 0 0 0 0 0 0 0 0 0.000 273 O 0.94135 0.32848 0.44621 0 0 0 0 0 0 0 0 0.000 274 O 0.05865 0.32848 0.55379 0 0 0 0 0 0 0 0 0.000 275 O 0.94135 0.67152 0.55379 0 0 0 0 0 0 0 0 0.000 276 O 0.05865 0.67152 0.44621 0 0 0 0 0 0 0 0 0.000 277 O 0.44135 0.94621 0.17152 0 0 0 0 0 0 0 0 0.000 278 O 0.55865 0.05379 0.17152 0 0 0 0 0 0 0 0 0.000 279 O 0.44135 0.05379 0.82848 0 0 0 0 0 0 0 0 0.000 280 O 0.55865 0.94621 0.82848 0 0 0 0 0 0 0 0 0.000 281 O 0.44621 0.82848 0.05865 0 0 0 0 0 0 0 0 0.000 282 O 0.55379 0.82848 0.94135 0 0 0 0 0 0 0 0 0.000 283 O 0.55379 0.17152 0.05865 0 0 0 0 0 0 0 0 0.000 284 O 0.44621 0.17152 0.94135 0 0 0 0 0 0 0 0 0.000 285 O 0.32848 0.94135 0.05379 0 0 0 0 0 0 0 0 0.000 286 O 0.32848 0.05865 0.94621 0 0 0 0 0 0 0 0 0.000 287 O 0.67152 0.94135 0.94621 0 0 0 0 0 0 0 0 0.000 288 O 0.67152 0.05865 0.05379 0 0 0 0 0 0 0 0 0.000 289 Si 0.50000 0.09316 0.68499 0 0 0 0 0 0 0 0 0.000 290 Si 0.50000 0.18715 0.59042 0 0 0 0 0 0 0 0 0.000 291 O 0.50000 0.11367 0.74663 0 0 0 0 0 0 0 0 0.000 292 O 0.50000 0.14459 0.64591 0 0 0 0 0 0 0 0 0.000 293 O 0.55379 0.05865 0.67152 0 0 0 0 0 0 0 0 0.000 294 Si 0.50000 0.90684 0.68499 0 0 0 0 0 0 0 0 0.000 295 Si 0.50000 0.81285 0.59042 0 0 0 0 0 0 0 0 0.000 296 O 0.50000 0.88633 0.74663 0 0 0 0 0 0 0 0 0.000 297 O 0.50000 0.85541 0.64591 0 0 0 0 0 0 0 0 0.000 298 O 0.44621 0.94135 0.67152 0 0 0 0 0 0 0 0 0.000 299 Si 0.50000 0.09316 0.31501 0 0 0 0 0 0 0 0 0.000 300 Si 0.50000 0.18715 0.40958 0 0 0 0 0 0 0 0 0.000 301 O 0.50000 0.11367 0.25337 0 0 0 0 0 0 0 0 0.000 302 O 0.50000 0.14459 0.35409 0 0 0 0 0 0 0 0 0.000 303 O 0.44621 0.05865 0.32848 0 0 0 0 0 0 0 0 0.000 304 Si 0.50000 0.90684 0.31501 0 0 0 0 0 0 0 0 0.000 305 Si 0.50000 0.81285 0.40958 0 0 0 0 0 0 0 0 0.000 306 O 0.50000 0.88633 0.25337 0 0 0 0 0 0 0 0 0.000 307 O 0.50000 0.85541 0.35409 0 0 0 0 0 0 0 0 0.000 308 O 0.55379 0.94135 0.32848 0 0 0 0 0 0 0 0 0.000 309 Si 0.68499 0.00000 0.59316 0 0 0 0 0 0 0 0 0.000 310 Si 0.59042 0.00000 0.68715 0 0 0 0 0 0 0 0 0.000 311 O 0.74663 0.00000 0.61367 0 0 0 0 0 0 0 0 0.000 312 O 0.64591 0.00000 0.64459 0 0 0 0 0 0 0 0 0.000 313 O 0.67152 0.05379 0.55865 0 0 0 0 0 0 0 0 0.000 314 Si 0.68499 0.00000 0.40684 0 0 0 0 0 0 0 0 0.000 315 Si 0.59042 0.00000 0.31285 0 0 0 0 0 0 0 0 0.000 316 O 0.74663 0.00000 0.38633 0 0 0 0 0 0 0 0 0.000 317 O 0.64591 0.00000 0.35541 0 0 0 0 0 0 0 0 0.000 318 O 0.67152 0.94621 0.44135 0 0 0 0 0 0 0 0 0.000 319 Si 0.31501 0.00000 0.59316 0 0 0 0 0 0 0 0 0.000 320 Si 0.40958 0.00000 0.68715 0 0 0 0 0 0 0 0 0.000 321 O 0.25337 0.00000 0.61367 0 0 0 0 0 0 0 0 0.000 322 O 0.35409 0.00000 0.64459 0 0 0 0 0 0 0 0 0.000 323 O 0.32848 0.94621 0.55865 0 0 0 0 0 0 0 0 0.000 324 Si 0.31501 0.00000 0.40684 0 0 0 0 0 0 0 0 0.000 325 Si 0.40958 0.00000 0.31285 0 0 0 0 0 0 0 0 0.000 326 O 0.25337 0.00000 0.38633 0 0 0 0 0 0 0 0 0.000 327 O 0.35409 0.00000 0.35541 0 0 0 0 0 0 0 0 0.000 328 O 0.32848 0.05379 0.44135 0 0 0 0 0 0 0 0 0.000 329 Si 0.59316 0.18499 0.50000 0 0 0 0 0 0 0 0 0.000 330 Si 0.68715 0.09042 0.50000 0 0 0 0 0 0 0 0 0.000 331 O 0.61367 0.24663 0.50000 0 0 0 0 0 0 0 0 0.000 332 O 0.64459 0.14591 0.50000 0 0 0 0 0 0 0 0 0.000 333 O 0.55865 0.17152 0.55379 0 0 0 0 0 0 0 0 0.000 334 Si 0.40684 0.18499 0.50000 0 0 0 0 0 0 0 0 0.000 335 Si 0.31285 0.09042 0.50000 0 0 0 0 0 0 0 0 0.000 336 O 0.38633 0.24663 0.50000 0 0 0 0 0 0 0 0 0.000 337 O 0.35541 0.14591 0.50000 0 0 0 0 0 0 0 0 0.000 338 O 0.44135 0.17152 0.44621 0 0 0 0 0 0 0 0 0.000 339 Si 0.59316 0.81501 0.50000 0 0 0 0 0 0 0 0 0.000 340 Si 0.68715 0.90958 0.50000 0 0 0 0 0 0 0 0 0.000 341 O 0.61367 0.75337 0.50000 0 0 0 0 0 0 0 0 0.000 342 O 0.64459 0.85409 0.50000 0 0 0 0 0 0 0 0 0.000 343 O 0.55865 0.82848 0.44621 0 0 0 0 0 0 0 0 0.000 344 Si 0.40684 0.81501 0.50000 0 0 0 0 0 0 0 0 0.000 345 Si 0.31285 0.90958 0.50000 0 0 0 0 0 0 0 0 0.000 346 O 0.38633 0.75337 0.50000 0 0 0 0 0 0 0 0 0.000

63

347 O 0.35541 0.85409 0.50000 0 0 0 0 0 0 0 0 0.000 348 O 0.44135 0.82848 0.55379 0 0 0 0 0 0 0 0 0.000 349 Si 0.09316 0.50000 0.81501 0 0 0 0 0 0 0 0 0.000 350 Si 0.18715 0.50000 0.90958 0 0 0 0 0 0 0 0 0.000 351 O 0.11367 0.50000 0.75337 0 0 0 0 0 0 0 0 0.000 352 O 0.14459 0.50000 0.85409 0 0 0 0 0 0 0 0 0.000 353 O 0.05865 0.55379 0.82848 0 0 0 0 0 0 0 0 0.000 354 Si 0.90684 0.50000 0.81501 0 0 0 0 0 0 0 0 0.000 355 Si 0.81285 0.50000 0.90958 0 0 0 0 0 0 0 0 0.000 356 O 0.88633 0.50000 0.75337 0 0 0 0 0 0 0 0 0.000 357 O 0.85541 0.50000 0.85409 0 0 0 0 0 0 0 0 0.000 358 O 0.94135 0.44621 0.82848 0 0 0 0 0 0 0 0 0.000 359 Si 0.09316 0.50000 0.18499 0 0 0 0 0 0 0 0 0.000 360 Si 0.18715 0.50000 0.09042 0 0 0 0 0 0 0 0 0.000 361 O 0.11367 0.50000 0.24663 0 0 0 0 0 0 0 0 0.000 362 O 0.14459 0.50000 0.14591 0 0 0 0 0 0 0 0 0.000 363 O 0.05865 0.44621 0.17152 0 0 0 0 0 0 0 0 0.000 364 Si 0.90684 0.50000 0.18499 0 0 0 0 0 0 0 0 0.000 365 Si 0.81285 0.50000 0.09042 0 0 0 0 0 0 0 0 0.000 366 O 0.88633 0.50000 0.24663 0 0 0 0 0 0 0 0 0.000 367 O 0.85541 0.50000 0.14591 0 0 0 0 0 0 0 0 0.000 368 O 0.94135 0.55379 0.17152 0 0 0 0 0 0 0 0 0.000 369 Si 0.00000 0.68499 0.90684 0 0 0 0 0 0 0 0 0.000 370 Si 0.00000 0.59042 0.81285 0 0 0 0 0 0 0 0 0.000 371 O 0.00000 0.74663 0.88633 0 0 0 0 0 0 0 0 0.000 372 O 0.00000 0.64591 0.85541 0 0 0 0 0 0 0 0 0.000 373 O 0.05379 0.67152 0.94135 0 0 0 0 0 0 0 0 0.000 374 Si 0.00000 0.68499 0.09316 0 0 0 0 0 0 0 0 0.000 375 Si 0.00000 0.59042 0.18715 0 0 0 0 0 0 0 0 0.000 376 O 0.00000 0.74663 0.11367 0 0 0 0 0 0 0 0 0.000 377 O 0.00000 0.64591 0.14459 0 0 0 0 0 0 0 0 0.000 378 O 0.94621 0.67152 0.05865 0 0 0 0 0 0 0 0 0.000 379 Si 0.00000 0.31501 0.90684 0 0 0 0 0 0 0 0 0.000 380 Si 0.00000 0.40958 0.81285 0 0 0 0 0 0 0 0 0.000 381 O 0.00000 0.25337 0.88633 0 0 0 0 0 0 0 0 0.000 382 O 0.00000 0.35409 0.85541 0 0 0 0 0 0 0 0 0.000 383 O 0.94621 0.32848 0.94135 0 0 0 0 0 0 0 0 0.000 384 Si 0.00000 0.31501 0.09316 0 0 0 0 0 0 0 0 0.000 385 Si 0.00000 0.40958 0.18715 0 0 0 0 0 0 0 0 0.000 386 O 0.00000 0.25337 0.11367 0 0 0 0 0 0 0 0 0.000 387 O 0.00000 0.35409 0.14459 0 0 0 0 0 0 0 0 0.000 388 O 0.05379 0.32848 0.05865 0 0 0 0 0 0 0 0 0.000 389 Si 0.18499 0.59316 0.00000 0 0 0 0 0 0 0 0 0.000 390 Si 0.09042 0.68715 0.00000 0 0 0 0 0 0 0 0 0.000 391 O 0.24663 0.61367 0.00000 0 0 0 0 0 0 0 0 0.000 392 O 0.14591 0.64459 0.00000 0 0 0 0 0 0 0 0 0.000 393 O 0.17152 0.55865 0.94621 0 0 0 0 0 0 0 0 0.000 394 Si 0.18499 0.40684 0.00000 0 0 0 0 0 0 0 0 0.000 395 Si 0.09042 0.31285 0.00000 0 0 0 0 0 0 0 0 0.000 396 O 0.24663 0.38633 0.00000 0 0 0 0 0 0 0 0 0.000 397 O 0.14591 0.35541 0.00000 0 0 0 0 0 0 0 0 0.000 398 O 0.17152 0.44135 0.05379 0 0 0 0 0 0 0 0 0.000 399 Si 0.81501 0.59316 0.00000 0 0 0 0 0 0 0 0 0.000 400 Si 0.90958 0.68715 0.00000 0 0 0 0 0 0 0 0 0.000 401 O 0.75337 0.61367 0.00000 0 0 0 0 0 0 0 0 0.000 402 O 0.85409 0.64459 0.00000 0 0 0 0 0 0 0 0 0.000 403 O 0.82848 0.55865 0.05379 0 0 0 0 0 0 0 0 0.000 404 Si 0.81501 0.40684 0.00000 0 0 0 0 0 0 0 0 0.000 405 Si 0.90958 0.31285 0.00000 0 0 0 0 0 0 0 0 0.000 406 O 0.75337 0.38633 0.00000 0 0 0 0 0 0 0 0 0.000 407 O 0.85409 0.35541 0.00000 0 0 0 0 0 0 0 0 0.000 408 O 0.82848 0.44135 0.94621 0 0 0 0 0 0 0 0 0.000 409 O 0.44621 0.94135 0.32848 0 0 0 0 0 0 0 0 0.000 410 O 0.55379 0.05865 0.32848 0 0 0 0 0 0 0 0 0.000 411 O 0.55379 0.94135 0.67152 0 0 0 0 0 0 0 0 0.000 412 O 0.44621 0.05865 0.67152 0 0 0 0 0 0 0 0 0.000 413 O 0.32848 0.94621 0.44135 0 0 0 0 0 0 0 0 0.000 414 O 0.32848 0.05379 0.55865 0 0 0 0 0 0 0 0 0.000 415 O 0.67152 0.05379 0.44135 0 0 0 0 0 0 0 0 0.000 416 O 0.67152 0.94621 0.55865 0 0 0 0 0 0 0 0 0.000 417 O 0.44135 0.82848 0.44621 0 0 0 0 0 0 0 0 0.000 418 O 0.55865 0.82848 0.55379 0 0 0 0 0 0 0 0 0.000 419 O 0.44135 0.17152 0.55379 0 0 0 0 0 0 0 0 0.000 420 O 0.55865 0.17152 0.44621 0 0 0 0 0 0 0 0 0.000 421 O 0.94135 0.44621 0.17152 0 0 0 0 0 0 0 0 0.000 422 O 0.05865 0.55379 0.17152 0 0 0 0 0 0 0 0 0.000 423 O 0.94135 0.55379 0.82848 0 0 0 0 0 0 0 0 0.000 424 O 0.05865 0.44621 0.82848 0 0 0 0 0 0 0 0 0.000 425 O 0.94621 0.32848 0.05865 0 0 0 0 0 0 0 0 0.000 426 O 0.05379 0.32848 0.94135 0 0 0 0 0 0 0 0 0.000 427 O 0.05379 0.67152 0.05865 0 0 0 0 0 0 0 0 0.000 428 O 0.94621 0.67152 0.94135 0 0 0 0 0 0 0 0 0.000 429 O 0.82848 0.44135 0.05379 0 0 0 0 0 0 0 0 0.000 430 O 0.82848 0.55865 0.94621 0 0 0 0 0 0 0 0 0.000 431 O 0.17152 0.44135 0.94621 0 0 0 0 0 0 0 0 0.000 432 O 0.17152 0.55865 0.05379 0 0 0 0 0 0 0 0 0.000 433 Si 0.50000 0.59316 0.18499 0 0 0 0 0 0 0 0 0.000 434 Si 0.50000 0.68715 0.09042 0 0 0 0 0 0 0 0 0.000

435 O 0.50000 0.61367 0.24663 0 0 0 0 0 0 0 0 0.000 436 O 0.50000 0.64459 0.14591 0 0 0 0 0 0 0 0 0.000 437 O 0.55379 0.55865 0.17152 0 0 0 0 0 0 0 0 0.000 438 Si 0.50000 0.40684 0.18499 0 0 0 0 0 0 0 0 0.000 439 Si 0.50000 0.31285 0.09042 0 0 0 0 0 0 0 0 0.000 440 O 0.50000 0.38633 0.24663 0 0 0 0 0 0 0 0 0.000 441 O 0.50000 0.35541 0.14591 0 0 0 0 0 0 0 0 0.000 442 O 0.44621 0.44135 0.17152 0 0 0 0 0 0 0 0 0.000 443 Si 0.50000 0.59316 0.81501 0 0 0 0 0 0 0 0 0.000 444 Si 0.50000 0.68715 0.90958 0 0 0 0 0 0 0 0 0.000 445 O 0.50000 0.61367 0.75337 0 0 0 0 0 0 0 0 0.000 446 O 0.50000 0.64459 0.85409 0 0 0 0 0 0 0 0 0.000 447 O 0.44621 0.55865 0.82848 0 0 0 0 0 0 0 0 0.000 448 Si 0.50000 0.40684 0.81501 0 0 0 0 0 0 0 0 0.000 449 Si 0.50000 0.31285 0.90958 0 0 0 0 0 0 0 0 0.000 450 O 0.50000 0.38633 0.75337 0 0 0 0 0 0 0 0 0.000 451 O 0.50000 0.35541 0.85409 0 0 0 0 0 0 0 0 0.000 452 O 0.55379 0.44135 0.82848 0 0 0 0 0 0 0 0 0.000 453 Si 0.68499 0.50000 0.09316 0 0 0 0 0 0 0 0 0.000 454 Si 0.59042 0.50000 0.18715 0 0 0 0 0 0 0 0 0.000 455 O 0.74663 0.50000 0.11367 0 0 0 0 0 0 0 0 0.000 456 O 0.64591 0.50000 0.14459 0 0 0 0 0 0 0 0 0.000 457 O 0.67152 0.55379 0.05865 0 0 0 0 0 0 0 0 0.000 458 Si 0.68499 0.50000 0.90684 0 0 0 0 0 0 0 0 0.000 459 Si 0.59042 0.50000 0.81285 0 0 0 0 0 0 0 0 0.000 460 O 0.74663 0.50000 0.88633 0 0 0 0 0 0 0 0 0.000 461 O 0.64591 0.50000 0.85541 0 0 0 0 0 0 0 0 0.000 462 O 0.67152 0.44621 0.94135 0 0 0 0 0 0 0 0 0.000 463 Si 0.31501 0.50000 0.09316 0 0 0 0 0 0 0 0 0.000 464 Si 0.40958 0.50000 0.18715 0 0 0 0 0 0 0 0 0.000 465 O 0.25337 0.50000 0.11367 0 0 0 0 0 0 0 0 0.000 466 O 0.35409 0.50000 0.14459 0 0 0 0 0 0 0 0 0.000 467 O 0.32848 0.44621 0.05865 0 0 0 0 0 0 0 0 0.000 468 Si 0.31501 0.50000 0.90684 0 0 0 0 0 0 0 0 0.000 469 Si 0.40958 0.50000 0.81285 0 0 0 0 0 0 0 0 0.000 470 O 0.25337 0.50000 0.88633 0 0 0 0 0 0 0 0 0.000 471 O 0.35409 0.50000 0.85541 0 0 0 0 0 0 0 0 0.000 472 O 0.32848 0.55379 0.94135 0 0 0 0 0 0 0 0 0.000 473 Si 0.59316 0.68499 0.00000 0 0 0 0 0 0 0 0 0.000 474 Si 0.68715 0.59042 0.00000 0 0 0 0 0 0 0 0 0.000 475 O 0.61367 0.74663 0.00000 0 0 0 0 0 0 0 0 0.000 476 O 0.64459 0.64591 0.00000 0 0 0 0 0 0 0 0 0.000 477 O 0.55865 0.67152 0.05379 0 0 0 0 0 0 0 0 0.000 478 Si 0.40684 0.68499 0.00000 0 0 0 0 0 0 0 0 0.000 479 Si 0.31285 0.59042 0.00000 0 0 0 0 0 0 0 0 0.000 480 O 0.38633 0.74663 0.00000 0 0 0 0 0 0 0 0 0.000 481 O 0.35541 0.64591 0.00000 0 0 0 0 0 0 0 0 0.000 482 O 0.44135 0.67152 0.94621 0 0 0 0 0 0 0 0 0.000 483 Si 0.59316 0.31501 0.00000 0 0 0 0 0 0 0 0 0.000 484 Si 0.68715 0.40958 0.00000 0 0 0 0 0 0 0 0 0.000 485 O 0.61367 0.25337 0.00000 0 0 0 0 0 0 0 0 0.000 486 O 0.64459 0.35409 0.00000 0 0 0 0 0 0 0 0 0.000 487 O 0.55865 0.32848 0.94621 0 0 0 0 0 0 0 0 0.000 488 Si 0.40684 0.31501 0.00000 0 0 0 0 0 0 0 0 0.000 489 Si 0.31285 0.40958 0.00000 0 0 0 0 0 0 0 0 0.000 490 O 0.38633 0.25337 0.00000 0 0 0 0 0 0 0 0 0.000 491 O 0.35541 0.35409 0.00000 0 0 0 0 0 0 0 0 0.000 492 O 0.44135 0.32848 0.05379 0 0 0 0 0 0 0 0 0.000 493 Si 0.09316 0.00000 0.31501 0 0 0 0 0 0 0 0 0.000 494 Si 0.18715 0.00000 0.40958 0 0 0 0 0 0 0 0 0.000 495 O 0.11367 0.00000 0.25337 0 0 0 0 0 0 0 0 0.000 496 O 0.14459 0.00000 0.35409 0 0 0 0 0 0 0 0 0.000 497 O 0.05865 0.05379 0.32848 0 0 0 0 0 0 0 0 0.000 498 Si 0.90684 0.00000 0.31501 0 0 0 0 0 0 0 0 0.000 499 Si 0.81285 0.00000 0.40958 0 0 0 0 0 0 0 0 0.000 500 O 0.88633 0.00000 0.25337 0 0 0 0 0 0 0 0 0.000 501 O 0.85541 0.00000 0.35409 0 0 0 0 0 0 0 0 0.000 502 O 0.94135 0.94621 0.32848 0 0 0 0 0 0 0 0 0.000 503 Si 0.09316 0.00000 0.68499 0 0 0 0 0 0 0 0 0.000 504 Si 0.18715 0.00000 0.59042 0 0 0 0 0 0 0 0 0.000 505 O 0.11367 0.00000 0.74663 0 0 0 0 0 0 0 0 0.000 506 O 0.14459 0.00000 0.64591 0 0 0 0 0 0 0 0 0.000 507 O 0.05865 0.94621 0.67152 0 0 0 0 0 0 0 0 0.000 508 Si 0.90684 0.00000 0.68499 0 0 0 0 0 0 0 0 0.000 509 Si 0.81285 0.00000 0.59042 0 0 0 0 0 0 0 0 0.000 510 O 0.88633 0.00000 0.74663 0 0 0 0 0 0 0 0 0.000 511 O 0.85541 0.00000 0.64591 0 0 0 0 0 0 0 0 0.000 512 O 0.94135 0.05379 0.67152 0 0 0 0 0 0 0 0 0.000 513 Si 0.00000 0.18499 0.40684 0 0 0 0 0 0 0 0 0.000 514 Si 0.00000 0.09042 0.31285 0 0 0 0 0 0 0 0 0.000 515 O 0.00000 0.24663 0.38633 0 0 0 0 0 0 0 0 0.000 516 O 0.00000 0.14591 0.35541 0 0 0 0 0 0 0 0 0.000 517 O 0.05379 0.17152 0.44135 0 0 0 0 0 0 0 0 0.000 518 Si 0.00000 0.18499 0.59316 0 0 0 0 0 0 0 0 0.000 519 Si 0.00000 0.09042 0.68715 0 0 0 0 0 0 0 0 0.000 520 O 0.00000 0.24663 0.61367 0 0 0 0 0 0 0 0 0.000 521 O 0.00000 0.14591 0.64459 0 0 0 0 0 0 0 0 0.000 522 O 0.94621 0.17152 0.55865 0 0 0 0 0 0 0 0 0.000

64

523 Si 0.00000 0.81501 0.40684 0 0 0 0 0 0 0 0 0.000 524 Si 0.00000 0.90958 0.31285 0 0 0 0 0 0 0 0 0.000 525 O 0.00000 0.75337 0.38633 0 0 0 0 0 0 0 0 0.000 526 O 0.00000 0.85409 0.35541 0 0 0 0 0 0 0 0 0.000 527 O 0.94621 0.82848 0.44135 0 0 0 0 0 0 0 0 0.000 528 Si 0.00000 0.81501 0.59316 0 0 0 0 0 0 0 0 0.000 529 Si 0.00000 0.90958 0.68715 0 0 0 0 0 0 0 0 0.000 530 O 0.00000 0.75337 0.61367 0 0 0 0 0 0 0 0 0.000 531 O 0.00000 0.85409 0.64459 0 0 0 0 0 0 0 0 0.000 532 O 0.05379 0.82848 0.55865 0 0 0 0 0 0 0 0 0.000 533 Si 0.18499 0.09316 0.50000 0 0 0 0 0 0 0 0 0.000 534 Si 0.09042 0.18715 0.50000 0 0 0 0 0 0 0 0 0.000 535 O 0.24663 0.11367 0.50000 0 0 0 0 0 0 0 0 0.000 536 O 0.14591 0.14459 0.50000 0 0 0 0 0 0 0 0 0.000 537 O 0.17152 0.05865 0.44621 0 0 0 0 0 0 0 0 0.000 538 Si 0.18499 0.90684 0.50000 0 0 0 0 0 0 0 0 0.000 539 Si 0.09042 0.81285 0.50000 0 0 0 0 0 0 0 0 0.000 540 O 0.24663 0.88633 0.50000 0 0 0 0 0 0 0 0 0.000 541 O 0.14591 0.85541 0.50000 0 0 0 0 0 0 0 0 0.000 542 O 0.17152 0.94135 0.55379 0 0 0 0 0 0 0 0 0.000 543 Si 0.81501 0.09316 0.50000 0 0 0 0 0 0 0 0 0.000 544 Si 0.90958 0.18715 0.50000 0 0 0 0 0 0 0 0 0.000 545 O 0.75337 0.11367 0.50000 0 0 0 0 0 0 0 0 0.000 546 O 0.85409 0.14459 0.50000 0 0 0 0 0 0 0 0 0.000 547 O 0.82848 0.05865 0.55379 0 0 0 0 0 0 0 0 0.000 548 Si 0.81501 0.90684 0.50000 0 0 0 0 0 0 0 0 0.000 549 Si 0.90958 0.81285 0.50000 0 0 0 0 0 0 0 0 0.000 550 O 0.75337 0.88633 0.50000 0 0 0 0 0 0 0 0 0.000 551 O 0.85409 0.85541 0.50000 0 0 0 0 0 0 0 0 0.000 552 O 0.82848 0.94135 0.44621 0 0 0 0 0 0 0 0 0.000 553 O 0.44621 0.44135 0.82848 0 0 0 0 0 0 0 0 0.000 554 O 0.55379 0.55865 0.82848 0 0 0 0 0 0 0 0 0.000 555 O 0.55379 0.44135 0.17152 0 0 0 0 0 0 0 0 0.000 556 O 0.44621 0.55865 0.17152 0 0 0 0 0 0 0 0 0.000 557 O 0.32848 0.44621 0.94135 0 0 0 0 0 0 0 0 0.000 558 O 0.32848 0.55379 0.05865 0 0 0 0 0 0 0 0 0.000 559 O 0.67152 0.55379 0.94135 0 0 0 0 0 0 0 0 0.000 560 O 0.67152 0.44621 0.05865 0 0 0 0 0 0 0 0 0.000 561 O 0.44135 0.32848 0.94621 0 0 0 0 0 0 0 0 0.000 562 O 0.55865 0.32848 0.05379 0 0 0 0 0 0 0 0 0.000 563 O 0.44135 0.67152 0.05379 0 0 0 0 0 0 0 0 0.000 564 O 0.55865 0.67152 0.94621 0 0 0 0 0 0 0 0 0.000 565 O 0.94135 0.94621 0.67152 0 0 0 0 0 0 0 0 0.000 566 O 0.05865 0.05379 0.67152 0 0 0 0 0 0 0 0 0.000 567 O 0.94135 0.05379 0.32848 0 0 0 0 0 0 0 0 0.000 568 O 0.05865 0.94621 0.32848 0 0 0 0 0 0 0 0 0.000 569 O 0.94621 0.82848 0.55865 0 0 0 0 0 0 0 0 0.000 570 O 0.05379 0.82848 0.44135 0 0 0 0 0 0 0 0 0.000 571 O 0.05379 0.17152 0.55865 0 0 0 0 0 0 0 0 0.000 572 O 0.94621 0.17152 0.44135 0 0 0 0 0 0 0 0 0.000 573 O 0.82848 0.94135 0.55379 0 0 0 0 0 0 0 0 0.000 574 O 0.82848 0.05865 0.44621 0 0 0 0 0 0 0 0 0.000 575 O 0.17152 0.94135 0.44621 0 0 0 0 0 0 0 0 0.000 576 O 0.17152 0.05865 0.55379 0 0 0 0 0 0 0 0 0.000 LTA.block: 8 0.0 0.0 0.0 2.94660 0.0 0.0 0.5 2.94660 0.0 0.5 0.0 2.94660 0.0 0.5 0.5 2.94660 0.5 0.0 0.0 2.94660 0.5 0.0 0.5 2.94660 0.5 0.5 0.0 2.94660 0.5 0.5 0.5 2.94660 AFI.cssr: 13.827 13.827 8.58 90.0 90.0 120.0 SPGR = 1 P 1 OPT = 1 72 0 0 AFI : AFI 1 O 0.4565 0.3333 0.0 0 0 0 0 0 0 0 0 0.00 2 O 0.1232 0.4565 0.0 0 0 0 0 0 0 0 0 0.00 3 O 0.3333 0.8768 0.0 0 0 0 0 0 0 0 0 0.00 4 O 0.6667 0.1232 0.0 0 0 0 0 0 0 0 0 0.00 5 O 0.8768 0.5435 0.0 0 0 0 0 0 0 0 0 0.00 6 O 0.1232 0.6667 0.5 0 0 0 0 0 0 0 0 0.00 7 O 0.5435 0.8768 0.5 0 0 0 0 0 0 0 0 0.00 8 O 0.5435 0.6667 0.0 0 0 0 0 0 0 0 0 0.00 9 O 0.3333 0.4565 0.5 0 0 0 0 0 0 0 0 0.00 10 O 0.6667 0.5435 0.5 0 0 0 0 0 0 0 0 0.00 11 O 0.8768 0.3333 0.5 0 0 0 0 0 0 0 0 0.00 12 O 0.4565 0.1232 0.5 0 0 0 0 0 0 0 0 0.00 13 O 0.3693 0.3693 0.25 0 0 0 0 0 0 0 0 0.00 14 O 0.0 0.3693 0.25 0 0 0 0 0 0 0 0 0.00 15 O 0.3693 0.0 0.25 0 0 0 0 0 0 0 0 0.00

16 O 0.6307 0.0 0.25 0 0 0 0 0 0 0 0 0.00 17 O 0.0 0.6307 0.25 0 0 0 0 0 0 0 0 0.00 18 O 0.6307 0.6307 0.25 0 0 0 0 0 0 0 0 0.00 19 O 0.6307 0.6307 0.75 0 0 0 0 0 0 0 0 0.00 20 O 0.0 0.6307 0.75 0 0 0 0 0 0 0 0 0.00 21 O 0.6307 0.0 0.75 0 0 0 0 0 0 0 0 0.00 22 O 0.3693 0.0 0.75 0 0 0 0 0 0 0 0 0.00 23 O 0.0 0.3693 0.75 0 0 0 0 0 0 0 0 0.00 24 O 0.3693 0.3693 0.75 0 0 0 0 0 0 0 0 0.00 25 O 0.42 0.21 0.25 0 0 0 0 0 0 0 0 0.00 26 O 0.21 0.42 0.25 0 0 0 0 0 0 0 0 0.00 27 O 0.21 0.79 0.25 0 0 0 0 0 0 0 0 0.00 28 O 0.79 0.21 0.25 0 0 0 0 0 0 0 0 0.00 29 O 0.79 0.58 0.25 0 0 0 0 0 0 0 0 0.00 30 O 0.58 0.79 0.25 0 0 0 0 0 0 0 0 0.00 31 O 0.58 0.79 0.75 0 0 0 0 0 0 0 0 0.00 32 O 0.79 0.58 0.75 0 0 0 0 0 0 0 0 0.00 33 O 0.79 0.21 0.75 0 0 0 0 0 0 0 0 0.00 34 O 0.21 0.79 0.75 0 0 0 0 0 0 0 0 0.00 35 O 0.21 0.42 0.75 0 0 0 0 0 0 0 0 0.00 36 O 0.42 0.21 0.75 0 0 0 0 0 0 0 0 0.00 37 O 0.5798 0.4202 0.25 0 0 0 0 0 0 0 0 0.00 38 O 0.1596 0.5798 0.25 0 0 0 0 0 0 0 0 0.00 39 O 0.4202 0.8404 0.25 0 0 0 0 0 0 0 0 0.00 40 O 0.5798 0.1596 0.25 0 0 0 0 0 0 0 0 0.00 41 O 0.8404 0.4202 0.25 0 0 0 0 0 0 0 0 0.00 42 O 0.4202 0.5798 0.25 0 0 0 0 0 0 0 0 0.00 43 O 0.4202 0.5798 0.75 0 0 0 0 0 0 0 0 0.00 44 O 0.8404 0.4202 0.75 0 0 0 0 0 0 0 0 0.00 45 O 0.5798 0.1596 0.75 0 0 0 0 0 0 0 0 0.00 46 O 0.4202 0.8404 0.75 0 0 0 0 0 0 0 0 0.00 47 O 0.1596 0.5798 0.75 0 0 0 0 0 0 0 0 0.00 48 O 0.5798 0.4202 0.75 0 0 0 0 0 0 0 0 0.00 49 Si 0.4565 0.3334 0.1874 0 0 0 0 0 0 0 0 0.00 50 Si 0.1231 0.4565 0.1874 0 0 0 0 0 0 0 0 0.00 51 Si 0.3334 0.8769 0.1874 0 0 0 0 0 0 0 0 0.00 52 Si 0.6666 0.1231 0.1874 0 0 0 0 0 0 0 0 0.00 53 Si 0.8769 0.5435 0.1874 0 0 0 0 0 0 0 0 0.00 54 Si 0.1231 0.6666 0.3126 0 0 0 0 0 0 0 0 0.00 55 Si 0.5435 0.8769 0.3126 0 0 0 0 0 0 0 0 0.00 56 Si 0.5435 0.6666 0.1874 0 0 0 0 0 0 0 0 0.00 57 Si 0.3334 0.4565 0.3126 0 0 0 0 0 0 0 0 0.00 58 Si 0.6666 0.5435 0.3126 0 0 0 0 0 0 0 0 0.00 59 Si 0.8769 0.3334 0.3126 0 0 0 0 0 0 0 0 0.00 60 Si 0.4565 0.1231 0.3126 0 0 0 0 0 0 0 0 0.00 61 Si 0.5435 0.6666 0.8126 0 0 0 0 0 0 0 0 0.00 62 Si 0.8769 0.5435 0.8126 0 0 0 0 0 0 0 0 0.00 63 Si 0.6666 0.1231 0.8126 0 0 0 0 0 0 0 0 0.00 64 Si 0.3334 0.8769 0.8126 0 0 0 0 0 0 0 0 0.00 65 Si 0.1231 0.4565 0.8126 0 0 0 0 0 0 0 0 0.00 66 Si 0.8769 0.3334 0.6874 0 0 0 0 0 0 0 0 0.00 67 Si 0.4565 0.1231 0.6874 0 0 0 0 0 0 0 0 0.00 68 Si 0.4565 0.3334 0.8126 0 0 0 0 0 0 0 0 0.00 69 Si 0.6666 0.5435 0.6874 0 0 0 0 0 0 0 0 0.00 70 Si 0.3334 0.4565 0.6874 0 0 0 0 0 0 0 0 0.00 71 Si 0.1231 0.6666 0.6874 0 0 0 0 0 0 0 0 0.00 72 Si 0.5435 0.8769 0.6874 0 0 0 0 0 0 0 0 0.00 MFI.cssr: 20.09 19.738 13.142 90.0 90.0 90.0 SPGR = 1 P 1 OPT = 1 288 0 0 MFI : MFI 1 O 0.5012 0.0699 0.7018 0 0 0 0 0 0 0 0 0.00 2 O 0.0012 0.4301 0.7982 0 0 0 0 0 0 0 0 0.00 3 O 0.4988 0.5699 0.2982 0 0 0 0 0 0 0 0 0.00 4 O 0.9988 0.9301 0.2018 0 0 0 0 0 0 0 0 0.00 5 O 0.4988 0.9301 0.2982 0 0 0 0 0 0 0 0 0.00 6 O 0.9988 0.5699 0.2018 0 0 0 0 0 0 0 0 0.00 7 O 0.5012 0.4301 0.7018 0 0 0 0 0 0 0 0 0.00 8 O 0.0012 0.0699 0.7982 0 0 0 0 0 0 0 0 0.00 9 O 0.3875 0.0743 0.8008 0 0 0 0 0 0 0 0 0.00 10 O 0.8875 0.4257 0.6992 0 0 0 0 0 0 0 0 0.00 11 O 0.6125 0.5743 0.1992 0 0 0 0 0 0 0 0 0.00 12 O 0.1125 0.9257 0.3008 0 0 0 0 0 0 0 0 0.00 13 O 0.6125 0.9257 0.1992 0 0 0 0 0 0 0 0 0.00 14 O 0.1125 0.5743 0.3008 0 0 0 0 0 0 0 0 0.00 15 O 0.3875 0.4257 0.8008 0 0 0 0 0 0 0 0 0.00 16 O 0.8875 0.0743 0.6992 0 0 0 0 0 0 0 0 0.00 17 O 0.3995 0.1366 0.6251 0 0 0 0 0 0 0 0 0.00 18 O 0.8995 0.3634 0.8749 0 0 0 0 0 0 0 0 0.00 19 O 0.6005 0.6366 0.3749 0 0 0 0 0 0 0 0 0.00 20 O 0.1005 0.8634 0.1251 0 0 0 0 0 0 0 0 0.00 21 O 0.6005 0.8634 0.3749 0 0 0 0 0 0 0 0 0.00 22 O 0.1005 0.6366 0.1251 0 0 0 0 0 0 0 0 0.00 23 O 0.3995 0.3634 0.6251 0 0 0 0 0 0 0 0 0.00 24 O 0.8995 0.1366 0.8749 0 0 0 0 0 0 0 0 0.00

65

25 O 0.3972 0.0034 0.6326 0 0 0 0 0 0 0 0 0.00 26 O 0.8972 0.4966 0.8674 0 0 0 0 0 0 0 0 0.00 27 O 0.6028 0.5034 0.3674 0 0 0 0 0 0 0 0 0.00 28 O 0.1028 0.9966 0.1326 0 0 0 0 0 0 0 0 0.00 29 O 0.6028 0.9966 0.3674 0 0 0 0 0 0 0 0 0.00 30 O 0.1028 0.5034 0.1326 0 0 0 0 0 0 0 0 0.00 31 O 0.3972 0.4966 0.6326 0 0 0 0 0 0 0 0 0.00 32 O 0.8972 0.0034 0.8674 0 0 0 0 0 0 0 0 0.00 33 O 0.329 0.0385 0.9722 0 0 0 0 0 0 0 0 0.00 34 O 0.829 0.4615 0.5278 0 0 0 0 0 0 0 0 0.00 35 O 0.671 0.5385 0.0278 0 0 0 0 0 0 0 0 0.00 36 O 0.171 0.9615 0.4722 0 0 0 0 0 0 0 0 0.00 37 O 0.671 0.9615 0.0278 0 0 0 0 0 0 0 0 0.00 38 O 0.171 0.5385 0.4722 0 0 0 0 0 0 0 0 0.00 39 O 0.329 0.4615 0.9722 0 0 0 0 0 0 0 0 0.00 40 O 0.829 0.0385 0.5278 0 0 0 0 0 0 0 0 0.00 41 O 0.3297 0.9555 0.8155 0 0 0 0 0 0 0 0 0.00 42 O 0.8297 0.5445 0.6845 0 0 0 0 0 0 0 0 0.00 43 O 0.6703 0.4555 0.1845 0 0 0 0 0 0 0 0 0.00 44 O 0.1703 0.0445 0.3155 0 0 0 0 0 0 0 0 0.00 45 O 0.6703 0.0445 0.1845 0 0 0 0 0 0 0 0 0.00 46 O 0.1703 0.4555 0.3155 0 0 0 0 0 0 0 0 0.00 47 O 0.3297 0.5445 0.8155 0 0 0 0 0 0 0 0 0.00 48 O 0.8297 0.9555 0.6845 0 0 0 0 0 0 0 0 0.00 49 O 0.2571 0.0663 0.8106 0 0 0 0 0 0 0 0 0.00 50 O 0.7571 0.4337 0.6894 0 0 0 0 0 0 0 0 0.00 51 O 0.7429 0.5663 0.1894 0 0 0 0 0 0 0 0 0.00 52 O 0.2429 0.9337 0.3106 0 0 0 0 0 0 0 0 0.00 53 O 0.7429 0.9337 0.1894 0 0 0 0 0 0 0 0 0.00 54 O 0.2429 0.5663 0.3106 0 0 0 0 0 0 0 0 0.00 55 O 0.2571 0.4337 0.8106 0 0 0 0 0 0 0 0 0.00 56 O 0.7571 0.0663 0.6894 0 0 0 0 0 0 0 0 0.00 57 O 0.291 0.1296 0.106 0 0 0 0 0 0 0 0 0.00 58 O 0.791 0.3704 0.394 0 0 0 0 0 0 0 0 0.00 59 O 0.709 0.6296 0.894 0 0 0 0 0 0 0 0 0.00 60 O 0.209 0.8704 0.606 0 0 0 0 0 0 0 0 0.00 61 O 0.709 0.8704 0.894 0 0 0 0 0 0 0 0 0.00 62 O 0.209 0.6296 0.606 0 0 0 0 0 0 0 0 0.00 63 O 0.291 0.3704 0.106 0 0 0 0 0 0 0 0 0.00 64 O 0.791 0.1296 0.394 0 0 0 0 0 0 0 0 0.00 65 O 0.2034 0.0455 0.0275 0 0 0 0 0 0 0 0 0.00 66 O 0.7034 0.4545 0.4725 0 0 0 0 0 0 0 0 0.00 67 O 0.7966 0.5455 0.9725 0 0 0 0 0 0 0 0 0.00 68 O 0.2966 0.9545 0.5275 0 0 0 0 0 0 0 0 0.00 69 O 0.7966 0.9545 0.9725 0 0 0 0 0 0 0 0 0.00 70 O 0.2966 0.5455 0.5275 0 0 0 0 0 0 0 0 0.00 71 O 0.2034 0.4545 0.0275 0 0 0 0 0 0 0 0 0.00 72 O 0.7034 0.0455 0.4725 0 0 0 0 0 0 0 0 0.00 73 O 0.2936 0.001 0.1564 0 0 0 0 0 0 0 0 0.00 74 O 0.7936 0.499 0.3436 0 0 0 0 0 0 0 0 0.00 75 O 0.7064 0.501 0.8436 0 0 0 0 0 0 0 0 0.00 76 O 0.2064 0.999 0.6564 0 0 0 0 0 0 0 0 0.00 77 O 0.7064 0.999 0.8436 0 0 0 0 0 0 0 0 0.00 78 O 0.2064 0.501 0.6564 0 0 0 0 0 0 0 0 0.00 79 O 0.2936 0.499 0.1564 0 0 0 0 0 0 0 0 0.00 80 O 0.7936 0.001 0.3436 0 0 0 0 0 0 0 0 0.00 81 O 0.1075 0.1265 0.0883 0 0 0 0 0 0 0 0 0.00 82 O 0.6075 0.3735 0.4117 0 0 0 0 0 0 0 0 0.00 83 O 0.8925 0.6265 0.9117 0 0 0 0 0 0 0 0 0.00 84 O 0.3925 0.8735 0.5883 0 0 0 0 0 0 0 0 0.00 85 O 0.8925 0.8735 0.9117 0 0 0 0 0 0 0 0 0.00 86 O 0.3925 0.6265 0.5883 0 0 0 0 0 0 0 0 0.00 87 O 0.1075 0.3735 0.0883 0 0 0 0 0 0 0 0 0.00 88 O 0.6075 0.1265 0.4117 0 0 0 0 0 0 0 0 0.00 89 O 0.0846 0.037 0.9443 0 0 0 0 0 0 0 0 0.00 90 O 0.5846 0.463 0.5557 0 0 0 0 0 0 0 0 0.00 91 O 0.9154 0.537 0.0557 0 0 0 0 0 0 0 0 0.00 92 O 0.4154 0.963 0.4443 0 0 0 0 0 0 0 0 0.00 93 O 0.9154 0.963 0.0557 0 0 0 0 0 0 0 0 0.00 94 O 0.4154 0.537 0.4443 0 0 0 0 0 0 0 0 0.00 95 O 0.0846 0.463 0.9443 0 0 0 0 0 0 0 0 0.00 96 O 0.5846 0.037 0.5557 0 0 0 0 0 0 0 0 0.00 97 O 0.1301 0.0781 0.767 0 0 0 0 0 0 0 0 0.00 98 O 0.6301 0.4219 0.733 0 0 0 0 0 0 0 0 0.00 99 O 0.8699 0.5781 0.233 0 0 0 0 0 0 0 0 0.00 100 O 0.3699 0.9219 0.267 0 0 0 0 0 0 0 0 0.00 101 O 0.8699 0.9219 0.233 0 0 0 0 0 0 0 0 0.00 102 O 0.3699 0.5781 0.267 0 0 0 0 0 0 0 0 0.00 103 O 0.1301 0.4219 0.767 0 0 0 0 0 0 0 0 0.00 104 O 0.6301 0.0781 0.733 0 0 0 0 0 0 0 0 0.00 105 O 0.0728 0.9589 0.7832 0 0 0 0 0 0 0 0 0.00 106 O 0.5728 0.5411 0.7168 0 0 0 0 0 0 0 0 0.00 107 O 0.9272 0.4589 0.2168 0 0 0 0 0 0 0 0 0.00 108 O 0.4272 0.0411 0.2832 0 0 0 0 0 0 0 0 0.00 109 O 0.9272 0.0411 0.2168 0 0 0 0 0 0 0 0 0.00 110 O 0.4272 0.4589 0.2832 0 0 0 0 0 0 0 0 0.00 111 O 0.0728 0.5411 0.7832 0 0 0 0 0 0 0 0 0.00 112 O 0.5728 0.9589 0.7168 0 0 0 0 0 0 0 0 0.00

113 O 0.2201 0.1313 0.6456 0 0 0 0 0 0 0 0 0.00 114 O 0.7201 0.3687 0.8544 0 0 0 0 0 0 0 0 0.00 115 O 0.7799 0.6313 0.3544 0 0 0 0 0 0 0 0 0.00 116 O 0.2799 0.8687 0.1456 0 0 0 0 0 0 0 0 0.00 117 O 0.7799 0.8687 0.3544 0 0 0 0 0 0 0 0 0.00 118 O 0.2799 0.6313 0.1456 0 0 0 0 0 0 0 0 0.00 119 O 0.2201 0.3687 0.6456 0 0 0 0 0 0 0 0 0.00 120 O 0.7201 0.1313 0.8544 0 0 0 0 0 0 0 0 0.00 121 O 0.4911 0.8548 0.7169 0 0 0 0 0 0 0 0 0.00 122 O 0.9911 0.6452 0.7831 0 0 0 0 0 0 0 0 0.00 123 O 0.5089 0.3548 0.2831 0 0 0 0 0 0 0 0 0.00 124 O 0.0089 0.1452 0.2169 0 0 0 0 0 0 0 0 0.00 125 O 0.5089 0.1452 0.2831 0 0 0 0 0 0 0 0 0.00 126 O 0.0089 0.3548 0.2169 0 0 0 0 0 0 0 0 0.00 127 O 0.4911 0.6452 0.7169 0 0 0 0 0 0 0 0 0.00 128 O 0.9911 0.8548 0.7831 0 0 0 0 0 0 0 0 0.00 129 O 0.3681 0.8311 0.7743 0 0 0 0 0 0 0 0 0.00 130 O 0.8681 0.6689 0.7257 0 0 0 0 0 0 0 0 0.00 131 O 0.6319 0.3311 0.2257 0 0 0 0 0 0 0 0 0.00 132 O 0.1319 0.1689 0.2743 0 0 0 0 0 0 0 0 0.00 133 O 0.6319 0.1689 0.2257 0 0 0 0 0 0 0 0 0.00 134 O 0.1319 0.3311 0.2743 0 0 0 0 0 0 0 0 0.00 135 O 0.3681 0.6689 0.7743 0 0 0 0 0 0 0 0 0.00 136 O 0.8681 0.8311 0.7257 0 0 0 0 0 0 0 0 0.00 137 O 0.4263 0.75 0.6431 0 0 0 0 0 0 0 0 0.00 138 O 0.9263 0.75 0.8569 0 0 0 0 0 0 0 0 0.00 139 O 0.5737 0.25 0.3569 0 0 0 0 0 0 0 0 0.00 140 O 0.0737 0.25 0.1431 0 0 0 0 0 0 0 0 0.00 141 O 0.3217 0.8611 0.9561 0 0 0 0 0 0 0 0 0.00 142 O 0.8217 0.6389 0.5439 0 0 0 0 0 0 0 0 0.00 143 O 0.6783 0.3611 0.0439 0 0 0 0 0 0 0 0 0.00 144 O 0.1783 0.1389 0.4561 0 0 0 0 0 0 0 0 0.00 145 O 0.6783 0.1389 0.0439 0 0 0 0 0 0 0 0 0.00 146 O 0.1783 0.3611 0.4561 0 0 0 0 0 0 0 0 0.00 147 O 0.3217 0.6389 0.9561 0 0 0 0 0 0 0 0 0.00 148 O 0.8217 0.8611 0.5439 0 0 0 0 0 0 0 0 0.00 149 O 0.241 0.8582 0.799 0 0 0 0 0 0 0 0 0.00 150 O 0.741 0.6418 0.701 0 0 0 0 0 0 0 0 0.00 151 O 0.759 0.3582 0.201 0 0 0 0 0 0 0 0 0.00 152 O 0.259 0.1418 0.299 0 0 0 0 0 0 0 0 0.00 153 O 0.759 0.1418 0.201 0 0 0 0 0 0 0 0 0.00 154 O 0.259 0.3582 0.299 0 0 0 0 0 0 0 0 0.00 155 O 0.241 0.6418 0.799 0 0 0 0 0 0 0 0 0.00 156 O 0.741 0.8582 0.701 0 0 0 0 0 0 0 0 0.00 157 O 0.2943 0.75 0.0579 0 0 0 0 0 0 0 0 0.00 158 O 0.7943 0.75 0.4421 0 0 0 0 0 0 0 0 0.00 159 O 0.7057 0.25 0.9421 0 0 0 0 0 0 0 0 0.00 160 O 0.2057 0.25 0.5579 0 0 0 0 0 0 0 0 0.00 161 O 0.1976 0.8313 0.0003 0 0 0 0 0 0 0 0 0.00 162 O 0.6976 0.6687 0.4997 0 0 0 0 0 0 0 0 0.00 163 O 0.8024 0.3313 0.9997 0 0 0 0 0 0 0 0 0.00 164 O 0.3024 0.1687 0.5003 0 0 0 0 0 0 0 0 0.00 165 O 0.8024 0.1687 0.9997 0 0 0 0 0 0 0 0 0.00 166 O 0.3024 0.3313 0.5003 0 0 0 0 0 0 0 0 0.00 167 O 0.1976 0.6687 0.0003 0 0 0 0 0 0 0 0 0.00 168 O 0.6976 0.8313 0.4997 0 0 0 0 0 0 0 0 0.00 169 O 0.0948 0.75 0.0205 0 0 0 0 0 0 0 0 0.00 170 O 0.5948 0.75 0.4795 0 0 0 0 0 0 0 0 0.00 171 O 0.9052 0.25 0.9795 0 0 0 0 0 0 0 0 0.00 172 O 0.4052 0.25 0.5205 0 0 0 0 0 0 0 0 0.00 173 O 0.0809 0.867 0.9275 0 0 0 0 0 0 0 0 0.00 174 O 0.5809 0.633 0.5725 0 0 0 0 0 0 0 0 0.00 175 O 0.9191 0.367 0.0725 0 0 0 0 0 0 0 0 0.00 176 O 0.4191 0.133 0.4275 0 0 0 0 0 0 0 0 0.00 177 O 0.9191 0.133 0.0725 0 0 0 0 0 0 0 0 0.00 178 O 0.4191 0.367 0.4275 0 0 0 0 0 0 0 0 0.00 179 O 0.0809 0.633 0.9275 0 0 0 0 0 0 0 0 0.00 180 O 0.5809 0.867 0.5725 0 0 0 0 0 0 0 0 0.00 181 O 0.1177 0.837 0.7406 0 0 0 0 0 0 0 0 0.00 182 O 0.6177 0.663 0.7594 0 0 0 0 0 0 0 0 0.00 183 O 0.8823 0.337 0.2594 0 0 0 0 0 0 0 0 0.00 184 O 0.3823 0.163 0.2406 0 0 0 0 0 0 0 0 0.00 185 O 0.8823 0.163 0.2594 0 0 0 0 0 0 0 0 0.00 186 O 0.3823 0.337 0.2406 0 0 0 0 0 0 0 0 0.00 187 O 0.1177 0.663 0.7406 0 0 0 0 0 0 0 0 0.00 188 O 0.6177 0.837 0.7594 0 0 0 0 0 0 0 0 0.00 189 O 0.2116 0.75 0.6913 0 0 0 0 0 0 0 0 0.00 190 O 0.7116 0.75 0.8087 0 0 0 0 0 0 0 0 0.00 191 O 0.7884 0.25 0.3087 0 0 0 0 0 0 0 0 0.00 192 O 0.2884 0.25 0.1913 0 0 0 0 0 0 0 0 0.00 193 Si 0.4214 0.0711 0.6898 0 0 0 0 0 0 0 0 0.00 194 Si 0.9214 0.4289 0.8102 0 0 0 0 0 0 0 0 0.00 195 Si 0.5786 0.5711 0.3102 0 0 0 0 0 0 0 0 0.00 196 Si 0.0786 0.9289 0.1898 0 0 0 0 0 0 0 0 0.00 197 Si 0.5786 0.9289 0.3102 0 0 0 0 0 0 0 0 0.00 198 Si 0.0786 0.5711 0.1898 0 0 0 0 0 0 0 0 0.00 199 Si 0.4214 0.4289 0.6898 0 0 0 0 0 0 0 0 0.00 200 Si 0.9214 0.0711 0.8102 0 0 0 0 0 0 0 0 0.00

66

201 Si 0.3259 0.0336 0.85 0 0 0 0 0 0 0 0 0.00 202 Si 0.8259 0.4664 0.65 0 0 0 0 0 0 0 0 0.00 203 Si 0.6741 0.5336 0.15 0 0 0 0 0 0 0 0 0.00 204 Si 0.1741 0.9664 0.35 0 0 0 0 0 0 0 0 0.00 205 Si 0.6741 0.9664 0.15 0 0 0 0 0 0 0 0 0.00 206 Si 0.1741 0.5336 0.35 0 0 0 0 0 0 0 0 0.00 207 Si 0.3259 0.4664 0.85 0 0 0 0 0 0 0 0 0.00 208 Si 0.8259 0.0336 0.65 0 0 0 0 0 0 0 0 0.00 209 Si 0.2792 0.0536 0.0655 0 0 0 0 0 0 0 0 0.00 210 Si 0.7792 0.4464 0.4345 0 0 0 0 0 0 0 0 0.00 211 Si 0.7208 0.5536 0.9345 0 0 0 0 0 0 0 0 0.00 212 Si 0.2208 0.9464 0.5655 0 0 0 0 0 0 0 0 0.00 213 Si 0.7208 0.9464 0.9345 0 0 0 0 0 0 0 0 0.00 214 Si 0.2208 0.5536 0.5655 0 0 0 0 0 0 0 0 0.00 215 Si 0.2792 0.4464 0.0655 0 0 0 0 0 0 0 0 0.00 216 Si 0.7792 0.0536 0.4345 0 0 0 0 0 0 0 0 0.00 217 Si 0.1246 0.0514 0.0481 0 0 0 0 0 0 0 0 0.00 218 Si 0.6246 0.4486 0.4519 0 0 0 0 0 0 0 0 0.00 219 Si 0.8754 0.5514 0.9519 0 0 0 0 0 0 0 0 0.00 220 Si 0.3754 0.9486 0.5481 0 0 0 0 0 0 0 0 0.00 221 Si 0.8754 0.9486 0.9519 0 0 0 0 0 0 0 0 0.00 222 Si 0.3754 0.5514 0.5481 0 0 0 0 0 0 0 0 0.00 223 Si 0.1246 0.4486 0.0481 0 0 0 0 0 0 0 0 0.00 224 Si 0.6246 0.0514 0.4519 0 0 0 0 0 0 0 0 0.00 225 Si 0.0721 0.036 0.8233 0 0 0 0 0 0 0 0 0.00 226 Si 0.5721 0.464 0.6767 0 0 0 0 0 0 0 0 0.00 227 Si 0.9279 0.536 0.1767 0 0 0 0 0 0 0 0 0.00 228 Si 0.4279 0.964 0.3233 0 0 0 0 0 0 0 0 0.00 229 Si 0.9279 0.964 0.1767 0 0 0 0 0 0 0 0 0.00 230 Si 0.4279 0.536 0.3233 0 0 0 0 0 0 0 0 0.00 231 Si 0.0721 0.464 0.8233 0 0 0 0 0 0 0 0 0.00 232 Si 0.5721 0.036 0.6767 0 0 0 0 0 0 0 0 0.00 233 Si 0.2034 0.0687 0.7197 0 0 0 0 0 0 0 0 0.00 234 Si 0.7034 0.4313 0.7803 0 0 0 0 0 0 0 0 0.00 235 Si 0.7966 0.5687 0.2803 0 0 0 0 0 0 0 0 0.00 236 Si 0.2966 0.9313 0.2197 0 0 0 0 0 0 0 0 0.00 237 Si 0.7966 0.9313 0.2803 0 0 0 0 0 0 0 0 0.00 238 Si 0.2966 0.5687 0.2197 0 0 0 0 0 0 0 0 0.00 239 Si 0.2034 0.4313 0.7197 0 0 0 0 0 0 0 0 0.00 240 Si 0.7034 0.0687 0.7803 0 0 0 0 0 0 0 0 0.00 241 Si 0.4195 0.8274 0.6805 0 0 0 0 0 0 0 0 0.00 242 Si 0.9195 0.6726 0.8195 0 0 0 0 0 0 0 0 0.00 243 Si 0.5805 0.3274 0.3195 0 0 0 0 0 0 0 0 0.00 244 Si 0.0805 0.1726 0.1805 0 0 0 0 0 0 0 0 0.00 245 Si 0.5805 0.1726 0.3195 0 0 0 0 0 0 0 0 0.00 246 Si 0.0805 0.3274 0.1805 0 0 0 0 0 0 0 0 0.00 247 Si 0.4195 0.6726 0.6805 0 0 0 0 0 0 0 0 0.00 248 Si 0.9195 0.8274 0.8195 0 0 0 0 0 0 0 0 0.00 249 Si 0.3152 0.8765 0.8361 0 0 0 0 0 0 0 0 0.00 250 Si 0.8152 0.6235 0.6639 0 0 0 0 0 0 0 0 0.00 251 Si 0.6848 0.3765 0.1639 0 0 0 0 0 0 0 0 0.00 252 Si 0.1848 0.1235 0.3361 0 0 0 0 0 0 0 0 0.00 253 Si 0.6848 0.1235 0.1639 0 0 0 0 0 0 0 0 0.00 254 Si 0.1848 0.3765 0.3361 0 0 0 0 0 0 0 0 0.00 255 Si 0.3152 0.6235 0.8361 0 0 0 0 0 0 0 0 0.00 256 Si 0.8152 0.8765 0.6639 0 0 0 0 0 0 0 0 0.00 257 Si 0.2733 0.8278 0.0402 0 0 0 0 0 0 0 0 0.00 258 Si 0.7733 0.6722 0.4598 0 0 0 0 0 0 0 0 0.00 259 Si 0.7267 0.3278 0.9598 0 0 0 0 0 0 0 0 0.00 260 Si 0.2267 0.1722 0.5402 0 0 0 0 0 0 0 0 0.00 261 Si 0.7267 0.1722 0.9598 0 0 0 0 0 0 0 0 0.00 262 Si 0.2267 0.3278 0.5402 0 0 0 0 0 0 0 0 0.00 263 Si 0.2733 0.6722 0.0402 0 0 0 0 0 0 0 0 0.00 264 Si 0.7733 0.8278 0.4598 0 0 0 0 0 0 0 0 0.00 265 Si 0.1185 0.8279 0.0183 0 0 0 0 0 0 0 0 0.00 266 Si 0.6185 0.6721 0.4817 0 0 0 0 0 0 0 0 0.00 267 Si 0.8815 0.3279 0.9817 0 0 0 0 0 0 0 0 0.00 268 Si 0.3815 0.1721 0.5183 0 0 0 0 0 0 0 0 0.00 269 Si 0.8815 0.1721 0.9817 0 0 0 0 0 0 0 0 0.00 270 Si 0.3815 0.3279 0.5183 0 0 0 0 0 0 0 0 0.00 271 Si 0.1185 0.6721 0.0183 0 0 0 0 0 0 0 0 0.00 272 Si 0.6185 0.8279 0.4817 0 0 0 0 0 0 0 0 0.00 273 Si 0.0657 0.8794 0.8087 0 0 0 0 0 0 0 0 0.00 274 Si 0.5657 0.6206 0.6913 0 0 0 0 0 0 0 0 0.00 275 Si 0.9343 0.3794 0.1913 0 0 0 0 0 0 0 0 0.00 276 Si 0.4343 0.1206 0.3087 0 0 0 0 0 0 0 0 0.00 277 Si 0.9343 0.1206 0.1913 0 0 0 0 0 0 0 0 0.00 278 Si 0.4343 0.3794 0.3087 0 0 0 0 0 0 0 0 0.00 279 Si 0.0657 0.6206 0.8087 0 0 0 0 0 0 0 0 0.00 280 Si 0.5657 0.8794 0.6913 0 0 0 0 0 0 0 0 0.00 281 Si 0.1947 0.8288 0.7092 0 0 0 0 0 0 0 0 0.00 282 Si 0.6947 0.6712 0.7908 0 0 0 0 0 0 0 0 0.00 283 Si 0.8053 0.3288 0.2908 0 0 0 0 0 0 0 0 0.00 284 Si 0.3053 0.1712 0.2092 0 0 0 0 0 0 0 0 0.00 285 Si 0.8053 0.1712 0.2908 0 0 0 0 0 0 0 0 0.00 286 Si 0.3053 0.3288 0.2092 0 0 0 0 0 0 0 0 0.00 287 Si 0.1947 0.6712 0.7092 0 0 0 0 0 0 0 0 0.00 288 Si 0.6947 0.8288 0.7908 0 0 0 0 0 0 0 0 0.00

SAS.cssr: 14.349 14.349 10.398 90.0 90.0 90.0 SPGR = 1 P 1 OPT = 1 96 0 0 SAS : SAS 1 O 0.2418 0.0 0.0 0 0 0 0 0 0 0 0 0.00 2 O 0.0 0.2418 0.0 0 0 0 0 0 0 0 0 0.00 3 O 0.0 0.7582 0.0 0 0 0 0 0 0 0 0 0.00 4 O 0.7582 0.0 0.0 0 0 0 0 0 0 0 0 0.00 5 O 0.7418 0.5 0.5 0 0 0 0 0 0 0 0 0.00 6 O 0.5 0.7418 0.5 0 0 0 0 0 0 0 0 0.00 7 O 0.5 0.2582 0.5 0 0 0 0 0 0 0 0 0.00 8 O 0.2582 0.5 0.5 0 0 0 0 0 0 0 0 0.00 9 O 0.169 0.169 0.0 0 0 0 0 0 0 0 0 0.00 10 O 0.831 0.169 0.0 0 0 0 0 0 0 0 0 0.00 11 O 0.169 0.831 0.0 0 0 0 0 0 0 0 0 0.00 12 O 0.831 0.831 0.0 0 0 0 0 0 0 0 0 0.00 13 O 0.669 0.669 0.5 0 0 0 0 0 0 0 0 0.00 14 O 0.331 0.669 0.5 0 0 0 0 0 0 0 0 0.00 15 O 0.669 0.331 0.5 0 0 0 0 0 0 0 0 0.00 16 O 0.331 0.331 0.5 0 0 0 0 0 0 0 0 0.00 17 O 0.5 0.8745 0.7865 0 0 0 0 0 0 0 0 0.00 18 O 0.1255 0.5 0.7865 0 0 0 0 0 0 0 0 0.00 19 O 0.8745 0.5 0.7865 0 0 0 0 0 0 0 0 0.00 20 O 0.5 0.1255 0.2135 0 0 0 0 0 0 0 0 0.00 21 O 0.5 0.8745 0.2135 0 0 0 0 0 0 0 0 0.00 22 O 0.5 0.1255 0.7865 0 0 0 0 0 0 0 0 0.00 23 O 0.8745 0.5 0.2135 0 0 0 0 0 0 0 0 0.00 24 O 0.1255 0.5 0.2135 0 0 0 0 0 0 0 0 0.00 25 O 0.0 0.3745 0.2865 0 0 0 0 0 0 0 0 0.00 26 O 0.6255 0.0 0.2865 0 0 0 0 0 0 0 0 0.00 27 O 0.3745 0.0 0.2865 0 0 0 0 0 0 0 0 0.00 28 O 0.0 0.6255 0.7135 0 0 0 0 0 0 0 0 0.00 29 O 0.0 0.3745 0.7135 0 0 0 0 0 0 0 0 0.00 30 O 0.0 0.6255 0.2865 0 0 0 0 0 0 0 0 0.00 31 O 0.3745 0.0 0.7135 0 0 0 0 0 0 0 0 0.00 32 O 0.6255 0.0 0.7135 0 0 0 0 0 0 0 0 0.00 33 O 0.3261 0.8629 0.8723 0 0 0 0 0 0 0 0 0.00 34 O 0.1371 0.3261 0.8723 0 0 0 0 0 0 0 0 0.00 35 O 0.8629 0.6739 0.8723 0 0 0 0 0 0 0 0 0.00 36 O 0.3261 0.1371 0.1277 0 0 0 0 0 0 0 0 0.00 37 O 0.6739 0.8629 0.1277 0 0 0 0 0 0 0 0 0.00 38 O 0.6739 0.1371 0.8723 0 0 0 0 0 0 0 0 0.00 39 O 0.8629 0.3261 0.1277 0 0 0 0 0 0 0 0 0.00 40 O 0.1371 0.6739 0.1277 0 0 0 0 0 0 0 0 0.00 41 O 0.6739 0.1371 0.1277 0 0 0 0 0 0 0 0 0.00 42 O 0.8629 0.6739 0.1277 0 0 0 0 0 0 0 0 0.00 43 O 0.1371 0.3261 0.1277 0 0 0 0 0 0 0 0 0.00 44 O 0.6739 0.8629 0.8723 0 0 0 0 0 0 0 0 0.00 45 O 0.3261 0.1371 0.8723 0 0 0 0 0 0 0 0 0.00 46 O 0.3261 0.8629 0.1277 0 0 0 0 0 0 0 0 0.00 47 O 0.1371 0.6739 0.8723 0 0 0 0 0 0 0 0 0.00 48 O 0.8629 0.3261 0.8723 0 0 0 0 0 0 0 0 0.00 49 O 0.8261 0.3629 0.3723 0 0 0 0 0 0 0 0 0.00 50 O 0.6371 0.8261 0.3723 0 0 0 0 0 0 0 0 0.00 51 O 0.3629 0.1739 0.3723 0 0 0 0 0 0 0 0 0.00 52 O 0.8261 0.6371 0.6277 0 0 0 0 0 0 0 0 0.00 53 O 0.1739 0.3629 0.6277 0 0 0 0 0 0 0 0 0.00 54 O 0.1739 0.6371 0.3723 0 0 0 0 0 0 0 0 0.00 55 O 0.3629 0.8261 0.6277 0 0 0 0 0 0 0 0 0.00 56 O 0.6371 0.1739 0.6277 0 0 0 0 0 0 0 0 0.00 57 O 0.1739 0.6371 0.6277 0 0 0 0 0 0 0 0 0.00 58 O 0.3629 0.1739 0.6277 0 0 0 0 0 0 0 0 0.00 59 O 0.6371 0.8261 0.6277 0 0 0 0 0 0 0 0 0.00 60 O 0.1739 0.3629 0.3723 0 0 0 0 0 0 0 0 0.00 61 O 0.8261 0.6371 0.3723 0 0 0 0 0 0 0 0 0.00 62 O 0.8261 0.3629 0.6277 0 0 0 0 0 0 0 0 0.00 63 O 0.6371 0.1739 0.3723 0 0 0 0 0 0 0 0 0.00 64 O 0.3629 0.8261 0.3723 0 0 0 0 0 0 0 0 0.00 65 Si 0.2662 0.8893 0.0 0 0 0 0 0 0 0 0 0.00 66 Si 0.1107 0.2662 0.0 0 0 0 0 0 0 0 0 0.00 67 Si 0.8893 0.7338 0.0 0 0 0 0 0 0 0 0 0.00 68 Si 0.2662 0.1107 0.0 0 0 0 0 0 0 0 0 0.00 69 Si 0.7338 0.8893 0.0 0 0 0 0 0 0 0 0 0.00 70 Si 0.7338 0.1107 0.0 0 0 0 0 0 0 0 0 0.00 71 Si 0.8893 0.2662 0.0 0 0 0 0 0 0 0 0 0.00 72 Si 0.1107 0.7338 0.0 0 0 0 0 0 0 0 0 0.00 73 Si 0.7662 0.3893 0.5 0 0 0 0 0 0 0 0 0.00 74 Si 0.6107 0.7662 0.5 0 0 0 0 0 0 0 0 0.00 75 Si 0.3893 0.2338 0.5 0 0 0 0 0 0 0 0 0.00 76 Si 0.7662 0.6107 0.5 0 0 0 0 0 0 0 0 0.00 77 Si 0.2338 0.3893 0.5 0 0 0 0 0 0 0 0 0.00 78 Si 0.2338 0.6107 0.5 0 0 0 0 0 0 0 0 0.00 79 Si 0.3893 0.7662 0.5 0 0 0 0 0 0 0 0 0.00 80 Si 0.6107 0.2338 0.5 0 0 0 0 0 0 0 0 0.00 81 Si 0.609 0.891 0.75 0 0 0 0 0 0 0 0 0.00

67

82 Si 0.109 0.609 0.75 0 0 0 0 0 0 0 0 0.00 83 Si 0.891 0.391 0.75 0 0 0 0 0 0 0 0 0.00 84 Si 0.609 0.109 0.25 0 0 0 0 0 0 0 0 0.00 85 Si 0.391 0.891 0.25 0 0 0 0 0 0 0 0 0.00 86 Si 0.391 0.109 0.75 0 0 0 0 0 0 0 0 0.00 87 Si 0.891 0.609 0.25 0 0 0 0 0 0 0 0 0.00 88 Si 0.109 0.391 0.25 0 0 0 0 0 0 0 0 0.00 89 Si 0.391 0.109 0.25 0 0 0 0 0 0 0 0 0.00

90 Si 0.891 0.391 0.25 0 0 0 0 0 0 0 0 0.00 91 Si 0.109 0.609 0.25 0 0 0 0 0 0 0 0 0.00 92 Si 0.391 0.891 0.75 0 0 0 0 0 0 0 0 0.00 93 Si 0.609 0.109 0.75 0 0 0 0 0 0 0 0 0.00 94 Si 0.609 0.891 0.25 0 0 0 0 0 0 0 0 0.00 95 Si 0.109 0.391 0.75 0 0 0 0 0 0 0 0 0.00 96 Si 0.891 0.609 0.75 0 0 0 0 0 0 0 0 0.00

68

Appendix B: Supplementary information for “Chapter 4.

Large-scale Screening of Zeolite Structures for CO2

Membrane Spearations”

Contents: Illustration of a CH4/CO2 membrane separation process Detailed derivation of an ideal membrane system for binary separation application Relationship between CO2 Henry coefficient vs self diffusion coefficient for large database of zeolite materials

69

Illustration of a CH4/CO2 membrane separation process

Figure B.1. An illustration of a simplified membrane separation process with the feed (CH4/CO2) gas separated into CO2 and CH4 utilizing a membrane comprised of zeolite materials. We assume a feed of

50% CO2 and 50% CH4 at T = 300K. As the membrane has a higher CO2 permeability in general, the permeate will be CO2 rich and the retentate, CH4 rich. We assume that the process specification sets the purity of the permeate. As a consequence of the mass balances for a given recovery of methane (ratio of

methane in the retentate and feed), the area of a given membrane is set.

70

Detailed derivation of an ideal membrane system for binary separation

application.

A detailed derivation of ideal membrane system is introduced below for CO2/CH4 separation, and the process configuration is shown in Figure SI 2. In Figure SI 2, φF, φF, and φP represent the molar flow rate of the feed, output of the retentate side, and output of the permeate site, respectively. Xi,j (i = F, R, P and j= CO2, CH4) is the molar concentration on different sides of the membrane and for different species. PR is the total pressure on the retentate side, which is assumed to be constant and typically has the same value as the feed condition. This process aims for obtaining certain purity of CH4 as a product on the retentate side. (i.e., and are given as a designed variables). In the process, ideal mixing

condition and negligible pressure (vacuum) on the permeate side are assumed. For a given material, molecular simulation was used to obtain the permeation selectivity (α) and the permeability of CO2 (

).

Figure B.2: Schematic figure of an ideal membrane process for CO2/CH4 separation. The vacuum condition is assumed on the permeate side. The objective of the process is to obtain certain purity of CH4 from the

retentate side as a product.

The definition of the permeation selectivity of the ideal membrane system is shown in Eq. (B.1)

. (B.1)

jP

jF

(XP,CO

2

, XP,CH

4

)

(XR,CO

2

, XR,CH

4

)

Permeate

Retentate

Feed (XF,CO

2

, XF,CH

4

)

Vacuum

PR

jR

71

where and

are the permeation of CO2 and CH4, respectively. The permeation

of a given molecule here is defined as the self-diffusion coefficient multiplies the ratio of the concentration difference to the fugacity difference across the membrane. In addition, based on the assumption that the permeate site is under vacuum, the fugacity of each species at the permeate side as well as the corresponding adsorbed concentration is zero. According to the definition shown in Eq. (B.1), the

can be further determined by

Eq. (B.2).

. (B.2)

The overall mass balance of CO2 in the process is shown in Eq. (B.3). From Eq. (B.3) to Eq. (B.4), we further define and determine the molar flow ratio of the flow rate on the permeate side to the feed flow rate (i.e., defined as θ).

(B.3)

(B.4)

(B.5)

Also, the governing equation of the permeation of CO2 through the membrane is given by the Eq. (B.6). After simple re-arrangement, the area of the membrane, A, is expressed explicitly in the Eq. (B.7).

(B.6)

(B.7)

where L is the thickness of the membrane material. As a result, one can combine the Eq. (B.2), Eq. (B.5), and Eq. (B.7), for a given material as well as the given specification of CH4 purity, to obtain the area required for the separation. Area of the membrane in the work is assumed to be a measurement of the cost of the whole process, which is proportional to the area. In addition to the area of the membrane, it is also very important to minimize the loss of the CH4 from feed to the CH4 on the retentate side in the separation process. The loss of the CH4 (i.e., η) is defined and calculated by the Eq. (B.8).

(B.8)

72

Relationship between CO2 Henry coefficient vs self diffusion coefficient for

large database of zeolite materials

Figure B.3. CO2 DS as a function of KH for the predicted zeolite structures. Most of the zeolite structures

like within 5x10-6

< KH < 5x10-5

mol/kg/Pa and 10-9

< DS < 10-8

m2/sec.

10-6

10-5

10-4

10-3

10-11

10-10

10-9

10-8

10-7

Se

lf d

iffu

sio

n c

oe

ffic

ient

(m2/s

)

Henry coefficient (mol/(kg*Pa))

00.10.20.30.40.50.60.70.80.91

Relative

fraction

Predicted zeolite structures;

Pure component CO2 ; 300K

73

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