COMPUTATIONAL STUDIES OF METHANE ADSORPTION IN NANOPOROUS CARBON

Transcription

1 COMPUTATIONAL STUDIES OF METHANE ADSORPTION IN NANOPOROUS CARBON A Thesis presented to the Faculty of the Graduate School at the University of Missouri-Columbia In Partial Fulfillment of the Requirements for the Degree Masters of Science by LINDSEY ORTIZ Dr. Carlos Wexler, Thesis Supervisor DECEMBER, 212

2 Copyright by Lindsey Ortiz 212 All Rights Reserved

3 The undersigned, appointed by the dean of the Graduate School, have examined the thesis entitled presented by Lindsey Ortiz, COMPUTATIONAL STUDIES OF METHANE ADSORPTION IN NANOPOROUS CARBON a candidate for the degree of Masters of Science and hereby certify that, in their opinion, it is worthy of acceptance. Professor Carlos Wexler Professor Carsten Ullrich Professor Yuyi Lin

4 ACKNOWLEDGEMENTS I would like to thank Dr. Carlos Wexler for his support. Through his mentoring, I have grown as a scientist and learned much. I also would like to thank him for putting in the time to have frequent meetings with me and reading through my thesis. I would like to thank my thesis committee for taking the time to read my thesis. I would also like to thank Bogdan Kuchta for helping me set up the computer simulations and answering all my questions. I would also like to acknowledge all team members of ALL- CRAFT for their support. I would also like to thank the California Energy Commission, for their financial support of this project. ii

5 Table of Contents ACKNOWLEDGEMENTS... ii TABLE OF ABBREVIATIONS... v LIST OF FIGURES... vi LIST OF TABLES... ix Chapter 1: Introduction Alternative Energy and Natural Gas... 1 Chapter 2: Adsorption of methane into graphitic carbon pores Monte Carlo Simulation Procedure Monte Carlo Simulation Setup Interaction Potentials Simulation Results Density profiles Adsorption in Different Regions of the Pore Adsorption Isotherms Isosteric Heat of Adsorption Comparison of Computational and Experimental Results Pore size distributions Synthetic methane adsorption isotherms iii

6 2.5.3 Synthetic methane isosteric heat of adsorption.51 Chapter 3: Summary and Outlook.53 REFERENCES..55 iv

7 TABLE OF ABBREVIATIONS Abbreviation Full Name Abbreviation Full Name AC activated carbon H simulation pore size ANG adsorbed natural gas LNG liquefied natural gas CH 4 methane LJ 12-6 Lennard-Jones 12-6 potential CNG compressed natural gas MC Monte Carlo CO 2 carbon dioxide MD Molecular Dynamics D experimental pore size NG natural gas Δh Isosteric heat of NOx nitrogen oxide adsorption/enthalpy of adsorption ε ff fluid-fluid interaction PSD pore size distribution parameter ε ac adsorabte-carbon interaction parameter q st isosteric heat f fraction of number of pores S Steele potential F Fugacity SOx sulfur oxide GCMC Grand Canonical Monte Carlo T c critical temperature v

8 LIST OF FIGURES Figure Page 1. Schematic of the GCMC algorithm Computational cell dimensions Slit-shaped pore Methane-methane potential Methane molecule a distance z above graphitic sheets LJ potential of methane with a single graphitic wall Methane-graphitic potential for slit-shaped pores of various sizes H Methane-graphitic potential, small z region Methane density at pressures of 1, 5, and 12 bar Methane density at pressures of 1, 5, and 12 bar, pore center Methane density at pressure of 1 bar, individual pores Methane density at pressure of 5 bar, individual pores Methane density at pressure of 12 bar, individual pores Methane density for the 3 Å pore Three regions of adsorption Total Excess adsorption by region vi

9 17. Total amount stored by region A desorption cycle from 12 to 5 bar and 5 to 3 bar classified by region Experimental versus simulation pore size Absolute and excess adsorption, 7 and 9 Å pores Absolute and excess adsorption, 1 and 12 Å pores Absolute and excess adsorption, 15 and 2 Å pores Absolute and excess adsorption, 25 and 3 Å pores Absolute and excess adsorption, 4 and 5 Å pores Absolute gravimetric adsorption for all pores Absolute gravimetric adsorption for all pores, highlighting low pressure region Gravimetric excess for all pores Gravimetric excess for all pores, highlighting low pressure region Volumetric storage for all pores Volumetric storage for all pores, highlighting low pressure region Isosteric heat for all pores Isosteric heat for all pores, highlighting low pressure regions PSD for experimental samples 4K and 3K fit with Gaussians Simulation data fit to experimental samples 4K and 3K... 5 vii

10 35. Synthetic isosteric heats of adsorption for sample 4K A TEM micrograph of AC and a "realistic view of AC" Possible model of AC to run the multi-pore size simulation at once viii

11 LIST OF TABLES Table Page 1. Lennard-Jones 12-6 and Steele interaction parameters Potential and density results from analyzing potential of 3 Å pore at 12 bar The potential and enhancement at z = 15 Å due to each region Experimental fitting parameters for samples 4K and 3K ix

12 Chapter 1: Introduction 1.1 Alternative Energy and Natural Gas The development of alternative energies has received significant attention recently due to climate concerns and rising gas prices [1]. There are a variety of alternative energy options that are currently being researched. Of these options, natural gas (NG) is a promising candidate. NG is primarily composed of methane (CH 4 ); as much as 95 mol%, depending on the source [1]. The other components of NG are carbon dioxide (CO 2 ), nitrogen (N 2 ), and small amounts of other alkanes such as ethane (C 2 H 6 ), propane (C 3 H 8 ), and butane (C 4 H 1 ) [1, 2]. Methane, and thus NG, is an attractive candidate for alternative energy because it has the highest hydrogen to carbon ratio of any primary fuel, thus its combustion has the lowest CO 2 emissions per unit energy. Furthermore, its combustion results in no emissions of NOx, SOx, or particulates, therefore eliminating a major source of pollution [3]. Additionally, the cost of methane is lower than the cost of gasoline [4]. Storage and transportation of NG is achieved in three different basic forms: as liquefied natural gas (LNG), compressed natural gas (CNG), and adsorbed natural gas (ANG). Since the critical temperature of methane is Tc = 191 K, methane is supercritical at room temperature, which makes it difficult to utilize LNG or CNG due to the very high 1

13 pressures and/or low temperatures that would be required [1]. ANG is therefore a feasible method to store methane at moderate pressures by taking advantage of the strong attractive potentials between methane and numerous porous materials [5, 6, 7]. This also makes ANG significantly safer by operating closer to standard room temperatures and pressures [7]. The lower pressures allow a safer tank to operate with thinner walls that is less expensive and can be more easily integrated into a personal vehicle [7]. In ANG, methane can be stored as a gas by adsorbing it into a suitable porous material [5, 6, 7]. The adsorbent is a key aspect to successfully commercializing an ANG tank [8]. Adsorption of NG occurs through physisosorption due to the Van der Waals attraction between the adsorbate, the fluid being adsorbed, and the adsorbent, the material on which adsorption takes place [5]. Unlike chemisosoprtion, physisosorption involves no chemical bonding between the adsorbent and adsorbate [5], which allows reversible adsorption/desorption cycling of the system [6, 7]. In order to better understand adsorption of methane, a computational study was carried out on the adsorption of methane into graphitic slit-shaped pores. The goal was to characterize the adsorption at the molecular level, something not possible experimentally, so that it can be better understood, aiding the engineering of an optimized material to use in an ANG tank, particularly for personal vehicular use. 2

14 Chapter 2: Adsorption of methane into graphitic carbon pores 2.1 Monte Carlo Simulation Procedure Grand Canonical Monte Carlo (GCMC) simulations are the most widely used method to study adsorption of fluids on solid surfaces and into porous materials because GCMC efficiently explores the phase space consistent with experimental conditions (fixed volume, temperature and chemical potential) [9]. The GCMC algorithm used (see Figure 1) starts with the initial positions of the molecules in the system. This is used to calculate the initial configuration energy of the system. From this point on, the standard Metropolis Monte Carlo algorithm [1] is used to generate a Markov trajectory. Trials are generated by translating, removing, or adding molecules to the system. Whether or not the new configuration is accepted or rejected is determined by the Metropolis algorithm [1]. If going to the new configuration lowers the energy of the system then the trial is always accepted, and if the new configuration raises the energy the trial is accepted with probability [1] 3

15 GCMC algorithm INPUT Initial configuration (positions and orientations of N molecules) bin step cycle INITIAL MAIN Calculate initial energy of the system Prepare the neighbor list. Trial translational and/or orientational moves. Attempts to insert and/or remove a molecule Calculate the energy difference Accept or reject the new configuration according to the Boltzmann distribution RECORD Store energies and positions after each bin. OUTPUT UNIVERSITE de PROVENCE Final configuration (positions and orientations of N molecules) UNIVERSITE MONTPELLIER II Figure 1 Schematic of the GCMC algorithm used in our simulations (courtesy of B. Kuchta [11]). Acceptance of states that increases the energy of the system allows exploration of the system beyond local minima. This is repeated many times in order to reach equilibrium (monitored by the stabilization of the number of molecules and energy of the system), at which point the program starts storing the position of each molecule after each bin is completed. The number of steps and cycles is chosen so that the positions of the molecules between subsequent bins can be considered statistically independent, and a sufficiently large number of bins are computed for later statistical analysis. Once the 4

16 simulations are complete the positions of the molecules in each bin are used to compute relevant averages (and, if desired, fluctuations) of all desired observables. An important idea to MC is the idea of Importance Sampling. As Metropolis et al. state [1], the most naïve method to simulate a system is to randomly place the particles within the system and then calculate the potential energy using ( ) where V is the potential between particles i and j and d ij is the distance between them. The potential energy for each configuration would then be weighted by the Boltzmann factor exp(-e/kt). In almost all cases, and in particular for a close packed system, as is the case with adsorption, this method is impractical because there are a large number of configurations where the potential is very repulsive, hence exp(-e/kt) is extremely small. Besides the impracticality to probe a potentially enormous sample space, it would be very inefficient to move to these high E configurations, since they add a negligible amount to any thermodynamic average. Metropolis et al. first showed how to implement importance sampling [1]. Instead of randomly moving to a configuration and then weighting by exp(-e/kt), you move to a configuration with a probability of exp(-e/kt) and then weight each configuration evenly [1]. This implementation allows the simulation to move to configurations that contribute appreciably to the average, thereby increasing the efficiency of the simulation drastically [1]. This is the main idea behind the Metropolis 5

17 algorithm, and what makes it a reasonable procedure capable of reproducing relevant statistical averages with a reduced number of (highly probable) samples. 2.2 Monte Carlo Simulation Setup Grand Canonical Monte Carlo simulations were carried out for the adsorption of methane into a graphitic nanoporous adsorbent [5, 6] with slit shaped pores (see Figures 2 and 3). Pore size, denoted by H, is defined as the length in the z-direction from carbon center in one wall to the carbon center in the other wall. The pore walls extend 42.6 Å in the x direction and 49.2 Å in the y direction. Boundary effects are minimized with the use of periodic boundary conditions in x and y [1]. Figure 2 shows the computational cell dimensions. Figure 3 shows a depiction of the slit shaped pore used with methane molecules (larger circles) adsorbed in a pore with size H. The smaller circles are the carbon atoms that make up the graphitic sheets. 6

18 z H y 42.6 Å x 49.2 Å Figure 2 Computational cell dimensions. Periodic boundary conditions are used in the x and y directions. Adsorbed methane Carbon in graphitic sheet H Left pore wall Right pore wall Figure 3 A slit shaped pore used in our simulations. 7

19 Pore sizes H of 7, 9, 1, 12, 15, 2, 25, 3, 4, and 5 Å were used. The simulations were run at pressures P of 1, 2, 5, 1, 2, 3, 5, 7, 1, 12, 18, 24, and 36 bar. The runs were done at a temperature of 33 K, comparable to experiments done in Prof. P. Pfeifer s laboratory [7]. Given the relative internal stiffness of methane molecules, the United Atom (UA) model was used. In the UA model, a methane molecule is represented as one super atom with a mass of 16.4 amu. The methanemethane interaction potential was modeled with a Lennard-Jones 12-6 potential and the methane-graphite interaction was modeled with a Steele potential (refer to section 2.3 for further information). The initial number of methane molecules in the system was 6, and a cut-off range of 15 Å for all interactions. The step size for the translational moves was.1 Å, which results in approximately 5% acceptance on average across the all simulations. For all simulations 5, bins were used, except when computing the isosteric heats, where 1, bins were used for better statistics since the fluctuation theorem used to calculate the isosteric heat see Section requires a larger data set. Each bin consisted of 2 steps, each step consisted of 5 cycles, and each cycle consisted of 5 attempts to translate, insert, or delete a molecule. Overall, when 5, bins were used, 2.5 billion (2.5 x 1 9 ) attempts to translate, remove, or add a molecule was considered for each P and H combination. To put things in perspective, if the system contained 1, molecules, each molecule was subjected to more than 1 million changes. Each simulation consists of two runs. The first run is a short run to get the system closer to equilibrium. The second run is the longer, production run. All data comes from the production runs. 8

20 2.3 Interaction Potentials There are two interactions that need to be accounted for in adsorption: Van der Waal forces, which are always attractive, and repulsive forces. Van der Waals forces, also known as dispersion or London forces, are short range, attractive interactions that arise from the fluctuation in a molecule s charge cloud. They have a quantum mechanical origin and were described by London in 193 as having the following form [12] ( ) The repulsive forces, which originate from the overlap of electronic orbitals, can be modeled by [12] ( ) ( ) for simple molecules. The exponential piece, however, can be cumbersome to use and is often replaced with, such that the repulsive force becomes [12] ( ) ( ). There are a variety of potentials that are commonly used to model the interactions in adsorption. Two of the common models are the Lennard-Jones 12-6 (LJ 12-6) and the Steele potentials (S 1-4-3) [5, 12]. These potentials are not exact, 9

21 V (r) (K) but have been demonstrated to be valid for modeling the adsorption interactions [13, 14]. Other models exist to describe the dispersive interaction between fluids, but the LJ 12-6 is the preferred model for adsorption on a carbon surface, such as graphene, because it is consistent with using the S model to describe the fluid-solid interactions [13]. The S potential is the model of choice for slit shaped pores consisting of parallel graphitic slabs, as is the case with this study [14]. In our simulations, the methane-methane interaction was modeled with the LG 12-6 potential [5, 13]: ( ) [( ) ( ) ] where the well depth ɛ = 148 K and the collision diameter σ = 3.73 Å (see Table 1) [13, 15]. Figure 4 shows the methane-methane potential, modeled with the LG 12-6 potential. 1 r min -1 Figure 4 Methane-methane potential r (Å)

22 We now consider the interaction potential between a methane molecule and the substrate, which is graphitic sheets (see Figure 5). Figure 5 Methane molecule a distance z above graphitic sheets (not to scale). Assuming all interactions to be additive, ( ) [ ( ) ( ) ] where ε ac is the well depth of the potential between the adsorbent and an individual carbon atom in the grapheme sheet, and σ ac is the point where the potential between the adsorbent and an individual carbon atom in the graphene sheet is zero. The i index corresponds to individual carbon atoms on a graphene plane, and the sum over k 11

23 represents the different parallel graphene planes. Substituting the i sum by an integral, and performing the sum over k, the adsorbate-adsorbent potential can be approximated quite well by [16, 17]: ( ) [ ( ) ( ) ( ) ] where n c is the density of carbon atoms per unit volume and Δ is the spacing between the parallel sheets of graphene (Δn c is the density of carbon atoms per unit area in one graphene sheet). The first two terms arise from the first graphene plane and the third accounts for the other layers [14]. The factor.61 was set by Steele as an empirical adjustment to improve the accuracy of the model [14]. This S potential is commonly used in computer simulations of graphitic materials and was used in our simulations. The interaction parameters are given in Table 1. Potential model Interaction parameter Lennard-Jones 12-6 Steele ɛ (K) (ɛ ac for S 1-4-3) σ (Å) (σ ac for S 1-4-3) Δ (Å) n/a 3.35 n c (1/Å 3 ) n/a.144 Table 1 Lennard-Jones 12-6 and Steele interaction parameters [7, 8, 13, 15]. Figure 6 shows the potential between methane and a single graphitic wall, which is modeled with the S potential. The methane-single graphitic wall interaction has a potential well depth of approximately 15 K. 12

24 V (z) (K) z (Å) Figure 6 LJ potential of methane with a single graphitic wall (right). For slit-shaped pores, the total methane-pore potential arises from the superposition of the potentials of the left and right walls, i.e. V(z) + V(H - z). Figure 7 shows the potentials for all pore sizes in our simulations and Figure 8 focuses on the small-z region. The 7 Å pore has the deepest potential well, and as pore size increases, the potentials from the two graphitic walls begin to separate. This allows for larger bulk region, which consists of methane molecules far from the pore walls that are not adsorbed. For pore sizes 25 Å and larger, the potential for the two walls are completely separated. 13

25 V (z) (K) V (z) (K) z (Å) H = Figure 7 Methane-graphitic potential for slit-shaped pores of various sizes H z (Å) H = Figure 8 Methane-graphitic potential, small z region. Notice that for H > 12 Å the region near the adsorption minima becomes H-independent. 14

26 2.4 Simulation Results In this section we present the results of the computer simulations. In order to ensure that data points are uncorrelated, and to expedite the processing of the data, the particles positions are processed every 5 th bin for each P, H combination. This can be used to determine density profiles, isotherms, isosteric heats, and other quantities of interest. All of the raw data comes from simulations run at a temperature of 33 K Density profiles Density profiles were computed using histograms with bin widths Δz ranging from.35 to.2 Å, chosen small enough to provide good resolution, but wide enough for reasonable statistics (each bin in the histogram had on the order of 1, counts). The densities shown below are normalized to that of methane at the corresponding pressure and temperature obtained from the National Institute of Standards and Technology (NIST) Fluid Database [18]. 15

27 Figure 9 shows the density profile for a pressure of 1 (top), 5 (middle), and 12 (bottom) bar. The simulations show high densities in the adsorption region. At a pressure of 1 bar the 7 Å pore has an adsorption density that is 45 times the gas density. For medium to large pores, the two-wall pore has effectively decoupled into 2 single wall systems near the pore walls. This explains why the peaks for the medium to large pores have the same densities. The situation in the bulk region is more complex and will be analyzed below. With the chosen normalization, one would expect the density of the gas to be 1 in the bulk region, since these molecules are far enough from the pore walls that any potential from the walls is negligible. Figure 1 zooms in to the methane density in the bulk region. It is remarkable that even far from the surfaces the calculated densities are substantially larger than those of unadsorbed gas, especially at higher pressures. This is due to the attractive interaction that the first (and sometimes second) layer(s) exert on the methane molecules in the bulk. This can be verified by the following observations: (i) the reduction of ρ/ρ gas as the pore becomes wider and the interaction between the central region and the adsorbed layers becomes larger; (ii) the fact that ρ/ρ gas in the bulk only becomes significant at higher pressures where the first (and sometimes second) layer(s) become saturated; and (iii) that for larger pores the ρ/ρ gas > 1 only when a second layer of adsorbed gas is formed. 16

28 5 H = gas z (Å) 1 H = gas z (Å) 4 H = gas z (Å) Figure 9 Methane density at pressure of 1 (top), 5 (middle), and 12 (bottom) bar. 17

29 gas H = z (Å) gas H = z (Å) gas z (Å) H = Figure 1 Methane density at pressure of 1 (top), 5 (middle), and 12 (bottom) bar, highlighting the region with ρ ρ gas near the center of the pore (bulk region). 18

30 To provide more detail, Figures show the gas density of all pores individually at pressures of 1 bar (Figure 11), 5 bar (Figure 12), and 12 bar (Figure 13). gas Å 9 Å z (Å) gas Å z (Å) 12 Å Å Å gas Å z (Å) gass z (Å) 3 Å Å gas Å z (Å) Figure 11 Methane density at pressure of 1 bar, individual pores. 19

31 gas Å 9 Å z (Å) gas Å 12 Å z (Å) gas Å Å z (Å) gas Å Å z (Å) gas Å Å z (Å) Figure 12 Methane density at pressure of 5 bar, individual pores. 2

32 gas Å Å z (Å) gas Å 12 Å z (Å) gas Å Å z (Å) gas Å 3 Å z (Å) gas Å 5 5 Å z (Å) Figure 13 Methane density at pressure of 12 bar, individual pores. Figure 14 shows the methane density for the 3 Å pore for all pressures in the region z = 3 to 5 Å. As pressure increases, the number of molecules adsorbed increases as well. The adoption peaks for all pressures above 7 bar are approximately the same height, indicating saturation of this region. As can be seen from the isotherm for the 3 Å pore in Figure 23 in 21

33 Section 2.4.3, the 3 Å pore has not yet reached saturation at 7 bar, essentially all further adsorption occurs outside this primary peak then (see Section for more information). gas P = z (Å) Figure 14 Methane density for the 3 Å pore at all pressures in small z region Adsorption in Different Regions of the Pore As discussed earlier (Section 2.4.1), for larger pores and at high pressures we observe an enhancement in the bulk density, and attribute this enhancement to the attractive interactions with the large densities in the first (and perhaps second) adsorbed layer(s). For the purposes of classification, it is of interest to split up the adsorption into three regions: 22

34 i. Primary peaks ii. Secondary peaks iii. Bulk region The primary peak corresponds to the first adsorbed layer and the secondary peak corresponds to additional adsorbed layers (bilayer adsorption). The bulk region corresponds to methane farther from the walls, with bulk excess adsorption caused by interaction with the primary and secondary peaks. The excess bulk adsorption therefore only exists significantly at higher pressures. These regions are depicted in Figure 15, using the 4 Å pore size as an example. The vertical axis shows the density of the methane in the pore normalized by the density of methane gas at a pressure of 12 bar, taken from NIST: ρ gas = 1.37 kg/m 3 [18]. Using these criteria, the 7 Å pore has only a single primary peak at all pressures (Figures 11-13). The 9, 1, and 12 Å pores each have two primary peaks at all pressures (Figures 11-13). Pores 15 Å and larger have two primary peaks and depending on the pressure and pore size may have one or two secondary peaks and a bulk region (Figures 11-13). 23

35 15 Primary peak gas 1 5 Bulk region Secondary peak z (Å) Figure 15 Three regions of adsorption for a H = 4 Å pore at P = 12 bar. We posit that adsorption in large pores can be better understood by looking at how much each region contributes to the total adsorption. Figures 16 and 17 show the total and excess adsorption broken up by region. Breaking the adsorption up in this way allows us to know where in the pore methane predominantly adsorbs or desorbs at various ranges of pressures. As Figures 16 and 17 show, at a pressure of 1 bar the adsorption occurs solely in the primary peaks for all pores. Once the pressure increases to 3 bar, adsorption continues in the primary peaks, but also begins to take place in the secondary peaks. At 5 bar, adsorption becomes significant also in the bulk. 24

36 N exc g/g N exc g/g N exc g/g N exc g/g bar Primary Secondary Bulk Total Excess bar Primary Secondary Bulk Total Excess H (Å) H (Å).3 5 bar.5 Primary Secondary Bulk Total Excess 12 bar.2.1 Primary Secondary Bulk Total Excess H (Å) H (Å) Figure 16 Total Excess adsorption by region at pressure of 1 bar (top left), 3 bar (top right), 5 bar (bottom left), and 12 bar (bottom right). 25

37 N tot g/g N tot g/g N tot g/g N tot g/g Primary Secondary Bulk Total amount stored 1 bar.4.3 Primary Secondary Bulk Total amount stored 3 bar H (Å) H (Å).5.4 Primary Secondary 5 bar Bulk Total amount stored Primary Secondary Bulk Total amount stored 12 bar H (Å) H (Å) Figure 17 Total amount stored by region at pressure of 1 bar (top left), 3 bar (top right), 5 bar (bottom left), and 12 bar (bottom right). 26

38 The formation of the three regions can be understood qualitatively by analyzing the interaction potentials. A 3 Å pore at a pressure of 12 bar is used to illustrate the formation of these three regions. If one neglects interactions within each region, it is reasonable to expect an exponential enhancement of the density due to the presence of attractive potentials according to Boltzmann s rule: For our qualitative analysis we average over each region both in terms of density and potentials. Table 2 shows a comparison of the Boltzmann factors above with the average ρ/ρ gas from the GCMC simulations (Section 2.4.1) Region Potential from wall (K) Potential from PP (K) Potential from SP (K) Potential from Bulk (K) Total Potential (K) PP -891 n/a n/a n/a -891 SP -174 n/a n/a n/a -174 pore center Region ρ / ρ gas from total potential ρ / ρ gas from GCMC ρ from total potential (kg/m 3 ) ρ from GCMC (kg/m 3 ) PP SP pore center Table 2 Potential and density results from analyzing potential of 3 Å pore at 12 bar. PP = primary peak ( < z < 5.6 Å), SP = secondary peak (5.6 Å < z < 9 Å), pore center (z = 15 Å) which is in the bulk region (9 Å < z < 2.4 Å). 27

39 Region Potential (K) ρ/ρ gas ρ (kg/m 3 ) Wall PP SP Bulk Table 3 The potential and enhancement at z = 15 Å due to each region. For the primary peaks, ρ/ρ gas is significantly smaller than the prediction from the Boltzmann factor. This is easy to understand as the peak is close to saturation (i.e., there is substantial repulsive interaction terms not considered in the simple analysis above). In the case of the secondary peak, the Boltzmann factor coming from the wall alone is far from the GCMC results, addition of the attraction due to the primary improves the result but still underestimates the attraction present (interactions within the SP, interactions with the bulk). For the bulk region (here we focus on the central part of it farthest from any influence from the wall) the agreement of including the potential from the walls, primary and secondary peaks is reasonable, though it still misses some contributions due to attractive forces within the bulk region. In both secondary peaks and bulk, the inhibitor effect of the repulsive part of the interaction is weak, due to the relatively low fluid density there. Figure 18 show desorption cycles from a pressure of 12 to 5 bar (top) and from a pressure of 5 to 3 bar (bottom). The y axis shows the percent of the total adsorption that occurs in each region for all pore sizes at 33 K. Desorption over high pressures ranges primarily occurs from the secondary peaks and bulk region for pores 28

40 15 Å and larger. Pore sizes 3 Å and larger primarily desorb from the bulk region when the desorption occurs over high pressure ranges. As the pressure range is lowered, more of the desorption begins to occur from the primary peaks. For pore sizes 2 Å and larger, desorption is split approximately evenly between the primary peaks and the total desorbed from both the secondary peaks and bulk region. The spike in desorption from the secondary peaks for the 5 Å pore is likely due to the fact that the three regions do not always have distinct boundaries. As the 7 and 9 Å pore do not have secondary peaks or a bulk region, desorption always occurs from the primary peaks. 29

41 Desorption by region (%) Desorption by region (%) Primary Secondary Bulk Total H (Å) Primary Secondary Bulk Total H (Å) Figure 18 A desorption cycle from 12 to 5 bar (top) and 5 to 3 bar (bottom) classified by region Adsorption Isotherms Adsorption isotherms are a key aspect of understanding adsorption as they are one of the basic tools used by experimental surface scientists to characterize the 3

42 adsorption process. Isotherms are normally presented as either coverage (relative to a monolayer) or amount adsorbed (e.g., mass of adsorbate per mass of adsorbent) versus pressure at a constant temperature. By analyzing adsorption isotherms, one can determine important information such as how many layers are adsorbed and when saturation is reached. Adsorption isotherms plotted as amount adsorbed versus pressure for all pores at 33 K are presented in Figures Each graph shows the absolute and excess adsorption normalized as mass of methane adsorbed divided by mass of carbon. The absolute adsorption isotherms are determined by first calculating N abs, which is the average number of molecules per bin from the simulation results. This is then converted to grams of CH 4 per grams of C using [ ] where = 16.4 g is the mass of a methane molecule in grams and N A is Avogadro s number, and m c is mass of the carbon in g. Excess adsorption isotherms are calculated by ( ) where H is the simulation pore size, A is the area of the graphitic sheet, and ρ is density of methane gas taken from NIST [18]. As previously stated, the simulation pore size H is defined as the distance from carbon center in one pore wall to carbon center in the other pore wall. The experimental pore size is defined differently as the distance from the edge of carbon atom in one wall to the edge of carbon atom in the other wall. The 31

43 m CH 4 / m C m CH 4 / m C quantity H 2t converts simulation pore size H to experimental pore size D by subtracting off the diameter of the carbon atom, which is denoted by 2t = 3.8 Å. Figure 19 shows the comparison of D and H, as well as the distance t. Carbon in graphitic sheet t D H Figure 19 Experimental versus simulation pore size: D and H, respectively Absolute Excess.1.5 Absolute Excess P (bar) P (bar) Figure 2 Absolute and excess adsorption results of the GCMC simulations for 7 Å (left) and 9 Å (right) pores. 32

44 m CH 4 / m C m CH 4 / m C m CH 4 / m C m CH 4 / m C Absolute Excess.1 Absolute Excess P (bar) P (bar) Figure 21 Absolute and excess adsorption results of the GCMC simulations for 1 Å (left) and 12 Å (right) pores Absolute Excess Absolute Excess P (bar) P (bar) Figure 22 Absolute and excess adsorption results of the GCMC simulations for 15 Å (left) and 2 Å (right) pores. 33

45 m CH 4 / m C m CH 4 / m C m CH 4 / m C m CH 4 / m C Absolute Excess.2 Absolute Excess P (bar) P (bar) Figure 23 Absolute and excess adsorption results of the GCMC simulations for 25 Å (left) and 3 Å (right) pores P (bar) Absolute Excess.5 Absolute Excess P (bar) Figure 24 Absolute and excess adsorption results of the GCMC simulations for 4 Å (left) and 5 Å (right) pores. As can be seen by the graphs for the 7 and 9 Å pores, the smaller pores have a larger initial rate of adsorption than the bigger pores due to the smaller pores very deep potential well. The small pores quickly reach saturation, however. These trends are well supported by the potential graphs. The number of molecules adsorbed in a monolayer is the same for smaller and larger pores. However, pore sizes 1 Å and larger can have multilayer adsorption as well as intake of molecules into the bulk region. The 34

46 Absolute gravimetric adsorption (m CH 4 /m C) adsorption isotherms for the larger pores consist of adsorption onto the graphitic walls as well as intake of molecules into the bulk. The adsorption isotherms for the larger pores are not yet saturated these pores can continue to take in molecules into the bulk region, even after the first adsorbed layers are full. The isotherms for the absolute gravimetric adsorption and the excess gravimetric adsorption for all pores at 33 K are presented in Figures H = P (bar) Figure 25 Absolute gravimetric adsorption for all pores. Looking at the total gravimetric adsorption isotherm in the region P = -- 4 bar shows interesting results (see Figure 26). In the region -- 1 bar, the average number of molecules adsorbed is greater for the smaller pores. Specifically, from most molecules adsorbed to least molecules adsorbed the trend by pore size is 7, 9, 1, and then 12 Å. Pores 15 Å and greater have the same number of molecules adsorbed. This trend is due to the fact that the smaller pores fill up quicker due to the deep potential 35

47 Absolute gravimetric adsorption (m CH 4 /m C) well depth, as previously discussed. For large pores, the potential from each pore wall is completely disassociated and there is no overlap. Therefore these pores will have the same number of molecules adsorbed because they have the same well depth. Additionally, the number of molecules that can be taken into bulk region increases as pore size increases. This is seen clearly by the fact that as pressure increases the adsorption isotherms for the larger pores separate. This separation occurs around 2 bar, except for the 4 Å pore which separates around 1 bar. The slope of each isotherm is related to the amount of particles in the bulk region. The number of molecules is the same for the 7 and 9 Å pores above 3 bar because both have saturated H = P (bar) Figure 26 Absolute gravimetric adsorption for all pores, highlighting low pressure region. 36

48 Gravimetric Excess (m CH 4 /m C) Gravimetric Excess (m CH 4 /m C).45 H = Figure 27 Gravimetric excess for all pores P (bar) H = P (bar) Figure 28 Gravimetric excess for all pores, highlighting low pressure region. 37

49 Volumetric Storage gch 4 /l Figure 29 shows the volumetric storage for all pores. Figure 3 shows the volumetric storage, zoomed in on the pressure region from P = to 4 bar. The volumetric storage is calculated using [ ] where v c is the volume of our simulation cell P (bar) H = Figure 29 Volumetric storage for all pores. 38

50 Volumetric Storage gch 4 /l P (bar) H = Figure 3 Volumetric storage for all pores, highlighting low pressure region Isosteric Heat of Adsorption The isosteric heat of adsorption, abbreviated as q st, is the heat released when one molecule is adsorbed [19]. The heat is calculated from the GCMC simulations using fluctuation theorem given by the following equation [19] [ ] As usual, < > are ensemble averages. N is the total number of particles and E is the total energy of the system. Simulation data results in absolute isosteric heats may be directly compared to experimental results using (incorrectly) the excess adsorption using [2] 39

51 ( ) ( ) Where q st, excess is the excess isosteric heat, q st, absolute is the absolute isosteric heat, P is pressure, T is temperature, N b is the number of moles of gas that would be present in the pore without adsorption at the bulk density, and N exc / P is the slope of the gravimetric excess isotherm. Isosteric heats are an important thermodynamic quantity to characterize the adsorption. For example, one can learn how heterogeneous the adsorptive-adsorbate interaction is by examining the isosteric heats as well as further understanding when a sample saturates [2]. Figure 31 shows (absolute) isosteric heats for all pores sizes at 33K. 4

52 q st (kj/mol) q st (kj/mol) q st (kj/mol) H = P (bar) Figure 31 Isosteric heat for all pores H = H = P (bar) P (bar) Figure 32 Isosteric heat for all pores, highlighting - 12 bar region (left) and - 2 bar region (right). 41

53 As expected from the reduced adsorption potentials, larger pore sizes have smaller isosteric heats. The isosteric heat comes from two interactions: interactions with the pore wall and interactions with other methane molecules [2]. As long as the molecules are close to the wall, the interaction with the wall is approximately constant as adsorption increases [2]. At low coverage, the isosteric heat is mainly due to the interaction of the methane molecules with the pore wall [21]. The interaction with other methane molecules increases as adsorption increases, however, because the adsorption layer becomes denser and there are more molecules to interact with [2]. Both of these interactions will be larger for smaller pores than larger pores, because the adsorbed molecules are closer to the pore walls and each other. As expected, therefore, the isosteric heat is largest for the 7 Å pore size and decreases in size as pore size increases. There is a larger difference in the isosteric heat between the smaller pores than there is between the larger pores. This is due to the fact that for the larger pores, the system has decoupled into 2 single pore walls except for in the bulk. Increasing from 3 to 5 Å, for example, does not change the strength of the interaction between the majority of the adsorbed methane molecules and the pore wall. That is why the isosteric heats for pore sizes of 2, 25, 3, 4 and 5 Å are much closer than the isosteric heats for pore sizes smaller than 2 Å. In fact, the isosteric heat is virtually independent of pore size for these larger pores, which was also shown by He and Seaton [2]. As adsorption increases, however, the 5 Å pore does not have as high a density of molecules as the 4, 3, 25, and 2 Å pores because of its larger size. Therefore, these larger pores will not have the exact same isosteric heat. The 5 Å pore will have the 42

54 smallest isosteric heat, the 4 Å pore will have the second smallest isosteric heat, the 3 Å pore will have the third smallest isosteric heat, and etc. Notice that in experimental samples the isosteric heats almost always decrease with pressure. This is due to heterogeneity of the samples (samples containing both large and small pores), not considered so far (see Section 2.5.3). 2.5 Comparison of Computational and Experimental Results After completing the numerical simulations of methane adsorption at many different pressures in slit-shaped pores of various pore sizes, it is important to review whether these simulations bear resemblance to experimental evidence such as adsorption isotherms and isosteric heats of adsorption of methane in high-performance nanoporous carbon from ALL-CRAFT [7, 22, 23] Pore size distributions A first important step is to consider the actual structure of the pores in activated carbon (AC). Generally, AC s are comprised of a variety of pore sizes. Subcritical nitrogen adsorption is used to determine pore size distributions (PSD s) that indicate how much pore volume can be understood as being formed by effective slit-shaped pores with a given size H [24, 25]. Figure 33 shows experimental PSDs for two ALL-CRAFT activated 43

55 carbons: samples 3K and 4K, where the number indicates the ratio KOH:C used during activation [23]. The more aggressive activation due to a higher concentration of KOH in sample 4K results in a more open pore structure (larger H s). To fit the results of our computer simulations of methane adsorption to experiments we need to determine the correct mix of different pore sizes that correspond to the samples. We first determine the pores to employ and their proportions by fitting the PSD s with a sum of Gaussian peaks of the form: ( ) ( ) where a is a weighting factor, H is the variable pore width, H is the characteristic pore, and the width of the Gaussian is σ H. Figure 33 also shows the PSD fits for each of the samples. The results of the fits are presented in Table 4. 44

56 dv / dh (cm 3 Å -1 g -1 ) dv / dh (cm 3 Å -1 g -1 ) K Sum H = H (Å) K Sum H = H (Å) Figure 33 Pore size distribution for experimental samples 4K (top) and 3K (bottom) and fit with Gaussians, see Table 4. 45

57 Sample 4K H (Å) σ H (Å) a (cm 3 Å -1 g -1 ) Pore volume (cm 3 /g) Volume Fraction Pore Number Fraction Sample 3K H (Å) σ H (Å) a (cm 3 Å -1 g -1 ) Pore volume (cm 3 /g) Volume Fraction Pore Number Fraction Table 4 Experimental fitting parameters for samples 4K (top) and 3K (bottom). The pore volume for a given pore size H, denoted by v H, is determined by integrating the Gaussian for each pore size. The volume fractions for a pore of size H, denoted by Z H, are calculated for each pore by dividing the pore volume v H by the total volume, which is 1.96 cm 3 /g for 4K and 1.49 cm 3 /g for 3K. The fraction of number of pores f is finally calculated for each pore size H by the following equation 46

58 Since only pore sizes of 7, 15, 2, and 3 Å are used to construct the fit, the general equation above becomes ( ) The values of v H, Z H, and f H are show in in Table Synthetic methane adsorption isotherms Once the pore number fraction has been determined from the nitrogen PSD s, a synthetic sample, formed by a linear combination of the different pores in the PSD can be calculated. For example, gravimetric adsorption isotherms can be constructed from: where the f i s are as determined above (Table 4), and the n i,ads are the adsorption isotherms determined from GCMC for each individual pore. Since only pore sizes of 7, 15, 2, and 3 Å are used to construct the fit, the general equation above becomes 47

59 When using the above equation, it is important to recall that the result of simulation is the absolute adsorption, whereas experiments determine excess adsorption. Excess adsorption is then converted to total adsorption and total amount stored using [26]: ( ) ( ). Here the porosity, Φ, is defined as the volume fraction that is occupied by open pores. The porosities for experimental samples 3K and 4K are.78 g/cm 3 and.81 g/cm 3, respectively [26]. The skeletal density, ρ skeletal, is assumed in experiments to be 2. g/cm 3 [26], and we take the same value for direct comparison. The apparent density, ρ apparent, is defined as the density of the sample when the open pore volumes and skeletal volume of the sample is taken into account [26]. The density of the gas at the same pressure, ρ gas, is taken from NIST s Thermophysical Properties of Fluid Systems database [18]. The mass of the sample is m sample, the total volume is v tot, the mass adsorbed in total adsorption is m stored, and the mass adsorbed in excess adsorption is m excess. If needed, volumetric storage can be calculated using 48

60 where the factors n 1,ads /v 1 and n 2,ads /v 2 are the volumetric amount adsorbed by each pore size per unit volume. This equation accounts for the fact that a 5-5 ratio number wise does not translate to a 5-5 ratio volume wise. Figure 34 shows the gravimetric total and excess fitting of simulation data to experimental samples 4K (top) and 3K (bottom). It shows reasonable agreement of the excess adsorption isotherms and of the total amount stored, especially for sample 4K. The fact that simulated curves for 3K come systematically above the experimental ones may be caused by the presence of pores with walls that are not a single sheet of grapheme, as assumed in the simulations. 49

61 Gravimetric total and excess (gch 4 /kgc) Gravimetric total and excess (gch 4 /kgc) K Total 4K Excess Fit Absolute Fit Total Fit Excess P (bar) P (bar) 3K Total 3K Excess Fit Absolute Fit Total Fit Excess Figure 34 Simulation data fit to experimental samples 4K (top) and 3K (bottom). 5

62 2.5.3 Synthetic methane isosteric heat of adsorption The enthalpy of adsorption, also called the isosteric enthalpy of adsorption and denoted by Δh, is used to futher understand the adsorption process. For ideal adosrption, where there are no defects or heterogenity, the isosteric heat is independent of loading [21]. As discussed above, adsorbents such as activated carbon, however, are not ideal and contain many energetic heterogenities, mostly due to the various pore sizes present. In the same way we used the Nitrogen PSD s to characterize which pore sizes are present and produce synthetic adsorption isotherms from GCMC results for various pore sizes, the enthalpy of adsorption for a realistic sample can be synthesized equivalently where Δh i is the enthalpy of adsorption for a particular type of pore. Figure 35 shows the synthetic isosteric heats for sample 4K. The rapid decrease of Δh with P or coverage can be understood from the fact that the narrowest pores (with stronger binding of methane) are filled first and quickly saturate, thus they dominate the low coverage region, whereas the wider pores with their lower binding take longer to fill but become more important at higher coverages. The general characteristics of these curves are typical of heterogeneous materials and very similar to those reported by Himeno, 51

63 h (kj/mol) h (kj/mol) Komastu, and Fujita for the adsorption of methane into the activated carbon Norit R1 Extra [21] P (bar) g CH 4 / g C Figure 35 Synthetic isosteric heats of adsorption as a function of pressure (left) and Δh as a function of the amount of methane stored gch 4 /gc (right) for sample 4K. 52

64 Chapter 3: Summary and Outlook In this thesis we have completed computational studies on the adsorption of methane into nanoporous carbon. We identified multi-layer adsorption at supercritical temperatures with excess amount even at large distances from the pore walls. We also determined that results could be used successfully to model methane adsorption from PSD s coming from N 2. This works for both the adsorption isotherms and isosteric heats. A future direction would be to analyze lower temperature adsorption. Simulations at 195 K, the temperature of dry ice, would be of interest since dry ice is deemed of possible importance for storage. Another future direction is to study more varied pore geometries. In this thesis, we have only studied slit shaped pores. As can be seen in Figure 36, AC contains more varied pore geometries. Analysis of more varied pore geometries would offer a greater understanding of adsorption in AC and is therefore of interest. Figure 37 shows a possible model that may be used to run simulations on multiple pore sizes at the same time. 53

65 HRTEM on activated carbon (Norit GSX) a-as received b- Heated Figure 36 A TEM micrograph (left) showing what AC actually looks like and a "realistic view of AC" (right), courtesy of J. Romanos [26]. Figure 37 Possible model of AC to run the multi-pore size simulation at once. 54