Nanoporous Carbons: Porous Characterization and Electrical Performance in Electrochemical Double Layer Capacitors

Transcription

1 Nanoporous Carbons: Porous Characterization and Electrical Performance in Electrochemical Double Layer Capacitors by Johnathon N. Caguiat A thesis submitted in conformity with the requirements for the degree of Masters of Applied Science Department of Chemical Engineering and Applied Chemistry University of Toronto Copyright by Johnathon N. Caguiat 2013

2 Nanoporous Carbons: Porous Characterization and Electrical Performance in Electrochemical Double Layer Capacitors Abstract Johnathon N. Caguiat Masters of Applied Science Department of Chemical Engineering and Applied Chemistry University of Toronto 2013 Nanoporous carbons have become a material of interest in many applications such as electrochemical double layer capacitors (supercapacitors). Supercapacitors are being studied for their potential in storing electrical energy storage from intermittent sources and in use as power sources that can be charged rapidly. However, a lack of understanding of the charge storage mechanism within a supercapacitor makes it difficult to optimize them. Two components of this challenge are the difficulties in experimentally characterizing the sub-nanoporous structure of carbon electrode materials and the electrical performance of the supercapacitors. This work provides a means to accurately characterize the porous structure of sub-nanoporus carbon materials and identifies the current limitations in characterizing the electrical performance of a supercapacitor cell. Future work may focus on the relationship between the sub-nano porous structure of the carbon electrode and the capacitance of supercapacitors, and on the elucidation of charge storage mechanisms. ii

3 Acknowledgments I would first like to thank Professor Charles Jia and Professor Donald Kirk, for giving me the opportunity to work on this project and for their patience in guiding me. I would like to thank Dr. Eric Morris for his all his efforts in keeping the lab running smoothly during the first year of my M.A.Sc. in addition to the sound advice and grounded perspectives he provided my moments of confusion or frustration. As for my fellow desk-mates, Bithun Sarkar and Jocelyn Zuliani, I appreciate the serious and fun conversations that lead to new insights and understandings in this thesis. I would also like to thank the University of Toronto, Chemical Engineering Department and The ILead / LoT group for the diverse community that exists in and is available to Graduate students. With regard LoT:G, the environment and people provided me the opportunity to develop my more complex forms of articulation by recognizing that I was still learning rather than assuming there existed some inherent ability to articulate all my thoughts. Lastly I would like to thank Catherine Lee for her support and patience throughout the time she s been with me. Through all my ups and downs she has only ever been supportive. With an open mind and inquisitive nature, she helped remind me of the basic knowledge which I take for granted on a daily basis that previously hindered my ability to provide context to any argument. iii

4 Table of Contents Acknowledgments... iii Table of Contents... iv List of Tables... viii List of Figures... x Nomenclature Alphabetical Order... xii Chapter Characterization of the sub-nanoporous structure in porous carbonaceous materials Introduction Experimental Measurement of Porous Properties Using Gas Adsorption Technique (Physisorption) Carbonaceous Materials Analyzed Material Preparation and Data Collection Reproducibility: Random Error Adsorption Models Brunauer-Emmett-Teller Model Dubinin-Radushkevich and Dubinin-Astakhov Models Capillary Condensation Models Horvath-Kawazoe and Saito-Foley Models Density Functional Theory Adsorption Models Models for Analysis Results and Discussion Exploring the Limitations of the Traditional Methods Considerations in the Use of Density Functional Theory Based Models iv

5 1.4.3 The Recommended Method Conclusions Chapter Characterizing the Electrical Performance of Activated Carbon in an Electrochemical Double Layer Capacitor (Supercapacitor) Introduction Measurement of Capacitance Basic Equations of Capacitance Interpretation of Total Capacitance Experimental Carbon Pore Characterization Conditions Cell Construction Capacitance Measurement Considerations Normalizing Capacitance Results and Discussion Carbon Pore Structure Challenges Current Sweep Results Testing the Result of the Current Sweep Reversing Polarity Apparatus Conclusions Chapter Future Work and Recommendations Future Work Recommendations v

6 4 5 References Appendices A IUPAC Isotherm Classes A.1 Adsorption Types A.2 Hysteresis Types B Reproducibility of the kernel data C DA PSD Fit D Comparison of BJH Values with QSDFT Slit/Cylindrical Kernel for nm E HK and SF PSD Fit F Detailed Dough Making Procedure F.1 Washing Dough of Soluble Contaminants F.2 Preparing Dough G Operation of the Autosorb G.1 Operation add Setup section for N 2 and CO G.1.1 Nitrogen G Software v G Software v G.1.2 Carbon Dioxide G Software v G Software v G.2 Maintenance and Operation G.2.1 Correcting Errors G Leaks and Seemingly Ineffective Outgassing G System Calibration G.2.2 Equipment selection: G.2.3 Choosing Sample Quantity vi

7 G.2.4 Outgassing conditions G.2.5 Data Point Settings vii

8 List of Tables Table 1: Carbon content of experimental materials... 3 Table 2: Summary of Brunauer-Emmett-Teller method to calculate specific surface areas... 7 Table 3: Uncertainties of Isotherms and BET-SSA Table 4: Uncertainties for DR "micropore" volume and average pore size... 8 Table 5: Summary of Adsorption Models. * - By definition of DR. ** - By extension of BJH. *** By the limitation of the kernels provided in the ASQiWin Software Table 6: BET parameters of the PM, CSC and APC Table 7: Tabulated DR derived micropore volumes Table 8: Comparison of surface texture (smooth-nldft vs. corrugated-qsdft) with a fixed pore shape (slit) Table 9: Comparison of average fitting error between surface textures (smooth-nldft vs. corrugated-qsdft) of slit-like pore Table 10: Comparison of the relative contribution of micropores (<2nm) to SSA and SPV between surface textures (smooth-nldft vs. corrugated-qsdft) for a slit-like pore Table 11: Comparison between different pore shapes (slit, cylindrical & hybrid) for a corrugated surface (QSDFT) Table 12: Comparison of the average fitting error between different pore shapes (slit, cylindrical & hybrid) for a corrugated surface (QSDFT) Table 13: Comparison of the relative contributions of micropores (<2nm) to SSA and SPV between different pore shapes (slit, cylindrical & hybrid) with corrugated surfaces Table 14: Comparison of the cumulative SSA and SPV between CO 2 and N 2 for raw petroleum coke viii

9 Table 15: Comparison of the SSA and SPV of sub-nanopore between CO 2 and N Table 16: Comparison of pores less than 1nm using different adsorbates with different surface textures Table 17: Comparison of the CO 2 vs. N 2 to quantify SSA and SPV below 1 nm Table 18: Compilation of the SSA and SPV of the recommended method that incorporates a division at 2nm (Definition of a micropore; Section 1.1) Table 19: The tabulated values resulting from the calculation of capacitance using the upper in comparison with that of lower half of a discharge CC plot Table 20: Initial capacitance test results Table 21: Capacitance results after reducing ESR Table 22: Results of varying total current to characterize the capacitance all three cells Table 23: Results after adding an additional fast CV to the original tests Table 24: Results after doing consecutive Fast CV + 5 Test Cycles Table 25: Total points in the discharge of capacitor cell and step size used to calculate the slopes in Figure ix

10 List of Figures Figure 1: Isotherms of the four carbon materials used in this study... 6 Figure 2 - ABCDEF: BET pots for all samples. Parts A, B and C represent show the linear middle section of a BET fit and a potential upper linear region. Plots D, E and F show a lower linear region in addition to the fitted middle region Figure 3: Plot of v*(1-p/po) to find the applicable BET upper relative pressure limit Figure 4: Fitted DR plots for each carbon Figure 5-ABC: Experimental isotherm in comparison to the modelled isotherm of smooth vs. corrugated textures for PM (A), CSC (B) and APC (C) Figure 6 -ABC: Comparison of the PSDs using a smooth (NLDFT) and corrugated (QSDFT) texture generated for PM (A), CSC (B) and APC (C) Figure 7-ABCDEF: Comparison of experimental isotherms with modeled isotherms of different pore shapes: slit (A & D), cylindrical (B & E), slit/cylindrical (C & F). The segments shown in D, E and F are marked with a box Figure 8-ABC: Comparison of the PSDs provided by different pore shapes for PM (A), CSC (B) and APC (C) Figure 9-AB: N 2 based experimental and modeled isotherms (A) and the CO 2 experimental and modeled isotherms for RPC (B) Figure 10-ABC: Comparison of PSDs using different adsorbates for PM (A), CSC (B) and APC (C) Figure 11: Summary of the combined CO 2 and N 2 PSDs for RPC, PM, CSC and APC Figure 12: Schematic of a supercapacitor cell Figure 13: Differential PSD of activated coconut shell carbon with respect to pore width using the recommended model from Chapter x

11 Figure 14: CC discharge curve at 1mA for Cell B Figure 15: Plot of calculated capacitance by starting at either end of a CC curve to illustrate the large variance in measured capacitance depending on the points used Figure 16: Graphical representation of current sweep results Figure 17: A CC plot of the data for Cell C at 50mA Figure 18: Comparison of the effects of preconditioning (specifically the fast CV) on measured capacitance Figure 19: Capacitance Measurements when doing 12 consecutive fast CC measurements Figure 20: The change in slope during the CC discharge of a cell at various total currents Figure 21: Variation of capacitance with number of polarity switches at 50mA Figure 22: Variation of capacitance with number of polarity switches at 1mA Figure 23: Select intermediate cycles of the consecutive charging and discharging cycles of when reversing the polarity of a cell to illustrate the change in capacitance with cycle xi

12 Nomenclature Alphabetical Order Negative of the free adsorption energy (kj/mol) for the DR equation; The surface area available to store electric charge within a capacitor (m 2 ) An intermolecular constant from the Kirkwood-Muller equation (N m/ molecule) The surface area of pore (m 2 ) Intermolecular constant (N m/ molecule) Condensation constant (unitless) The ratio of the average change in radius of from the pore size to that of pore (unitless) The capacitance of typical 2 plate capacitor (F or F/g), The capacitance of opposing films/electrode 1 and 2 within a supercapacitor (F or F/g) The capacitance of a single electrode (F or F/g) The sum of the adsorbate and adsorbent radii (nm) for the HK or SF Method; the distance between opposing charges in a capacitor (m) Generalized variable replacing (kj/mol) from the DR equation; The total energy stored within a capacitor device (J) Characteristic Energy of Adsorption (kj/mol) for DR; ( ) Ideal component of the weighted hard sphere system described by Tarazona ( ) Pore volume as a function of R ( ) Pore size distribution as a function of pore size [{ ( )}] Intrinsic Helmholtz free energy [{ ( )}] Ideal component of the Helmholtz free energy [{ ( )}] Attractive component of the Helmholtz free energy [{ ( )}] Repulsive component of the Helmholtz free energy [ ( )] A differential component of the [ { ( )}] Free energy of adsorption; by derivation the same as, but used differently than with the DR method (kj/mol) Pore size (nm) Either the smallest pore size in a sample or smallest pore size quantifiable and the largest pore size present, respectively (nm) An index of pore size for capillary models; either represents the fluid, solid, fluid-fluid interaction, solid-fluid interaction or solid interaction respectively with regard to the fluid density or interactions energies for DFT; the current used to charge or discharge a capacitor (A) An index ranging from 1 to Empirical proportional constant (kj nm/mol) for the DR equation; Planck s Constant for DFT models Avogadro s number (atoms/mol) Distance between two opposing walls bounding a slit-shaped pore (m) Pore Index ( ) Variable for the mathematical representation of an isotherm(i.e. volume xii

13 adsorbed at some pressure) The number atoms per unit surface area (atoms/m 2 ) Number of molecules per unit area of adsorbate (molecules/m 2 ) for the HK or SF method; Total System Pressure (units of pressure); the power output of an electrical storage device (W) Adsorbate Vapor Pressure (units of pressure) The free energy that represents the adsorbate-adsorbent-adsorbate interactions ( ) The weighted hard sphere system pressure as described by Tarazona s work After integration used as a summation index or dummy variable for integration for CI equation; the centre to centre distance of a gas atom from the surface (nm) for HK or SF methods; Position relative to the pore wall (nm) for DFT models, The lower and upper limits used to integrate in to represent some average pore range (nm) Maximum pore size for capillary filling (nm) Radius of Pore Ideal gas constant ( kj/[mol K]); the total equivalent series resistance (Ω) of a capacitor Ratio of radii that relates to the total volume of gas desorbed inclusive of pore (unitless) Differential change in pore radius (nm) Difference of and (nm) Thickness of the adsorbate adsorbed to the wall of a pore (nm), The respective thicknesses of the adsorbate on the pore wall corresponding to and, respectively (nm) Thickness of the adsorbate adsorbed the wall of pore Absolute Temperature (K) Differential change in the thickness of the adsorbate adsorbed to the wall of a pore (nm) The change in the thickness of adsorbate adsorbed to the wall of the pore relative to that of the previous desorption step (cm or [cm The free energy that represents the adsorbate-adsorbent interaction Volume of gas adsorbed (cm or [cm Adsorbate adsorbed or desorbed from the capillary between and (cm or [cm Volume of gas contained within the adsorbed monolayer (cm or [cm Microporous volume occupied (cm or [cm for the DR method; The potential held within a capacitor cell (V) Total volume available for adsorption (cm or [cm Total pore volume from to (cm or [cm Volume of some pore of size or index (cm or [cm xiii

14 Differential change in pore volume (cm or [cm Scan rate; the change in voltage with respect to time (V/s) The change volume of gas desorbed from the capillary segment found in pore as a result of the previous desorption step (cm or [cm Average micropore radius (nm) Dummy variable for integration in lieu of Affinity Coefficient (unitless) The dielectric constant of a vacuum (Electric permittivity constant) The dielectric constant for of material Chemical potential of component ( ) The weighted hard sphere chemical potential as described by Tarazona s work ( ) The mean density of confined gaseous fluid as a function of pressure and pore size ( ) The fluid density as a function of distance from the pore size ( ) Number density of component as a function of relative position ( ) The weighted density as defined by Tarazona s work The centre to centre distance at zero interaction energy of a gas atom from the planar surface (nm) The net interaction energy of a single atom interacting with an infinite plane (J) External Potential of component (can represent those inexplicitly or explicitly accounted for) A term used specifically for NLDFT to take into account the potential effect from the adsorbate surface to the fluid [{ ( )}] Grand thermodynamic potential as a function of system components xiv

15 Chapter 1 1 Characterization of the sub-nanoporous structure in porous carbonaceous materials 1.1 Introduction Porous activated carbons can be used for the adsorption of gases [1], to adsorb contaminants in aqueous processes, as electrode materials in supercapacitors [2] or as a catalyst support [3]. In each case, the degree to and speed at which a porous material can adsorb species within the pores is typically a key performance factor. Logically, it is understandable that smaller pores result in slower diffusion of species into a porous particle, but depending on the process, smaller pores can also facilitate a stronger uptake of materials [4,5]. The adsorption capacity has been quantified using by various techniques. One method, physisorption, uses the isothermal adsorption of gas (physisorption) at various pressures below the vapor pressure of the gas. From the physisorption data, traditional adsorption models have been used to infer the total available surface area or approximate the total pore volume available in a sample. However, the traditional models have been shown to be inadequate for pores near or below 2nm that are present in a porous material [6]. These pores are referred to in literature as micropores, but perhaps more aptly should be referred to as nanopores. In some works, materials that contain pores primarily below 1nm have been referred to as ultramicroporous which in this work will be termed sub-nanoporous. These micropores, although difficult to be characterized, can be very important. As an example, consider an electrochemical double layer capacitor (supercapacitor). These energy storing devices depend critically upon micropores materials with pore sizes similar to the electrolyte ion, ~0.8nm, and have been observed to have large energy densities (in particular, using an ionic liquid with ion size) [2,7]. 1

16 Even with the newest of adsorption modelling techniques, the determination and characterization of specific surface area (SSA), specific pore volume (SPV) and pore size distribution (PSD) of an activated carbons is challenging. Landers et al. [6] provide a brief account of the development of density functional theory (DFT) for the modelling of adsorption isotherms over the past 20 years. Two particular models are emphasized because of their ability to characterize a variety of materials: non-local density functional theory (NLDFT) and quenched solid density functional theory (QSDFT). Landers et al. compare separate results from NLDFT and QSDFT to the results of traditional methods for surface characterization, but with little detail of the traditional models [6]. There are significant differences between the derivations of these models which are important to understand before applying them. Landers et al. [6] report that there is now a library of model variants that can be used to determine SSA and SPV or convert a physisorption isotherm into a PSD. However, they do not discuss how the NLDFT and QSDFT models differ in their variation and the significance of using carbon dioxide as an adsorbate in the characterization of activated carbons. The goal of this chapter is to illustrate the significance of using a procedure that combines NLDFT and QSDFT. In general, the procedure can be considered a more accurate and reliable quantification of the sub-nanoporous structure of carbonaceous materials. The description begins with a brief summary of the adsorption models which is followed by a discussion of the limitations of the most widely used BET and DR methods. Afterwards, the importance of the DFT model considerations is demonstrated using the determination of systematic errors that may result from the improper choice of different models or model parameters. 2

17 1.2 Experimental Measurement of Porous Properties Using Gas Adsorption Technique (Physisorption) Carbonaceous Materials Analyzed For this chapter, four carbon materials are used as experimental examples. Activated peat moss (PM), coconut shell carbon (CSC) and petroleum coke (APC) have been selected because of their distinctive pore structures. The PM and CSC were purchased from Fischer Scientific and subsequently modified using an oxidizing gas such as CO 2. The APC is an activated product of raw petroleum coke (RPC). The RPC contains about 7wt% sulphur and has an onion-like layered structure [8]. It has been activated using an RPC to KOH weight ratio of 5:2. The RPC- KOH mixture was preheated for 1 hour at 350 C and then activated at 850 C for 2 hours. The RPC is used both as a base line comparison to APC and because it exhibits an interesting behaviour when tested using two different adsorbents. As shown in Table 1, the carbon content of PM, RPC, and APC were measured while that of RPC was found in literature [9]. Table 1: Carbon content of experimental materials Material Carbon wt% RPC ~80% PM >90% CSC >90% APC >90% 3

18 1.2.2 Material Preparation and Data Collection A two-step process is necessary to gather data for surface characterization. The first step is outgassing and the second is isotherm generation or physisorption for short. For each of these steps, the Autosorb-1 by Quantachrome Inc. was used. The carbon materials, if not already in powdered form, are ground to a particle size range of approximately um and placed within a 6mm small bulb glass sample cell. The outgassing process occurs under a vacuum and at elevated temperatures (using a heating jacket placed overtop the sample cell in order to vaporize volatile materials. In doing so, there will be no volatiles to interfere with pressure measurements during analysis. The specific outgassing conditions for the RPC differed slightly from that of the porous materials, but results in volatile free powder. The RPC was outgassed at 200 C for 1-2 hours. The other materials were outgassed at 70 C for a half hour to remove the large moisture content (~20wt%) and then at 200 C for an hour. The samples were weighed in a sample cell before and after outgassing using a Sartorius CPA225D weigh scale. The weight after outgassing is used for reporting area per mass. The physisorption step requires that a sample is held at constant temperature and placed under a series of increasing or decreasing relative pressure steps ranging from 0 to near 1 (i.e ). The relative pressure of the system is pressure normalized by the adsorbate vapor pressure at the operating temperature. At each of these steps the amount of gas that is adsorbed by a solid can be calculated using the difference between the total volume of gas added a system to achieve a desired pressure and the known system volume. The resulting data set is called the isotherm. Specifically two adsorbates were used: nitrogen (N 2 ) and carbon dioxide (CO 2 ). The N 2 adsorption was done at 77.4 K in a dewar containing liquid N 2. The CO 2 adsorption was done using a dewar filled with coolant that was kept at 273 K using a Fisher Scientific 9105 hot/cold recirculating bath. IUPAC classifications can be used to describe general trends found in isotherms. The classes of isotherms can be found in [10] and are provided as sketches in Appendix F. Activated carbon adsorption isotherms do not specifically follow one of the 6 general isotherm types, but show the 4

19 sudden jump characteristic of a Type I isotherm at low relative pressures less than Nonmicroporous materials show a jump characteristic between 0.05 and 0.35 relative pressure [11]. In some cases, an activated carbon can have the gradual increase of a Type II isotherm representing multilayer adsorption and a hysteresis loop (in general most similar to Type H3 or H4 Type). Typically for non-carbon materials, the region just before a linear increase found between 0.05 and 0.35 of the relative pressure indicates the formation of the first adsorption layer (as seen at point B in a Type II or IV isotherm in [10]). This step increase is much larger and occurs at much lower relative pressures (<0.04) in the presence of micropores and would appear on the left hand side of a Type I isotherm. Materials with Type I like isotherms require a high resolution set of data points below 0.1 relative pressure and those with hysteresis require a wide range data points from 0.1 to The N 2 based isotherms were collected using 85 data points following a logarithmic step size: 28 data points for adsorption from 4x10-6 to 0.1 relative pressure; 37 data points for adsorption from 0.1 to relative pressure and 21 for desorption from to 0.1 relative pressures. For N 2 adsorption, the data from 4x10-6 to 0.1 relative pressures is used to provide the necessary detail quantify the micropores present the materials and data from 0.1 to is used to identify the presence of hysteresis. As for the CO 2 data, it was collected using 36 points for adsorption from 4x10-6 to relative pressures. The SSA, SPV and PSD for higher degrees of adsorption were characterized with N 2 and because high pressures would be impractical to characterize with CO 2 (the vapor pressure of CO 2 at 273k is ~34 atm). Because the porosity of materials can vary significantly, so can the total amount of gas adsorbed. To make sure that the total amount of gas adsorbed onto a sample is similar (for the purpose of being measureable) similar quantities of total surface area are added the sample cell when testing. For N 2 data, the carbon materials were weighed out such that the equivalent of approximately 50 m 2 was present in the sample cell (to minimize the effect of limited diffusion at 77K). Limited diffusion occurs because the relatively high amount of N 2 that will need to be adsorbed by microporous materials. For the CO 2 data, the equivalent of approximately 100 m 2 was added to each sample cell to increase the gas volume measurement accuracy since diffusion at 273 K is much less of an issue and less total gas adsorbed per m 2. In both cases, the lower 5

20 Volume Adsorbed STP) pressure points (<0.01) are given a longer amount of time to equilibrate before recording the volume measurement than those at higher relative pressures. The collected data that compose an isotherm is plotted with relative pressure on the horizontal axis and total volume of gas adsorbed in cubic centimeters per gram at STP on the vertical axis. The SSA, SPV and PSD data were then generated using the ASQiWin software that was supplied by Quantachrome. All PSDs are presented with characteristic pore width (short-form: pore width) on the horizontal axis and differential (rather than cumulative) pore volume or differential specific surface area on the vertical axis to make it easier to visualize the position of any key pore ranges. The N 2 adsorption branches of the isotherms for all four materials of this study (i.e. excluding the desorption branches for ease of viewing), in Figure 1, to provide general sense of the relative porosity of the materials used in this study. On the left hand side of the figure for PM, CSC and APC, the step-like Type I characteristic can be seen. More so for PM, a gradual upward curve approaching the right hand side of the isotherm is similar to that of a Type II isotherm. For RPC it is clear that there is little N 2 adsorption relative to the other materials PM RPC CSC APC Relative Pressure (P/Po) Figure 1: Isotherms of the four carbon materials used in this study 6

21 Provided in Table 2 is a summary of the Brunauer-Emmett-Teller method of specific surface area for later comparison (and those familiar with the method). This data is provided here, but analyzed later because surface area it provides a singular value that can be used to then illustrate of the reproducibility of the data provided in the following section. It is important to note that the values found in Table 2 follow the limitations described in Section Table 2: Summary of Brunauer-Emmett-Teller method to calculate specific surface areas Carbon Material BET SSA (m 2 /g) RPC 29 PM 926 CSC 1176 APC Reproducibility: Random Error There are two types of errors that affect the precision and accuracy of measurements: systematic error and random error. Systematic error is a bias that results from the regular application of inaccurate measurements or calculations (known or unknown). Random error is the variability in a measurement as a result of uncontrollable parameters disturbing the measurement. While the focus of this work is on the systematic errors that may result from improper choices of models and/or model parameters, this section is to quantify the uncertainty resulting from random error. Although regular performance checks are done using a material of known surface area, a set of data was gathered using PM for the purpose of quantifying random error. The PM sample was tested using three different weights with three replicates each to this end (nine samples). Three sets of values are used to assess random error in the collected data for PM: the average relative standard deviation of the isotherms, the standard deviation of the Brunauer-Emmett-Teller specific surface area (BET-SSA) and the standard deviation of pore volume and pore size determined with the DR model. Table 3 summarizes the standard deviations in isotherms and BET-SSA values, while Table 4 gives the standard deviations in micropore volume and average pore size. 7

22 Table 3: Uncertainties of Isotherms and BET-SSA. Number of isotherms Average Relative Standard Deviation of Adsorbed Volume (% of STP) Average BET SSA (m 2 /g) Standard Deviation of BET SSA (m 2 /g) Relative Standard Deviation of BET SSA (% of m 2 /g) Data Set Overall st Weighing nd Weighing rd Weighing As shown in Table 3, the relative standard deviations in both cases are less than 10%. As expected the relative standard deviations of BET-SSA are similar to those of isotherms. The collection of isotherms was better controlled for weighing 1 and 2. In these two weighings, the average relative standard deviation of the isotherms is 0.36 and 0.65 %, respectively, and the relative standard deviations for the calculated BET SSA are 0.21% and 0.18%, respectively. Table 4, in terms of DR microporous volume, shows a similar trend with regard to the standard error in BET SSA. However, the relative standard deviations are much smaller for the DR mean pore size, which is only 0.4 %. This seems to suggest that the DR mean pore volume is not as sensitive to the change in isotherms, which may make the pore size value calculated with the DR model less reliable. Table 4: Uncertainties for DR "micropore" volume and average pore size DR Microporous Volume DR Mean Pore Size Number of Rel. Rel. Samples Average StdDev StdDev Average StdDev StdDev Overall % % Weighing % % Weighing % % Weighing % % Similar data for other models used in the later sections is provided in Appendix B. 8

23 1.3 Adsorption Models Presented in this section are the adsorption models that are found in the ASQiWin Software related to porous characterization (two exceptions are two of the capillary models which were presented to illustrate the simplicity of the Barett-Joyner-Halenda model). This section is intended to provide enough detail of adsorption models to describe their applicability, but also differentiate them from each other. In each of the models, the terms potential or adsorption potential are used to refer to the free energy difference between the adsorbed state and the gas or fluid state of an adsorbate. This nomenclature is common in adsorption literature Brunauer-Emmett-Teller Model The Brunauer-Emmett-Teller (BET) model was created to estimate the surface area of a porous material [12]. This method relies on the physical adsorption (physisorption) of a gas onto a solid by assuming that there exists a pressure at which a single layer of adsorbate molecules forms across the surface of an adsorbent. The linearized form of the BET equation is as follows: ( ) Eq. 1-1 The total system pressure is denoted by, vapor pressure of adsorbate by, volume of adsorbate adsorbed by, adsorbate molecular monolayer volume by and a constant that relates the kinetic constants of condensation and vaporization by. Two important points: isolated on the right of the equation is which is referred to as relative pressure, a normalized variable and parameter which must be positive (typically a large value relative to 1). After an isotherm is replotted with a vertical axis representing ( ) and the horizontal axis remaining as, the determination of and can be made. This is accomplished by calculating the slope and intercept of an increasing set of linear data points within the range of 0.05 and 0.35 relative pressure. Once calculated, the monolayer volume,, can be converted to a total number of atoms and subsequently equated to a surface area under the assumption of an adsorbate-specific atomic packing factor. 9

24 1.3.2 Dubinin-Radushkevich and Dubinin-Astakhov Models The Dubinin-Radushkevich (DR) model is an adsorption model that is used to characterize microporous materials [3,4]. The DR method functions under the assumption that the adsorbate adsorption potentials follow a Rayleigh distribution [13] within a microporous material. The volume of adsorbate adsorbed in a micropore is related to the total microporous volume using the following equation: ( ) Eq. 1-2 The volume of gas occupying the micropores of a material is denoted by, the maximum volume (micropore volume) that could be occupied by said gas is denoted by, the free energy of adsorption by, the affinity coefficient (a correlative constant) by, and the characteristic adsorption energy by. The negative of the free energy of adsorption is a thermodynamic value that represents difference between the chemical potential of adsorbate in liquid form and that of the adsorbate in the adsorbed state at constant temperature: ( ) Eq. 1-3 The universal gas constant is denoted by, temperature by, and the reciprocal of relative pressure. Substituting Equation 1-3 into Equation 1-2 and linearizing the resulting equation linearized results in: ( ) ( ) ( ) ( ( )) Eq. 1-4 Equation 1-4 can be used on a linear region found on a plot of ( ) versus ( ( )). Then, the characteristic adsorption energy (an empirical value),, can be calculated from the slope and the micropore volume,, from the intercept of a linear fit. 10

25 In addition to the micropore volume,, that can be calculated from Equation 1-4, Dubinin and Stoeckli [14] proposed that could be used to approximate a characteristic (mean) pore width,, using a relationship where the pores radius of an adsorbent using x-ray based measurement was suggested to be inversely proportional for a slit shaped pore. The resulting equation from this inference was found to be: Eq. 1-5 where the half-pore width is denoted by and a proportionality constant (representing the spread of the adsorption distribution [13]) by. Using data collected for industrial activated carbons and those with sieve like properties, k was empirically found to have values of 13 and 10 kj. nm/mol, respectively [14]. Sometime after Equation 1-2 was published, the generalized case of the DR method was revisited and renamed the Dubinin-Astakhov (DA) method [4]: ( ) Eq. 1-6 where Dubinin had used =2 and as the product of and [15]. From this equation, could be adjusted or empirically selected such that could be calculated Capillary Condensation Models Outlined in [3], there are three well-known models that aim to quantify pore size by using the desorption process: Barett-Joyner-Halenda (BJH), Cranston-Inkley (CI) and Dollimore-Heal (DH). These models strictly assume that the capillary within a porous material is cylindrical. As described in [16], the desorption process has three major influences: the capillary condensate, adsorbed layer and the pore walls of varying pore sizes. The condensate is found within the centre of a pore and kept there by the adsorbed layer, an ordered multilayer directly adsorbed to the pore wall. The desorption process begins with a material held at a relative pressure near 1 (i.e ) where the pores would be completely filled with condensate. When lowering the relative pressure, it is conceived that the largest pore would have its capillary volume emptied along with a portion of the adsorbed layer therein. In following decreases of relative pressure, the evacuation of the 11

26 capillary volume would continue for subsequently lower pore sizes; alongside the evaporation of condensate from said capillaries, a fraction of the adsorbed layer therein will evaporate in addition to further evaporation (thinning) of adsorbed layers in previous (larger) pore sizes. The relative pressure is then lowered in a step-wise fashion while summing the adsorbate which is forced to evaporate in response. As such, each of the models (BJH, CI, DH) aim to relate the total volume desorbed to corresponding changes in capillary diameter (the space left behind once the condensate evaporates) and adsorbed layer thickness Barett-Joyner-Halenda Model The BJH method correlates the pore volume of some pore size to the volume of the capillary and adsorbed layer evaporated during the desorption process. The equation that represents a stepwise account of said mechanism, for a cylindrical pore, is: ( ) Eq. 1-7 The pore volume is denoted by ; the ratio of radii that relates to the sum of the evaporated amounts from the capillary and adsorbate layer found within pore by ; the total volume of gas desorbed such that the capillary of pore could become empty by ; and the volume of condensate that evaporated due to the thinning of the adsorbed layer in all previous pores by where represents the change in the thickness of the adsorbed layer relative to the previous desorption step, represents the ratio of the average change in radius between the capillary radius of the previous desorption step ( ) and that of the current one ( ) to the capillary radius shared across all emptied capillary volumes and represents the surface area of the pore. The parameters in Equation 1-7 that are not inherently related to relative pressure are the capillary radius (in the constant ) and thickness of the adsorbed layer,, such that a relationship with and relative pressure can be formed. To do so, at the time the paper was published, the generalized Kelvin equation was used to create a correlation for capillary radius and data from [17] to create a correlation for adsorbed layer thickness. Even with functions for and capillary radius, due to the derivation of BJH there exists the constant which has no known value or relationship and had to be approximated for various pore sizes in [16]. 12

27 Cranston-Inkley Model As for the CI method, from [4], Crankston takes a differential approach to modelling the capillary pore filling mechanism. The differential form of said equation is: ( ) ( ) ( ) Eq. 1-8 where a differential change in the volume of gas adsorbed or desorbed corresponding to a differential change in relative pressure denoted by ; the pore radius surrounding a capillary at some relative pressure by where represents some differentially different pore size due to a differential change in relative pressure; adsorbed layer thickness at some relative pressure across all emptied pores by ; the pore volume distribution by ( ) where ( ) would be the specific volume of some pore between and ; and some differential change in adsorbed layer thickness resulting from some differential change in relative pressure by. Given that the generation of a physisorption isotherm with infinitely small changes in relative pressure would be impractical, the differential equation can be integrated between some radius (corresponding to some pressure ) and some other pore radius (corresponding to some pressure ) to create a finite equation for some given step change in relative pressure. The result of such integration is as follows: ( ) [ ( ) ] [ ( ) ( ) ( ) ] Eq. 1-9 For clarity, the volume of the pores between the radius of and is denoted by ; the total volume of gas adsorbed or desorbed from the capillary due to a change relative pressure by ; the thickness of the adsorbed layer at some pressure by and at some by. As with BJH, relationships for adsorbate thickness and capillary radius are needed to complete the relationship between and relative pressure Dollimore-Heal Model The derivation of DH begins with the same basic steps as BJH: the evaporation of a capillary volume in addition to some loss in the total thickness of wall adsorbed adsorbate. Specifically, DH uses differentiation and integration to get explicit terms to represent the changes in the 13

28 thickness of the adsorbed layer rather than averages. The results of these changes to the derivation are an equation of similar form to BJH, but with an extra term: ( ) Eq where the thickness of the adsorbed layer for the current desorption step is denoted by and is the pore radius. Again, as with the previous methods, each variable requires [18] additional relationships with relative pressure to complete the analysis Horvath-Kawazoe and Saito-Foley Models The Horvath-Kawazoe (HK) and Saito-Foley (SF) methods were independently derived from the Kelvin equation. These methods relate the free energy of adsorption to molecular interaction potentials in order to manage what the capillary methods could not: a pore modelling when a pore is without a condensate phase. The HK method uses the molecular interaction potentials presented in [18] for nitrogen and graphite (Equation 1-11) and the Gibbs free energy of adsorption (Equation 1-12; a variation of Equation 1-3) as a basis to create a relationship between relative pressure and pore width: [ ( ) ( ) ] Eq ( ) Eq In Equation 1-11, the net interaction energy of a single atom interacting with an infinite plane is denoted by ; the atoms per unit of surface area by ; an intermolecular constant from the Kirkwood-Muller equation by ; the distance between a gas atom and the centre of a surface at net zero interaction energy; and the distance of a the centre of a gas molecule from the centre of the surface by [4]. As for Equation 1-12, the Gibbs free energy of adsorption is denoted by ; the adsorbateadsorbent potential by ; and the adsorbate-adsorbate-adsorbent potentials by. 14

29 Equation 1-11 was then modified to take into account multiple adsorbate molecules adsorbed to one of two parallel graphite surfaces. After doing so, the resulting equation can be equated to Equation 1-12 and then integrated to take into account the average potential within the system: ( ) ( ) [ ( ) ( ) ( ) ( ) ] Eq Avogadro s number is denoted by ; the number of molecules per unit area of adsorbate by ; another intermolecular constant by ; the distance between the centres of graphite sheets by ; and the sum of the adsorbate and adsorbent radii by. In Equation 1-13, there are only two unknown values: the relative pressure,, and the spacing between graphite planes,. The SF method used a similar process to create a similar relationship between the characteristic pore size and relative pressures, but with a cylindrical pore shape. There were two ways that [19] proposed to average the adsorption potential within their modeled system: using a lineaverage or an area average of the interaction energy function. The former has been described to represent the movement of an adsorbate atom being restricted along the diameter of a pore and the latter restricted to movement along the surface [19]. For brevity the SF method is not shown Density Functional Theory Adsorption Models Density function theory (DFT) is a statistical mechanics approach used to determine the interactions between molecules such that higher level parameters (nanoscopic, microscopic or macroscopic) can be calculated or approximated. Seaton et al. presented a mathematic representation of a point on an experimental isotherm, ( ) [20]: ( ) ( ) ( ) Eq where the pressure is denoted by ; pore width by ; the upper and lower pore size bounds by and, respectively; the mean density of the fluid within and by ; and the PSD by. In general, it is this equation which needs to be solved in order to reduce an isotherm to an SSA, SPV or a PSD. A set of values of are calculated for a pre-determined number of pore range (a kernel), such that each point on an isotherm can be used to determine the statistical presence of a pore. 15

30 For later use, an equation that can be used to provide a bulk metric for the purpose of comparing the material s is the average pore volume: ( ) ( ) Eq In this equation, is the maximum pore size present in some pore size distribution ( ). The pore size distribution ( ) can be generated as either as the differential contributions of pore size to specific pore volume or specific surface area. From here Seaton et al. used statistical mechanics to determine and a statistical fitting algorithm to fit the PSD to a positive and continuous distribution such the log-normal or gamma distribution. However, the following discussion will be limited to describing the processes used to derive and calculate from non-local density functional theory (NLDFT) and quenched solid density functional theory (QSDFT). The general form of is: ( ) Eq Where the fluid density profile within some pore wide and averaged to create is denoted by and a dummy variable for pore width by z. In general, this equation provides the determination the average density for a given pore size [20]. Specifically, ( ) is the function that is solved by minimizing the grand thermodynamic potential function. However, to use the grand potential function implies that the grand canonical ensemble is being used to model a system. To understand what the grand canonical ensemble implies, an ensemble shall be defined: a system of finite volume containing a finite quantity of matter that is held at constant temperature by some surrounding thermal reservoir (similar to a system as described in thermodynamics or 16

31 control volume). Following this, a canonical ensemble can be created by replicating an ensemble to any large number of neighboring units to model a single system where the grand canonical ensemble is a canonical ensemble with following properties: the volume of each unit is of the same fixed size and the matter within each ensemble is able to move across the boundaries therein. Consequently, the latter property allows for flexibility by allowing the composition or components within any given unit to fluctuate. More details can be found in [21]. Recognizing the use of the grand canonical ensemble the grand thermodynamic potential,, can be used for the determination of the average fluid density within the pores during adsorption [20,22]. The grand potential is of the general form: [{ ( )}] [ { ( )}] ( )[ ] Eq where the intrinsic Helmholtz free energy is denoted by ; the number (molecular) density, external potential and chemical potentials of component by, and respectively. For NLDFT and QSDFT, the is divided into three components: [{ ( )}] [ { ( )}] [ { ( )}] [ { ( )}] + ( )[ ] Eq where, with respect to intrinsic Helmholtz free energy, is a calculable ideal component; is the long range attractive (van der Waals) component of the non-ideal component; and is the short range repulsive component of non-ideal (excess) component [22]. Importantly, after the determination of each component and, the minimization of Equation 1-18 with respect to provides an equation that represents the equilibrium density profile to substituted into Equaiton 1-16 to generate isotherms for the fitting of experimental data. In both NLDFT and QSDFT, is calculated using an equation for the free energy of an ideal gas and incorporated using a mean field approximation particle position [20,22]. In both cases, each parameter becomes a function of to allow the in the minimization of with respect to. Additionally, (a component of ) is related to some hard sphere parameter using the bulk equation of state. Discussed in the following two subsections, the major 17

32 differences between the NLDFT and QSDFT models are found in how and are modeled which also provides a solution for as a function of. A noteworthy concept is that the minimization of can result in two minima, where the higher one correlates with the desorption process and the lower one correlates with the adsorption process [6]. The extent of the presence of hysteresis in an isotherm would suggest which of these two would be most important [6] Non-Local Density Functional Theory Non-local density functional theory (NLDFT) takes [{ ( )}] to be solely a function of ( ). Then, Equation 1-17 becomes: [{ ( )}] [ { ( )}] [ { ( )}] [ { ( )}] ( )[ ] Eq In this case, is based on Tarazona s work [23]. Lastoskie presents the result of Tarazona s work using the following equation: ( ) [ ( )] Eq Where is Planck s constant; and some distribution is a function of some smoothed (weighted) number density,, that is selected based on the distance between adsorbed layers found in atomic modeling which results a function with respect to. Then, is derived to be: ( ) ( ) ( ) Eq where and are, respectively, the chemical potential and system pressure for a weighted hard sphere system; and is some ideal component of the weighted system. The former two parameters are related to the (thus, indirectly to ) using the Carnahan-Starling equation of state (a model intended for dense hard sphere fluids [24]), whereas the latter parameter is simply a function of. 18