Analysis of Catalyst Surface Structure by Physical Sorption

CHAPTER ONE

Analysis of Catalyst Surface Structure by Physical Sorption Karl D. Hammond*, Wm. Curtis Conner Jr.†

*Department of Nuclear Engineering, University of Tennessee, Knoxville, Tennessee, USA † Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts, USA

Contents 1. Introduction 2. The Phenomenon of Physical Adsorption 2.1 History of adsorption in catalyst characterization 2.2 Choice of adsorbate 2.3 Presentation of adsorption data 2.4 The Langmuir isotherm 2.5 Monolayers to multilayers 2.6 BET theory 3. A Tour of the Adsorption Isotherm: From Vacuum to Saturation and Back 3.1 The micropore-filling region: 10
8 < P/P < 10
2 3.2 Monolayer region: 0.05 < P/P < 0.2 for nitrogen at 77 K or argon at 87 K 3.3 The multilayer region: 0.2 < P/P < 0.4–0.98 for nitrogen at 77 K (no hysteresis) 3.4 Mesopore-filling region: 0.2 < P/P < 0.45 (no hysteresis) and 0.4 < P/P < 0.99 (hysteresis present) 3.5 Adsorption on exterior surfaces: 0.45 < P/P < 1 4. High-Resolution Adsorption: P/P < 0.01 4.1 Measurement of HRADS isotherms 4.2 Collection and presentation of HRADS isotherms 4.3 Position of the inflection point 4.4 Estimating surface areas in microporous materials 5. Adsorption–Desorption Hysteresis 5.1 Network effects 5.2 Comparison with hysteresis in mercury porosimetry 5.3 Metastability 5.4 Cavitation and the tensile strength effect 6. Assessing Porosity 6.1 Statistical thickness: The Halsey equation 6.2 Standard isotherms: The t- and as-plots 7. Pore Size Distributions 7.1 Traditional sorption-based techniques

Advances in Catalysis, Volume 56 ISSN 0360-0564 http://dx.doi.org/10.1016/B978-0-12-420173-6.00001-2

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Karl D. Hammond and Wm. Curtis Conner Jr.

7.2 Sorption analysis techniques derived from simulations 8. Details of Adsorption Apparatus 8.1 Volumetric adsorption systems 8.2 Start to finish: Acquiring an adsorption isotherm 9. Common Pitfalls in Adsorption Experiments and Analyses 9.1 Definitions of standard temperature and pressure 9.2 Measurement of the saturation pressure 9.3 Drift in bath temperatures/compositions and equipment calibration 9.4 Reference state for argon at 77 K 9.5 Misuse of the BET equation 9.6 Failure to degas properly 9.7 Inaccurate or nonequilibrium pressure readings 9.8 Correcting (incorrectly) for thermal transpiration 9.9 Interpreting hysteresis at low pressures to be porosity 9.10 Reporting 17–20 Å pores 10. Summary References

69 72 72 75 85 86 86 88 88 89 91 91 92 94 94 95 97

Abstract Heterogeneous catalysis usually takes place by sequences of reactions involving fluidphase reagents and the exposed layer of the solid catalyst surface. Estimation of the total catalyst surface area, its potential accessibility to gas- or liquid-phase reactants, and general catalytic activity are initially based on the morphology of the catalyst. Universally, measurements of adsorption and their interpretation are used to estimate the surface area and porosity relevant to catalytic reactions. We provide here a description of many traditional and recent techniques in adsorption-based catalyst characterization intended for experimental practitioners of adsorption. Our chapter includes descriptions of which regions of the isotherm correspond to micropore filling, mesopore filling, surface coverage, and saturation, supplemented by discussions of model isotherms, from the Langmuir isotherm and the Brunauer–Emmett–Teller theory to the Halsey equation. Pore size distribution methods include the Barrett–Joyner–Halenda and related methods for mesopores, empirical methods developed for micropores, and simulation-based methods that have finally resolved the differences between adsorption (increasing loading) and desorption (decreasing loading). This chapter also includes a discussion of hysteresis and metastability, both of which “trip up” experimentalists from time to time. We finish with a description of data acquisition methods and equipment, which are often obscured behind the facade of automation, and a discussion of what users should be aware of and what can go wrong.

NOMENCLATURE [A] activity of species A as normalized adsorption isotherm on a standard, nonporous material such as silica or carbon; as ¼ Vads/Vads(P/P ¼ 0.4) g surface tension

Analysis of Catalyst Surface Structure by Physical Sorption

3

u loading: y ¼ n/nm, interpreted as the fraction of unoccupied sites in the first layer ui loading in layer i k proportionality constant between flux and pressure for an ideal gas; k ¼ (2pMakT)
1/2 m chemical potential ((partial) molar Gibbs free energy) n vibrational frequency along the normal mode that results in desorption from a layer r density of adsorbate s atomic/molecular diameter V grand potential, O ¼
PV þ gA A surface area AA–A force constant between the adsorbate and itself in the HK and Cheng–Yang models AA–E force constant between the adsorbate and the adsorbent in the HK and Cheng–Yang models Adsorbate the atom/molecule that is adsorbing onto a surface Adsorbent the solid species onto which the adsorbate adsorbs Adsorptive a synonym for adsorbate Am area of one molecule, typically reported in nm2 or A˚2 per molecule or atom but can also be in m2/mol or m2/cm3 STP (i.e., surface area per unit volume adsorbed) Ap surface area of a pore ai sticking coefficient: probability of adsorbing given a collision with the surface b Langmuir’s parameter, which has units of inverse pressure AEI International Zeolite Association framework code for materials such as AlPO-18, SAPO-18, SIZ-8, and SSZ-39 (the code is derived from AlPO EIghteen) ATS International Zeolite Association framework code for materials such as AlPO-36, MAPO-36, FAPO-36, and SSZ-55 (the code comes from AlPO Thirty-Six) BET Brunauer–Emmett–Teller theory BJH the Barrett–Joyner–Halenda method of pore size distribution analysis for mesopores CBET or C second BET fitting parameter (the other being the monolayer capacity), a positive number related to the difference in the heat of adsorption between layers Chemisorption chemical adsorption, the process of atoms or molecules chemically reacting with a solid surface, in contrast to physisorption CH empirical Halsey constant in the Frenkel–Halsey–Hill equation CK constant of proportionality between inverse pore size and the base-10 logarithm of relative pressure: CK ¼ ð2gVl Þ=ðRT logð10Þ CPDFT classical potential density functional theory D inner diameter of the glass tube used in physical adsorption experiments d0 arithmetic mean diameter of adsorbate and adsorbent atoms/molecules in the HK model Degassing the process of heating a sample to remove adsorbed gases, such as water, from the surface in preparation for an adsorption experiment; also called outgassing DFT density functional theory DH the Dollimore–Heal method of pore size distribution analysis for mesopores Dp pore diameter, Dp ¼ 2rp f fugacity F Helmholtz free energy Fr density functional FAU International Zeolite Association framework code for materials such as faujasite, zeolite X, zeolite Y, ECR-30, LZ-210, and SAPO-37 (the code comes from FAUjasite) G Gibbs free energy of adsorption

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Karl D. Hammond and Wm. Curtis Conner Jr.

h height of the bath H enthalpy of adsorption H1 hysteresis hysteresis classification in which the loop has similar shape on adsorption as it does on desorption H2 hysteresis hysteresis classification in which the loop has a different shape on adsorption than it does on desorption; the adsorption process is gradual, whereas desorption is sudden HEU International Zeolite Association framework code for materials such as heulandite, clinoptilolite, and LZ-219 (the code is derived from HEUlandite) HK Horva´th–Kawazoe model for adsorption in slit-like micropores HRADS high-resolution adsorption, a term applied to techniques that measure adsorption below about 1 Torr (10
3 atm) IBET intercept of the BET plot Jm molecular flux hitting the surface k Boltzmann’s constant ka rate coefficient for adsorption kd rate coefficient for desorption K equilibrium constant ℓ length of the neck of the sample tube LTA International Zeolite Association framework code for materials such as zeolite A, ITQ-29, SZ-215, SAPO-42, and ZK-21, ZK-22, and ZK-4 (the code is derived from Linde Type A, the original name for zeolite A as synthesized by the Linde group at Union Carbide) log natural logarithm (note that “common” (base 10) logarithms are written explicitly (e.g., log10(x))) m mass of the adsorbent Macropore pore with a radius larger than 50 nm (such pores fill near the saturation pressure and are generally not resolvable on adsorption isotherms) Manifold the chamber directly above a valve that connects it to the sample tube in a volumetric adsorption system Ma molecular (or atomic) mass of the adsorbate MEL International Zeolite Association framework code for materials such as ZSM-11, SSZ46, silicalite-2, and TS-2 (the code is derived from Mobil ELeven) Mesopore pore with a radius between 2 and 50 nm MFI International Zeolite Association framework code for materials such as ZSM-5, silicalite-1, EU-13, ISI-4, mutinaite, and KZ-1 (the code is derived from Mobil FIve) Micropore pore with a radius of 2 nm or less (important: the word “micropore” has nothing to do with the word “micrometer”) MTT International Zeolite Association framework code for materials such as ZSM-23, EU-13, ISI-4, and KZ-1 (the code is derived from Mobil Twenty-Three) MTW International Zeolite Association framework code for materials such as ZSM-12, CZH-5, NU-13, TPZ-12, and VS-12 (the code derives from Mobil TWelve) NA areal density of adsorbate atoms/molecules Nanopore name sometimes used in place of the word “pore,” especially by authors attempting to draw analogies to nanotechnology (all micropores, mesopores, and macropores are nanopores, despite the illogic of the prefixes)

Analysis of Catalyst Surface Structure by Physical Sorption

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NE areal density of adsorbent atoms/molecules nads or n number of molecules (or moles) adsorbed, usually per gram adsorbent NA am number of moles in the adsorption manifold before the sample valve is opened NB am number of moles in the adsorption manifold after the sample valve is opened Ngas number of moles currently in the gas phase inside the adsorption manifold and over the sample Nin total number of moles added to the adsorption manifold since the start of the experiment NLDFT nonlocal density functional theory, a type of CPDFT in which the functional includes the contributions of density gradients to the free energies nm monolayer capacity: the number of molecules or moles of adsorbate in one “layer” on the surface, typically normalized per unit mass of adsorbent NMR nuclear magnetic resonance spectroscopy Outgassing synonym for degassing P absolute pressure PA pressure in the adsorption manifold before the sample valve is opened PB pressure in the adsorption manifold after the sample valve is opened P/P° relative pressure P° saturation pressure Pg pressure on the vapor side of a vapor–liquid meniscus Physisorption physical adsorption, the process of atoms or molecules adhering to a surface without forming chemical bonds, in contrast to chemisorption Pℓ pressure on the liquid side of a vapor–liquid meniscus Ps standard pressure (usually 1 atm. ¼ 101,325 Pa ¼ 760 Torr) Q heat of adsorption (negative enthalpy change of adsorption); subscript i indicates adsorption on the ith molecular layer QL heat of condensation QSDFT quenched solid density functional theory, a type of CPDFT in which gradients in both the adsorbate and adsorbent densities are factored into the free energy calculation; the net effect is more flexible pore walls than NLDFT R universal gas constant (in J mol
1 K
1) RHO International Zeolite Association framework code for materials such as zeolite Rho, ECR-10, LZ-214, and pahaspaite rc rate of condensation re rate of evaporation rK or hri mean radius of curvature of the meniscus inside a filled or filling pore, often called the Kelvin radius Re ratio of the volume of the empty tube to the manifold volume Rf ratio of the volume of the filled tube to the manifold volume Rn square of the ratio of the volume of a cylinder representing the pore and another cylinder representing the fluid added to/removed from the pore; used in BJH analysis rp mean pore radius S entropy SBET slope of the BET plot SABET surface area as extracted from the BET equation

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SANS small-angle neutron scattering SBA material code designating unique materials first made at the University of California, Santa Barbara, such as SBA-15 SEM scanning electron microscopy SF Saito–Foley model of adsorption in cylindrical micropores STP standard temperature and pressure, typically defined as 273.15 K and 1 atm t statistical thickness of the adsorbed layer T absolute temperature (in Kelvins) Tb temperature of the bath TEM transmission electron microscopy TON International Zeolite Association framework code for materials such as zeolite Theta-1, ISI-1, KZ-2, NU-10, and ZSM-22 (the code comes from Theta-ONe) Tr room temperature, typically about 20–25  C Ts standard temperature (usually 273.15 K¼0  C, but others are also commonly used) Tt triple-point temperature (in Kelvins) Type I isotherm an isotherm, such as the Langmuir isotherm, that is concave and has no apparent multilayer adsorption Type II isotherm an isotherm, such as the BET isotherm, that has an inflection point and shows multilayer adsorption but no hysteresis Type IV isotherm an isotherm that begins like a Type II isotherm but exhibits hysteresis Vads or V standard volume adsorbed, in cm3 STP/g adsorbent. Vads ¼ nadsRTs/Ps Vam volume of the adsorption manifold, typically in cm3 Vbulb volume of the “bulb” at the bottom of a typical glass adsorption cell VDS apparent volume of the chamber containing the sample from the sample valve onward, including effects due to temperature gradients, typically in cm3 Ve volume of the empty calibration tube, typically in cm3 Vf volume of the calibration tube after the known-volume insert is in place, typically in cm3 Vg molecular or molar volume of the vapor phase Vgas volume, at STP, of adsorbate currently in the gas phase inside the sample and manifold volumes Vin total volume, at STP, of adsorbate added to the adsorption manifold since the start of the experiment VET International Zeolite Association framework code for the zeolite VPI-8 (the code is derived from Virginia Polytechnic Institute EighT) VFI International Zeolite Association framework code for materials such as VPI-5, AlPO-54, H1, and MCM-9 (the code is derived from Virginia Polytechnic Institute FIve) Vℓ liquid molar or molecular volume Vm monolayer volume, or the volume all adsorbed molecules would take up if desorbed and returned to STP; Vm ¼ nmRTs/Ps Vp pore volume Vstd volume of a known standard, such as a cylindrical tube of precision-bore glass w width of slit-like pores in the HK model W weighting function for each isotherm in the simulated “kernel” of isotherms; this becomes the pore size distribution function when fit to an experimental isotherm xm areal density of surface sites

Analysis of Catalyst Surface Structure by Physical Sorption

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1. INTRODUCTION Physical adsorption results from interactions between subcritical fluid species and nearly any solid surface. The measurements are made by a variety of well-developed techniques and interpreted by using ever more sophisticated models. Physical adsorption experiments probe thermodynamic phase equilibria between bulk fluid phases and adsorbed phases, which progress from single, isolated molecules to a single layer of molecules on the surface (a monolayer) to multilayers to condensation (or sublimation). Analyses of equilibrium data characterizing the adsorption of physisorbing gases are commonly employed to estimate the morphology of the sample, including the total surface area, the distribution of the dimensions of any pores (ranging in diameter from about 0.1 to 50 nm), and the total pore volume/void fraction. These analyses are employed to guide understanding of the influence of morphology on sorption, separations, and catalysis. Considerable progress has been made in the last several decades in investigations of physical adsorption on high-surface-area solids, both experimentally and theoretically (1–8), such that we now understand the phenomena associated with sorption far better than we ever have. Furthermore, materials synthesis has developed to such an extent that we can now produce materials possessing very high surface areas (>1000 m2 g
1 of solid) or with uniform pores in the range 1–20 nm, or even solids with multiscale porosity comprising a network of pores of one size embedded within a network of pores of another dimension and/or connectivity. Physical adsorption (physisorption) is then employed to characterize, design, and optimize the morphology of the material for specific applications. The purpose of this chapter is to provide an understanding of what is known about physisorption with respect to analyses and interpretation as these relate to the morphology of high-surface-area solids. To these ends, we begin by describing the sequence of phenomena associated with physisorption, its history, and simple modes of adsorption on relatively flat surfaces. We then begin our tour, considering the adsorption isotherm, region by region (Section 3). Adsorption on materials having pores less than 2 nm in fundamental dimensions (called “micropores”) exhibits unique challenges, as their pores fill at extremely low pressures, before the surface is completely covered; these challenges are discussed in Section 4. For larger pores, called mesopores and macropores, there is often hysteresis between the adsorbing

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and desorbing trajectories along an adsorption isotherm; the phenomenon of adsorption hysteresis is discussed in Section 5. We also discuss methodologies involved in determining porosity and pore size distributions and close the chapter with discussions of the experimental aspects of adsorption: how adsorption apparatus works and how to avoid some common mistakes in measuring and interpreting adsorption data.

2. THE PHENOMENON OF PHYSICAL ADSORPTION 2.1. History of adsorption in catalyst characterization It was recognized well over a century ago that solid surfaces could enable gases or liquids to react under conditions in which they would not react in the absence of the surface. These observations were quickly understood to occur when molecules or atoms “stuck” to the surface, changing both their relative reaction energies and their local concentrations. This process—atoms and molecules adhering to a surface—was termed “adsorption” by Kayser in 1881 (9). Irving Langmuir (10,11) later expressed the kinetics of steps associated with the individual adsorption and desorption processes by assuming that each atom/molecule reacted with an array of “sites” on the surface, ka ½A½S

A þ S ! AS kd ½AS

AS ! A þ S,

ð1:1Þ ð1:2Þ

where [A] represents the activity of an adsorbed molecule, the adsorbate, and [S] the activity of a site on the solid, the adsorbent. The adsorbate can also be called the adsorptive, especially in situations in which “adsorbent” and “adsorbate” may be confused. At equilibrium, an equilibrium constant, K, reflects the ratio of adsorption and desorption rate coefficients, ka/kd. Langmuir was attempting to provide a quantitative analytical background for heterogeneous catalytic reactions, and the sorption processes to which he was referring were generally exothermic and activated—what we would now call chemical adsorption, or chemisorption. However, it was soon understood that less exothermic processes could occur under the general heading of “adsorption,” and indeed, adsorption could occur without forming chemical bonds with the surface. Such nonchemical adsorption is called physical adsorption, or physisorption.

Analysis of Catalyst Surface Structure by Physical Sorption

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2.2. Choice of adsorbate The choice of adsorbate (vapor to be adsorbed) is largely dictated by the type of information desired and the adsorbent that one wishes to characterize. For physical adsorption, the adsorbate must be chemically inert with respect to all compounds present on the surface. Nitrogen and argon are common choices for this reason—and several other (very important) considerations: they are readily available, inexpensive, and relatively safe to handle. Nitrogen and argon also have another distinct advantage when it comes to porous materials: they are very small molecules and can thus penetrate much smaller pores and cover smaller surface features than larger molecules such as cyclohexane. Krypton has been employed as an adsorbate, particularly for materials with very low surface areas. The reason for this application is that krypton’s vapor pressure at liquid nitrogen temperature is very low (2 Torr (12)), meaning that errors in the “dead space” (Section 8.2.1) are less important. Xenon is also an inert probe for low-surface-area materials; like krypton, it is typically used at liquid nitrogen temperatures. This choice is one of practicality: liquid nitrogen is much cheaper than liquid krypton or xenon. It has the drawback that 77 K is well below the triple points of both krypton and xenon. Xenon NMR spectroscopy has also been employed to probe the texture and chemistry of surfaces; see the many studies by Fraissard et al. (13–18). Water can be used as an adsorbate, but its highly polar nature provides some interesting analysis quirks. The most important of these is that interactions between water molecules are sometimes stronger than interactions of water with surface atoms/molecules, meaning that the isotherm (see next section) is convex to the pressure axis. This is particularly true for graphite and other carbonaceous (i.e., nonpolar) adsorbents. Water adsorption therefore holds its own niche in the adsorption community and requires completely different interpretations than those characterizing more typical adsorbate–adsorbent interactions. We do not discuss water adsorption in this chapter. Carbon dioxide is an increasingly common choice as an adsorbate. However, it poses challenges in the interpretation of adsorption data: carbon dioxide sublimes under the conditions of most adsorption experiments, which implies that any concept of monolayer adsorption is complicated by the differences between the surface phase, a supercooled liquid reference phase, and the solid phase that actually occurs at saturation. It has also been

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known to chemisorb on many materials (12), and it is also known to intercalate between layers of carbonaceous materials (19,20), a process that is more akin to dissolution or absorption than adsorption. Hydrocarbons such as methane or butane are commonly used as adsorbates, but only for the purpose of estimating adsorption capacities of that specific hydrocarbon, such as for interpretation of diffusion or catalytic activity data. We emphasize, however, that the surface area and porosity accessible to larger hydrocarbons such as benzene or nonane may be smaller than those available to small molecules such as nitrogen or argon. Indeed, materials with multiscale porosity may have many small pores connecting larger ones; if the adsorbing molecule is too large to diffuse through the small pores at appreciable rates, large fractions of the porosity may be inaccessible. Furthermore, if the surface is atomically rough, the adsorbate might not adsorb on the rougher areas, obscuring the surface area. For the purpose of surface area determination and assessment of porosity, which are the primary topics of this chapter, the use of nitrogen and argon is nearly universal for the reasons discussed earlier in this section. Other inert gases, such as krypton or xenon, are also used in specialized adsorption experiments, such as determining “occluded” porosity (e.g., pores accessible to nitrogen but not xenon), assessing surface roughness (21), or determining surface areas of very low-surface-area materials.1 A list of common adsorbates and their properties is included in Table 1.1. It might be noted that nitrogen, oxygen, and carbon dioxide have nonzero quadrupole moments, whereas argon, krypton, and xenon do not. Quadrupolar interactions between adsorbing species and a surface have been invoked to explain several differences in adsorption phenomena (22). These include smaller molecular surface areas for nitrogen as a consequence of nitrogen’s ability to “stand up” on the surface because of interactions between its quadrupole moment and ions or polar functional groups on the surface, such as oxides or hydroxides. The coverage per molecule is less when the nitrogen molecule is perpendicular to the surface than it is when the molecule lies flat on the surface with its axis parallel to the surface. Quadrupolar interactions have also been suggested as the reason why nitrogen adsorbs on zeolites and other microporous silicates at lower pressures than argon (which lacks a quadrupole moment). However, we note that silicalite-1 exhibits the same differences between nitrogen and argon 1

Krypton and xenon are often used in low-surface-area analyses because their vapor pressures at liquid nitrogen temperatures are extremely low, minimizing errors in the “dead” volume (Section 8.2.4).

Table 1.1 Common adsorbates and their properties under the conditions of common adsorption experiments Vapor T (K) Am (Å2) Am (m2/cm3 STP) g (N/m) Vℓ (cm3/mol) P (Torr)

N2

77.36

N2

87.30

N2

90.20

Ar

77.36

16.2

a

4.30

17.0 b

18.0

4.51

34.7

760

4.16

(23–25)

0.00667

36.9

2130

2.95

(23,24)

0.00606

37.7

2750

2.65

(23–25)

27.5

230

c

c

4.78

a

4.40

0.01242

28.6

760

4.25

(12,27)

4.00

(24–26)

4.30

(25,28,29)

87.30

16.6

Ar

90.20

14.4

3.82

0.01186

29.1

1020

O2

90.20

14.1

3.74

0.0132

28.1

760

CO2

194.7

References

0.00888

Ar

b

CK (Å)

28.1

1320

(12,26)

d

14.1

3.74

Kr

77.36

b

(28,30)

20.8

5.52

1.78

(31,32)

Xe

77.36b

23.3

6.18

0.00187

(33,34)

There are two proposals (12): use 16.6 A for adsorption on both carbons and oxides, meaning Am(N2, 77 K) ¼ 16.2 A for adsorption on carbons and 19.3 A˚2 for ˚ 2 for adsorption on everything for nitrogen and use 13.8 A˚2 for argon adsorption on carbons and 16.6 A˚2 for adsorption on oxides. adsorption on oxides; or use 16.2 A Keeping the value for nitrogen constant and changing the value for argon is typical, although not necessarily motivated by any theoretical considerations. b Temperature is below the triple-point temperature. c The value of 230 Torr refers to the supersaturated liquid; the sublimation pressure at 77.36 K is 200 Torr (26). The value Am ¼ 18.0 A˚2 corresponds to BET plots using the sublimation pressure rather than the vapor pressure (12). d The value of 1320 Torr refers to the supersaturated liquid; the sublimation pressure is of course 760 Torr (30). Symbols: T, temperature of bath; Am, area of one monolayer for use with Equation (1.22); g, surface tension; and Vℓ, molar volume for the Kelvin equation (Equation 1.23); CK ¼ ð2gVl Þ=ðRT log10Þ, the constant for the reduced Kelvin equation (Equation 1.35); and P , saturation pressure. Rows in boldface indicate the most common and/or recommended bath temperature for the adsorbate in question. a

˚2

˚2

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adsorption as ZSM-5, even though the former has no aluminum or internal hydroxyl groups. We are unaware of any independent spectroscopic evidence of significant quadrupolar interactions giving rise to differences in orientation or energetics under the conditions employed in physisorption measurements, but we do not discount the possibility.

2.3. Presentation of adsorption data Adsorption data are largely presented in two ways: isotherms, or plots of quantity adsorbed against pressure or a similar abscissa at a fixed temperature; and isobars, or plots of quantity adsorbed against temperature or inverse temperature at constant pressure. Isotherms are much more common, as it is typically much more difficult to control pressure at non-atmospheric values than it is to control temperatures. A list of common low-temperature baths is given in Table 1.2. The vast majority of adsorption isotherms for the purpose of catalyst characterization are nitrogen isotherms recorded at 77 K (liquid nitrogen at its normal boiling point). Adsorption isotherms are plots of quantity adsorbed against pressure, fugacity, activity, or chemical potential. Pressure is nearly always the preferred abscissa in experiments: measuring the chemical potential is (unfortunately) rather difficult. Theoretical or simulated isotherms, however, often are based on chemical potential or activity instead. There is little difference Table 1.2 Common (and not-so-common) low-temperature baths used in physical and chemical adsorption Bath Bath temperature Adsorbates typically used Tt (K)d

Nitrogen

77.36 K (boiling)

N2, Ar, Kr, Xe

63.1526

Argon

87.30 K (boiling)

Ar, N2, Kr, Xe

83.8058

Oxygen b

90.20 K (boiling)

Ar,

N2a

54.36

194.7 K (subliming) Hydrocarbons, CO2

216.55

Ammonia

240 K (boiling)

195.40

Water/ice

273.15 K (freezing) Hydrocarbons, carbohydrates, CO2 273.16

Ambient air

292–300 K

CO2

Boiling water 373.15 K (boiling) a

Ammonia H2,c COc

N/A 273.16

Liquid oxygen baths are typically used only in specialized experiments because of safety concerns. Carbon dioxide “baths” are usually dry ice in a low-freezing liquid such as acetone or alcohol. c For chemisorption. d The symbol Tt indicates the triple-point temperature. b

Analysis of Catalyst Surface Structure by Physical Sorption

13

between these choices: the vapor phase obeys the ideal gas law at the pressures involved in most adsorption experiments, and thus the pressure and chemical potential are related via the following result, derived from the Gibbs–Duhem equation: m ¼ m þ kT log a ¼ m þ kT log f =f   m þ kT log P=P 

ð1:3Þ

where P is the saturation pressure, the pressure at which the bulk liquid is in equilibrium with the vapor at this temperature, and the quantity P/P is a dimensionless quantity called the relative pressure. Strictly speaking, the reference pressure is not necessarily equal to the saturation pressure, but the difference in chemical potential between two points on the isotherm will still be equivalent to a difference in the logarithm of the relative pressure. The quantity adsorbed is usually expressed in moles or the equivalent (see following paragraph) and normalized by the mass of the adsorbent. This normalization is dubious in some cases: the quantity adsorbed per gram of a very dense adsorbent may be quite high per unit volume of adsorbent, for example, despite low values of the number of moles per gram on the isotherm. Comparing the quantity adsorbed per gram of ceria (r  7 g/cm3) would not be a good comparison to the quantity adsorbed per gram of alumina (r  4 g/cm3), for example, because of the difference in density. When comparing quantity adsorbed across different materials, one should take care to normalize the plots in such a way that the resulting comparison makes logical sense. When measuring adsorption isotherms, it is common to substitute the number of molecules (or moles) adsorbed, n, for standard volumes adsorbed, V. The standard volume is simply the volume that the molecules would take up in an ideal gas at standard temperature and pressure (STP). What precisely it means to be at STP is somewhat varied; we discuss this in Section 9.1. Because these quantities differ by a constant, it makes very little difference which one is chosen. Because using standard volumes typically requires one fewer set of conversions (see Section 8), doing so is typical. From this point on, we make no distinction between the number of molecules or moles and standard volumes, and, in all instances when n appears in the equations in the preceding text, it can be replaced by V without changing the meaning of the equations. The astute reader will note readily that it is the logarithm of pressure that is proportional to changes in chemical potential, the driving force behind

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adsorption. Using logarithmic pressure axes, however, is hindered by the fact that taking the logarithm obscures most of the “interesting” parts of the isotherm related to porosity (Section 3), which occur in the decade between P/P ¼ 10
1 and P/P ¼ 1. Examples of adsorption isotherms can be found in figures throughout this document. We consider some model isotherms in the remainder of this section. We emphasize that none of these models predicts observed adsorption isotherms in perfect detail in all regions of the isotherms. Indeed, they tend to be good models of adsorption only for idealized systems or in relatively narrow regions of the adsorption isotherm.

2.4. The Langmuir isotherm The Langmuir adsorption isotherm (11) is the simplest model of adsorption that yields useful results. The Langmuir isotherm is based on the following assumptions: 1. The surface consists of a uniform two-dimensional array of identical adsorption sites. 2. The probability of adsorbing on or desorbing from a site is independent of the number of nearby molecules (the loading, y). 3. The activation energy for desorption is equal to the heat of adsorption, Q. 4. The vapor phase obeys the ideal gas law. 5. A site may not adsorb more than one adsorbate species at a time (no “layering”). With these assumptions, the number of molecules striking the surface per unit area (the flux, Jm) is the following (35): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ¼ Pk ð1:4Þ Jm ¼ P 2p Ma kT where P is the pressure of the gas, Ma the molecular mass of the adsorbate, k Boltzmann’s constant, and T temperature. If we define the loading (fraction of occupied sites) as y, then the fraction of empty sites is 1
y. If we assume that there is a probability, a, of a molecule sticking to the surface, then the rate of condensation is rc ¼ aJm ð1
yÞ ¼ aPkð1
yÞ:

ð1:5Þ

By the third assumption, the rate of evaporation is re ¼ xm ne
Q=kT y,

ð1:6Þ

15

Analysis of Catalyst Surface Structure by Physical Sorption

where xm is the number of surface sites per unit area and n is the frequency of vibration along the reaction coordinate that results in desorption. At equilibrium, rc ¼ re and, thus, aPkð1
yÞ ¼ xm ne
Q=kT y:

ð1:7Þ

Solving for the loading, we obtain the Langmuir adsorption isotherm, y¼

akP xm

ne
Q=kT

þ aLP

¼

bP 1 þ bP

ð1:8Þ

where b ¼ ak/(xmne
Q/kT) has units of inverse pressure and is often a fitted parameter. If we define Ps as the standard-state pressure, then the quantity K ¼ bPs is the equilibrium constant of the reaction, that is, ∘

∘

∘

∘

K ðT Þ ¼ e
DGads =kT ¼ e
DSads =k e
DHads =kT ¼ e
DSads =k eQ=kT ,

ð1:9Þ

where G is Gibbs free energy, S entropy, and H enthalpy. These results are summarized in Figure 1.1. The primary characteristic of the Langmuir isotherm is the last assumption: only a single layer forms, At equilibrium Kinetics

A+ ·

ka

→

kaP(1 – q) = kdq ⇒ q ka = P = bP ⇒ 1 – q kd bP q= 1 + bP

A·

kd

A· → A+ · q 1

b = 10 atm–1

0.75

b = 1 atm–1

0.5

0.25

b = 0.1 atm–1 1

2

P (atm)

Figure 1.1 The Langmuir (Type I) adsorption isotherm. In this model, a single layer, or monolayer, forms when adsorbate particles adhere to specific sites on the surface, resulting in a horizontal asymptote at unit loading. The loading, y, is the quantity adsorbed divided by the quantity adsorbed at saturation (infinite pressure). Higher values of b indicate stronger adsorbate–adsorbent interactions.

16

Karl D. Hammond and Wm. Curtis Conner Jr.

meaning there is saturation (a horizontal asymptote) once monolayer capacity is reached. Equation (1.8) is the basis for most theories of heterogeneous catalysis and chemical adsorption.

2.5. Monolayers to multilayers The Langmuir adsorption isotherm (Equation 1.8) is based on the assumption that adsorption proceeds from zero up to saturation, y ! 1. The adsorbate species in the gas or liquid phase are in equilibrium at a specific temperature and pressure with the species adsorbed on the surface. All species adsorbed are presumed to have equal chemical potentials, which do not depend on the presence of other adsorbed species. Thus, it is assumed that the interactions of each adsorbing species with the surface are identical and that interactions between adsorbate atoms/molecules on the surface are much weaker than the interactions with the surface. When applied to reversible chemical adsorption (chemisorption), the Langmuir adsorption isotherm applies to both uniformly distributed sites, in which case the loading is proportional to the fractional coverage of the surface, and sites distributed on only a (possibly small) fraction of the surface. In the latter case, saturation of the Langmuir adsorption isotherm represents the covering of only a part of the surface. In physical adsorption, in contrast to chemisorption, the entire surface accessible to the adsorbate is involved. Thus, to a good approximation, the surface is entirely composed of active sites, and saturation would be achieved when the adsorbing species completely covers the surface. This would represent a monolayer of adsorbed species. Intuitively, the maximum coverage would reflect the closest packing of adsorbing species on the surface. In physisorption, the forces between an adsorbed species and a surface site are relatively weak, and the adsorbed species are relatively free to move across the surface and to change the surface sites with which they primarily interact. This facile, two-dimensional mobility also differentiates physical from chemical adsorption. Physisorption occurs as a consequence of the interactions between any surface and molecules as the temperature approaches the boiling (or dew) point of the molecules in the gas. It begins at a temperature or pressure substantially below the actual pressure or temperature at which bulk

Analysis of Catalyst Surface Structure by Physical Sorption

17

condensation would occur. As an analogy, we can feel the effects of humidity even though it is not raining, or even “damp,” outside. When an atom or molecule approaches any surface, it is influenced by forces of attraction (e.g., van der Waals forces). A molecule is also influenced by such forces when it approaches other adsorbed molecules. In physical adsorption, whereby adsorbate–adsorbent interactions are relatively weak, a molecule that encounters a molecule already adsorbed will be influenced by similar (albeit weaker) forces of attraction than those involving a surface site. It will have a probability of adsorbing on top of a group of already adsorbed molecules, which will probably be less than the probability of adsorbing if it had encountered the uncovered surface. The difference in the probabilities between adsorbing on top of already adsorbed species and on exposed solid surface is directly related to the difference between the energies of attraction between the surface and an adsorbing molecule and between an adsorbed molecule and an adsorbing molecule. The surface also contributes to the forces of attraction for adsorption of the molecules in the second and higher layers, but the forces are reduced because the molecules are at a larger distance from the surface. Physical adsorption will therefore involve the formation of more than a single layer of adsorbed molecules as the pressure increases. Thus, multilayer adsorption is primarily a property of physical adsorption. It can, however, be found for chemisorption if subsequent layers differ in composition, as in atomic layer deposition (36). If we wish to interpret the relationship between the quantity adsorbed and pressure under isothermal conditions (or the quantity adsorbed and temperature under isobaric conditions), it is necessary to understand multilayer adsorption—specifically, the relationship between adsorption of the first layer (the monolayer) and adsorption of subsequent layers. Probability (and thus entropy) leads one to conclude that the second layer should start to fill before the first layer is completed, provided there is not an extremely large difference in the heat of adsorption between these layers. As the number of adsorbed layers increases, it is also reasonable to assume that the heat of adsorption will eventually approach the heat of condensation of the adsorbate. Several relationships have been proposed to express the changes in the amount adsorbed and the pressure and temperature for adsorption up to and in excess of a monolayer. We discuss several of these throughout this chapter.

18

Karl D. Hammond and Wm. Curtis Conner Jr.

2.6. BET theory 2.6.1 The BET equation By far, the best known model of multilayer adsorption is that developed by Brunauer, Emmett, and Teller (37), universally known in the adsorption community as BET theory. This theory was developed to describe the initial adsorption of a monolayer and the simultaneous adsorption of multilayers. It starts with the premise that more than a single layer can be formed on a surface. It is further based on the assumption that the energy of interaction between the adsorbing species and the surface is strongest in the first layer and decreases for subsequent layers. To simplify the analyses, Brunauer, Emmett, and Teller made a further assumption: the energy of interaction (heat of adsorption) between Nth and N þ 1st layers for N  2 and as N ! 1 is the same as the heat of condensation. The BET theory is also based on the assumption that the corresponding sticking coefficients and attempt frequencies for the second and higher layers are the same as for the second layer. Only the forces of interaction (and sticking coefficients and attempt frequencies) between the surface and the first layer are different in the BET theory. Furthermore, it is assumed that the volume of each adsorbed layer is identical. This is equivalent to assuming that the surface is flat (smooth on an atomic scale). Moreover, at P ¼ P , the saturation pressure, the number of layers is infinite, and the adsorbate density becomes identical to that of the bulk liquid. Just as in the Langmuir expression (Equation 1.8), it is possible to express the formation of a monolayer by considering the rate of adsorption onto empty sites and their rate of desorption. We express the fraction of empty sites as y0 and the concentration of those sites covered in the first layer as y1 (and so on for higher layers). The heat of adsorption in the first layer is Q1, and xm is the number density of “sites” in the sample, as before. By analogy to Equation (1.7), the rate of ad/desorption for each layer i at equilibrium is as follows: ai Pkð1
yi Þ ¼ xm ni e
Qi =kT yi :

ð1:10Þ

The attempt frequency ni is, at the microscopic level, the vibrational frequency of the normal mode of the adsorbed complex that, if sufficiently excited, will result in desorption of a molecule. It is never actually measured, nor is it necessary to do so.

19

Analysis of Catalyst Surface Structure by Physical Sorption

By definition, 1 X

yi ¼ 1,

ð1:11Þ

i¼0

P and the number of molecules adsorbed on the surface is n ¼ nm 1 i¼0iyi, where nm is the monolayer capacity (total number of sites in the sample). From Equation (1.10) and the assumption that the second and higher layers have identical properties, we find the following set of equations: y1 ¼ y2 ¼

a1 Pe
Q1 =kT y0 xm n1

a2 a1 a2 2
ðQ1 þQL Þ=kT Pe
QL =kT y1 ¼ P e y0 xm n2 xm n1 xm n2

 2 a2 a1 a2
QL =kT
Q1 =kT
QL =kT y3 ¼ Pe y2 ¼ Pe Pe y0 xm n2 xm n1 xm n2 . ..  i
1 a2 a a2 1
QL =kT
Q1 =kT
QL =kT yi ¼ Pe yi
1 ¼ Pe Pe y0 ð1:12Þ xm n2 xm n1 xm n2 These equations can be written more concisely if we define a ¼ y1/y0 and b ¼ y2/y1. We define another constant, C, as their ratio,   a a1 n2 Q1
QL C¼ ¼ exp ð1:13Þ kT b a2 n1 and write the fractional coverage of each layer as follows: yi ¼ abi
1 y0 ¼ bi Cy0 :

ð1:14Þ

C is positive and dimensionless. From Equation (1.11), we can write " # 1 1 X X y0 ¼ 1
yi ¼ 1
bi Cy0 : ð1:15Þ i¼1

i¼1

P i Because b < 1, we recognize the geometric series b/(1
b) ¼ P 1 i¼1b , 1 2 meaning that y0 ¼ 1
b/(1
b þ Cb). Using the series x/(1
x) ¼ i¼1ixi in the result for the total number of molecules adsorbed, n, and rearranging, we get the following result:

20

Karl D. Hammond and Wm. Curtis Conner Jr.

n Cb ¼ : nm ð1
bÞð1
b þ CbÞ

ð1:16Þ

If we now factor in the assumption that n ! 1 as P ! P , then we know, from the definition of b, that lim b ¼ 1 ¼

P!P 

a2  QL =kT P e , xm n2

ð1:17Þ

which means that b ¼ P/P and thus n CP=P  : ¼  nm ð1
P=P Þð1 þ ðC
1ÞP=P  Þ

ð1:18Þ

Equation (1.18) is the BET adsorption isotherm. To find the number of molecules in one monolayer, which is proportional to the surface area, it is convenient to rearrange this equation into something easily plotted, such as P=P  1 C
1 ¼ þ P=P  : nð1
P=P  Þ nm C nm C

ð1:19Þ

Equation (1.19) is called the BET equation. A plot of the left-hand term, P/ n(P
P), versus P/P yields (if the model assumptions are accurate, at least) a straight line with slope (C
1)/nmC and intercept 1/nmC. The surface area, SABET, and the value of C (often written CBET) are therefore given in terms of the slope, SBET, and the intercept, IBET, by the following: SABET ¼

Am SBET and CBET ¼ 1 þ SBET þ IBET IBET

ð1:20Þ

where Am is the area one molecule occupies on the surface. The general approach employed in the BET theory is depicted in Figure 1.2. 2.6.2 The constant C in the BET equation The value of C in Equation (1.19) reflects the differences between the formation of the first layer and the formation of subsequent layers (i.e., a/b as in Equation (1.13)). In the BET formulation, this is viewed as the difference between the first and second layers, with all layers from 2 to 1 regarded as being similar. Differences between the reflection coefficients and attempt frequencies between the first and second layer are presumably small (of order unity), so that the value of C is most sensitive to the difference in interaction energy (heat of adsorption) between the surface and the first layer and

21

Analysis of Catalyst Surface Structure by Physical Sorption

Liquid-like second and higher layers Monolayer Adsorbent

kd,i qi = ka,iPqi–1

Each layer is assumed Langmuir-like on top of prior layers, but the first layer differs from all higher layers.

BET Equation P/ P ° C−1 1 + P /P ° = Vads (1 − P/P °) CV m CV m

P /P ° Vads (1 − P / P °)

C = Ae

Q1−Q2 kT

BET Plot Vm = slope =

C−1 CVm

intercept =

1 slope + intercept

1 CVm P / P°

>0

C = 1+

slope intercept

Figure 1.2 Schematic representation of the BET adsorption isotherm and its assumptions. The monolayer volume, Vm, from the BET plot is often used to estimate the surface areas of catalysts, provided that the value of C is reasonable and the assumptions of the model apply.

between the second and subsequent layers (these latter differences tend to the heat of condensation). That is,     a1 n2 Q1
Q2 Q1
QL SBET C¼ exp  exp ¼1þ a2 n1 kT kT IBET

ð1:21Þ

The value of C therefore reflects the difference in the heat of adsorption for the first layer compared with the heat of condensation. The value of C is thus sensitive to the enhancement resulting from adsorption in comparison with bulk condensation. Large values of C reflect high adsorption energies

22

Karl D. Hammond and Wm. Curtis Conner Jr.

for the first layer, whereas small values of C reflect small differences in adsorption compared with condensation. The ability to calculate a monolayer volume from an adsorption isotherm depends on the nature of the isotherm and, thus, on the difference in the energy for the interaction between the surface and the first layer and the energy of interaction between the first and subsequent layers. Low values of C calculated from the BET equation can mean that the first layer is not significantly enhanced in adsorption compared with subsequent layers, and it will therefore be difficult to determine a proper value of the monolayer volume. A rule of thumb is that C must be greater than 50 for the BET theory to give rise to a reasonable calculation of the monolayer volume and, thus, the surface area (38). A C value of 20 corresponds to a difference of greater than 1.92 kJ/mol in the heat of adsorption of nitrogen at 77 K, for example; the heat of condensation is 5.56 kJ/mol at this temperature. At the other extreme, large values of C reflect (in the BET theory) large differences in the energy of interaction for the first layer compared with subsequent layers. The theory was developed to represent physical adsorption on a flat (or nearly flat) surface, not chemisorption or adsorption in micropores (i.e., pores less than approximately 2 nm in radius, for which the assumption of a flat surface is no longer valid). Thus, there is an upper limit to the value of C for which BET analysis is reasonable to employ. Sing et al. (12,38) suggested that values of C greater 200 found in BET analyses would make the analysis questionable, and therefore the surface areas calculated from such data should be used only with reservation. A C value of 200 corresponds to a difference in the heat of adsorption of 19.2 kJ/mol for nitrogen adsorption at 77 K. The shape of the BET isotherm as a function of the C parameter is shown in Figure 1.3. It is apparent from Figure 1.3 that isotherms with low values of C do not exhibit a definite transition between the first and subsequent layers (this transition occurs at a relative pressure of 0.1), whereas isotherms characterized by higher values of C (>200) exhibit a transition at much lower relative pressures. Intermediate values of C (and thus the difference between the adsorption in the first layer and that in subsequent layers) give an easily distinguishable transition from monolayer to multilayer adsorption. Higher values of C imply strong adsorption (i.e., more than simple physical adsorption onto a flat surface). We emphasize that the calculation suggested earlier (C ¼ 1 þ slope/intercept) can be extremely sensitive to the value of the intercept. This point is

2 10,000 1000 200

1.5

Loading (n / nm)

100

50

1

10 10,000 1000 200 50

0.5

C=1

10 1 0 10−6

10−5

10−4 10−3 10−2 Relative pressure (P/P°)

10−1

0.1

0.2

0.3

0.4

0.5

Relative pressure (P/P °)

Figure 1.3 The shape of the BET isotherm varies significantly as a function of the C parameter. Values of C between 50 and 200 (shaded region) are generally considered “reasonable”; values outside this range are found in situations for which the assumptions underlying the BET theory are likely invalid. The plot on the left is represented with the pressure on a logarithmic scale; such plots are typical for high-resolution (micropore) adsorption isotherms.

24

Karl D. Hammond and Wm. Curtis Conner Jr.

particularly problematic because the BET surface area parameter is often not particularly sensitive to the intercept. The slope is always positive (unless C < 1, in which case the theory decidedly does not apply), and so, the intercept must be positive. If a negative intercept is found or the intercept is close to zero, it may indicate that the assumptions of the BET isotherm are not satisfied. This result could also indicate significant noise in the data, suggesting the need for collection of more data in the same pressure regime and the use of more sophisticated methods of determining C, such as nonlinear regression. 2.6.3 Surface area from the BET isotherm By far, the most important use of physisorption data in the catalysis community is in the determination of catalyst surface areas. This determination is done through use of Equation (1.19) and the use of a value for the area of a single molecule: SA ¼ nm Am

ð1:22Þ

where Am is the area of a molecule or atom of adsorbate and nm is the monolayer capacity (quantity adsorbed in one monolayer). The value of Am varies with temperature and adsorbate. For nitrogen at 77 K, the nearly universally accepted value is 16.2 A˚2 per molecule (12); this corresponds to 4.30 m2/cm3 STP. For adsorbates other than nitrogen at its normal boiling point, the value of the specific surface area is less agreed upon, which is one reason for the complete dominance of nitrogen for this purpose. Argon at 77 K has an illdefined molecular area, as 77 K is below argon’s triple point of 83.8 K. Thus, solid argon forms at saturation. Consequently, there are two possible reference states: the solid, which may not actually cover the surface or even be the “correct” thermodynamic state so close to a surface, and the supercooled liquid, which is of course a metastable phase. When solid argon is used as ˚ 2 per atom is often used (38). When the reference, the value Am ¼ 18.2 A  supercooled liquid argon (P ¼ 220 Torr at 77.2 K (12)) is used as the refer˚ 2 per atom. The assumption of close-packed ence, the value of Am is 17 A ˚ 2 per atom (12). Some instruments default argon results in a value of 13.8 A 2 ˚ per atom for argon (39). Because of the uncerto a value of Am ¼ 14.2 A tainty, it is in general best to avoid surface area measurements with argon at 77 K. Argon at 87 K is above its triple-point temperature, and so the liquid phase is the only realistic reference state. In principle, the gas should take up slightly more area at 87 K than it does at 77 K, but in practice, the

Analysis of Catalyst Surface Structure by Physical Sorption

25

difference is smaller than the uncertainty in the measurement of Am. The value of the area for a close-packed liquid is thus often used, meaning ˚ 2 per atom. A value of 14.2 A˚2 per atom is used in some cases Am ¼ 13.8 A as well (39). In general, surface areas can be determined from argon isotherms, but the value of the specific surface area assumed in the calculation should be specified along with the C constant in the BET equation. Values for other temperatures and other adsorbates are presented in Table 1.1. If the assumptions underlying the model hold, Vads(1
P/P ) is a strictly increasing function of relative pressure in the range in which the BET equation is applied, and if the value of C is reasonable, then measured values of the BET surface area are typically repeatable within 5%. 2.6.4 Rough estimates: The single-point BET surface area The value of Am ¼ 4.30 m2/cm3 STP for nitrogen at 77 K lends itself to an easy estimate of the BET surface area of a non-microporous material. Pick a point on the isotherm that is above monolayer coverage but below any mesopore filling; P/P ¼ 0.2 is usually a good choice, although values anywhere in the range from 0.1 to 0.25 have been used by various authors, depending on their guess of “Point B” defined by Brunauer et al. (37). Now, multiply the volume adsorbed (in cm3/g) by 4.3 (or 4 for a rougher estimate). The result is a very rough estimate of the surface area. 2.6.5 Weaknesses of the BET theory The BET theory was formulated on the basis of a series of assumptions (Section 2.6.1) that may or may not be too restrictive for a particular system. Fortunately, for a large fraction of solids, these assumptions are appropriate at relative pressures below P/P ¼ 0.3 (i.e., an average of approximately one monolayer of adsorption on a smooth surface). One problem with the formulation of the BET theory is that each individual molecule added on top of another molecule in a partial layer is viewed as being adsorbed with the same energy as found for bulk condensation. The interactions between molecules in a given layer are also disregarded. Thus, the n þ 1st and n þ 2nd layers may begin to form before the nth layer is complete. This picture also does not fully account for the entropy of adsorption: it accounts for changes in configurational entropy (ways of arranging molecules on the surface), but neglects entropy arising from molecular mobility, as the molecules are fixed in position in the BET model (12). It is also difficult to interpret rough surfaces in the context of BET theory, as such surfaces violate the assumption of an array of nearly identical adsorption sites.

26

Karl D. Hammond and Wm. Curtis Conner Jr.

Halsey (40) observed that the BET theory includes “the quite untenable hypothesis that an isolated adsorbed molecule can adsorb a second molecule on top, yielding the full energy of liquefaction, and that in turn the second molecule can adsorb a third....” One would expect that the linear picture of columns of molecules would not be formed, but instead layers would more closely approximate close-packed layers, in which subsequent adsorbing molecules can interact with more than one molecule in the layer(s) below. These effects, if accounted for, would all add small corrections to the BET equation, some of which become more important for specific systems. The assumption that the second and subsequent layers all have adsorption energies that are equal to the energy of condensation neglects the possibility that the second layer may be influenced by the solid surface, which is only a ˚ ngstro¨ms distant. In many cases, the second layer will be influenced by few A the presence of the surface below and thus interact with the surface. This net interaction energy in the second layer will fall somewhere between that of the first layer and the energy of condensation. As layers above the second layer are formed, the differences between the first and the second and the second and the third layers become evident. The BET theory therefore overestimates the rate at which multilayers form and does not account for the adsorption entropy. It also simplifies the energy of interaction between layers. However, these problems occur primarily at loadings above an average of one monolayer of adsorption. At loadings up to one monolayer average coverage, the BET theory has been shown to provide the most consistent approach for the estimation of the exposed surface area for surfaces for which appropriate values of C are found (12,38,41)—that is, for C > 50 (reasonably strong forces of adsorption) and C < 200 (not too strong adsorption or samples containing micropores)— such that a transition between monolayer and multilayer adsorption can be found. Recent investigations have suggested that the BET surface area may be estimated—by use of a different region of the isotherm—for materials that contain micropores. We reserve a full discussion of these proposals for Section 4.4, after we discuss adsorption in microporous materials (Section 4). We mention one further criterion, however, that was first suggested by Rouquerol in the late 1960s (38): the region to which the BET equation is applied should always have the denominator of the left-hand side of the BET equation (Equation 1.19) increasing with relative pressure; that is, Vads(1
P/P ) increases with P/P .

Analysis of Catalyst Surface Structure by Physical Sorption

27

3. A TOUR OF THE ADSORPTION ISOTHERM: FROM VACUUM TO SATURATION AND BACK A physical adsorption isotherm can be analyzed to determine a variety of morphological characteristics of a solid. No single theory is able to reflect all physical interactions for sorption (adsorption and desorption): from the first few sorbing molecules, to a monolayer, to multilayers, to condensation of a liquid (or even a solid) throughout the system. Theories have been developed to represent each of the sequential processes associated with the measurement of sorption. In this section, we offer a tour of the physical adsorption isotherm, starting at the lowest pressures that can be obtained by conventional vacuum equipment (typically P  10
8 P or higher), progressing in order through the following regions: micropore filling, surface coverage (monolayer formation), mesopore filling, macropore filling, saturation, macropore emptying, and mesopore emptying. The astute reader will recognize that these regions often overlap—because the transition between them is often unclear—and thus analysis is typically restricted to ensure applicability of the given model analyses. We use Figure 1.4 as a guide.

3.1. The micropore-filling region: 10
8 < P/P < 10
2 The first range of relative pressures (P/P , where P is the saturation pressure at the measurement temperature) in which significant sorption can be found is 10
8 < P/P < 10
2. We call this the micropore-filling region, as it is the region in which the smallest pores (0–1 nm in radius) fill. For such pores, the heats of adsorption are most often significantly greater than the heat (enthalpy) of condensation as a consequence of the interactions between the adsorbing molecules and the pore walls, which are close enough together that more than one solid surface interacts with each adsorbate molecule at the same time. Models with which this region of the isotherm can be analyzed are described in Sections 7.1.3 and 7.2.2. Neither of the isotherms depicted in Figure 1.4 enables one to determine how much microporosity is present, as the micropores fill before the plot starts to deviate visibly from the vertical axis. It is standard practice to express the data in an expanded form (Vads vs. the logarithm of P/P ) in order to discriminate among the differences in adsorption for different microporous systems. This practice is often referred to in the context of high-resolution

28

Karl D. Hammond and Wm. Curtis Conner Jr.

1000

Volume adsorbed (cm3 STP/g)

800

600

400

200

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.8

0.9

1

Relative pressure (P/P°) 500

Volume adsorbed (cm3 STP/g)

400

300

200

100

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Relative pressure (P/P°)

Figure 1.4 Nitrogen isotherms at 77 K on SBA-15 silica samples that incorporate both micropores and mesopores, suggesting different regions of the isotherm as discussed in the text. The top plot indicates a solid containing larger mesopores than the lower plot; the material represented in the lower plot has pores that lie right on the transition between micropores and mesopores.

Analysis of Catalyst Surface Structure by Physical Sorption

29

Figure 1.5 Schematic representation of pore filling in micropores. The pores “fill” with adsorbate before the exterior surface is covered. Reprinted with permission from Ref. (43). Copyright © 2005, American Chemical Society.

adsorption, or HRADS (a term coined by Venero and Chiou (42)), as discussed in Section 4. A representation of the phenomena associated with this region of the adsorption spectrum is shown in Figure 1.5. The pores fill at pressures well below those required to give a monolayer on the exterior surface because of

30

Karl D. Hammond and Wm. Curtis Conner Jr.

the three-dimensional interactions between the sorbing molecule and the surface. It is not even clear what is the density of the sorbed species when the pores are filled, as this depends on the molecule-to-surface and molecule-to-molecule interactions, which can differ even for physisorption.

3.2. Monolayer region: 0.05 < P/P < 0.2 for nitrogen at 77 K or argon at 87 K All relatively flat surfaces for which there is an attractive interaction (exothermic, negative sorption enthalpy) will progress from partial surface coverage through an average of one atomic sorbent layer, a monolayer, to the formation of multiple layers on the surface. These isotherms are most often first analyzed (via the BET theory, Section 2.6) to estimate the amount of a sorbing species comprising the monolayer, from which the surface area of the solid can be estimated. The relative differences between the energies of interaction, contained in the BET C parameter, are central to one’s ability to estimate the monolayer capacity. This point was discussed in Section 2.6.1. The surface areas from conventional analyses of the isotherms in Figure 1.4 are 880 m2/g for Figure 1.4A and 820 m2/g for Figure 1.4B; the calculated CBET parameters are 118 and 72.5, respectively. These results indicate that the synthesis of this particular SBA-15 sample did not create micropores, as microporosity often causes the BET constants to take on unreasonable, even negative, values. This point is discussed further in Section 4.4.

3.3. The multilayer region: 0.2 < P/P < 0.4–0.98 for nitrogen at 77 K (no hysteresis) There is usually a region above the monolayer formation pressure (P/P  0.2) in which the adsorption and desorption processes are coincident. If the slope of the isotherm continues to decrease, pores are not being filled, in which case it is perceived that this is the relative pressure region at which a monolayer has already been formed while further sorption proceeds to form multilayers. This picture reflects a “thickening” of the adsorbed layer. It is normally analyzed by empirical methods such as a “t-plot” (Section 6.2.1), an “as-plot” (Section 6.2.2), or a “Halsey relationship” (Equation 1.27). These analyses—both by themselves and as they pertain to the estimation of pore dimensions—are discussed in Section 6. Conventional analyses disregard this region of the isotherm.

3.4. Mesopore-filling region: 0.2 < P/P <  0.45 (no hysteresis) and 0.4 < P/P < 0.99 (hysteresis present) If there are any pores between 2 and 50 nm in radius/width (whether cylindrical, spherical, slit-like, or some combination in a

Analysis of Catalyst Surface Structure by Physical Sorption

31

three-dimensional network), these void spaces will gradually fill with condensing adsorbate as the pressure increases. The smaller pores fill and empty at lower relative pressures. At the same time, more is adsorbed as the exposed surface becomes thicker. The existence of hysteresis—a difference in quantity adsorbed at the same relative pressure between the adsorption and desorption branches of the isotherm—is discussed in Section 5. We stress that mesoporosity often is present when a sample consists of an agglomerate of particles: such porosity is created between the particles, and the voids created by the agglomeration are often similar in size to the dimensions of the primary particles (Figure 1.6).

Figure 1.6 Simplified representation of the process of adsorption and desorption in mesopores, showing surface coverage, pore filling, pore emptying (both by cavitation and otherwise), and saturation. Note that the surface is covered before the pores fill. Reprinted with permission from Ref. (43). Copyright © 2005, American Chemical Society.

32

Karl D. Hammond and Wm. Curtis Conner Jr.

3.5. Adsorption on exterior surfaces: 0.45 < P/P < 1 For a porous sample, adsorption will continue after the internal surfaces (those inside pores) are occluded by the filling of the pores with condensing adsorbate. Often, the vast majority of the accessible surfaces are internal (and the exterior surface is small by comparison). If this is the case, the slope of the adsorption isotherm after the pores fill will be minimal. This phenomenon is evident in Figure 1.4B in which the slope of the isotherm is relatively small for relative pressures above 0.47.

4. HIGH-RESOLUTION ADSORPTION: P/P < 0.01 HRADS is both a fancy name for adsorption in the micropore-filling region (10
8 < P/P < 10
2) and a descriptive term for the process of acquiring measurements in that pressure regime. Performing HRADS typically requires more precise equipment, more patience, and more time. We discuss several aspects of HRADS in this section.

4.1. Measurement of HRADS isotherms The process of measuring adsorption in the range 10
8 < P/P < 10
2 has now become more common with the advent of automated, computercontrolled sorption apparatus. Nitrogen or argon is employed as the adsorbate, usually at its normal boiling temperature—that is, using liquid nitrogen (77 K) or liquid argon (87 K) baths for temperature control. The difficulty in these experiments is accurate measurement of the pressure over the range of 10
5 Torr2 up to 103 Torr. Most non-micropore analyses (BET surface area, mesopore size distribution, and thickness of surface layers) are performed in the last two decades of relative pressure (10
2–100), and thus, the use of a single pressure transducer with a range of 0–1000 Torr or similar is usually sufficient. However, no single pressure transducer is able to cover the whole range of HRADS with the necessary precision. The most precise pressure measurement devices over a range of pressures near 1 Torr are diaphragm transducers, which measure the capacitance between a flexible metallic membrane and a solid plate as a function of the external stress (i.e., pressure) on the plate. The useful 2

The torr is a unit of pressure equal to 1 mm Hg at 273.15 K and 45 latitude (i.e., the pressure exerted by a 1-mm column of mercury at 0  C at 45 latitude), which is about 133.3 Pa. The standard atmosphere is defined to be 1 atm ¼ 760 Torr ¼ 101.325 kPa.

Analysis of Catalyst Surface Structure by Physical Sorption

33

range of pressures that can be measured with such transducers is four decades at best, with accuracy increasing near the top of the range. Thus, a 1000 Torr transducer (e.g., one that translates a pressure in the range 0–1000 Torr into a voltage in the range 0–10 V) is most accurate in the range 100–1000 Torr, is marginally accurate from 1 to 10 Torr, and is not accurate at all below about 0.1 Torr (about 10
4 atm). Consequently, at least two transducers are required to cover the required range of pressures, preferably a combination that includes a transducer that is accurate (1%) at approximately 10
5 Torr. Not all adsorption instruments that claim to measure microporosity by HRADS employ pressure measurement systems (transducers) with this precision, and in the ones that do, the low-pressure transducer is often optional equipment. In addition to the required pressure measurement accuracy, the measurements must be performed over a sufficiently long time that adsorption equilibrium is achieved. This concern is often not readily apparent: if one watches the pressure drop, it may appear stable for several minutes before dropping as little as 10
4 Torr, but over a longer period (say, 30 min), the pressure will drop by more than two or three times that, and neglect of the continuing change can lead to significant errors in the determination of pore sizes. The experimental problem is that the heats of adsorption in microporous solids are unusually high, often significantly higher than the heat of vaporization. Furthermore, the rates of heat and mass transfer to and from the micropores are low because of the low pressures involved with samples that are essentially thermal insulators and held in glass (a good thermal insulator itself ). It therefore takes a considerable time for sorption equilibrium to be reached: as much as an hour or more between points may be necessary at the lowest pressures at which adsorption takes place. It is extremely important that the measurements be performed properly—consequences of not doing so range from inaccurate determinations of pore sizes to nonphysical results (such as oscillating isotherms (44)). The measurement of an HRADS isotherm can easily take more than 18 h, during which the liquid nitrogen or argon baths must be maintained at constant levels (see Section 8.2.1, where we discuss dead space calculations, for why). Thus, those wishing to perform HRADS must choose when to perform HRADS measurements so as to minimize the inconvenience, and it is therefore important to assess, quickly, whether such an analysis is required. If one has a zeolite, clay, carbon black, or other sample suspected

34

Karl D. Hammond and Wm. Curtis Conner Jr.

of being microporous, one probably needs to perform such measurements. For an unknown sample, the first measurement would be a 5–7 point BET measurement and calculation. If the amount adsorbed at the lowest pressure (usually P/P ¼ 0.05) is significant (>20 cm3/g), then HRADS experiments might be warranted.

4.2. Collection and presentation of HRADS isotherms The very low pressures involved in micropore adsorption mean the isotherm usually will not appear to deviate from the vertical axis within the thickness of the lines on most plots until the micropores have all filled and a significant portion of a monolayer already covers the external surface (Figure 1.4). Thus, it is customary to plot high-resolution isotherms either on a very narrow domain of relative pressure (e.g., P/P in the range 0 to 0.001) or on a semilogarithmic plot. The semilogarithmic plot has an added bonus: differences in the logarithm of relative pressure are proportional to differences in chemical potential. Results of HRADS on several microporous zeolites are shown in Figure 1.7, which indicates several points unique to high-resolution adsorption. Such materials typically show a Langmuir-like isotherm with a transition in the range of 300

Volume adsorbed (cm3 STP/g)

250

ATS MTT MTW TON VET

200

150

100

50

0 10−7

10−6

10−5

10−4

10−3

−2

10

10−1

100

Relative pressure (P/P°)

Figure 1.7 Low-pressure (high-resolution) argon isotherms at 87 K, characterizing zeolites with the frameworks MTT, VET, TON, ATS, and MTW. Lines are shown to guide the eye.

Analysis of Catalyst Surface Structure by Physical Sorption

35

10
6–10
3 relative pressure, followed by a gradual rise in the quantity adsorbed as a result of monolayer and multilayer formation on the external surface and further densification of the adsorbed fluid inside the micropores. We emphasize that desorption is usually not measured for microporous materials. The reason for this limitation is one of practicality: in measuring adsorption (a non-activated process), it can take more than an hour to collect one datum. Most of this time is spent waiting for thermal equilibrium to be reached between a low-pressure gas (often in equilibrium with fluid adsorbed on a thermal insulator) through a poor heat conductor (glass) over a temperature gradient of a few tenths or even hundredths of a Kelvin. Measuring desorption would in turn require the transfer of the heat of adsorption to the adsorbate when the temperature gradients are even smaller. In the authors’ experience, attempts to obtain desorption data at relative pressures below 10
4 require the order of 6 h under constant bath height to gather one point on the desorption branch of the isotherm. The possibility of acquiring a few dozen points, each requiring about a quarter of a day, is impractical. We also stress that it is quite difficult to maintain a cryogenic (liquid nitrogen, argon, or oxygen) bath for long times, as oxygen and nitrogen from the atmosphere will dissolve into the bath after extended periods, changing its boiling point. The bath itself will also boil off and have to be replenished, usually more than once, during this period of experimentation.

4.3. Position of the inflection point The position of the inflection point (roughly the center of the sharp rise in quantity adsorbed) is fairly well correlated with pore size in one-dimensional zeolites, as shown for argon adsorption in Figure 1.8. Similar data for nitrogen adsorption are presented in Table 1.3. One-dimensional zeolites are those incorporating pores that are largely connected in straight or possibly “zigzag” fashion and do not directly intersect with each other. Some zeolites, exemplified by FAU, have three-dimensional structures in which large “cage” pores are connected by relatively small “window” pores. The position of the inflection point itself changes with adsorbate. The most common comparison is nitrogen at its normal boiling temperature (77 K) with argon at its normal boiling temperature (87 K). In this case, the actual pressure (e.g., in pascals or torr) is the same for each value of relative pressure, meaning the comparison is relevant to the accuracy of measurement (Section 4.1). The difference is striking: almost a decade in relative pressure for many microporous materials. Examples of this difference are shown in Figures 1.9–1.11, which

36

Karl D. Hammond and Wm. Curtis Conner Jr.

1.4

1.2

1D Pore network zeolites 3D Pore network Dp = 3.046 nm (P/P°)0.15169 Dp = 16.1823 nm [−log(P/P°)]−1.39845

Pore diameter (nm)

1

0.8

0.6

0.4

0.2

0 10−6

10−5

10−4 10−3 Relative pressure (P/P°)

10−2

Figure 1.8 Correlation between crystallographic pore size and the inflection point (center of steep region) of the adsorption isotherms characterizing various zeolite structures (listed in Table 1.3) for argon adsorption at 87 K. Note that a similar correlation exists for nitrogen at 77 K, although the inflection points are shifted to lower relative pressures (Table 1.3). Consequently, argon isotherms obtained with liquid argon baths are typically more accurate in this range.

show a comparison of adsorption of nitrogen and argon on small-, medium-, and large-pore zeolites at both 77 and 87 K. The isotherms for nitrogen at 87 K are omitted, as measuring them requires a special pressure transducer accurate in the range of approximately 2000 Torr, with which the instrument used to obtain the data was not equipped.

4.4. Estimating surface areas in microporous materials Microporosity is typically a barrier to determination of the accurate surface area of a material: the prevailing method of surface area determination is that of the BET plot (Section 2.6.1), which is based on the assumptions underlying the BET isotherm (Equation 1.18). The fact that the pores fill before the monolayer forms clearly violates multiple assumptions made in deriving the model isotherm. This limitation has not stopped people from applying it anyway, of course, some with more care than others. Naive application of the BET equation to the isotherm of a microporous material in the usual range of relative pressures usually results in a BET plot

37

Analysis of Catalyst Surface Structure by Physical Sorption

Table 1.3 Crystallographic pore sizes, channel types, and low-pressure isotherm “step” locations for nitrogen at 77 K Mean Low-P/P IZA Channel pore size step for N2 at 77 K References Zeolite code type Pore size (Å) (nm)

Silicalite

MFI

3D

5.1 5.5 and 5.3 5.6

0.57

2.0 10
6

(45)

ZSM-5

MFI

3D

5.1 5.5 and 5.3 5.6

0.56

2.0 10
6

(45)

ZSM-11

MEL 3D

5.4 5.3

0.54

2.0 10
6

(45)

Y

FAU 3D

7.4

0.74

9.0 10
5

(45)

5A RHO VPI-5

LTA 3D RHO 3D VFI

1D

12.7

0.36 1.27

8.0 10 1.0 10


5

3.0 10


4
4

(45) (45)

3.0 10

3D

5.1 5.5 and 5.3 5.6

0.56

4.0 10
4

(46)

3D

5.1 5.5 and 5.3 5.6

0.56

2.0 10
5

(42)

4.1

0.41

4.0 10
5

(42)

NaZSM-5

MFI

ZSM-5

MFI

CaA

LTA 3D FAU 3D FAU 3D

7.4 7.4

0.74 0.74

4.0 10


5

3.0 10


5
4

Clinoptilolite HEU 2D

7.5 3.1, 4.6 3.6, 4.7 2.8

0.44

1.0 10

AlPO-18

3.8 3.8

0.38

2.0 10
6

AEI

(45)

0.74

FAU 3D

NaY

3.6

0.41

7.4

NaY

Y

4.1 4.1


7

3D

(46)

(47) (42) (48)

(49)

The step locations for argon adsorption at 87 K on the same materials are included in Figure 1.8. Please note that all zeolites are in the hydrogen-exchanged form unless otherwise designated. Pore sizes are the “nominal” size from crystallography.

that is slightly convex to the relative pressure axis. The intercept is always near zero for microporous materials, meaning that the value of C is highly sensitive to the error in the experiment. In many cases, the intercept will actually be negative, resulting in a negative value of C; if we were to plot

38

Karl D. Hammond and Wm. Curtis Conner Jr.

Volume adsorbed (cm3 STP/g)

200

150

100

50

0 10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Relative pressure (P/P°)

Figure 1.9 Adsorption isotherms characterizing Linde Type 5A (LTA), a small-pore zeolite with a crystallographic pore size of 0.41 nm, showing the shift in the microporefilling region of the isotherm providing a comparison of nitrogen and argon adsorbates. The lowest points, near 10
7 atm (10
5 Torr), are at or below the accuracy limit of the 0.01 Torr pressure transducer used in the experiments. The lines are to guide the eye. The data are from Refs. (45,50).

the actual isotherm predicted by such a C value, we would see an asymptote appear in the predicted isotherm (as shown in Figure 1.12). Unfortunately, most adsorption equipment and software currently available do not print the value of C on the BET plot. With this point in mind, the value of C should always be reported anytime the BET surface area of a material is published. Such values should always be positive. Recently, suggestions have been made that the usual range of use of the BET equation can be modified in such a way as to allow calculation of a proper surface area as a basis for comparison of microporous materials with more conventional catalytic materials for the purpose of determining catalytic activity or adsorption capacity per unit surface area. There are conceptual and practical concerns with these extensions to the BET theory. Adsorption at pressures below P/P ¼ 10
3 fills the voids of highsurface-area solids before the external surface is covered by a monolayer.

39

Analysis of Catalyst Surface Structure by Physical Sorption

200

Volume adsorbed (cm3 STP/g)

Argon, 77 K Argon, 87 K Nitrogen, 77 K 150

100

50

0 10−7

10−6

10−5

10−4

10−3

10−2

Relative pressure (P/P°)

Figure 1.10 Adsorption isotherms characterizing zeolite Y (FAU), a medium-pore zeolite with a crystallographic pore size of 0.74 nm, showing the shift in the microporefilling region providing a comparison of nitrogen and argon adsorbates. The lines are to guide the eye. The data are from Refs. (45,50).

Notwithstanding this fact, many investigators report surface areas for microporous solids including zeolites by naive applications of the BET equation. This procedure disregards the assumptions of the model and ignores obvious constraints on CBET (such as the requirement that it must be positive and not too large; see Equation 1.13). There is no question that the quantity adsorbed includes the amount of adsorbate that would physically cover the walls in the pores of such solids; the questions are the following: 1. What fraction of the quantity adsorbed as a function of relative pressure corresponds to that which might represent a “monolayer” adsorbed on the walls of the micropores? 2. How should one convert this fraction to an equivalent surface area for adsorption in a fully three-dimensional network inside a microporous solid? In 2007, Snurr and coworkers (51,52) proposed an extension of the suggestions by Rouquerol and coworkers (53) in which two criteria should be applied to adsorption data to identify a region of the isotherm to which to apply the BET analysis:

40

Karl D. Hammond and Wm. Curtis Conner Jr.

125

Volume adsorbed (cm3 STP/g)

100

Argon, 77 K Argon, 87 K Nitrogen, 77 K

75

50

25

0 10−6

10−5

10−4

10−3

10−2

10−1

Relative pressure (P/P °)

Figure 1.11 Adsorption isotherms characterizing VPI-5 (VFI), a large-pore zeolite with a crystallographic pore size of 1.27 nm, showing the shift in the micropore-filling region providing a comparison of nitrogen and argon adsorbates. The lines are to guide the eye. The data are from Refs. (45,50).

1. The region of the isotherm chosen for the BET plot must have a positive intercept (i.e., the region must be chosen so that CBET must be positive). 2. The pressure range should be chosen so that Vads(1
P/P ) always increases with relative pressure (P/P ). The first criterion is used to determine the lower bound of the domain used to determine the surface area (i.e., remove low-pressure points from the BET plot until the first criterion is satisfied), and the second criterion is suggested as the criterion for the upper bound. Rouquerol et al. (38) suggested the second of these criteria for all BET plots as early as 1967. Both of these criteria are based on the assumption that the adsorption data have been properly collected (i.e., equilibrium has been reached). To an extent, the results of these analyses are subjective. This statement is particularly true for the first criterion related to selecting data for which the extrapolated intercept of the BET plot will be positive: it is clear that excluding data sacrifices the robustness of the estimate of the amount adsorbed in the monolayer.

Analysis of Catalyst Surface Structure by Physical Sorption

41

Another question that must be addressed is how to interpret the resulting estimate of a “monolayer volume” (usually based on the value determined in units of cm3 STP/g adsorbed for nitrogen or argon) to an equivalent surface area of a highly curved surface and/or of a three-dimensional pore network. Conventional BET analyses of surface areas employ a calculation of the surface area covered by each molecule on a flat (two-dimensional) surface. For nitrogen, the area per molecule is estimated to be 0.162 nm2, and for argon, it is 0.138–0.182 nm2, depending on the source of the estimate (Section 2.6.3 and Table 1.1). Basic estimates of these areal densities come from liquid or solid densities and atomic/molecular volumes. When applied to the truly three-dimensional metal organic framework (MOF) systems, the surface covered by each molecule is probably far less: six molecules will surround a single molecule of similar size in a plane, but this does not make the surface of that molecule six times the flat covered surface of a single molecule! We emphasize that the methods discussed in this section are different from the standard BET analysis and emphasize that use of the “standard” BET range of 0.05 < P/P < 0.3 for microporous materials should be avoided. An example illustrating why this statement is appropriate is presented in Figure 1.12, which shows four different BET isotherms overlaid on the measured isotherm for zeolite Y. The first through third isotherms correspond to naive applications of the BET equation to the “standard” region, and the fourth isotherm corresponds to the procedure outlined in this section. Note that the CBET values determined from the naive applications of the “standard” method are negative and give ridiculous isotherms with asymptotes in them. We hereby dub the procedure outlined in this section for estimating a surface area from the BET plot by using low-pressure adsorption data from the Rouquerol–Llewellyn–Rouquerol–Snurr modification of the Brunauer– Emmett–Teller theory, or RLRS–BET method. This is pronounced “rollers-bee-ee-tee,” in an allusion to the Snurr group’s method of “rolling” a probe molecule along the surface to estimate the accessible surface. Most important, the quantity adsorbed for the data points giving rise to the negative intercepts in the BET plot should be subtracted from the amount adsorbed before calculation of the monolayer volume from the plot. The reason for this statement is shown in Snurr and coworkers’ simulations (51,52): the initial quantity adsorbed is strongly adsorbed, and only then is a “monolayer” assembled on that exposed surface. Furthermore, the molecular area of argon or nitrogen used in the calculations should be specified, as

42

Karl D. Hammond and Wm. Curtis Conner Jr.

Measured isotherm BET isotherm (SA = 561.1 m2/g, C = −66.4) BET isotherm (SA = 564.1 m2/g, C = −94.1) BET isotherm (SA = 554.4 m2/g, C = −68.2) BET isotherm (SA = 653.5 m2/g, C = 5289)

P/P° / [Vads(1−P/P°)] (mg / cm3 STP)

Volume adsorbed (cm3 STP/g)

350 300 250 200 150 100 50 0 −6 10

−5

10

−4

−3

−2

−1

10 10 10 10 Relative pressure (P/P°)

2 1.5 1 0.5 0 0

10

0.05 0.1 0.15 0.2 0.25 0.3 Relative pressure (P/P°)

0.35

200 V(1−P/P°) (cm3 STP/g)

Volume adsorbed (cm3 STP/g)

2.5

0

250 200 150 100 50 0

3

0

0.05 0.1 0.15 0.2

0.25 0.3

Relative pressure (P/P°)

0.35

150

100

50

0

0

0.05

0.1

0.15

0.2

Relative pressure (P/P°)

Figure 1.12 Comparison of adsorption data recorded at 77 K for nitrogen on zeolite Y (FAU), with the BET isotherms derived from several sets of parameters. All parameters were derived from BET plots of different domains of relative pressure; the first three (with negative C values) use points between 0.05 and 0.3, whereas the thick solid line was derived using the RLRS–BET (rollers BET) recommendations discussed in the text and using points between 10
7 and 0.03. The two plots on the left are the isotherms (with linear and logarithmic pressure axes), the upper right figure is the BET plot itself, and the lower right plot is the denominator of the left-hand side of the BET equation. The RLRS–BET procedure requires this plot to be an increasing function for all data used in the analysis. Note that the intercept in the RLRS– BET plot is 1.26 10
6 g cm
3 STP (i.e., extremely close to zero).

it should for all BET-derived surface areas. Authors should call surface areas obtained by this method RLRS–BET surface areas, not simply BET surface areas. Many new materials exhibit combinations of micropores and mesopores. Such materials are designed to enable species to access the

43

Analysis of Catalyst Surface Structure by Physical Sorption

micropores readily without the limitations of diffusion through large micropore networks. The results of BET analysis applied to such mixedpore materials are often the same as they are for purely microporous materials—that is, negative or otherwise unrealistic values of the BET C parameter. To obtain the area of the mesopores, the data used in BET analysis can be adjusted by subtracting the volume adsorbed in the micropores from the total volume adsorbed, leaving an estimate of the adsorption isotherm in the absence of micropores. The relative pressure at which the micropores fill will typically be some value less than 10
3, possibly less than 10
5; a few examples based on BET data in the “standard” region (0.05 P/P 0.3) with the volume adsorbed at several different points along the isotherm are shown in Table 1.4. The negative C values determined for the zeolites become slightly more realistic (although still small) positive values when micropore adsorption is removed. The pores of MFI, ˚ , fill below which are three-dimensional with a nominal diameter of 5.6 A

Table 1.4 BET fitting parameters from conventional BET analysis of nitrogen adsorption at 77 K in the region 0.05 P/P 0.3 for zeolites MFI and VET and the mixed micro– mesoporous material SBA-15 Sample BET Conventional Subtract Subtract Subtract parameters Vads(10
5) Vads(10
4) Vads(10
3)

C-MFI

SABET (m2g
1)

47.1

254.2

232.4

251.5


98.0

20.6

6.8

4.1

497.3

220.5

174.4

149.4

CBET


83.9

78.7

28.8

14.2

SABET (m2g
1)

548.4

539.8

246.8

185.8

CBET


76.7


72.6

89.5

23.3

SABET (m2g
1)

814.9

799.3

755.5

658.5

CBET

130.0

121.5

87.3

45.6

CBET MW-MFI SABET (m2g
1)

VET

SBA-15

Also shown are BET parameters calculated from the modified isotherm, where the quantity adsorbed at specific pressures has been subtracted from the total volume adsorbed (the remaining adsorption is intended to represent adsorption on the exterior surface and/or on mesopore walls). The expressions C-MFI and MW-MFI indicate silicalite-1 (MFI structure) as synthesized by conventional (C) or microwave (MW) heating; microwave heating generally produces smaller crystal sizes.

44

Karl D. Hammond and Wm. Curtis Conner Jr.

P/P ¼ 10
5; the 6.6-A˚ one-dimensional VET pore network fills at a slightly higher relative pressure (Figure 1.7). The larger pores of VTI fill a decade higher, near P/P ¼ 10
4. The volume of these materials’ micropores is largely responsible for the unrealistic values of C from the BET plot, and subtraction of the micropores’ contribution to the adsorption isotherm yields small positive values of C; the resulting BET isotherm is probably consistent with adsorption on the external surface. In SBA-15, BET analyses for samples in which micropores are present do not necessarily yield unrealistic (e.g., negative) C values. In analyses of data characterizing these materials, it may be more difficult to isolate the micropore contributions to the adsorption isotherm from those of the mesopores. In general, we recommend analyzing the various areas of the isotherms separately, as discussed in Section 3.

5. ADSORPTION–DESORPTION HYSTERESIS Adsorption and desorption isotherms are not coincident in the range 0.4 < P/P < 0.99 for adsorption of nitrogen or argon at their normal boiling points on mesoporous solids. In such materials, the quantity adsorbed at a given pressure will be less along the adsorption branch (increasing pressure) than it would be along the desorption branch (decreasing pressure). As a consequence, the adsorption and desorption isotherms form a closed loop, as shown in Figure 1.4A. The loop thus formed is called “hysteresis,” after a Greek word meaning “to be behind; come short” (54). Interpreting these values with the Kelvin equation (Section 7.1.1) and the statistical thickness models presented in Section 6.1 yields pore sizes in the range of 2 to 50 nm—by the IUPAC definition, these are mesopores (55). There are several proposed reasons for adsorption–desorption hysteresis: 1. Differences in the effective radius of curvature of a meniscus when it forms in an empty pore compared with that when it evaporates from a filled one (see, e.g., Gregg and Sing (12), p. 127). 2. Delayed nucleation of the adsorbing fluid during adsorption, resulting in a vapor-like spinodal (limit of stability) on the adsorption branch and a liquid-like binodal (equilibrium) curve on the desorption branch (56). 3. Pore-blocking effects, such as would occur in “ink-bottle” pores and pore networks: small constrictions block the way for larger pores to empty, causing the desorption branch to appear delayed until the onset of cavitation (57,58).

Analysis of Catalyst Surface Structure by Physical Sorption

45

4. Swelling of the pores during adsorption, producing a higher quantity adsorbed at the same relative pressure during desorption than during adsorption. This situation usually results in an isotherm that does not “close” properly. 5. Extraordinarily slow diffusion, causing what appear to be stable equilibrium states but which are actually slightly unstable states that decay to equilibrium states on very long timescales (59). As has been pointed out by Monson (60), the reproducibility of adsorption experiments and observed hysteresis loops, combined with their stability under thermal shocks and vibrations (61), strongly suggests that the slow diffusion hypothesis (what has been called the “impatient experimentalist” hypothesis (62)) is incorrect. The others—delayed adsorption (metastability), network effects, and cavitation—are discussed in the remainder of this section.

5.1. Network effects 5.1.1 Network sorption Most high-surface-area solids are porous, and the solid and void phases are bicontinuous. This statement means that both solid and void phases span any particle with relatively uniform macroscopic density in all directions and throughout the volume. For microporous crystalline solids, such as zeolites (and some mesoporous solids, such as MMS), both the void and solid structures form a network of varying dimensions along any pathway into or out of the solid particles. The filling or emptying of the pores with a physical adsorbate can thus proceed by different sequences of events, depending on the nature of the porous network. A simplified depiction is shown in Figure 1.13 for a two-dimensional solid-void network. Lord Kelvin (Section 7.1.1) showed that a pore filled with a continuum fluid can form a thermodynamically favored state that depends on the dimension of the pore and the (relative) pressure of an adsorbing gas as it condenses (Equation 1.23). The smallest pores would therefore fill first during the adsorption process and empty last during the desorption process. Because the pore space is most often a noncrystalline network, the dimensions vary, and different pores throughout a single particle fill in a different sequence. The pressures at which a pore fills as a function of the partial pressure of the adsorbate and the state of the pore (volume already adsorbed) are employed to calculate a distribution of pore sizes within the pore network (Section 7.1.2). However, there is a difference between the processes involved in pore filling during adsorption and the processes involved in pore

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Figure 1.13 Simplified representation of the process of adsorption, saturation, and desorption in the voids between particles of a larger solid. Desorption can (and often does) occur by cavitation (lower right), allowing larger pores to empty even though smaller pores “block” their access to the bulk.

emptying during desorption in a single pore, even if there is no network. If there is a network of pores within a sample, the sequence of pores that fill or empty and/or the pressures at which those pores fill during adsorption can be different from those during desorption. Networked pores give rise to “poreblocking” effects in which large pores (which typically empty at high pressure) are unable to desorb their contents as a consequence of being blocked by the small pores that connect them. These “network effects” are one mechanism that has been proposed to give rise to the hysteresis between adsorption and desorption. One can picture a void network as an interconnected space of constrictions, the minimum-dimensional spaces between larger openings, and “pores.” These should be envisioned as three-dimensional networks, not solely as pores connected by narrowing constrictions to the gas phase, such as the “ink-bottle” pore (single openings connected by a single constriction), as often used as model pores for simulations. A three-dimensional network is far more realistic and probable. Figure 1.13 illustrates the network concerns associated with sorption hysteresis. The figure represents a two-dimensional (2D) network formed by the void spaces between nine solid spheres (shown as circles in 2D).

Analysis of Catalyst Surface Structure by Physical Sorption

47

Openings are formed between sets of spheres (C < A < B < D) and are connected to other openings by constrictions of varying sizes (9 < 1 < 2 < 8 < 6 < 11 < 10 < 12 < 3 < 5 < 4 < 7). Similar network processes occur in three- dimensional networks, although more spheres are often required to form a constriction or occlude a void volume. 5.1.2 Pore blocking in networks As discussed earlier, physical adsorption on surfaces is a very common phenomenon. The process is progressive, as more is adsorbed as the partial pressure of the adsorbing species is increased. The average thickness of the adsorbed layer increases with the relative pressure. The schematic represented in Figure 1.13, top left, shows the initial solid and void spaces. As the pressure of a physisorbing gas is increased, the adsorbing molecules adhere to the exposed surfaces and eventually progress to form an average of a monolayer covering the surface at a relative pressure of approximately P/P  0.2 for nitrogen or argon adsorption at their normal boiling points. The BET surface area is typically calculated from these data (Section 2.6.3). At higher relative pressures, multiple layers are formed, and the mean thickness of the adsorbed layer increases. However, even a model as simple as that represented by the equation named for Lord Kelvin (see Section 7.1.1 for derivation),   P 2gVl 1 log  ¼
, ð1:23Þ kT rK P shows that fluid could condense within a confined space such as a capillary with increasing relative pressures (P/P ) corresponding to pores of increasing dimension (rp ¼ rK þ t). Thus, in addition to forming an increasingly thick adsorbed layer, fluid could condense to fill the spaces between subparticles within the solid, and the smallest spaces would fill first. This situation is depicted in the upper right of Figure 1.13 with constrictions 9 ! 2 ! 1 ! 6 ! 11 ! 10 filled. As this happens, the surface area exposed to the gas phase decreases. The process continues to fill the constrictions, as does the thickening of the adsorbed layers on all surfaces. As complete liquid behavior is approached, occluded gaseous spaces remain as bubbles (shown on the bottom right in the figure) that eventually collapse sequentially, with the smaller ones filling at lower pressures. Recall that these are equilibrium processes and the pressures and temperatures are constant at any point throughout the sequential processes.

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As the adsorbent pressure reaches saturation, all of the void space fills with condensing gas. Liquid fills all voids, while the liquid–gas interfaces decrease in area and are occluded as the voids fill. Eventually, all void space is filled with condensed liquid and the total exposed surface represents the exterior surface (if this schematic depiction is interpreted as an essentially isolated particle composed of an agglomerate of the subparticles depicted in Figure 1.13). The desorption process proceeds from a fluid-saturated void such as that represented on the bottom left of Figure 1.13. As the pressure is reduced, the formerly collapsed voids do not reform into bubbles spontaneously. The surfaces in contact with the gas, the pressure of which is being reduced, are now those at the constrictions in the void network, although the total pressure remains uniform throughout the system at any time during the measurement. The constriction size distribution thus controls the desorption process, unless cavitation occurs within the voids as is discussed in Section 5.4. However, the delay in desorption attributable to these constrictions gives rise to hysteresis between the adsorption and desorption isotherms. Thus, the figure proceeds from the lower left at saturation to the upper right representation in the figure in the absence of cavitation.

5.2. Comparison with hysteresis in mercury porosimetry Hysteresis in adsorption–desorption is similar to the phenomenon associated with mercury porosimetry. However, mercury is a nonwetting fluid on most surfaces, so the process is reversed with respect to pore size. Mercury intrusion is controlled by the constrictions in the void network, and mercury extrusion is controlled by openings in the network (63). The intrusion process is sequential from the exterior to the interior of a porous particle, in contrast with adsorption, for which the pressure of the adsorbing gas is uniform at all positions and fluid is therefore able to condense anywhere at any time. Thus, the void dimension distributions are shifted by shadowing and conductivity issues (63–66). In general, however, the dimensions measured during adsorption correspond to the dimensions measured during extrusion porosimetry, whereas the dimensions measured during desorption correspond to those measured during intrusion porosimetry (63–66). In general, the surface areas estimated during mercury intrusion porosimetry are significantly higher than those estimated by BET analyses (63).

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5.3. Metastability Another cause of hysteresis is metastability. There is mounting evidence from simulations (58,60,67–69) of single pores and small networks of pores and experiments on materials with known, uniform, and/or tailored porosity (22,70) that hysteresis is present both for isolated pores and pore networks. These same simulations and experiments strongly suggest that adsorption is “delayed” as a consequence of nucleation barriers, making the adsorption branch of the isotherm closer to a spinodal (limit of stability) rather than the true binodal (equilibrium) curve. Metastability is evident from simulations of adsorption that employ the so-called “gauge-cell” method. A simple description of this method is that it simulates a closed system with a finite number of gas particles confined to a volume in the presence of a pore (or pores). This situation, in a sense, represents what is done experimentally, except that the systems involved are significantly smaller. The chemical potential is measured in such simulations by a technique known as Widom sampling; an excellent description of this technique is found in Frenkel and Smit (71). The finite size of the gauge cell means that the density fluctuations normally present in large systems are damped out, which implies that states that are normally unstable can actually be observed in the simulation (72,73). Such observations are obviously not possible during an experiment, but the resulting S-shaped adsorption isotherms make it relatively easy to see the limits of stability of the interaction potentials. Although the models involved are typically simple Lennard-Jones fluids (74), the Lennard-Jones potential is often a good enough representation of actual host–guest and guest–guest interactions in porous materials that the predictions are quantitatively and qualitatively consistent with the results of many experiments (70). Gauge-cell simulations are often compared with another common type of adsorption simulation, namely, grand canonical Monte Carlo (GCMC) simulations. This type of simulation involves (a) random movements of adsorbate particles and acceptance/rejection of those movements based on the Metropolis sampling criterion (75), which mimics thermal vibrations and the resulting equilibria, and (b) random particle insertions and deletions, sampled with a modification to Metropolis sampling that is designed to fluctuate the chemical potential around a desired average throughout the simulation. Both canonical Monte Carlo (often called Metropolis Monte Carlo after the seminal paper by an author with that name (75)) and GCMC techniques are discussed in Frenkel and Smit (71).

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The result of gauge-cell simulations is an S-shaped curve with a vapor-like spinodal point (limit of stability) near the adsorption branch and a liquid-like spinodal at low pressures (below those at which the isotherm would usually close). GCMC simulations typically show a similar shape above and below the spinodal points, but the adsorption and desorption branches typically close suddenly, in the manner of spinodal decomposition. These abrupt transitions are the result of (a) using rigid walls in the simulation (real walls will have more “give” or flexibility to them, causing a very small change in the pore dimensions and rounding of the corners of the isotherm); (b) using a single, definite pore size (experiments will, of course, indicate a possibly narrow distribution around a peak in the pore size distribution because of grain boundaries and other defects in the material); and (c) the extremely small (to an experimentalist, anyway) size of the system being simulated. A diagram showing results typical of a gauge-cell simulation is shown in Figure 1.14. The actual branches of the adsorption isotherm (i.e., the adsorption branch and the desorption branch) lie somewhere between the spinodal points (dashed lines in Figure 1.14): as the fluid’s state approaches the spinodal, the magnitude of the density fluctuations required to “jump” to either of the points on the curve with the same chemical potential becomes smaller and smaller. This jump is to the condensed (liquid-like) state during adsorption and to the vapor-like state during desorption. As particles are added to the system and the state of the fluid approaches the spinodal more and more closely, the barrier between the stable state and the metastable state becomes smaller and smaller, until finally the phase transition occurs and vapor–liquid equilibrium is established. In an experiment, the slightly varied sizes of the various pores in the sample (which are obviously ignored in a single-pore simulation) and the flexibility of the pore walls (which is often neglected or poorly modeled in simulations) mean that the transition is less abrupt. Moreover, the experimental system is much larger, increasing the chances for density fluctuations that might circumvent the barrier between the metastable state (region between binodal and spinodal points) and the stable state, meaning that (a) each pore will have a very slightly different spinodal decomposition point and (b) each pore may fill/empty before reaching the spinodal. In short, changes in the adsorption isotherm are typically much more gradual in experiments than in simulations of a single pore. Simulation results typically indicate that the desorption branch of the isotherm is closer to the binodal curve (i.e., thermodynamic equilibrium), whereas the adsorption branch is closer to the vapor-like metastable state until the relative pressure approaches the spinodal point (20,60,70). This statement does not mean that global thermodynamic equilibrium and the

Density (or number of particles)

exp

m–m° kT

» P/P °

50

Density (mol/L)

40

30

20

10

0 -3000

Ideal Gas Law van der Waals Peng–Robinson -2000

-1000 Pressure (bar)

0

1000

Figure 1.14 Example of the S-shaped density–pressure plot often found in gauge-cell simulations of adsorption–desorption. The dashed lines represent spinodal decomposition points; all points inside this region correspond to unstable equilibria and are observable in these simulations only because of the very small size of the system and the constrained number of particles. The plot on the bottom shows a similar curve representing argon at 77 K as modeled by the van der Waals (76) and Peng and Robinson (77) equations of state. Upper figure adapted from Ref. (73) with permission. Copyright © 2010, American Chemical Society.

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desorption branch of the isotherm are synonymous at all times, although it is often assumed that they are. We stress, however, that the equilibrium pressure at which a pore empties may have little or nothing to do with the pore’s size: large pores connected to small pores (or “necks”), either in pore networks or in small “ink-bottle” pores, can empty before the necks empty, via cavitation. This phenomenon is discussed in Section 5.4. It has also been noted that the phase transition in Lennard-Jones fluids confined to cylindrical pores shifts to lower pressures than in the bulk, beyond that predicted by the Kelvin equation (78). At higher temperatures (but still below the critical temperature), the isotherm of such a fluid is continuous, suggesting that there exists a capillary critical temperature, Tcc, such that Tcc < Tc. At temperatures below Tcc, capillary condensation occurs, and hysteresis is observed. As is evident from the simulations of Monson and coworkers (57,79), the qualitative picture gained from the Kelvin equation—that slit pores fill all at once during adsorption but form a meniscus (with a different radius of curvature) that slowly retreats along the length of the slit—is accurate. It is not, however, quantitatively accurate in the limit of small pores. If the nitrogen (77 K) or argon (87 K) isotherm closes above P/P ¼ 0.5, an analysis based on the Kelvin equation may give reasonable results when applied to the desorption branch of the isotherm. Analysis of the adsorption branch can yield appropriate results if a model based on metastable fluid states is used, such as is employed in many DFT-based calculations (20,70). We emphasize that although early studies suggested that analyses of desorption be employed to analyze mesopores: “When hysteresis is observed, the desorption branch of the experimental isotherm is to be used for calculation of pore size distributions” (69), current wisdom is that analyses of the desorption branch often meet with problems associated with the tensile strength effect (Section 5.4), especially when traditional analyses are used. The adsorption branch is shifted because of nucleation barriers, but its location is more uniformly linked to pore structure as determined by other methods of pore size characterization (e.g., XRD and microscopy (80)), particularly in pore networks in which pore blocking and the tensile strength effect are important.

5.4. Cavitation and the tensile strength effect In 1965, Harris (81) made the observation that many nitrogen isotherms at 77 K reported for mesoporous materials contained hysteresis loops that closed at or near P/P ¼ 0.42, but none ever closed at lower relative

Analysis of Catalyst Surface Structure by Physical Sorption

53

pressures (we refer to this as the “lower closure point”). In fact, the number of isotherms with a lower closure point of 0.41–0.45 for nitrogen at 77 K is a significant fraction of the total number of reported adsorption isotherms, ever. If interpreted naively through a Kelvin equation analysis (Section 7.1.1) or BJH and similar analyses (Section 7.1.2), this observation would imply that all materials with this lower closure point have a narrow distribution of pores with radii in the range of 1.7 nm < rp < 2.0 nm. Harris and others suspected that this situation was highly unlikely, and they were correct. The phenomenon has been dubbed the “tensile strength effect,” and it has recently been understood to be related to the onset of cavitation of the fluid inside larger pores whose exits to the bulk (or other large pores) are “blocked” by smaller pores. The name “tensile strength effect” comes from the continuummechanics (i.e., the Kelvin equation and related analyses) interpretation of the effect, which takes in the concept of a “meniscus” and the maximum tension that the surface can withstand before blowing apart and venting the (pressurized) contents of the pore. The stress on the surface is given by the Young–Laplace equation,   1 1 2g Pl
Pg ¼
g ¼
þ r1 r2 hri

ð1:24Þ

where Pℓ is the pressure in the liquid phase, Pg is the pressure in the gas phase, and r is the radius of curvature along the meniscus. As the pressure outside the meniscus (Pg) decreases, the stress on the meniscus itself increases. When that stress exceeds the tensile strength (so the conventional interpretation goes), the meniscus will collapse and the fluid will rush out of the pore. This occurrence corresponds to a specific value of the Kelvin radius and thus a specific pore size. This “minimum” pore size would depend only on gVℓ/kT and would thus be a constant for a given adsorbate at a given temperature. It would not depend on the porous material. The present-day interpretation of the tensile strength effect is that it corresponds to the onset of cavitation in pore networks (i.e., “blocked” pores, Section 5.1). This interpretation means that larger pores, which if isolated would have emptied at much higher relative pressures, instead empty at the pressure at which cavitation commences (which is, obviously, somewhere around P/P ¼ 0.42 for nitrogen at 77 K). The cavitation pressure has been found to be relatively close to the value for the “bulk” adsorbate (72), as shown in Figure 1.15.

54

0.5

2.03

0.48

1.93

0.46

1.85

0.44

1.77

Cylindrical pores (SLN326, SE3030) Spherical Pores (KLE) Spherical pores (SLN-326) Spherical pores (SBA-16)

0.42

0.4

0

10

20 Pore diameter (nm)

30

Apparent BJH pore radius (nm)

Lower closure point (P/P °)

Karl D. Hammond and Wm. Curtis Conner Jr.

1.69

1.61 40

Figure 1.15 Dependence of the cavitation pressure (lower loop-closure point) on pore diameter for several mesoporous silicas with porosity known by independent methods. The right-hand vertical axis is based on the assumption of the Kelvin equation and the thickness model of de Boer (Equations 1.23 and 1.29, respectively). Adapted from Ref. (72) with permission. Copyright © 2010, American Chemical Society.

Cavitation, when it occurs, creates an isotherm in which the pressure at which desorption occurs has very little to do with the pore size (22,70) (also see Figure 1.15). This occurrence is a frequent source of errors in analysis, particularly when traditional methods such as the BJH method are used for the desorption branch. If one obtains BJH pore radii in the range of 1.7–2 nm, it should raise a red flag: such a result is likely uncorrelated with pore size, as shown on the right-hand vertical axis in Figure 1.15. It has been suggested (70) that there is a critical width, Wc, for which cavitation occurs, in much the same way that there is a critical temperature below which hysteresis can occur. If the pore width is larger than Wc, hysteresis is thought to be a consequence of pore blocking or metastability. If W < Wc, cavitation also plays a role. There is also a critical pore size: pores greater than 2.0 nm in radius showed hysteresis, whereas pores less than 1.8 nm in radius did not (82–84). The position of the lower closure point is also a function of temperature for a given adsorbate. This result is shown for argon in Figure 1.16.

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55

Figure 1.16 Normalized isotherms characterizing adsorption on MCM-41, showing the change in the lower closure point of the isotherm as a function of temperature (top) and adsorbate (bottom). Filled symbols are adsorbing points; open symbols are desorbing points.

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This effect underscores the problem associated with interpreting the tensile strength effect as being a consequence of pores emptying: using values from Table 1.1, we find that the Kelvin radius at the closure point is about 1.15 nm for argon at 90 K but only 1.04 nm for argon at 87 K. Although we do not have a value for the surface tension of argon at 77 K (as it is a solid at that temperature), if we assume that it is an unrealistic value of 50% greater than it is at 87 K, we obtain an estimated Kelvin radius of 0.69 nm, which is clearly even more unrealistic than the values estimated earlier. We reiterate that lower closure points do not correlate well with pore sizes if the loop closes at approximately 0.42 (nitrogen, 77 K), 0.27 (argon, 77 K), 0.39 (argon, 87 K), or 0.45 (argon, 90 K), as shown in Figure 1.15.

6. ASSESSING POROSITY Sorption data are analyzed to estimate the presence and dimensions of pores within solid samples on the basis of two assumptions: (1) the progress of sorption for a nonporous solid and (2) the size of pores that will fill at a given relative pressure. We discuss the first assumption in this section and reserve discussion of the second for Section 7. The first assumption is commonly addressed by the use of standard isotherms—that is, adsorption isotherms characterizing materials with surface chemistry that is similar to that of the adsorbate but known to be nonporous—or by models of such isotherms based on fitted parameters. Isotherms for nonporous solids depend on the forces of attraction between the adsorbing species in the first adsorbed layer (the monolayer) and the underlying solid compared with the forces between the second and subsequent layers of adsorbate with each other and with the surface. Intuitively, these data will have a constant or slightly decreasing slope when the volume adsorbed is plotted against the pressure, that is, d2 Vads  0, dP 2

ð1:25Þ

with the value usually being slightly negative. The trend inferred in this physical adsorption process is that the adsorbed species are nominally mobile in two dimensions. Species on the surface will tend to find their lowest energy states, and subsequent adsorption will therefore exhibit the same or decreasing energies of interaction. The strength of attraction between adsorbing molecules and the surface will depend on van der Waals

Analysis of Catalyst Surface Structure by Physical Sorption

57

interactions, which depend not only on the chemical composition of the surface but also on the nature of the solid just below the surface. Thus, not all “standard” isotherms will be identical. If surfaces are atomically rough, they could be covered unevenly because of a larger number of atoms with initially higher and nonuniform interaction energies in the first layer. These phenomena would mean that the second, smoother, layer would require fewer atoms to cover. There are two basic approaches to realize the standard isotherm for the solid being investigated. The first approach is based on knowledge (or estimation) of the area of the exposed surface and the relationship between the thickness of the adsorbed layer and the partial pressure of the adsorbing gas. This relationship between the thickness of the adsorbed layer and the relative adsorbate pressure is also useful in calculating the pore dimensions when the pore fills. The second approach is to assume a standard isotherm from prior empirical data for solids of similar composition, particularly on the surface. This procedure depends on having the needed standard data for a nonporous adsorbent, which are often tabulated in the literature (Section 6.2.2).

6.1. Statistical thickness: The Halsey equation The BET isotherm (Section 2.6) can be used to estimate the thickness of the adsorbed layer. However, this estimate is typically too high once the isotherm gets much beyond monolayer capacity. A better-accepted statistical thickness of the adsorbed layer can be determined by any of several models. In 1948, Halsey (85) suggested an empirical equation relating the coverage and the relative pressure, which was discussed further in Advances in Catalysis (Volume 4) in 1952 (40). He found a fair spectrum of relationships depending on the surface being analyzed but suggested a common relationship, often called the Frenkel–Halsey–Hill equation:   P CH log  ¼
EH P y

ð1:26Þ

where CH and EH are the (empirical) Halsey constant and Halsey exponent, respectively, and y is nads/nm, the loading. Halsey found a spectrum of values for these two variables. With a value of the thickness of an atomic layer, s, one can then estimate the coverage and thus the average thickness, t, at any relative pressure once the Halsey constants are known:

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Karl D. Hammond and Wm. Curtis Conner Jr.

 t ¼ sy ¼ s


CH logðP=P  Þ

1=EH

:

ð1:27Þ

Hill (86) was able to derive a similar relationship with EH ¼ 3. More commonly, values for the exponent in the range 2.61–2.76 have been determined empirically for various oxides (12). Many automated sorption systems first employ the following equation as a “Halsey” equation for t in their calculation of the pore size distribution:   P
1=EH : ð1:28Þ t ¼ aH
log  P ˚ and EH ¼ 3 for nitrogen at 77 K (41). However, most where aH is 6.0533 A automated sorption systems also allow one to vary the constants in the Halsey equation and thus tune the analysis. Another model for the statistical thickness is that of de Boer (87), which he derived on the basis of the thin-film model of Harkins and Jura (88): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˚2 13:99 A ð1:29Þ t¼ 0:034
log10 ðP=P  Þ where the values given are for nitrogen at 77 K. The combination of the Halsey equation (Equation 1.27) or another model and the Kelvin equation (Section 7.1) enables one to estimate the dimensions of the pores being filled during adsorption or emptied during desorption. As the pores are filled or emptied, the accessible surface area changes progressively. During adsorption, the accessible surface decreases relative to the BET area. During desorption, the theoretically uncovered area increases from near zero at saturation back to near the BET area at low loadings. Unfortunately, these associated calculations are commonly inconsistent (i.e., total surface area calculations during adsorption or desorption analyses do not agree, often by factors of greater than two).

6.2. Standard isotherms: The t- and as-plots 6.2.1 t-plots The determination of pore size distributions from sorption data depends on the relationship between the quantity adsorbed for the sample being

59

Analysis of Catalyst Surface Structure by Physical Sorption

investigated and the amount adsorbed on an equivalent nonporous sample. This procedure requires an understanding of the increase in the amount adsorbed, nads, as a function of relative pressure expressed as an increase in the average thickness of the adsorbed layer, t. Thus, de Boer and others suggested that one plot the amount adsorbed versus the thickness of the adsorbed layer, t(P/P ) (87–92). These plots are readily available on most automated adsorption equipment. Significant increases in sorption relative to a linear relationship with t imply the existence of microporosity at low thicknesses or mesoporosity at high thicknesses. These trends are shown in “t-plots,” such as that presented in Figure 1.17. That particular plot is for the isotherm in Figure 1.4A. As the figure shows, the models for thickness

1200

Volume adsorbed (cm3 STP/g)

1000

t from de Boer (1966) t from Halsey with EH = 3 t from Halsey with EH = 2.75

800

600

400

200

0 0

5

10

20

15

25

30

35

t (Å)

Figure 1.17 Example of a t-plot for the adsorption branch of the isotherm in Figure 1.4A (nitrogen at 77 K on SBA-15). The models for t are Equation (1.29), which is from de Boer and coworkers (87), and the Frenkel–Halsey–Hill equation (Equation 1.28) with different values of the exponent. Note that the “straight” lines between the abscissa values of 5 and 10 Å or so are not straight as a consequence of inaccuracies in the models for t. These can be corrected by using standard isotherm data, similar to an as-plot.

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are often an imperfect approximation to the actual thickness even in “flat” regions, causing deviations from linearity in the t-plot. 6.2.2 as-plots Sing (93,94) suggested that one should compare the relationship between the volume adsorbed, Vads, as a function of relative pressure with the normalized isotherm of a standard, nonporous solid. This normalized quantity, dubbed as, is the quantity adsorbed on the standard solid, which should have surface chemistry similar to that of the desired adsorbent, divided by the amount adsorbed by the standard solid at a relative pressure of 0.4. The amount adsorbed at this relative pressure corresponds to slightly more than one layer of adsorption for a majority of solids and is therefore a reasonable point at which to normalize. Furthermore, this relative pressure is most commonly less than the relative pressure at which pores are being filled. This value is appropriate for mesoporous solids with isotherms similar to those in Figure 1.4A, according to which no pores fill near the region of P/P ¼ 0.4. It would not be inappropriate, however, for isotherms similar to those in Figure 1.4B, for which pores are being filled in this range of relative pressure (P/P ¼ 0.4). Standard isotherms, however, are typically selected to be those characterizing nonporous materials. To construct the as-plot, first, construct as, which is usually defined as the volume adsorbed on a “standard” isotherm divided by the volume adsorbed on the standard at P/P ¼ 0.4. Now, plot the volume adsorbed from your isotherm against as of the standard isotherm with relative pressure as a parameter. This plot will yield a straight line for nonporous materials with surface chemistry similar to that of the standard. Deviations from a linear relationship represent deviations from the standard, such as mesoporosity, microporosity, and differences in surface chemistry. Standard isotherms for nonporous silica have been published by Gregg and Sing (12) (Tables 2.14 and 2.15) and by Jaroniec and coworkers (95); such data are readily available on most automated adsorption systems for use in asplots. An example of an as-plot for the isotherm from Figure 1.4A is shown in Figure 1.18. The intercept of the as-plot can be used to estimate the total micropore volume by extrapolating the volume adsorbed along a straight line to zero pressure. The volume (or the number of moles) adsorbed extrapolated to zero pressure is converted to a pore volume by assuming that the adsorbate is at the bulk-liquid density (Table 1.1).

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Analysis of Catalyst Surface Structure by Physical Sorption

1000

Vads (cm3 STP/g)

800

600

400

Start of pore-filling

200 V = (318.908 cm3 STP/g)as − 5.35229 cm3 STP/g 0

0

1

2 as

3

4

Figure 1.18 Example as-plot for the isotherm in Figure 1.4A (SBA-15, adsorption branch) determined by using standard nitrogen adsorption data from Jaroniec and coworkers (95) for nonporous silica. The intercept is the gas volume of the micropores (which is slightly negative here because of experimental error); to convert to liquid volume, use values from Table 1.1.

7. PORE SIZE DISTRIBUTIONS Methods of determining pore size are important to the characterization of many materials. There are several tools that can, directly or indirectly, be used to measure pore sizes, including scanning or transmission electron microscopy (SEM or TEM), 129Xe NMR spectroscopy, small-angle neutron scattering (SANS), mercury porosimetry, and, of course, physical adsorption. Techniques such as SEM are typically useful for relatively large pores on the surface of the material, typically classified as macropores. Xenon-129 NMR spectroscopy is useful because the xenon chemical shift is extremely sensitive to the local electronic environment. Although this statement is true of all nuclei, xenon is useful for the characterization of porous materials because its electrons are much farther from the nucleus than they are in lighter elements, and they are thus affected much more strongly by the weak dispersion forces associated with physical adsorption.

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Karl D. Hammond and Wm. Curtis Conner Jr.

Mercury porosimetry, which determines pore sizes by the process opposite of adsorption (i.e., forcing a nonwetting liquid into progressively smaller pores at progressively higher pressures), is extremely useful for investigating macropores (pores larger than 50 nm in radius) and also mesopores. The reader is referred to the relevant sections of Gregg and Sing (12) and Lowell, Shields, Thomas, and Thommes (41), as well as Refs. (96) and (97) for more information on porosimetry. The main topic of this section, however, is the use of physical adsorption in determining pore size distributions.

7.1. Traditional sorption-based techniques Most adsorption instruments today host a variety of “traditional” pore size distribution methods3 based on the derivative of the isotherm. These techniques are separate and distinct from more recent methods based on molecular simulation and similar theoretical methods, which are discussed in Section 7.2. 7.1.1 The Kelvin equation Most traditional techniques derived for mesoporous materials are based on interpretation via the Kelvin equation, which we now derive. We start from the Gibbs–Duhem equation, which states that dm ¼
SdT þ V dP

ð1:30Þ

in all phases, where m is the chemical potential, S is the molecular entropy, V is the molecular volume, P is the pressure, and T is the temperature. If the vapor phase obeys the ideal gas law,4 then ð  ml ¼ m þ Vl dPl ð1:31Þ 

ðP



ðP

mg ¼ m þ Vg dPg ¼ m þ P

P

  kT P dPg ¼ m þ kT log  Pg P

ð1:32Þ

where the subscripts ℓ and g denote liquid and gas, respectively. We now call upon the Young–Laplace equation, which relates the pressure drop across a vapor–liquid interface (a meniscus) to the radii of curvature of the meniscus (ri) and the surface tension, g, as follows and repeated from earlier: 3 4

By “traditional,” we mean “not based on results from computer simulations.” The reader should be aware that “obeys the ideal gas law” and “is ideal” are different statements. Truly ideal gases would not adsorb, as they have no interactions between atoms. However, nearly all adsorbates used in adsorption measurements obey, to good approximation, the ideal gas law at the relatively low pressures that are used during such experiments.

Analysis of Catalyst Surface Structure by Physical Sorption

  1 1 2g ¼
Pl
Pg ¼
g þ r1 r2 hri:

63

ð1:24Þ

Taking the differential of this equation, we can write the following:   2g : ð1:33Þ dPl ¼ dPg
d hr i We can then write the chemical potential of the liquid in terms of the external (gas) pressure and the mean radius of curvature of the interface. At saturation, the mean radius of curvature is infinite; at the pressure in question, the mean radius of curvature is the Kelvin radius, rK; for a cylindrical pore, this is the radius of the pore with the thickness of the adsorbed layer subtracted off (i.e., rK ¼ rp
t). A similar subtraction must be done for other pore geometries. The chemical potential in the liquid phase is now   ðP ð 2g=rK 2g 2gVl  ml ¼ m þ Vl dPg
Vl d : ð1:34Þ  m
rK r P 1 The approximation we have made here is that the liquid is incompressible, and thus the first integral vanishes.5 At equilibrium, the chemical potentials in both phases are equal, meaning that   P 2gVl 1 : ð1:23Þ log  ¼
kT rK P Equation (1.23) is the Kelvin equation. The constants on the left are often lumped together (and the logarithm changed to base 10), and so the equation reads   P
CK : ð1:35Þ log10  ¼ rK P where CK has dimensions of length. For nitrogen at its normal boiling point ˚ ; values at other temperatures and for other adsorbates of 77 K, CK ¼ 4.16 A are given in Table 1.1. Remember that the Kelvin radius, rK, is the mean radius of curvature of the meniscus inside the pore, not the pore itself. The astute reader may immediately notice that the Kelvin equation represents a continuum model and ask the question, when does the 5

This assumption is equivalent to the approximation that the molar volume of the liquid is much smaller than that of the vapor. Some authors leave the liquid volume or density in the equation and express it as a difference between two densities.

64

Karl D. Hammond and Wm. Curtis Conner Jr.

continuum hypothesis break down? Indeed, the Kelvin equation has been shown to be correct in the limit of large pores (98), with the maximum “safe” pore width being about 7.5 nm in slit-like pores (about 15 or 16 molecular diameters) (99). In general, the Kelvin equation overestimates the pore-filling/emptying pressure at equilibrium and thereby underestimates the pore radius. The Kelvin equation was first applied to the analysis of adsorption isotherms characterizing porous materials by Zsigmondy in 1911 (100). Cohan (101) in 1938 first offered an explanation of hysteresis based on the Kelvin equation and the idea of different radii of curvature upon adsorption versus desorption. Although this hypothesis has logical appeal, its predictions (such as that the effective Kelvin radius should be twice as large upon adsorption as it is upon desorption in open-ended cylindrical pores) have proven to be incomplete (98). We emphasize that the simulation-based methods presented in Section 7.2.2 are generally considered to be more accurate and/or more widely applicable than methods based on the Kelvin equation or other empirical models. In particular, they often give similar pore size distributions when applied to the adsorption and desorption branches of the isotherm (22), whereas Kelvin-based methods show a shift in mean pore size between adsorption and desorption or a completely different pore size distribution. The simulationbased methods are usually significantly harder to implement, however. 7.1.2 Mesopores By far, the most common method used for mesopore size distributions is the method of Barrett, Joyner, and Halenda (102), universally referred to as simply the BJH method and present on all modern automated adsorption equipment. In this model, the volume of gas that would desorb (or adsorb) as a result of a decrease (or increase) in relative pressure in a cylindrical pore is computed for each point along the isotherm via the following equation:   n
1 X rp
hti   Vp,n ¼ Rn DVn
Rn Dtn ð1:36Þ j¼1 rp Ap, j where Vp,n is the pore volume of the pores that empty or fill during the nth ad/desorption step, Ap,n ¼ 2 Vp,n/rp,n is the area of the pores’ walls, DVn is the observed change in volume adsorbed for the nth point on the isotherm, Dtn is the change in the statistical thickness, and Rn ¼ r2p,n/(rK,n þ Dtn)2 is the ratio of cylinder volumes between the pore (radius rp) and the inside of the adsorbed layer (radius rK þ Dt), that is, the ratio of the volume of the fluid

65

Analysis of Catalyst Surface Structure by Physical Sorption

before step n and the volume after step n. The first term represents the total increase in volume, whereas the second represents adsorbate that caused thinning/thickening of the adsorbate layers in otherwise empty pores (i.e., those not yet filled with condensing fluid). Barrett, Joyner, and Halenda’s values of t came from experimental values obtained by Shull (103) in the original paper, but more recent estimates of t (Section 6.1) are typically used in modern implementations of the BJH method. According to the method of Dollimore and Heal (DH) (104), the pore size distributions are calculated by a method similar to the BJH method, by use of the following equation: Vp,n ¼ Rn DVn
Rn Dtn

n
1 X j¼1

Ap, j
2pRn Dtn

n
1 X

Lp, j

ð1:37Þ

j¼1

where Lp, j ¼ Ap, j/(2prp, j) is the length of the pore. The DH method gives pore size distributions that are nearly indistinguishable from BJH pore size distributions, and thus the two are used interchangeably. Each of these analyses is done incrementally, that is, between successive data points in an increasing or decreasing measurement of the isotherm. The calculations require an evaluation of the instantaneous thickness of the adsorbed layer, tn, in each pore, such as the one obtained from the models presented in Section 6.1. In terms of measurements, one measures only the amount adsorbed (or desorbed) over an increment in relative pressure. If one knows the exposed surface area of the sample and the change in the thickness of the adsorbed layer (Dt), one can calculate the incremental amount adsorbed over the pressure interval DVn. The difference between the measured change in the amount adsorbed and the incremental change in the amount adsorbed in the thickening surface layer is interpreted as the amount that fills any pores over the increment during adsorption or empties pores during desorption. The pore size in both of the aforementioned models is dictated by the Kelvin equation (Equation 1.23) added to the estimated thickness of the adsorbed layer to give the solid pore dimension (e.g., from Equation 1.28). Knowing the pore size and the volume of liquid used to fill it (from the gaseous volume or weight measured), one can then correct the exposed surface area for the subsequent interval calculation. Most of these calculations are performed upon request by software in modern sorption instruments. However, such software often does not inform the operator when something is inconsistent in the calculations. For example, if the actual estimate of surface area were too high at any point

66

Karl D. Hammond and Wm. Curtis Conner Jr.

in the measurement, the result would be a negative estimated pore volume over the pressure increment. Similarly, if the wrong standard isotherm (and thus Dt(P/P )) were employed, incorrect pore volumes would be calculated. Also, the fact that the estimated total surface areas employed in the calculations during ad- or desorption analyses can differ from each other and from the BET area is never noted for the operator. This situation might pertain to atomically rough surfaces. It is useful to examine the actual calculation worksheets periodically to be sure that these inconsistencies are identified. Adjustments to experimental procedure and the standard isotherm employed for the analyses can usually correct these analyses, although Murphy’s law often dominates. It should be noted that the BJH and related methods for analyses of desorption data do not require an estimate of the total surface area, such as from the BET equation: the areas involved are areas of cylindrical pores that fill/empty at a particular relative pressure. As a result, these methods may give strange or unexpected results if applied to a region of the isotherm in which significant thickening on flat surfaces is involved (i.e., during monolayer formation). In general, these methods are applied at values of P/P below 0.4 for nitrogen or argon adsorption. Note that BJH analysis using adsorption data requires knowledge of the surface area, as the pore areas are not yet known. Naive application of the BJH, DH, and similar analysis methods to some systems gives erroneous results, particularly for microporous materials. These errors stem from (a) inaccurate values of the statistical thickness, especially below monolayer capacity; (b) the assumption that pore filling/ emptying and the associated surface area changes are responsible for changes in the isotherm; and (c) the assumption that the Kelvin equation gives the correct pore size, especially for pores with radii smaller than 2 nm. In general, the methods described in this section should be used only for relative pressures at which a monolayer has formed and there are no micropores. Analyses such as BJH are further complicated by a phenomenon known as the tensile strength effect (Section 5.4). This effect usually results in a sharp ˚ , but only in the peak in the BJH pore size distribution at around 17–20 A desorption branch. The adsorption branch of such isotherms will not indicate such a peak; thus, we strongly recommend analyzing both adsorption and desorption isotherms for the purpose of determining pore size distributions in mesoporous materials. If the shapes are inconsistent between the two branches, interpret the results (particularly for the desorption branch; see Section 5.4) with care.

Analysis of Catalyst Surface Structure by Physical Sorption

67

7.1.3 Micropores Micropores (pores < 2 nm in width/radius) require analyses different from those applied to mesopores: in micropores, the pores themselves fill with adsorbate before the external surface is covered (Figure 1.5), meaning that any estimate of the “surface area” of a pore will be questionable at best. This point means that BJH and other techniques for mesopores must be abandoned in favor of more direct models of micropore adsorption that do not require the invocation of concepts like surface area. Micropore size distributions for graphite and other carbonaceous porous materials can be calculated by the slit-pore model of Horva´th and Kawazoe (HK) (105), which results in the following equation: " #   P NE AA
E þ NA AA
A s4 s10 s4 s10 log  ¼

þ P kT s4 ðw
2d0 Þ 3ðw
d0 Þ3 9ðw
d0 Þ9 3d03 9d03

ð1:38Þ

where NA is the adsorbate density per unit area, NE the adsorbent density per unit area, AA–A the force constant for adsorbate–adsorbate interactions, AA–E the force constant for adsorbate–adsorbent interactions, d0 the arithmetic mean diameter of adsorbate and adsorbent atoms/molecules, and w the width of the slit. Saito and Foley (106) introduced a modification to the HK model that takes into account surface curvature in cylindrical pore geometries, which is an improvement for materials that have cylindrical pores. According to the Saito–Foley model, the micropore radius, rp, is calculated by fitting the isotherm to the following equation: 2 !2j 0 " #10 " #4 13  1 P 3 p NA AA
A þ NE AA
E X 1 d 21 d d 4 @ aj 0 log  ¼
bj 0 A5 1
0 4 P 4 kT j þ 1 32 r r rp d p p j¼0 

0

ð1:39Þ

The constants aj and bj are defined as follows:  2  2 Gð
4:5Þ Gð
1:5Þ aj ¼ bj ¼ , Gð
1:5
jÞGðj þ 1Þ Gð
1:5
jÞGðj þ 1Þ

ð1:40aÞ

where G(x) is the gamma function, 1 ð

GðxÞ ¼ tx
1 e
t dt; 0

or by the following recursive definitions:

ð1:41Þ

68

Karl D. Hammond and Wm. Curtis Conner Jr.




4:5
j aj ¼ j

2




1:5
j bj ¼ j

aj
1

2 bj
1

ð1:40bÞ

with a0 ¼ b0 ¼ 1. Cheng and Yang (107) made similar modifications for spherical pores: "      6ðNA AA
A þ NE AA
E Þrp3 P d0 6 1 1 log ¼
T1 þ T2

3 rp P0 12 8 kT rp
d0  6  # ð1:42Þ d0 1 1 T3 þ T4 þ rp 90 80 where T1–T4 are as follows: 1 1 T1 ¼  3
 3 1
r
d 1 þ r
d r r p

0

p

p

ð1:43aÞ

0

p

1

1

T2 ¼  2
 2 r
d 1 þ r
d 1
r r

ð1:43bÞ

1 1 T3 ¼  9
 9 r
d 1
r
d 1 þ r r

ð1:43cÞ

p

0

p

p

p

p

0

p

p

T4 ¼ 

0

0

p

1

1 8
 8 : r
d 1 þ r
d 1
r r p

0

p

p

ð1:43dÞ

0

p

The original slit-pore model was developed with carbonaceous adsorbents in mind. Zeolites, aluminum phosphates, and other silica-like materials generally form pores that are more spherical or cylindrical, and the modifications are usually used in such cases. All three of these formulations have been criticized for the (sometimes questionable) assumptions that underlie them: (1) the mean adsorbent– adsorbate distance is the same as the mean adsorbate–adsorbate distance, (2) the available adsorption sites form a continuous distribution, (3) all interaction energies are the same (with no spectrum of energies), and (4) the isotherm obeys Henry’s law in the limited regime in which the equations are presumed to hold. These and possibly other limitations cause the HK model ˚) to underestimate pore sizes for relatively large micropores (rp ¼ 8–20 A (108). The assumption of Henry’s law is often “corrected” by approximating the isotherm as a Langmuir isotherm (Equation 1.8) instead; this correction

69

Analysis of Catalyst Surface Structure by Physical Sorption

1.25

Loading (n/nm)

1

Henry’s law isotherm Langmuir isotherm BET isotherm

0.75

0.5

0.25

0 10-6

10-5

10-4

10-3

10-2

10-1

100

Relative pressure (P/P °)

Figure 1.19 Comparison (semilogarithmic plot) of Henry's law [y ¼ CP/P ], Langmuir [y ¼ CP/P /(1 þ CP/P )], and BET [y ¼ CP/P /[(1
P/P )(1þ (C
1)P/P )] isotherms showing the successive layers of approximation that each model removes. All are based on the value C ¼ 200. The Horváth–Kawazoe, Saito–Foley, and Cheng–Yang models, in their original form, assume Henry's law behavior; a simple improvement is to apply the Langmuir correction.

gets larger as the pore size increases (Figure 1.19). Rege and Yang (108) proposed a modification that addressed several more of these assumptions through the introduction of a parameter specifying the number of “layers” of atoms or molecules that can be accommodated in a slit pore (a similar procedure is also given for cylindrical and spherical pores). The number of allowed layers is determined via an iterative procedure. Both the Saito–Foley and HK methods underestimate the pore sizes for pores greater than 5 A˚ in diameter by up to more than 50% for the largest ˚ pores. However, they are good approximations for pores in the 4.5–6 A diameter range (109). DFT, as described in Section 7.2, typically gives a much better estimate.

7.2. Sorption analysis techniques derived from simulations 7.2.1 Theory Most work in the field of pore size distributions in recent years has involved what we refer to as classical potential density functional theory

70

Karl D. Hammond and Wm. Curtis Conner Jr.

(DFT).6 Classical potential DFT is an extension of the Hohenberg–Kohn theorem (110), which was developed with electron densities in mind, to nonzero temperatures (111) and applied to the density of molecules rather than electrons (112). All types of DFT have one thing in common: they fit an energy of a system of electrons, atoms, molecules, etc., as a functional of the density, which is itself a function of position. Classical potential DFT, which we hereafter call P CPDFT,7 attempts to find the Helmholtz free energy (F ¼
PV þ gA þ jmjNj) and the grand potential (O ¼
PV þ gA) (where P is pressure, V volume, g surface tension, and A surface area) as a functional ! of the adsorbate density profile, rð r Þ, ð

!

!

!

!

O ¼ V ð r Þrð r Þd r þFr ½rð r Þ:

ð1:44Þ

Note that the adsorbate density, like the electron density in electronic DFT, ! is often written nð r Þ by other authors. The subject of classical potential DFT and its applications to adsorption has been reviewed by Gubbins (113) and is discussed in a more recent review by Monson (60). This section includes only the briefest of overviews. There are several subcategories of CPDFT that have been developed in the last two and a half decades in efforts to predict isotherms and pore size distributions—particularly those for micropores—with a more solid theoretical foundation than is possible by use of methods derived from the Kelvin equation or even models specific to small pores such as HK and Saito–Foley, as discussed in Section 7.1.3. The terms generally invoked by the methods’ authors are local DFT (114,115), nonlocal density functional theory (NLDFT) (116), and quenched solid density functional theory (QSDFT) (117). Strictly speaking, it is the density functional that is local or nonlocal, not the theory, but this is a matter of semantics. Local DFT is the simplest family of density functionals and is analogous to the local density approximation of electronic DFT in that it uses only the 6

7

The “classical potential” part is to distinguish it from the density functional theory that is used to find approximate solutions for the electronic energy of electrons and nuclei. The term “functional” is what mathematicians use for a function, the argument of which is another function. We use classical potential DFT, or CPDFT, here so as not to confuse it with CDFT, which generally stands for current density functional theory. That kind of DFT uses the charge and also current densities in computing the electronic energy. This usage is not the same as is meant in the adsorption community, except that CPDFT and DFT are based on the same underlying concept: predicting the (free) energy on the basis of density.

Analysis of Catalyst Surface Structure by Physical Sorption

71

local density of the adsorbate to calculate the Helmholtz free energy and grand potential, with no information about density fluctuations (gradients) near the pore walls. This type of functional was originally fit to the hard sphere potential (114,115), which is not a wholly inappropriate model of liquids. Nonlocal DFT, which began with Tarazona and coworkers’ publication of a “smoothed density approximation” (SDA) (116), accounts for oscillations of the density near the pore wall, yielding a much more accurate description of the free energies as a function of pore size. The original SDA was found to agree with the Kelvin equation for rp ≳ 5s, where s is the diameter of the adsorbate molecule. The recently introduced quenched solid functional (117) further accounts for oscillations not only in the adsorbate density but also along the pore walls—in NLDFT, the walls are rigid, whereas in QSDFT, they are softer, allowing the adsorbate to overlap the wall slightly in simulation of wall roughness. QSDFT was developed specifically with micropores in mind, although it is generally applicable. Although none of these models may be claimed to be a perfect representation of confined fluids, it is worth emphasizing that there is no such thing as an error-free experiment either. All experimental adsorption isotherms are, at some level, amalgamations of the adsorption isotherms of many pores of differing widths. Even when a given material contains pores that are “identical,” there are still subtle differences in the pore size that serve to broaden the measured isotherm. The singular strength of theoretical isotherms is the fact that they are derived from known model systems, which is very difficult for empirically derived models. The differences between GCMC or grand canonical molecular dynamics simulations and DFTderived isotherms are small enough that we can consider them nearly perfect descriptions of the model systems, in this case the potential energy functions and model pores used in the simulations (60). These, in turn, are becoming increasingly excellent models of real materials. 7.2.2 Simulation-based pore size calculation methods DFT-based methods allow calculation of a pore size distribution by fitting the isotherm to a linear combination of simulated adsorption isotherms. The set of isotherms to which the measured one is fit is often termed the “kernel,” which we will call {nk(P/P )}, where n denotes the quantity adsorbed in pores of a certain size (we could call it rk or wk and write {nk(P/P jrk)}, if desired). The measured isotherm—quantity adsorbed as a function of relative pressure, nads(P/P )—can therefore be expanded as follows:

72

Karl D. Hammond and Wm. Curtis Conner Jr.

nads ðP=P  Þ ¼

ð1

nðr,P=P  ÞW ðr Þdr 

0

N X

nk ðP=P  ÞWk

ð1:45Þ

k¼1

where nk is the amount adsorbed by pores with width rk as a function of pressure and W is a weight function (which is continuous in a “perfect” isotherm but discrete in practical implementations). The actual kernels used in this procedure are generally not published explicitly,8 but generally, more than 100 different isotherms are simulated with r values spaced between very small (0.1–1 nm, depending on the expected pore sizes) pores and small macropores in the 50–75 nm range. These are then fit to the experimental isotherm via any comfortable curve-fitting procedure. The exact curvefitting procedure varies from author to author, with quasi-Newton nonlinear solution techniques and nonlinear least-squares fitting being two examples (118). Although the currently available DFT analyses are excellent, they make one essentially dependent on the software installed on one’s automated adsorption system.

8. DETAILS OF ADSORPTION APPARATUS In this section, we walk users through the innards of an adsorption experiment. These details are often hidden inside a proverbial black box or obscured by computer automation. However, there are many details of the apparatus that are important—even critical—to the collection or interpretation of data.

8.1. Volumetric adsorption systems There are two varieties of adsorption equipment in general use: gravimetric systems and volumetric systems. Gravimetric systems are used to measure the change in mass of the sample as gas is added and adsorbs. If it is added slowly enough (read: if gas is added slowly relative to the rate of adsorption), a near-continuous adsorption isotherm can be plotted. Gravimetric systems are limited by the sensitivity of the balance, however, and thus they tend to be limited to mesopore and surface area characterization, 8

The authors submit that publishing only the (highly technical) procedure used to calculate the “kernel” of isotherms used to determine pore size distributions without publishing the model isotherms themselves obfuscates the process of implementing the procedure and explaining it to new practitioners. This practice effectively creates a divide between people who use adsorption equipment and those who study it and/or sell software for it.

Analysis of Catalyst Surface Structure by Physical Sorption

73

for which pressures are relatively high, changes in mass are pronounced, heats of adsorption are relatively modest, and rates of adsorption are relatively high. Volumetric systems serve to measure only changes in pressure, and the adsorption isotherm is measured as the user fills a chamber of known volume with vapor (and thereby determines the number of moles added to the system), then opens a valve and allows that known quantity of gas to interact with the sample until steady state is attained. This procedure is used by most modern equipment, as it requires only one sensitive instrument (the pressure transducer) rather than two (those to measure pressure and mass), and it is typically much more sensitive than the gravimetric systems in the lowpressure regions of the isotherm. Volumetric adsorption systems for both “standard” and high-resolution nitrogen/argon adsorption can be constructed from the following equipment: 1. Two pressure transducers with outputs in the range of 1–1000 Torr or similar: one on the “front” end (the adsorption manifold ) to make pressure measurements and one in the “back” end to estimate dose sizes. 2. One pressure transducer with outputs in the range of 0–1 Torr or 0–0.1 Torr (preferable for HRADS) over the same or similar voltage range. 3. A “roughing” vacuum pump; this pump can be an oil-based, rotary pump or a diaphragm pump, for example. 4. A turbomolecular pump (high-vacuum pump). 5. At least six valves, with at least three of them rated to high-vacuum. We described two of these systems in previous publications (119–121); a generic design of a working apparatus is shown in Figure 1.20. All examples of volumetric adsorption apparatus (possibly with major modifications) are variants of this design. In particular, the use of “needle” valves allows elimination of the second 1000 Torr pressure transducer and dosing valves if implemented properly, and the inclusion of a “liquid” port is typically done only when adsorption is performed with hydrocarbons or other adsorbates that can be transported as liquids at room temperature. The use and investigation of liquid–vapor sorption in systems in which HRADS is also possible can be extremely problematic. High temperatures and extended pumping are required to purge the system to remove remaining traces of adsorbates on system walls from adsorbed liquids.

74

Karl D. Hammond and Wm. Curtis Conner Jr.

to helium tank

to adsorbate tank

9 10

8

to vacuum pump

7

1

3

4

Manifold (69.21 cm 3 )

Dosing (19.80 cm3)

6

2.1 cm3

2

5

liquid port

O-ring joint Ballast (345 cm3)

Dewar

Figure 1.20 Schematic representation of a volumetric adsorption apparatus capable of measuring high-resolution adsorption isotherms with a variety of adsorbates. Many designs are possible; this is only an example. The volumes of each chamber correspond to one particular (working) system and are present only to give an order-of-magnitude estimate of the volumes involved: any change to the fittings, including replacement of valve seats, valves, and piping, requires recalibration of at least the main manifold against a known volume.

The basic idea behind volumetric systems is simple. We use Figure 1.19 as an example; the basic procedure is the following: 1. Open a valve to a tank (valve 8 or 9), behind which pressures are in the 100–500 bar range, filling a relatively small volume (dosing) with gas. 2. Close the tank valve, and then open the valve(s) (#4 and #7) between the small volume and an even smaller volume, equalizing the pressure between the two chambers. Use of two such chambers allows the pressure to be lowered more quickly, but is not strictly necessary.

Analysis of Catalyst Surface Structure by Physical Sorption

75

3. Close all valves, and open a “dosing” valve (#3) connecting the small volume (which is merely the volume of the tubing) to the adsorption manifold (the chamber of known volume right above the sample valve), equalizing the pressure to a much lower value than was in the original dosing chamber (or the tank). 4. If necessary, evacuate the back end of the chamber again (valves #7 and #10) and remove gas from the manifold by manipulating the dosing valves (#3 and #4) repeatedly. When a relatively low pressure (corresponding to the number of moles in the desired first dose) is obtained in the manifold, record the pressure. 5. Open the sample valve and wait for equilibrium. This process could take from five minutes to an hour or more, depending on the adsorbate, the adsorbent, the relative pressure, and the temperature of the bath. When equilibrium (constant pressure to within instrument error; see Section 9.7 for warnings) is reached, record the pressure. 6. Repeat this procedure for all points in the sample, as described in the next section.

8.2. Start to finish: Acquiring an adsorption isotherm There are six steps involved in the measurement of an adsorption isotherm: 1. Calibration of the manifold volume 2. Preparation of a sample 3. Degassing 4. “Dead” volume measurement 5. Measurement of the saturation pressure 6. Measurement of adsorption (the actual experiment) Step (1) needs to be done only if the adsorption manifold has changed since the last calibration. The adsorption manifold has changed if any of the following has happened since the last calibration took place: • The sample valve has been replaced (valve 2 in Figure 1.20) or tubing attached to it has been replaced or moved. • Any valves connected directly to the adsorption manifold have been replaced (valves #1, #2, or #3 in Figure 1.20). • The adsorption manifold has been dented or otherwise damaged. • Any fittings have been redone or valve seats replaced. • Any pressure transducers have been replaced or reseated. Calibration of the manifold is described in Section 8.2.1. Steps (2)–(6) must be done every time, for every sample, and are described in the sections that

76

Karl D. Hammond and Wm. Curtis Conner Jr.

follow. The manifold temperature should be maintained at slightly above ambient temperature by a thermostat in case the room’s temperature changes. 8.2.1 Calibrating the adsorption manifold Calibrating the adsorption manifold, as stated earlier, needs to be done only if something has changed in the chamber(s) above the sample; if all that has changed is that another sample has been placed on the O-ring fitting, this procedure may be skipped. Note that most automated adsorption equipment does the procedure in the succeeding text for you, but it still follows a variant on this same procedure. You should outgas your adsorption manifold any time it is exposed to the air. This outgassing drives off water, carbon dioxide, and any other surface contaminants. Complete the calibration of the manifold volume only after the manifold has been heated under vacuum for several hours and then allowed to cool to its operating temperature. You should also degas your manifold any time it becomes contaminated; in this circumstance, you do not need to recalibrate anything unless you had to take something apart in the process of decontaminating the manifold. The volume of the adsorption manifold is very important: measure it as accurately as possible, and store it to as many digits of precision as are warranted by the pressure readings and the accuracy of the standard volume. Calibration is typically done with helium, although any gas that obeys the ideal gas law at room temperature may be used, such as nitrogen or argon. First, acquire a sample of known volume. If one is not available, the easiest way is to purchase one. If this is not an option, one can fill a sample of a liquid of known density (such as mercury) and use that volume. A safer, mercury-free method is to take a piece of precision-bore glass (the volume of which is known to good accuracy) and use it as an insert in another tube. We walk through the method using an insert in this section. Let us denote the pressure in both the manifold and the empty tube as PB,0 in both chambers. Fill the manifold with gas; the quantity of gas in the manifold added is now the following: A Nam ,1 ¼

PA,1 Vam RT

ð1:46Þ

Now, open the sample valve and wait for the pressure to stabilize (because there is no adsorption taking place and no temperature gradients,

Analysis of Catalyst Surface Structure by Physical Sorption

77

this stabilization should be fast, of the order of a few seconds at most). The number of moles now in the adsorption manifold is the following: B Nam ,1 ¼

PB,1 Vam RT

ð1:47Þ

and the quantity in the empty volume is the following: NeB,1 ¼

PB,1 Ve RT

ð1:48Þ

where in this case, both Ve and Vam are unknown. Because no moles entered the system between step A and step B (all we did was open a valve internal to the system), we have the following equality: A A B B A B Nam ,1 þ Ne,1 ¼ Nam,1 þ Ne,1 ¼ Nam,1 þ Ne,0

ð1:49Þ

where the right-hand equality takes account of the fact that no moles were added to the empty tube when the sample valve was closed and thus A N Be,0 ¼ Ne,1 . Therefore,



ð1:50Þ PA,1
PB,1 Vam ¼ PB,1
PB,0 Ve : If we repeat this procedure a number of times, we have NB,n ¼ NA,n
1 inside the empty volume each time, and so, the formula holds generally:

PA,n
PB,n

Vam : Ve ¼ ð1:51Þ PB,n
PB,n
1 Repeat this sequence a few times (about 10 is usually sufficient, although more is of course better) to determine an average of the ratio Ve/Vam, which for reference we call Re. Now, repeat the previously mentioned sequence again, only this time do it with the insert in place, therefore replacing Ve by Vf in Equation (1.51). You will determine another ratio, this one Rf ¼ Vf/ Vam. Because we know that Ve ¼ Vf þ Vstd, and Vstd is known, we infer the following: Vstd ¼ Ve
Vf ¼ Vam ðRe
Rf Þ:

ð1:52Þ

Now, you should be able to find Vam. In practice, it is often easier (and probably more accurate) to determine Vam by a self-consistent iteration procedure rather than trying to calculate ratios.

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8.2.2 Sample cells and preparation Sample containers for adsorption experiments are typically made of glass. There are several reasons for this choice: • Glass is relatively inert to both samples and the air over appropriate ranges of temperature. • It is transparent, allowing one to be certain the sample is actually underneath the temperature bath (failure to meet this criterion results in nonisothermal conditions9). • It is easily worked into shape. • It has relatively low thermal conductivity, allowing it to support a steep temperature gradient near the height of the (usually cryogenic) bath and reducing water contamination of the bath caused by “sweating” and the associated temperature changes resulting from condensation along the walls of the sample tube. The last point may seem as though it would cause experiments to take longer as a consequence of the increased time to reach equilibrium, and it does. The alternative, however, is to conduct an experiment in which the entire apparatus (including pressure transducers) is at the same temperature as the sample, an obviously impractical suggestion. The glass tube itself should have a narrow neck and minimal internal volume, to minimize changes in the dead volume (Section 8.2.4) resulting from changes in the bath height—especially in nitrogen, argon, or oxygen baths, which boil off continuously—and minimize the dead volume itself. This point is explained by the fact that the change in height is multiplied by the room temperature to bath temperature ratio; the dead volume can be estimated by assuming a cylindrical tube with a bulb of volume Vbulb on the end via (122)   pD2 pD2 h Tr ð‘
hÞ þ Vbulb þ : ð1:53Þ VDS ¼ 4 Tb 4 where h is the height of the bath above its “normal” position, ℓ the total length of the tube, VDS the dead volume (or dead “space”), Vbulb the volume of the bulb at the end of the tube (fully immersed), Tr the absolute temperature of the room, and Tb the absolute temperature of the bath; labels are shown in Figure 1.21. As is clear from Equation (1.53), the change in the dead volume (and thus the error in the volume adsorbed) is proportional to the square of the diameter of the tubing. Use of 1/4 in. or 7 mm tubing is fairly standard; use of tubing with larger diameters is not recommended 9

Measurement of isotherms under nonisothermal conditions is generally frowned upon.

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because of the considerations discussed, and use of tubing that is much narrower increases the need for corrections associated with thermal transpiration (Section 9.8). First, weigh out a small mass of sample. For zeolites and other microporous materials, 50 mg is likely more than enough. For other samples, 100 mg or more may be necessary. The preferred sample mass depends on the total surface area and pore volume of the sample. Adding more sample means that more gas adsorbs at each step, and thus, each dose will be smaller relative to the volume adsorbed (which is normalized on a per-gram basis). Using too little means the signal-to-noise ratio will be too high, and results will be untrustworthy. Next, weigh the glass tube to be used for the experiment. This step is done separately because one cannot guarantee that all of the sample will be transferred to the glass cell. Note that your fingerprints may transfer oil to the glass, which may in fact be comparable in mass to the sample. Add your sample to the tube. We also recommend adding a small amount of glass wool or similar packing material to the neck of the tube—make sure this material is below the fitting and well above the bath level, or the wool itself may start to adsorb gas. The glass wool lessens the probability that your sample will be lifted into the adsorption manifold by mistake during the evacuation process.10 Now weigh again; record both masses carefully. You should weigh your sample a third time after the experiment: when degassing (the next step), your sample will lose water, changing its mass.11 Adsorption on samples that are not readily available as powders presents special challenges. Some authors have analyzed zeolite membranes, for example, by scraping off the zeolite layers and grinding them into powders (123–126); others have attempted to cut their samples up so that they fit down the neck of the tube (but are obviously useless for future separation experiments) (127–129). Hammond and coworkers presented a modified adsorption cell specifically designed for zeolite membranes (122,130,131), but such cells may be inadequate for nonmicroporous materials because of the high dead volume associated with large sample containers and in some

10

11

If powder is blown into the adsorption manifold or into one of the valve seats, the system will leak and/or start adsorbing, throwing off your results. If this happens, you must take the valve or adsorption manifold apart, clean it, reassemble it, outgas the manifold, and recalculate the manifold volume. This is a painful procedure that we recommend avoiding. Keep your manifold clean! In the authors’ experiences, the amount of water desorbed by the sample is, generally speaking, smaller than or comparable to the error in both the measurement of the mass of the sample and the error in the volume adsorbed itself. Nevertheless, it is important to be aware of how much error is being made due to preadsorbed material.

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cases the relatively high surface area of the container compared with that of the sample being analyzed. 8.2.3 Degassing Once the sample is prepared and weighed, it must be heated under vacuum for several hours to remove adsorbed water, carbon dioxide, trace volatile organics, and so on. This process is called “degassing” or “outgassing.” If this procedure is skipped or truncated, the surface analyzed will be the adsorbent as covered by adsorbed material, not the bare surface of the adsorbent. In the case of microporous materials, such as zeolites, we recommend degassing under full vacuum for at least 12 h, preferably 24 h, before conducting the rest of the experiment. For non-microporous materials, 4–8 h may be sufficient. If you have the ability to check the pressure rise, do so; the rate of pressure rise should be negligible, preferably less than 10
5 Torr min
1. The pressure rise will never be precisely zero: there is always residual pressure. When the sample is cooled to the desired temperature, often 77 or 87 K, the rate of residual degassing will decrease even further. Incidentally, if you suspect your sample to have residual oils, soot, or other contaminants (e.g., left over from synthesis), we strongly recommend two precautions: (1) use a cold trap between the sample and the vacuum line to “catch” any liquids or solids that come off before they stick to the inside of the adsorption system, and (2) if possible, outgas your sample using a different pump, then transfer it to your clean system and go through the normal degassing procedure. Important: if the manifold gets contaminated with grease, organic compounds, or other contaminants, you will have to heat the entire manifold and outgas it before starting future experiments. 8.2.4 Dead volume measurements The next step in the process is measurement of the “dead volume,” also called the “dead space.” The dead volume is not a physical volume; instead, it is a hypothetical volume that an equivalent amount of gas would take up were it at standard temperature instead of the temperature of the bath, as shown schematically in Figure 1.21. Measuring the dead volume involves a procedure similar to that applied for the measurements of the manifold volume, except that this time the manifold volume is a known quantity and the volume below the sample valve (the dead volume) is unknown. For this experiment, helium must be used, as it is essentially an ideal gas in every sense of the word at temperatures such as 77 K and higher. By “every sense of the word,” we mean

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Analysis of Catalyst Surface Structure by Physical Sorption

293 K

293 K

D h

Vbulb 77 K

VDS

Figure 1.21 Schematic representation showing the difference between the actual volume and the apparent volume, or “dead volume,” that is used when computing the isotherms. The two are equal when the bath temperature is the same as the manifold's temperature, although it is not necessary to calculate this ratio; only the apparent volume is used. Drawings are to scale for the temperatures shown.

that helium does not adsorb: this point is crucial, as the dead space calculation must be done with the temperature bath in place, which means that gases such as nitrogen or argon would adsorb on the available surface. Strictly speaking, helium does adsorb—but only to a very minor extent (132)—and the amount is assumed to be negligible as far as most adsorption experiments are concerned. We stress that helium is far above its critical temperature (5.2 K (28)) even at 77 K. One problem associated with helium, however, is that it is a very small atom and therefore can penetrate small pores that larger atoms or molecules, such as nitrogen or argon, cannot. However, unless the volume of such pores is expected to be a large fraction of the total pore volume, the “extra” dead space in the sample (i.e., space accessible to helium but not to the adsorbate) is likely negligible. If one is concerned with the differences in accessible volume between helium and the adsorbate, one can attempt to measure the dead space at two higher temperatures (both above the critical temperature of the adsorbate) and correct with something akin to Equation (1.53), as suggested by Bhat and Narayan (133). To find the dead space, VDS, we begin with the same procedure used to derive Equation (1.51) except that this time we must use helium as the “filler” gas. Indeed, we get the same answer, except for the volumes involved, so we cut to the chase:

PA,n
PB,n

: VDS ¼ Vam PB,n
PB,n
1

ð1:54Þ

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Karl D. Hammond and Wm. Curtis Conner Jr.

Again, Vam is the (known) volume of the adsorption manifold, PA,n is the pressure in the manifold at step n before the sample valve (#2 in Figure 1.20) is opened, and PB,n is the pressure in both the manifold and the sample tube at step n after the sample valve is opened and the system has reached steady state. These measurements should be averaged over 10–20 different doses. It is essential to realize that the dead volume depends on the temperature profile in the sample container. Thus, the level of the bath, if the bath is not at room temperature, must be controlled as closely as possible to its initial height, both during the dead space calculation and during the measurements themselves. It is acceptable for the bath to be below or above the level during periods when one is merely waiting for the pressure to drop. However, adjusting the bath height changes the pressure, which in turn requires one to wait (again) for steady state to be achieved; plan accordingly. Automated equipment typically employs a computerized controller to adjust the level, with the assumption that the temperature profile resulting from the cold air above the bath (the height of which relative to the height of the Dewar lip will change over time for the case of cryogenic baths) does not significantly influence the dead space. In practice, neglect of this change can be a significant source of error in the adsorption isotherm. The magnitude of the effect depends on several considerations, such as the humidity in the laboratory. Another source of error in the dead volume is impatience, particularly with the first point (if one is averaging as described earlier): because the sample has typically been evacuated for outgassing purposes, the first point will typically be at very low pressures, meaning that the system has to reach thermal equilibrium with a relatively hot gas at low pressure by transmitting heat through a poor thermal conductor. Thus, we recommend waiting a substantial amount of time for the first point and going up more quickly after that. If one is careful, one can fill the sample with helium first and then work one’s way down in pressure to compute the dead space, which does not suffer from the low-density problem. Measurement of the highestpressure dead space value first also has the advantage of cooling the sample to the bath temperature, which can take considerable time, before HRADS experiments. It is extremely important to add or withdraw gases slowly at all times. Pulling full vacuum on a sample at atmospheric pressure, for example, will likely suck the entire sample up into the manifold, or worse, into the turbopump itself. If you have never heard the sound of a sand-like substance hitting blades of metal spinning at 1500 rpm, then consider yourself fortunate. Do not take shortcuts to the evacuation of powdered samples!

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8.2.5 Collecting an isotherm We are finally ready to measure the adsorption isotherm! Using Figure 1.20 as our guide, we do the following: 1. Assuming there is still helium left over from the previous dead volume measurement (Section 8.2.4), pump the sample down, slowly, either through a needle valve or by repeated pressure drops across valves (e.g., open #4, #2 open, #10 open/close, #7 open/close, #3 open/ close, #7 open/close, #10 open/close, #7 open/close, #3 open/close, #7 open/close, and #10 open/close). Continue until the pressure is low enough (perhaps approximately 1–5 Torr) to open the sample and vacuum valves (#1 and #2 in Figure 1.20) at the same time. 2. If possible, lower the cryogenic bath: this step will increase the rate at which helium leaves the sample chamber. Make certain you are pulling vacuum on your sample before doing this, or your sample may explode as the gas expands rapidly. 3. If you have just connected the adsorbate line, purge it. In the setup in Figure 1.20, this means closing all valves; run adsorbate through the line to purge the line of air; attach the line; purge the remaining volume by cycling valves 8 and 10 (do not open the tank directly to the pump!) repeatedly about 20 or 30 times. This procedure ensures that the next “belch” of gas to come out of the adsorbate line is adsorbate, not air. 4. Leaving the sample valve (#2) closed, fill each chamber in the manifold with a moderate amount of adsorbate. Purge the system completely, and then repeat this process once or twice. This procedure ensures that all helium has been purged. Evacuate the system, including the sample. 5. If you have a separate chamber with which to measure P , or are using a method that does not involve exposing your sample to saturated nitrogen, use it now to measure the saturation pressure (Section 9.2). If not, do it at the end of the experiment. 6. Fill the manifold with a small amount of nitrogen. If you anticipate large amounts of adsorption and/or are not interested in microporosity, you might start with 25–50 Torr. If your first point will be at 10
7 atm, choose something lower, like 0.1–1 Torr. Record the pressure in the manifold. 7. Open the sample valve and wait. This step will take a while; if you are doing this manually, engage in another task for an hour or so. If a computer is doing the waiting, make sure the computer is programmed to wait an appropriate length of time. In high-resolution adsorption, the first few points, especially when significant fluid adsorbs, can take

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45–90 min per datum. Check the bath level; if the level is the same as it was when you measured the dead volume, you are finished. If not, adjust the bath height so that it is and wait (less time) for the pressure to stabilize with the bath at the right height. Record the equilibrium pressure. 8. Compute the quantity adsorbed (see the succeeding text) and close the sample valve. 9. Repeat steps (6)–(8) until you reach a point near saturation. If possible, you should try to measure P periodically throughout the experiment as well. You are now finished with the adsorption branch and ready to measure the desorption branch. 10. Modify step (6) earlier to remove gas from the manifold. Now, repeat the same steps, each point dropping in pressure. The calculations involved in computing the volume adsorbed are straightforward and similar to those encountered for the dead space. The total quantity of vapor added to the manifold at step n is the following: ðnÞ ðn
1Þ A,ðnÞ B,ðn
1Þ
Nam þ Nin : Nin ¼ Nam

ð1:55Þ

where A and B refer to before and after, respectively, the sample valve has been opened after a dose. For simplicity, we substitute standard volumes for numbers of moles, and so, we make the substitution Nin ¼ PsVinRTs, where Ps is standard pressure (760 Torr ¼ 1 atm; see Section 9.1 for other definitions) and Ts is standard temperature (273.15 K; see Section 9.1 for other definitions). If we now divide through by standard pressure and cancel the universal gas constants, we get the following: ðnÞ

ðn
1Þ

Vin ¼ Vin

þ Vam

P A,ðnÞ
P B,ðn
1Þ Ts : Ps Tr

ð1:56Þ

where Tr is room temperature (more specifically, the temperature of the adsorption manifold). Note that if standard temperature is defined to be 293.15 or 298.15 K, which it often is (Section 9.1), then the ratio on the right is approximately unity, and the standard volume added to the system can be calculated directly from measured quantities. We mentioned before that using volume adsorbed instead of moles or grams adsorbed eliminates a conversion factor; this is one step to which that statement refers. Our next step is to calculate the quantity of adsorbate that is still in the vapor phase. The measured dead space actually incorporates the temperature gradient already, so the amount in the gas phase is simply the following:

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Analysis of Catalyst Surface Structure by Physical Sorption

ðnÞ Ngas RTs

Ps

ðnÞ ¼ Vgas ¼ ðVam þ VDS Þ

P B,ðnÞ Ts P B,ðnÞ  ðVam þ VDS Þ : Ps Tr Ps

ð1:57Þ

The volume adsorbed is the difference between the volume of gas in the system and the amount still in the gas phase. So the standard volume adsorbed per unit mass of adsorbate for a sample of mass m is the following: ðnÞ

V^ads ¼

ðnÞ

ðnÞ Vin
Vgas

m

:

ð1:58Þ

The adsorption isotherm is simply the plot of this quantity against PB,(n)/P . We emphasize that adsorption of different adsorbates or in different regions of the adsorption isotherm requires different lengths of time to measure a point on the adsorption isotherm. High-resolution nitrogen or argon isotherms on zeolites and other microporous materials may take 45–90 min per point until the pressure is greater than 0.01 atm. The “standard” BET range takes about 3–8 min per point in most cases for nitrogen and argon isotherms. Adsorption in mesopores may still fall in the 3–8 min per point range because of the increased pressures (and resulting faster heat/mass transfer). Adsorption of hydrocarbons takes considerably longer, even at high pressures, as a consequence of the higher heats of adsorption and slower molecular diffusion; 90–120 min is not unheard of. When you have finished measuring your isotherm, measure the saturation pressure if you have not already done so. Then carefully evacuate the sample by removing small doses of gas, taking care not to suck powder into the adsorption manifold. When the sample is evacuated, you may remove the bath. Do not remove the bath when the sample is under pressure or it may explode as the adsorbate (which may now be above its critical temperature) rapidly vaporizes. Once the sample has returned to room temperature, you should backfill the sample with adsorbate or helium and remove it from the system (keep the sample valve closed to prevent contamination). If desired, reweigh your sample to determine water loss.

9. COMMON PITFALLS IN ADSORPTION EXPERIMENTS AND ANALYSES In this section, we discuss several phenomena that may trip up the unwary practitioner and result in “measured” isotherms that do not represent equilibrium sorption processes. Some of them relate to inconsistent

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definitions; others are easy mistakes to make or simply points of which to be aware.

9.1. Definitions of standard temperature and pressure The term STP means different things to different people and can cause problems in the presentation of adsorption data, because the most common units for the ordinates of such plots are standard volumes (i.e., milliliters of gas at STP) per unit mass of adsorbent. The International Union of Pure and Applied Chemistry (IUPAC) used to define STP as 0  C (273.15 K, or 32  F) and 1 atm (101.325 kPa ¼ 760 Torr) (134). This definition is now obsolete. The preferred definition, since 1982, is 273.15 K and 1 bar (100 kPa ¼ 750.06 Torr) (134,135). The National Institute of Standards and Technology, on the other hand, defines STP as 1 atm and 20  C (293.15 K, or 68  F). In practical applications of adsorption equipment, STP often refers to the pressure of the room and the temperature of the adsorption manifold, which is often kept above room temperature to prevent changes in room temperature from changing the apparent manifold or dead volumes (Section 8). Room temperature, at least in the summer months in many places, is usually close to 25  C, which is a de facto standard in gas flow controllers. For the sake of comparison, we recommend specifying the definitions of STP being used if there is any concern that they will affect the results. The difference between using a standard temperature of 0  C and 25  C will, in the authors’ experience, produce errors in the volume adsorbed that are less than the error in the measurements themselves. In commercial equipment, the volume adsorbed is usually reported by defining STP as 0  C (although the manifold is maintained well above this temperature) and 760 Torr (1 atm) (41). It is possible to convert volumes adsorbed to moles or even grams of adsorbate per unit mass of adsorbent, eliminating these (very small) errors, but such conversions are unnecessary as long as one knows the relative magnitudes of the sources of error involved.

9.2. Measurement of the saturation pressure There are several methods of measuring the saturation pressure, all of which are subject to errors. Most of the time, these errors are small except when dealing with very large mesopores, in which case a small error in P can

Analysis of Catalyst Surface Structure by Physical Sorption

87

translate to a large error in the computed pore size as P/P ! 1 (see Section 7.1 and Equation (1.23)). One method of measuring P is simply to use a barometer. There are two corrections that should be applied: a relatively important one for the effect of temperature (causing thermal expansion of the mercury and the scale) and a smaller one correcting for latitude and altitude (accounting for deviations in the measured value of the Earth’s gravity associated with its rotation). These are necessary corrections because the torr is defined as the pressure required to raise a column of mercury at 0  C and 45 latitude by one millimeter. The corrections typically applied for temperature include the thermal expansion coefficient of mercury and the scale (usually made of brass). Analogously, one can pressurize the adsorption manifold and then vent it to the air and measure the pressure. One must be careful in this scenario not to allow any backwash of air into the manifold, as this will contaminate the apparatus with water (which will rapidly adsorb onto the manifold’s surface and require heating to outgas properly). Another method is to fill the sample tube with the adsorbate to pressures far above saturation, and then read the pressure once the pressure reading stabilizes. This procedure typically requires several attempts to ensure that liquid adsorbate forms in the sample tube. This technique is obviously useful only when the adsorbate forms a saturated liquid at the temperature of the bath. This method suffers from the problems discussed in Section 9.3, more so than some others because the measurement is only done once. Another way to determine P is to use a separate vessel, containing no adsorbent, and saturate that vessel with adsorbate, forming a liquid or solid inside. This method is typically the most accurate if P is sampled throughout the experiment. However, doing so requires either a separate adsorption manifold and transducer (creating problems with transducer calibration), or it requires the experimenter to be extremely careful about getting the pressure in the manifold after a P measurement back to just slightly higher/ lower than the manifold pressure corresponding to the previous data point. If one can regulate the pressure in the manifold in fine gradations with a needle valve, for example, this method becomes much easier to implement. Commercial equipment typically makes use of a variant on this method. Like the previous method, this is one that requires saturated liquid to form at the temperature in question.

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9.3. Drift in bath temperatures/compositions and equipment calibration All pressure transducers eventually drift over time, especially if the barometric pressure changes suddenly, as they nominally operate as differential transducers. Although the authors do not recommend operating any piece of equipment during severe weather, these types of drift in instrumentation would be exacerbated during storms because of the rapid changes in atmospheric pressure that accompany them. This problem is unavoidable even in the absence of severe weather and grows worse the longer the experiment is run. A similar problem arises because adsorption experiments must be run in the air. As the experiment progresses, oxygen begins to dissolve in the nitrogen or argon bath, causing the temperature of the bath to rise. The vapor pressure of oxygen is 140 Torr at 77 K, which is below the partial pressure of oxygen in the atmosphere (159 Torr). The rise in bath temperature is not typically very important, but it amplifies errors in both the dead space and the saturation pressure.12 Important: The larger the area of liquid that is exposed to the air, the faster the bath will become contaminated with oxygen.

9.4. Reference state for argon at 77 K The normal boiling point of argon is 87.3 K. Although running argon adsorption experiments at this temperature is possible—indeed, we recommend doing so, especially for micropore adsorption—it is relatively common to employ argon as the adsorbate but to use liquid nitrogen as the cryogenic bath (primarily because it is cheaper and easier to obtain than liquid argon). Because the normal boiling point of nitrogen (77.4 K) is below the triple points of argon (83.8 K), the saturated state at 77 K for argon is a solid phase. It stands to reason that a solid would not be expected to cover a surface or fill a pore as efficiently as a liquid, and the presence of the surface may in fact make the supercooled liquid the appropriate reference state. The saturation pressure for solid argon at 77 K is approximately 200 Torr, whereas supercooled liquid argon has a saturation pressure P ¼ 230 Torr (Table 1.1). This correction is extremely significant for the determination of surface areas: using solid argon gives a surface area per atom of ˚ 2 (38); supercooled liquid argon gives a value of Am ¼ 17 A ˚ 2, Am ¼ 18.2 A 2 ˚ . Some and close-packed liquid argon as a reference yields Am ¼ 13.8 A 12

The normal boiling point of oxygen is 91 K, so the temperature rise in the bath could become as much as 14 K—causing a drastic increase in saturation pressure (Table 1.1)!

Analysis of Catalyst Surface Structure by Physical Sorption

89

˚ 2 for argon at 77 K. For this reainstruments (39) use a value of Am ¼ 14.2 A son, we recommend against using argon at 77 K for surface area measurements. If argon at 77 K is used, the value of the molecular area should always be reported, and authors should be aware of the magnitudes of errors involved (which are, at worst, about 25%).

9.5. Misuse of the BET equation The BET equation (Equation 1.19) is often misused—for example, the BET model is used with points in the wrong area of the isotherm or used when its assumptions clearly do not apply (e.g., Figure 1.12). Recall from Section 2.6 that these assumptions are as follows: • The only source of increased quantity adsorbed is the thickening of adsorbed layers (i.e., no pores fill in or below the region to which the equation is applied). • The heats of adsorption of the second and higher layers are all equal to the heat of condensation of the bulk liquid. • The sticking probabilities and attempt frequencies of the second and higher layers are all equal. • At saturation, the number of layers becomes infinite, and the liquid condenses. The BET equation includes two constants, usually written CBET and SABET.13 Recall in particular Equation (1.13), reprinted from the preceding text:   a1 n2 Q1
Q L SBET CBET ¼ exp þ1 ð1:21Þ ¼ a2 n1 kT IBET It is obvious from this definition that CBET must be positive in order for the BET equation to make any sense. It is also clear from the assumptions stated earlier that the equation must not be applied in any region or above any region in which pores fill. Thus, the BET equation is typically applied at relative pressures between approximately 0.05 and 0.25, as this is the region in which a monolayer forms (136). There are many materials for which the BET equation gives a very small, possibly negative, intercept and thus either a very large or a negative value of CBET. This occurrence is frequent with zeolites, for example: their micropores fill before the external surface starts to be covered significantly, leading to an intercept very close to zero (and often negative, attributed in part to 13

We recommend using CBET over simply C and SABET instead of SA because it makes it very clear to all readers that these are determined from the BET equation.

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experimental error and in part to systematic errors in the model). With this point in mind, the value of CBET should always be provided whenever a value of the BET surface area is quoted. Moreover, we recommend reporting the region of the isotherm that was used to determine the surface area. For example, “BET surface areas were calculated from the nitrogen adsorption isotherms at 77 K by using data in the range 0.05 < P/P < 0.25. Plots of Vads(1
P/P ) were found to be increasing functions of P/P in this region.”

We also recommend placing the actual BET plot in the text of a publication or in supplementary material. At the very least, this information should be provided to reviewers. As discussed in Section 2.6.2, the value of CBET should usually be roughly in the range of 50–200. Values such as 210 are probably acceptable; values such as 400 are not. If CBET < 50, the isotherm is too shallow, and the monolayer is not well enough defined to apply the model (i.e., the assumption of sequential adsorption of multiple layers is inapplicable). If CBET > 200, the isotherm is too steep, and there is either significant micropore filling or localized adsorption (i.e., the assumption of a uniform grid of similar sites is not valid) (38). As discussed in Section 4.4, it has been suggested (51,52) that the BET equation yields appropriate agreement with surface areas calculated by other means for some microporous materials, provided that the data used for the BET plot show values of Vads(1
P/P ) to be an always increasing function of relative pressure. We call this the RLRS–BET method, or “rollers” BET. We encourage the reader to consider the following when attempting to apply the BET equation to microporous systems (for which we know that the underlying assumptions are invalid): 1. Read Section 4.4 before making any judgments. 2. Regardless of the material, and the analysis method, CBET should never be negative. 3. In the works of Snurr and coworkers (51,52), the “standard” region over which the BET equation is typically applied (0.05 < P/P < 0.3) nearly always yields incorrect results when micropores are present, regardless of the interpretation. 4. The data used for the BET plot should always have Vads(1
P/P ) increasing as a function of relative pressure. 5. “A model’s ability to predict experimental results in no way indicates the validity of that model.”14 14

The authors apologize for their apparent plagiarism of this quotation. We are unaware of its original source, but its use here seems apropos.

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In any case, we strongly recommend that one plot the predicted isotherm (from Equation 1.18) alongside the measured one to ensure they are both appropriate and overlap each other to a good extent (Figure 1.12). The BET plot itself (a plot of Equation 1.19) must also actually be linear; the previously cited criteria are designed to ensure that that very criterion is satisfied, as it can be difficult to see the deviations from linearity (Figure 1.12C).

9.6. Failure to degas properly This point seems obvious, but it frequently trips people up. Degassing is an essential step: it cleans the sample and prevents hydrocarbons and other contaminants from entering your (very clean) adsorption system. The contaminants can cover the surface, block pores, desorb during the analysis, contaminate the apparatus, and generally wreak havoc with all adsorption equipment and analyses. If you ever see black sludge in your sample tube, either prepare another sample or apply more uniform heating to your sample and degas for longer. If you have a cold trap (e.g., a liquid nitrogen bath placed somewhere along the vacuum line with a liquid trap to catch anything that condenses/sublimes), use it. Protect your pump, and your manifold, or be the victim of leaks and bad data.

9.7. Inaccurate or nonequilibrium pressure readings Measuring high-resolution adsorption isotherms requires special equipment, as mentioned in Section 4.1. It also requires extreme patience. Neglect of either of these points is a common recipe for strange or erroneous results. It can take up to 90 min (45–60 min is typical) per point at low pressures before equilibrium is reached. This statement is particularly true near the relative pressure at which the isotherm levels off (e.g., P/P  1.10 10
4 in Figure 1.10), as this part of the isotherm fills all micropores, even those occluded by other pores nearer the exterior of the particle. Once the relative pressure is greater than about 0.2 on a nitrogen or argon isotherm, equilibrium is reached relatively quickly—perhaps 3–5 min per point is all that is needed. Hydrocarbon adsorption, on the other hand, often requires long times to reach equilibrium even at higher relative pressures because of the much lower rates of diffusion and/or higher heats of adsorption involved.

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Pressures must also be measured very accurately, particularly for highresolution isotherms. High-resolution isotherms require transducers or equivalent apparatus capable of measuring low pressures (10
5 to 1 Torr for nitrogen and argon isotherms). If one attempts to measure an adsorption isotherm with a 1000 Torr transducer, for example, the results in the extreme low-pressure range (say, 0.001 Torr) are basically going to be electrical noise. This noise is sometimes erroneously interpreted as a real phenomenon (44).

9.8. Correcting (incorrectly) for thermal transpiration Thermal transpiration is a phenomenon first identified by Osborne Reynolds in 1879 (137); James Maxwell’s last paper (138), published the same year, was also about thermal transpiration. The phenomenon occurs when a rarefied (low-pressure) gas in the presence of a temperature gradient does work by circulating around a hot/cold interface. This cycle of work, which is done because of the temperature differential between the air in the room and the bath, supports a (small) pressure differential between the two chambers. Thus, the pressure over the sample is slightly smaller than the pressure measured by the transducer in the (relatively) hot manifold. Knudsen (139,140) showed in 1909–1910 that the pressure between the two chambers in the limit of infinitely small tubing is P1 ¼ lim D!0,P2 !0 P2

rffiffiffiffiffiffi T1 : T2

ð1:59Þ

The limit for large tubing is, of course, that the pressures are equal. For liquid nitrogen baths, the implication is that this ratio approaches 1.97 or so in the limit of low pressure and narrow tubing. Although this value might seem to be large, it bears remembering that the pressures involved are very low (say, 10
7 atm), so that a factor of, for example, two is less significant than it might be at higher pressures. The ratio is also rarely near the minimum; for example, hydrogen at liquid nitrogen temperatures (77 K) has a pressure ratio about eight percent higher than the limit imposed by Equation (1.59) at a cold-side vapor pressure of 6.6 10
7 atm, which implies that the measured pressure was about 1.3 10
7 atm (141). Edmonds and Hobson (142) observed that Equation (1.59) is strictly valid only for apertures, which they attributed to this fact: a “warm” molecule is much more likely to pass through a tube than a “cold” molecule.

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In the intervening century, several empirical correlations have been published that attempt to estimate the thermal transpiration ratio. The oldest is that of Liang (143), pffiffiffiffiffiffiffiffiffiffiffiffiffi P1 aðfP2 DÞ2 þ bðfP2 DÞ þ T1 =T2 ¼ : ð1:60Þ P2 aðfP2 DÞ2 þ bðfP2 DÞ þ 1 where D is the inner diameter of the connecting tube, a and b are empirical parameters fitted for helium, and f is a pressure-shifting factor that is 1 for helium and larger for other gases. For nitrogen, f ¼ 3.28; for argon, f ¼ 2.93. Other factors are given in Liang’s Table 1.1 (143). The ratio, as expected, approaches unity as P2 or D increases without bound. Liang’s correlation was the first, and is still in use, but the model of Takaishi and Sensui (144) is often used in its place. Their model, with T1 > T2, is expressed as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffi T2 =T1
1 P1 pffiffiffiffi ¼1þ , ð1:61aÞ 2 P2 AX þ BX þ C X þ 1 where X¼

2P2 D : T1 þ T2

ð1:61bÞ

The values of A, B, and C were fit empirically to the equations A ¼ 1:4 104 e0:507dc

ð1:62aÞ

B ¼ 5:6e C ¼ 110=dc
14

ð1:62bÞ ð1:62cÞ

0:607dc

where dc is the molecular or atomic collisional diameter. For nitrogen, A, B, and C are 1.2 104 K2 mm/Torr2, 1000 K mm/Torr and 14 K1/2 mm/Torr1/2, respectively; for argon, they are 10.8 105, 808, and 15.6, in the same units. Values for other gases are given in the original paper (144). All thermal transpiration corrections are small and only important for very low-pressure gases across narrow-diameter tubing. In the authors’ experience, changes in pressure caused by bath contamination and measurement error are comparable to the thermal transpiration factor. Although these corrections are small, the default may be to apply them or at least to keep the previous user’s settings. If the previous user applied the thermal transpiration correction for 2 mm tubing, and the current user

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leaves the same settings in place despite using 9 mm tubing, the correction for thermal transpiration will in fact be a miscorrection! Our advice: always know, and report, whether thermal transpiration corrections have been added. One should always report (a) which correlation is used and (b) the inner diameter of the tube. Thermal transpiration corrections are typically not significant at values of P/P exceeding approximately 10
4.

9.9. Interpreting hysteresis at low pressures to be porosity There is one oft-studied class of zeolite, with the MFI framework, which often displays low-pressure hysteresis in the nitrogen isotherm between P/P  0.08 and 0.18. This hysteresis was first observed by Carrott and Sing (145) for nitrogen adsorption and was subsequently observed for CO adsorption as well by Llewellyn and coworkers (146). This hysteresis phenomenon is believed to be specific to MFI and is thought to be linked to the phase transition between MFI’s orthorhombic phase and its monoclinic phase. This phase transition happens as a function of temperature and usually as the material is calcined (as-synthesized ZSM-5 or silicalite-1, e.g., starts out orthorhombic and becomes monoclinic when they cool down, after the organic structure-directing agents are calcined away). This phase transition has also been proposed to occur as a result of adsorbate–adsorbent interactions, causing hysteresis in the adsorption isotherm. This hypothesis is supported by the observation that boron-MFI, which does not typically undergo this transition, also does not display low-pressure hysteresis (131). The shape of this low-pressure hysteresis in MFI is also related to particle size (147) and has been shown to disappear for silicalite membranes, which incorporate relatively large crystals (130). This hysteresis is thus, as pointed out by Tao and coworkers (148), not characteristic of porosity.

9.10. Reporting 17–20 Å pores ˚ There are many reports of narrow distributions of pores in the 17–20 A range (we do not cite the authors of these reports, who are numerous). The reports are not an error per se—there is almost certainly some material that has a sharp peak in its pore size distribution in the 17–20 A˚ range—but in practice, it is nearly always a mistake in interpretation. Reports of narrow distributions of pores in this size range are almost universally drawn from nitrogen isotherms at 77 K characterizing samples that

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display Type H2 hysteresis (see Sing (149)) with a loop-closure point at P/P  0.42 for nitrogen at 77 K. The erroneous result comes from applying the BJH or similar method (Section 7.1.2) to the desorption branch of the isotherm in the presence of the tensile strength effect (Section 5.4), which leads to the inference of a very narrow distribution of pores at about 20 A˚ radius. The same pore size distribution analysis applied to the adsorption branch will typically yield a much broader distribution at higher radii. In materials with H1 hysteresis (in which the loop is largely symmetrical), analysis of the desorption branch is often the better choice. If the tensile strength effect is present, however, this analysis of pore size distribution from desorption will give the wrong impression. Recent improvements in DFT-based pore size distributions (see Thommes (22) for some excellent examples) have closed the gap between the pore size distributions determined from adsorption and desorption data, giving plots that are largely identical for both even in the presence of the tensile strength effect.

10. SUMMARY Physical sorption as a function of adsorbent pressure is progressive (micropore filling ! monolayer ! multilayer ! mesopore filling). In the representation of adsorption on a flat surface, the BET theory (37) extended Langmuir’s theory (10,11) for monolayer chemisorption to express the formation of physically adsorbed layers. The BET theory, applied in a range of relative pressures of 0.05 < P/P < 0.3, enables one to estimate the monolayer volume and thus the surface area in a nonporous or a mesoporous/ macroporous material. The theory is applicable if the heat of adsorption in the first layer is roughly 50–70 percent higher than heat of condensation (e.g., QL(N2) ¼ 5.564 kJ/mol at 1 bar (37)) and CBET is in the range 50 < CBET < 200 (note that CBET ¼ 75 for Q1 ¼ 1.5QL for nitrogen at 77 K). For a porous solid, the interactions between adsorbate and adsorbent become three-dimensional, and the heats of adsorption increase significantly as the pore size decreases. For example, DHads ¼
17.6 kJ/mol for nitrogen on silicalite-1 (150). Micropores are filled at relative pressures below 10
4 for nitrogen or argon; less than one percent of a flat surface would be covered at this relative pressure. Thus, measurement of micropore isotherms requires specialized equipment. CPDFT (Section 7.2) is an effective method to analyze the adsorption phenomenon in micropores.

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Estimation of surface areas for microporous systems by application of the BET equation is generally inappropriate, as the presence of micropores violates the assumptions underlying the model. In particular, the second fitting parameter in the BET equation, called CBET, is frequently out of the acceptable range and/or negative for microporous materials. However, the RLRS–BET approach (Section 4.4) has been suggested as a method to compare microporous solids for adsorption at very low pressures. Although the assumptions underlying this method are not irrefutable—especially the assumption that nitrogen takes up the same surface area on a highly curved surface as it does on a flat one—it provides another basis for analysis. We stress that the equivalent CBET value for a first-layer heat of adsorption of 17.6 kJ/mol (i.e., nitrogen on silicalite-1) is approximately CBET ¼ exp[(17.6 kJ/mol
5.564 kJ/mol)/RT] ¼ 1.3 108; this value is far greater than what one would ever find from a BET plot. One particular RLRS–BET analysis on a silicalite isotherm from Hammond and coworkers (130) yields CBET  26,000, corresponding to Q1 ¼ 12.1 kJ/mol (data not shown). Estimates of the pore volumes for porous solids depend on the interpretation of sorption isotherms as the amount of liquid required to fill the pores, assuming that the liquid density is the same as that of a bulk liquid under the same conditions. The amount of gas adsorbed is measured volumetrically (or gravimetrically) and interpreted as a bulk liquid in equilibrium with a vapor obeying the ideal gas law. For mesopores, the gas volume is measured at a relative pressure exceeding P/P ¼ 0.2, and proper analyses account for the thickening of the adsorbed layer during the pore filling, as is attempted in BJH analyses (102) (Section 7.1.2). For micropores, the t-plot (Section 6.2.1) is commonly employed, by which the volume adsorbed is extrapolated back to zero pressure and the volume adsorbed per gram is interpreted as the bulk-liquid volume, yielding an estimate of the pore volume. This analysis depends on an invariant Vads (as related to the thickness of the adsorbed layer), independent of relative pressure as commonly expressed in models such as that represented by the Halsey equation (Equation 1.27). A similar analysis that does not depend on a model of the thickness is available through the alpha plot (Section 6.2.2). Recently, classical potential density functional theory, called CPDFT or simply DFT, has been applied extensively to adsorption in microporous materials (Section 7.2). This theory predicts that adsorbates such as nitrogen or argon adsorb extensively at pressures below P/P < 10
4, largely in quantitative agreement with experiments. The pressure at which pores of a given

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dimension fill depends on the pore geometry and can involve several energetic minima in, for example, a cylindrical pore (113). As adsorbate pressure increases, the pores would first fill these minima and then continue to densify toward the value for a bulk liquid (42,151). Thus, DFT projects that micropores fill at a density less than the bulk-liquid density and continue to fill as pressure increases. The pressure at which bulk-liquid densities are achieved is still unclear and the increase in density as a function of relative pressure might even continue well through and above the relative pressure at which a monolayer would be found on a flat surface. Two theories are available to explain sorption hysteresis for mesoporous solids: network effects and metastability during adsorption. Care must be taken to avoid desorption analyses when pores approximately ˚ in radius are calculated, as cavitation from within pore spaces 17–20 A occluded by small pores (the tensile strength effect) causes the BJH analysis to lead to the inference that there is a narrow distribution of pores when in fact the closure of the loop says little about the size of the pores. We recommend that both adsorption and desorption data be analyzed to gain additional insight into porous solid morphologies. DFT-based techniques have become sophisticated enough in modern analyses that there is often very little difference in the pore size distribution as reported on the basis of data from both branches of the isotherm. The influence of quadrupolar interactions on physical sorption phenomena requires further investigations to analyze its effect.

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