Accurate Determination of Pore Size Distributions

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1 Accurate Determination of Pore Size Distributions Of Catalysts and Supports: Emphasis on Alumina Calvin H. Bartholomew, Kyle Brunner, and Baiyu Huang Departments of Chemical Engineering and Chemistry, Brigham Young University

2 Outline Review fundamentals of pore size measurement Discuss isotherm hysteresis and selection of branch Address methods for analyzing data Illustrate with real data

3 How is Pore Size Distribution of Supports & Catalysts Important? A basic property of supports & supported catalysts-- relates directly to surface area and pore volume, i.e. d pore 2 V pore /S BET (Wheeler). Important parameter in catalyst preparation and design. In preparation: indicates uniformity of active phase distribution Reactions involving large reactant or product molecules require large pores, e.g. HDS, Fischer-Tropsch. Pore size distribution (PSD) can affect activity & selectivity Important parameter in modeling catalytic reactions and catalyst deactivation; determines pore-diffusional resistance; measure of distribution of coke.

4 Measurement of Pore Size Distribution in Porous Materials Physical adsorption of N 2, Ar, H 2 O, CO 2 or hydrocarbons at condensation temperature. Volume of gas adsorbed (condensed) is measured as a function of relative pressure, P/P o where P o is the standard is the vapor pressure of the absorbate at the adsorption temperature. Method is based on the fact that gas condenses to liquid in narrow pores at P < P o and in larger pores at progressively higher P/P o. N 2 adsorbs & condenses mainly in mesopores. Capillary condensation is modeled by Kelvin equation which relates P/ P o to pore radius r pore ln P P o = 2γ V cosθ r p liq RT Where γ is the surface tension of the liquid (contact angle θ is assumed to be zero).

5 V o (pore volume) Multilayer Region all pores are filled desorption large pores are filled adsorption Henry's Law Region 0 Monolayer Region Linear Region 0.05 < P/P o < very small pores filled (r < nm) Hysteresis Region (1 < r < 100 nm) P/P o 1.0 Full range Type IV adsorption isotherm with adsorption and desorption branches. Kelvin Equation predicts hysteresis region, since adsorption and desorption are limited by larger and smaller pores respectively.

Hysteresis Explained by Bottle Neck Effect In a mesoporous solid, void spaces between nanoparticles can be modeled as tubular passages of varying cross-section similar to the body and neck of a water

6 Hysteresis Explained by Bottle Neck Effect In a mesoporous solid, void spaces between nanoparticles can be modeled as tubular passages of varying cross-section similar to the body and neck of a water bottle. Molecules adsorb on the bottle walls and fill the bottle at Pa: = Po exp rw RT Desorption of molecules is limited by the neck radius and 2 γv P = Po exp occurs at Pd: d rn RT Since rn < rw, Pa > Pd In addition to the bottle neck effect, there are several other explanations for hysteresis of capillary condensation in pores. P a 2 γv

7 Explanations of Hysteresis & Implications for Equilibrium Bottle neck theory equilibrium is assumed for ads & des. Difference in contact angles of condensing and evaporating liquid. May delay formation of meniscus, i.e. condensation during adsorption isn t at equilibrium. Rate of evaporation is retarded if its only exit is a narrow channel or if entrapment of escaping gas occurs; thus, desorption is not at equilibrium. Modifications of meniscus formation, e.g. can be delayed in slit-shaped pores where the meniscus is formed at only high P/P o ; hence condensation is not at equilibrium during adsorption.

8 One Versus Two Open Ends a. Pore with one open end: a hemispherical meniscus is formed on adsorption Kelvin equation applies to adsorption and desorption; b.&c. With two open ends: two cylindrical (annular) menisci are formed during adsorption and the exponential in the Kelvin equation becomes exp(-k/r) rather than exp(-2k/r). Thus, the radius during adsorption is twice that during desorption!

9 Packed Spheres Meniscus formation is delayed during adsorption; occurs rapidly when the pore is filled (see steep adsorption branch). Thus, adsorption process is not described well by the Kelvin equation; desorption is. Condensation occurs initially in the crevices between spheres; liquid toruses work their way in and coalesce.

10 Slits Between Plates Since the mean radius of curvature of a plane is infinite, capillary condensation cannot occur at any pressure below saturation Thus, adsorption process is not described by Kelvin equation; desorption is because it involves capillary evaporation.

11 Types of Hysteresis Tubular capillaries, packed beds Parallel plates, slits Spheroidal pores Parallel plates, slits Four identified IUPAC hysteresis classifications. Each type may relate to specific types of nanostructures as shown Liquid formation may be unstable in either adsorption or desorption branch. Detailed analysis of the processes occurring in each branch can help identify unstable paths, e.g. for slitlike pores between slabs or closepacked spheres, expect instability during adsorption.

12 Analysis of PSD Data Two types of approaches: Methods based on area of pore walls e.g. PODV (Pierce, Orr, and DallaValle) Methods based on length and area of pore walls e.g. BJH (Barrett, Joyner and Halenda) All methods based on essentially same fundamental equations All methods use a spreadsheet with a number of corrections PODV is relatively simple and more fundamentally based (details in Thomas & Thomas) BJH is quite complicated and somewhat empirical; most widely used Most methods use outdated calculation procedures and the reported results are very crude.

13 Fundamental Equations Kelvin equation (contact angle assumed zero) in terms of r p : r p 2γ Vliq = + RTln P/ P First term accounts for condensation is the Kelvin or core radius second for thickness t of adsorbed layer. for N 2 Thickness t = σ (V/V m ) where σ the effective height of the adsorbed later is based on either hexagonal (σ = 0.36 nm) or cubic packing (σ = 4.3 nm). t 5 = 3.6 ln P / P Structural factor relating surface area A to V p and r p Δ A = p r p k α2 ΔV r p o r = r + t k = 9.53 ln P/ P o p t o 1/3 Based on the Wheeler model of non-intersecting, cylindrical parallel pores where α = 1. Alpha should be calculated or determined experimentally for intersection of pores and for other geometries (T&T).

14 Recommended Approach Determine structure of support or catalyst and type of isotherm and hysteresis; provides basis for Selecting appropriate branch for analysis (ads or des) Determining appropriate structure factor Emphasize data collection in hysteresis region to maximize number of data points in the PSD Smooth isotherm data and use BJH or other equations to fit cumulative pore volume versus r p. Then take derivative numerically to get dv p /dr p and dv p /dlogr p values. Use log normal volume density Gaussian function [Chu et al., 1971]; provides excellent fit of PSD data and provides a σ g which is the width of the distribution. f 1 [ln( d ) μ] pore vol = exp (2π) σdpore 2σ 2

Application to Alumina Supports Structure

This basic structure will predominate as

15 Application to Alumina Supports Structure Most aluminas are made via Gibbsite which is converted to Boehmite in turn consisting of layered slabs. This basic structure will predominate as the alumina is heated to higher temperatures. Courtesy of Steve Baxter, Sasol

18 Typical PSD Literature Data for Al 2 O 3 (micro-emulsion Al 2 O 3 ), Zhang et al. (2004)

19 Typical PSD Literature Data for Al 2 O 3 (micro-emulsion Al 2 O 3 ), Zhang et al. (2004) dvp/d(dp) d p and dv p data were smoothed d pore,arith = 9.61 nm Adsorption or desorption? NOT reported! Might be bi- or trimodal distribution but data are too scattered to draw any useful conclusions. d pore,lgmn = 17.6 nm σ PSD = 4.85 nm Pore Diameter d p (nm) Authors report pores sizes in the range of 5-25 nm. But very different pore sizes are estimated from simple vs log distribution (10 or 18?). dv/d(log rp) Experiment Calculation log (pore radius) (nm)

20 dvp/dr PODV PSD for 3%La/Catalox Al 2 O 3 3La/Catalox-Al 2 O 3 (Sasol), Calc 800, 4h, Desorption PODV method developed by Pierce, Orr, and DallaValle in the early 1950s is not as widely used as the BJH method but is advocated in classical textbooks because of its simplicity and straightforward use of basic equations. Log normal Gaussian distribution was originally proposed by Chu et al. (1971) and was recently 200 applied to PSDs by the author d pore,arith = 12.1 nm Only slight asymmetry and tailing for dv pore /d(d pore ) vs d pore for a number of aluminas Pore Radius (nm) dv/d(log rp) d pore,log = 12.5 σ PSD = 2.6 Log normal distribution provides (a) best fit of PSD data and (b) mean and width parameters. Calculation PODV line shape is close to log Gaussian. Experiment log (pore radius) (nm)

21 PODV PSDs for 3%La/Catalox Al 2 O 3 3La/Catalox-Al 2 O 3 (Sasol), Calc 800 or 900, 4h, Adsorption Adsorption branch is, as expected, less stable; nevertheless, data can be obtained for large pore aluminas. Pore diameters are much larger ink bottle effect! Greater asymmetry r p avg and (nm) instability. 0 V p/0r p avg 70.0 adsorption, clustering, and liquid 60.0 formation during adsorption V p/ r p avg Suggests simultaneous unstable r p avg (nm) Normalized V p /ln( r p avg) Normalized V p /ln( r p avg) dp = 17.9 ± 2.7 ln(r p avg) Sintering at 900 is evident by increase in d pore. dp = 19.7 ± ln(r p avg)

22 BJH PSD for 3%La/Catalox Al 2 O 3 3La/Catalox-Al 2 O 3 (Sasol), Calc 800, 4h, BJH-Desorption BJH PSD below is presently the standard for journal literature and surface-area analyzer software d pore,arith = 12.2 nm d pore,arith > d pore,log-mean (12.2 vs 11.6) d pore,lgmn = 11.6 nm σ PSD = 1.9 nm dvp/d(dp) Typical asymmetry and tailing for dv pore /d(d pore ) vs d pore Pore Diameter d p (nm) dv/d(log dp) BJH line shape is not quite log Gaussian distribution is artificially narrow. Calculation Experiment BJH method is very complicated and adds a fitting parameter log (pore diameter) (nm)

23 Method/ adsorption branch Pore diameter d pore,log-mean Pore width σ pore,log-mean BJH/adsorption PODV/desorption PODV/adsorption Summary of PSD Parameters for 3%La/Catalox Al 2 O % LaCatalox-800 Desorption d p = 12.5 ± 2.6 3% LaCatalox-800 Adsorption d p = 17.9 ± 2.7 dv/d(log rp) Sum of two distributions log (pore radius) (nm)

24 PSD for Small-pore Al 2 O 3 Possible Complications 0.60 Example of high quality data. PSD 0.50 is very well defined. dv/d(log rp) 3% La-Cosmas, calc.700, 2 h Desorption Experiment d pore,lgmn = 4.44 nm ± 2.21 Calculation Very good fit using log normal distribution; however, line shape is slightly off not enough data in the critical range log (pore radius) (nm) Micropores pore size is not predicted by Kelvin equation.

25 Effects of Data Scatter and Smoothing on PSD Parameters for Small-pore Al 2 O 3 3% La-Cosmas, calc.700, 2 h--desorption dv/d(log rp) d pore,lgmn = 4.09 nm ± 2.23 Calculation Experiment dv/d(log rp) log (pore radius) (nm) 3% La-Cosmas, calc.700, 2 h Desorption 0.00 d pore,lgmn = 4.07 nm ± 2.27 Calculation log (pore radius) (nm) Experiment (7-pt smoothing)

26 dvp/dr 3% La-Cosmas, calc.700, 2 h Desorption da p = 2*dV p /r p Negative dv/dr da p = alp1*2*dv p /r p Pore Radius (nm) Constrnt MesoSA V pore d pore m 2 /g cm 3 /g nm No Yes BET SA = 267; V pore(bjh) = d pore,arith = 4.03 nm dvp/dr Effects of Negative dv/dr and Structure on PSD, SA, & V pore 3% La-Cosmas, calc.700, 2 h Desorption alpha1 = 0.68 Constrain dv/dp 0 d pore,arith = 4.00 nm Pore Radius (nm)

27 PSD from Adsorption on Small-pore Al 2 O 3 [Same 3% La-Cosmas, calc.700, 2 h Desorption] dvp/dr Raw data were smoothed d pore,arith = 3.99 nm Data OSCILLATIONS are evident in log normal distribution! Evidence that liquid is not formed until pore is almost full. Gas adsorption and cluster formation are not governed by the Kelvin equation hence data/are are suspect (even if fit looks good) Experiment Pore Radius (nm) Data instability is apparent in the simple dv/dr versus r distribution, even after smoothing. dv/d(log rp) Calculation d pore,lgmn = 3.94 nm σ PSD = 3.06 nm log (pore radius) (nm)

28 PSD for FeCuK/Al 2 O 3 Considerable tailing for dv/dr distribution; d p,ave is too large. 100 V p/ r p avg d p,ave = 17.5 nm r p avg (nm) Normalized V p /ln( r p avg) Very good fit using log normal distribution; plenty of data. d p,log = 15.9 nm σ g = 2.9 nm ln(r p avg)

29 Conclusions Most of the previously reported studies of PSD used high quality instrumentation and relatively crude methods of analysis results are of questionable value. Methods are available for high quality analysis of the data. This presentation provides an approach that has worked well for our group. Keys are to understand the basic structure of materials being used and the type of hysteresis behavior to enable appropriate choices of isotherm branch and structural factor. to use numerical methods to improve the data to use a log normal Gaussian distribution to determine average particle diameters and distribution width

30 Willi Holdman Thanks for listening!

3.10. Capillary Condensation and Adsorption Hysteresis

3.10. Capillary Condensation and Adsorption Hysteresis We shall restrict our attention to the adsorption behavior of porous solids. Hysteresis: two quantities of adsorbed material for each equilibrium