Adsorptive Capacity and Evolution of the Pore Structure of Alumina on

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Adsorptive Capacity and Evolution of the Pore Structure of Alumina on Reaction with Gaseous Hydrogen Fluoride Grant J. McIntosh,*,†,‡ Gordon E. K. Agbenyegah,†,§ Margaret M. Hyland,†,§ and James B. Metson†,‡ †

Light Metals Research Centre, ‡School of Chemical Sciences, and §Department of Chemical and Materials Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand S Supporting Information *

ABSTRACT: Brunauer−Emmet−Teller (BET) specific surface areas are generally used to gauge the propensity of uptake on adsorbents, with less attention paid to kinetic considerations. We explore the importance of such parameters by modeling the pore size distributions of smelter grade aluminas following HF adsorption, an industrially important process in gas cleaning at aluminum smelters. The pore size distributions of industrially fluorinated aluminas, and those contacted with HF in controlled laboratory trials, are reconstructed from the pore structure of the untreated materials when filtered through different models of adsorption. These studies demonstrate the presence of three distinct families of pores: those with uninhibited HF uptake, kinetically limited porosity, and pores that are surface blocked after negligible scrubbing. The surface areas of the inaccessible and blocked pores will overinflate estimates of the adsorption capacity of the adsorbate. We also demonstrate, contrary to conventional understanding, that porosity changes are attributed not to monolayer uptake but more reasonably to pore length attenuation. The model assumes nothing specific regarding the Al2O3−HF system and is therefore likely general to adsorbate/adsorbent phenomena.

1. INTRODUCTION Efficient physisorption and chemisorption of pollutants plays a fundamental role in a number of important environmental technologies such as water filtration and wastewater treatment,1−3 as well as dry scrubbing systems for industrial exhaust gas treatment centers.4−7 Appropriate adsorbent materials require high specific surface area (SSA) to maximize the number of adsorption sites and a pore structure conducive to adsorbate contact with the surface over time scales relevant to the particular system. However, the aging of treatment materials likely also has considerable impacts on the efficiency and viability of these technologies. The aluminum reduction industry provides an excellent example of the application of porous media trapping molecules from fluid streams. Reduction cells generate a variety of fluoride-based emissions including hydrogen fluoride, produced by the reaction of water or hydrogen (some residual in the structure of the alumina fed to the cell) with the cryolite electrolyte. Most smelters operate with a dry scrubbing system, contacting alumina with the emissions as a means of both capturing reactive gases and recycling lost fluoride (this alumina is eventually fed to the cells). Dry scrubbing efficiency depends heavily on the rate of fluoride adsorption capacity of the alumina. However, smelter grade alumina (SGA) is an extremely complex material, formed by calcination of agglomerated Bayer Gibbsite, Al(OH)3, particles. Differences in the calcination process can manifest in considerable variations in phase between SGAs or even within a single grain. Hexagonal phases of Gibbsite transform ultimately to fully calcined α-Al2O3 (corundum), via a range of © XXXX American Chemical Society

defect spinel structures collectively termed the transition aluminas.8 The γ-, γ′-, δ-, and θ-Al2O3 structures are dominant in SGAs, listed here in order of decreasing retention of structural hydroxide, which acts as an HF source. Complicating matters, each also possesses unique pore structure and surface area characteristics from γ- to α-Al2O3, with SSAs dropping while average pore size increases.9 Intuitively, higher SSAs should be desired from a dry scrubbing point of view; however, residual hydroxyls in a high SSA material such as γ-Al2O3, with a Brunauer−Emmet−Teller (BET) SSA of ∼300 m2 g−1, are also a potent source of HF.9,10 Similarly, there is evidence that the smallest pores may pose kinetic limitations during dry scrubbing;11−13 thus, the presence of these phases may provide inflated estimates of scrubbing capacity. The pores in the SGAs tend to be at the lower end of the mesopore range, with pore size distribution (PSD) maxima rarely exceeding ∼20 nm in diameter. The pore structure of SGA in particular is presumed to be fairly uniform as it is believed that a large fraction of mesoporosity arises during calcination by c-axis shrinkage; hydroxyl rich planes are stacked facing one another in the (001) direction in Gibbsite, with H2O loss across these planes on calcination leaving parallel planes along this axis.14,15 This will lead to the finest channels, susceptible to blocking (the plane spacing is ∼0.95 nm);16 it is likely that later collapse and cracking also generates larger scale porosity. To illustrate this, several ESEM images are presented in Figure 1 for a number of Received: February 18, 2015 Revised: April 23, 2015

A

DOI: 10.1021/acs.langmuir.5b00664 Langmuir XXXX, XXX, XXX−XXX

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Figure 1. ESEM images: (left) whole SGA grain, (middle) Bayer Gibbsite grain calcined to α-Al2O3, and (right) cross-sectioned SGA grain demonstrating uniform porosity on multiple scales parallel to the Gibbsite c-axis.

representative grains. Gibbsite tends to form grains of ∼150 μm across by the aggregation of crystals ∼20−40 nm in size,16,17 which tend to be hexagonal prismatic. Striations appear parallel to the c-axis of these grains, consistent with the accepted dehydration mechanism, with much larger fissures in the fully calcined α-Al2O3 grains, consistent with increased collapse with calcination. Cross-sectioned images reveal clear aggregate boundaries, with all fissures mutually parallel but perpendicular to the outside edge and running deep into a primary grain edge. All features are consistent with porosity over multiple scales arising from a common mechanism of increasing water loss across opposing Gibbsite (001) planes.9 For optimal dry scrubbing, a compromise between SSA and average pore size is therefore sought; however, mechanistic insights and quantification of the ranges where pores become effectively inaccessible (if they are indeed inaccessible) is lacking, and exactly where the best compromise lies is still unknown. While the SSA and pore structure of fresh materials have been widely considered, the evolution of the porosity after adsorption has been essentially unaddressed in spite of the likely importance of such phenomena (many dry scrubbing systems in smelters recycle alumina through the waste gas stream multiple times, and unlike other adsorbentsmolecular sieves, for examplethe material is not regenerated). Knowledge of the aging of the postgas-contacted alumina (referred to as secondary alumina, sec-Al2O3, as distinct from the fresh primary alumina, prim-Al2O3) requires knowledge of the adsorption product. Laboratory tests have confirmed that adsorption is irreversible18 and therefore consistent with a chemisorption/reaction process. However, the nature of the phase formed is still contentious. Early works have demonstrated that AlF3 is not formed as the primary product under smelter-relevant conditions,19 or does so only if the sample is predried (as revealed by detailed XPS analyses of judiciously chosen samples, HF-dosed under controlled laboratory settings).20 Most works propose that surface-bound water is involved in adsorption. Several models exist which consider the formation of uniform mono- or multilayer adsorbate films hydrogen bonded to surface hydroxyl groups, or to the oxide surface in interactions mediated by adsorbed water molecules.19,21,22 Previous work has even provided an estimate of monolayer site areas, at 3.3−5.4 Å2, corresponding to layer thicknesses (the diameter of an assumed spherical adsorbate) of 0.65−0.83 nm.21

This work attempts to understand changes in the pore structure of aluminas on interaction with HF from which insights into the aging of these materials and nature of the alumina−HF interaction may be gained. We reconstruct the PSDs of a number of sec-Al2O3 samples, fluorinated in an industrial dry scrubber or to saturation in a laboratory-scale apparatus, from the prim-Al2O3 PSDs prior to reaction under a variety of commonly assumed models. Our results confirm pore blocking of fine pores and the presence of a family of kinetically inaccessible pores, allowing for measurement of the critical size delineating each family. However, we also demonstrate that the accessible pores do not show evidence of uniform monolayer or multilayer formation as is generally accepted; rather they undergo pore length attenuation consistent with pore blocking of nonuniform pores, or equivalently nonuniform adsorbate deposits on a uniform pore wall. These results have considerable impacts on our understanding of dry scrubbing as it pertains to the aluminum industry but are of wider relevance in understanding the aging of scrubbing/filtration materials and the evolution of pore structure in any system involving adsorption-based pollutant removal from fluid streams. In fact, preliminary results in our laboratory indicate that analogous pore structure changes occur during fluoride uptake by sol−gel synthesized aluminas in water, important model systems for the defluorination of drinking water.

2. EXPERIMENTAL AND THEORETICAL METHODOLOGY 2.1. Experimental Apparatus and Methodology. A 2 g amount of smelter grade alumina was reacted with HF gas in a fluidized bed reactor. The reactor is a cylindrical stainless steel column (16.5 cm long × 1.5 cm i.d.) lined with an HF-resistant fluoropolymer. Porous Teflon retaining frits were installed at the reactor inlet and outlet to prevent alumina entrainment in the gas stream. 1400 ppm of HF in nitrogen was diluted with preheated nitrogen gas in a mixing tee to attain 800 ppm of dry HF/N2 via high precision mass flow controllers. Dry nitrogen is used throughout as water may play competing roles in pore constriction while enhancing chemical bonding; future studies are underway with humidified gas streams to investigate these points. A reaction temperature of 120 °C was maintained by installing the reactor, tubes (copper and PTFE), and all fittings in an oven. Temperature was measured using an N-type thermocouple installed along copper tubes and at the reactor outlet. HF concentration was continuously measured in a Teflon gas cell (20 cm path length) with a Boreal Laser GasFinder 2.0 HF monitor and the gas subsequently bubbled through a solution of Ca(OH)2 for B


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Figure 2. PSD curves of unimodal (solid lines) and bimodal (dashed lines) primary and secondary SGAs after (left) industrial and (right) laboratory fluorination. disposal. The GasFinder 2.0 was calibrated by measuring the fluoride concentration with a fluoride ion selective electrode after capture of the gas in a total ionic strength adjustment buffer solution. Experiments were run until the inlet HF concentration equaled the outlet concentration (saturation). Surface area and pore size distributions were analyzed with a Micrometritics Tristar 3000 instrument. The BET and Barrett− Joyner−Halenda (BJH) methods were utilized to extract surface areas and pore size measurements, respectively, from 96-point N 2 physisorption isotherms. Approximately 2 g of material was used for each analysis; samples were predried at 120 °C for 12 h immediately prior to analysis. 2.2. Theoretical Methodology: The BJH Calculation of Pore Size Distributions. The BJH calculation23 has been used throughout this work to analyze changes in pore characteristics as dry scrubbing proceeds. The BJH calculation is generally performed assuming desorption from pores. During a physisorption experiment, the pressure of the adsorbate is decreased in a series of steps; at each pressure decrement from Pi to Pi+1, a volume ΔVi of gaseous adsorbate is liberated. At a partial pressure of 1.0, all pores are filled; every successive pressure decrement Pi (potentially) leads to the exposure of new pores. The radius, in angstroms, of the largest pore still filled with adsorbate at Pi can be computed as a function of pressure by the modified Rayleigh equation:

ri =

Therefore, at any given step from point i to i + 1, a change in wall thickness, Δtw, can be found, and one can compute (cylindrical pores are generally assumed) the total volume lost by pore wall desorption:

Vd, i =

() P0 Pi

where Lp,j represents the pore length (discussed shortly), a generally ignored term which will play an essential role in this work. Vd,i represents the volume desorbed from the walls of all open pores at the ith pressure decrement, with rj the pore diameter. The summation in eq 3 is carried out over all open pores; note that rj of open pores are corrected for this desorption after every pressure decrement. From the gas volume desorbed due to wall thinning in this step and the total desorbed liquid volume recorded for this pressure decrement, ΔVi, the remaining volume is that desorbed from the cores of a range of newly opened pores (Vnew). Treated as a single representative pore, this allows the derivation of the underlying pore geometry of the newly exposed pores: ΔV − Vd, i = Vnew, i = CSA new, iLnew, i

(1)

rav,new, i =

() Pi P0

(ri + ri + 1)rri i + 1 ri 2 + ri + 12

(5)

It follows from eqs 4 and 5 that the effective pore length in this decrement can be computed: Lnew, i =

Vnew, i π[(rav,new, i + Δtw,av, i)2 − rav,new, i 2]

(6)

Therefore, with pore radii computed through eqs 1 and 5, and the pore length in eq 6, we have a discretized set of radii and lengths which represent a complete description of the geometries that describe the underlying pore size distribution. As a consequence, one may introduce perturbations to the pore geometries of a primary alumina to simulate potential changes brought on during HF dry scrubbing, or any adsorptive uptake process for that matter. These may be compared to the pore size distributions obtained experimentally for materials following exposure to the adsorbate as a means of testing the adsorption models.

13.99 0.034 − log

(4)

That is, this approach models the range of pores exposed between volume decrements from Vi to Vi+1 as a single cylindrical pore with a representative cross-sectional area, CSAnew,i, an associated average pore radius, rav,new,i, and a pore length, Lnew,i, which is comparable to the sum of all of the pore lengths in this volume decrement. The average pore radius can be computed by25

P0 represents the saturation pressure of the bulk liquid adsorbate. This radius is not the intrinsic pore size but includes the wall thickness of the remaining adsorbate. While the first decrement exposes new pores only, there are in general two potential contributions to the desorbed gas volume thereafter: a thinning of the adsorbate layer on the walls of all previously exposed pores and the emptying of the core liquid from newly exposed pores. The thinning of the pore walls is, like the Kelvin radius defined in eq 1, a function of pressure, and there are several empirically derived equations to compute this quantity. The Halsey equation24 is commonly used; however, we employ the Harkins−Jura equation to give the statistical thickness of the adsorbate, ti, in this study as this was parametrized for alumina.

ti =

(3)

j

4.15 log

∑ π[(rj + Δtw)2 − rj 2]Lp,j

(2)

3. EXPERIMENTAL RESULTS As discussed in the Introduction, smelters do not demand the highest SSA aluminas, materials which tend to retain residual

All pertinent results in this study are reproduced with the Halsey equation (see the Supporting Information); however, no phenomenologically different features are noted. C


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Table 1. Average Surface Area, Pore Diameter, and Pore Volume Parameters of Industrially and Laboratory Fluorinated Unimodal and Bimodal Aluminas industrially fluorinated prim

sec

a

BET SSA/(m2 g−1) pore diam (4 V/A)/ nm pore vola/(cm3 g−1) BET SSA/(m2 g−1) pore diam (4 V/A)/ nm pore vola/(cm3 g−1)

laboratory fluorinated

unimodal

bimodal

unimodal

bimodal

74.1 8.20 0.1754 61.9 8.26 0.1530

65.3 11.50 0.1803 57.0 11.58 0.1656

64.2 10.38 0.1949 43.54 12.30 0.1481

65.3 11.50 0.1803 39.7 15.43 0.1592

Pore volume summed over pores from 1.7−300 nm.

materials is unchanged after gas contactthat is, pore geometry changes are negligible in the largest pores and adsorption preferentially changes the pore structure of the finest pores. Controlled dosing of alumina with HF leads to significant changes in PSDssee Figure 2 (right), with key surface properties also given in Table 1. Regardless of the initial PSDs’ shapes, they are almost identical after fluorination. There appear to be no open pores in either material up to ∼2 nm, and both exhibit a new small pore mode with a peak maximum centered at 3−4 nm. A dominant second peak exists shifted toward larger pore diameters. As with the less fluorinated industrially treated materials, the large pore limit of the PSD exhibits essentially no change. Thus, while the large pore peak position varies between aluminas, similarities of the pore distributions in pores < 5 nm in sec-Al2O3 suggest a mechanism intrinsic to the adsorption process itself and not the nature of the scrubbing material.

hydroxide (generating additional HF) and are dominated by fine pores suspected to be kinetically prohibited from participating in adsorption. Typically, a material with a BET SSA of ∼70−80 m2 g−1 is considered an optimal compromise. While this is ideally achieved by a uniform calcination to an intermediate transition alumina, a blended material (consisting of over- and undercalcined fractions with distinctive bimodal regions of larger and finer pore, respectively) will achieve the same average surface area. As a consequence, two materials of nominally similar SSA can in fact possess very distinct pore size distributions. This provides an interesting testing ground for two materials of similar SSA but very different pore size characteristics. We have chosen two representative prim-Al2O3 materials illustrative of both scenarios. Figure 2 demonstrates a unimodal SGA typical of a range of phases clustered around an intermediate transition alumina, along with, for contrast, a bimodal PSD representative of a blended over- and undercalcined material. Key properties are given in Table 1 with isotherms in Figure 3; clearly, both possess very similar BET

4. ADSORPTION MODEL RESULTS Elucidation of an appropriate model of adsorption has undergone several developmental stages. These are summarized in Table 2 outlining the progression of development. Table 2. Overview of Adsorption Model Development section

pore treatment

three-zone model?

4.1 4.2 4.3

monolayer model [4.1]a monolayer model pore attenuation model [4.3]

no yes [4.2] yes

a

The section where the submodel (monolayer, three-zone, or pore attenuation model) is first derived.

Figure 3. Isotherms of primary aluminas studied. Isotherms are shifted by P/P0 = 0.15 and 0.3 for clarity.

4.1. Monolayer-Only Models. The simplest and most widely hypothesized (indeed, perhaps the only) model considered in the literature regarding HF adsorption on alumina is the formation of a simple uniform mono- or multilayer. This is applied across all pores. There is no published agreement on the details of this layer, or layers, prompting treatment here as a monolayer of an as yet unspecified nature and thickness. Accumulation of a chemisorbed monolayer of thickness tm will decrease the diameter of the ith pore to di → di + 2tm, regardless of pore shape, decreasing pore surface area (in line with experiment). We assume there is negligible change in pore length, which follows from the assumption that pores are much longer than they are wide, inherent in the BJH model. In applying the model, monolayer thickness has been varied until

SSAs, a parameter which is used to quantify the adsorption capacity of a range of adsorbates, including smelter grade alumina.7,22,26,27 Similarly, average pore diameters are also quite similar in spite of the clearly different pore size distributions. PSDs of each alumina following recycling through an industrial dry scrubber are also provided. Gas contacts are dictated by keeping outlet emissions of HF below prescribed threshold levels, and adsorption does not in general approach the scrubbing capacity of the aluminas. We therefore expect only subtle changes, and indeed in both materials the underlying distribution shape appears to be retained, although the peak heights are lowered and surface area is lost (see Table 1). It is also noteworthy that the large pore limit of the PSD of both D


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Langmuir the closest correspondence between the fitted and experimental data was attained. Most results from this model are deferred to the Supporting Information as it is quickly apparent that the fits are, in general, very poor (see Figure 4, for example). The sole exception is the

Figure 5. Three-zone model for monolayer formation inside larger pores (or, in section 4.3, pore blocking), lack of reactivity in midrange mesopores, and blocking of the finest pores.

distribution function, Φ(x), which also defines the error function:

Figure 4. Measured PSD curves of a unimodal SGA before and after laboratory-based HF dosing employing a monolayer model both with and without the three-zone concept.

Φ(a) =

unimodal industrially fluorinated alumina, due solely to the very subtle changes in the PSDs with fluorine contact. Generally, no consensus on the monolayer thickness is obtained (see Figure S1 in the Supporting Information) even though this value should be intrinsic to the adsorbate/adsorbent pair. Further, estimates do not agree with the findings of other workers.21 Finally, regarding fitting, the large pore peak in the bimodal materials generally could not be fit without poisoning matching in the small pore peak and vice versa. Further, the ultimately bimodal character of heavily dosed unimodal prim-Al2O3 could not be reproduced at all. It should also be noted that while pore shape could affect absolute pore lengths, the PSDs reflect surface area changes which for a slit pore (of width w) is 2wl; that is, monolayer formation in slit pores does not change the PSD, at odds with measurement. 4.2. Monolayer Formation with the Three-Zone Model. A more refined approach accounts for differing families of pores that react uniquely with the adsorbatethis is the motivation behind the “three-zone model” (see Figure 5). It is suspected that there should be a family of pores which do not form a fluoride monolayer over the time scales present in practical scrubber operation. Gas flow through porous media can be extremely complicated; however, phenomena such as Knudsen diffusion (where the mean free path, and hence diffusion coefficient, is proportional to pore diameter, thereby making diffusion increasingly slow in the finer pores) may play a role.28−30 Indeed, such principles are behind a number of materials designed for gas separation.30−32 We therefore postulate that there are three distinctive regions of pore size; a large pore region which will obtain a monolayer coverage of fluoride, a region of unreacted kinetically inaccessible pores (treated as a set of pores unchanged from the original primary alumina), and a subset of pores that are blocked and therefore hidden in the reanalysis by N2 physisorption (there appear in the model as a subset that is simply removed from the primary alumina PSD when simulating that of the secondary). These regions are unlikely to have a strict cutoff in terms of pore diameter. We have used the following general cumulative

1 2π

a

⎛ −y 2 ⎞ ⎛ a ⎞⎤ 1⎡ ⎟⎥ ⎟ dy ≡ ⎢1 + erf⎜ ⎝ 2 ⎠⎦ 2 ⎠ 2⎣

∫−∞ exp⎜⎝

(7)

to partition the pores into the various regions (as opposed to a less realistic distribution such as the Heaviside function, for example). The use of this form of the cumulative distribution function, assuming an underlying normal probability distribution, is justified in section 4.3; however, in practice this allows the fractions of the prim-Al2O3 pore lengths Lp(D) to be split into two regions on either side of some threshold pore size Dthresh (the dashed vertical lines in Figure 5) as Lp(D≥Dthresh ) =

Lp(D) ⎡ ⎛ D − Dthresh ⎞⎤ ⎟⎥ ⎢1 + erf⎜ ⎝ 2 ⎣ 2 σ ⎠⎦

Lp(D≤Dthresh ) = Lp(D) − Lp(D≥Dthresh )

(8) (9)

The terms Dthresh and σ are fitting parameters which we will interpret later. This breaks the PSD into three separate PSDs which can be transformed independently (see Figure 5); then the data are rebinned to simulate the sec-Al2O3 PSDs. Because we are applying a model to the output of the BJH method, computational difficulties arise in applying a monolayer to one subset of pores and not another. Our rebinning procedure for monolayer treatment is deferred to the Supporting Information. The overall model has five adjustable parameters, with four from the three-zone model: monolayer thickness, threshold pore sizes between accessible/inaccessible pores and inaccessible/blocked pores, and the associated broadening parameters (represented as the standard deviation, σ). In principle we expect the results to reflect only four parameters as it is physically most reasonable to assume that Dthresh for the poreblocking region occurs at 2tm (justified in section 4.3) and therefore should be identical to the parameter used to fit the monolayer. In practice, we explored both this restriction and independent variation of all parameters. Again, most data are deferred to the Supporting Information because, despite closer fits, the model is still poor in general. Again, fitting both peaks in the bimodal dosed aluminas E


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Figure 6. Model of pore blocking and length attenuation. (a) A near-uniform pore of diameter D interacting with a nonuniform adsorbate is thought of as a nonuniform pore with a diameter D ± δ(D) (and a uniform adsorbate); deviations δ(D) are normally distributed around an average pore diameter of D. (b) Adsorbate (in this work, HF) molecules (purple) enter the material from the exterior and attach to the surface, forming an adsorbed phase (in the current study, a hydroxyfluoride/fluoride phase (green)) of monolayer thickness tm in a stochastic process which may eventually lead to (c) pore blocking if a pore narrows to within 2tm and diametrically opposing walls of the pore are both involved in adsorption.

The term σ includes a variety of unknown influences in addition to terracing, subsuming such effects as ranges of pore geometries and connectivity, and even the ranges of actual pore lengths encountered. This function appears physically reasonable, in particular predicting the average pore diameter is Dp:

(regardless of the original shape of the primary material) is found to be problematic as better fitting to one of the peaks in general leads to poorer fitting of the othersee Figure 4, for example. This figure does, however, illustrate that the threezone model leads to the correct step-like shape in the unimodal primary alumina not achieved otherwise. Thus, the three-zone model appears necessary, but the monolayer approach to describing adsorbate/adsorbent interactions cannot model experiment correctly and needs to be replaced. Finally, it is important to note that while monolayer transformations are only applied to the “monolayer region” in Figure 5, a physically incorrect model in this zone ultimately contaminates the entire model. 4.3. Pore Length Attenuation with the Three-Zone Model. Section 4.2 demonstrates that while a three-zone model approach appears necessary, the monolayer model is clearly inadequate. Here we develop a more sophisticated view of the change in pore structure of an adsorbent to supersede the monolayer-based models, with the trial replacement of the blocking and shortening of pores. The pore length attenuation model, derived in this section, simply replaces the monolayer approach (derived in section 4.1) in the pores labeled monolayer region in Figure 5. In the derivation of this model, we will also justify some of the model considerations described in section 4.2. The origin of the dominant porosity in SGAs is generally thought to be due to OH loss across hydroxide rich planes in Gibbsite. These uniform slit-shaped pores are evidenced by parallel cracks revealed by microscopy (Figure 1) and the H3-type hysteresis loops in the SGA isotherms (Figure 3) indicative of slit-shaped pores. However, the possible terracing of exposed oxygen atoms in (001) planes may lead to loss of uniformity, as too might contributions from porosity with a different physical origin. Further, with the nature of the adsorbate phase in this case being unknown, and with a view to more general adsorbent−adsorbate systems, it is not necessarily the case that the adsorbate size is uniform either. To effectively model all scenarios, consider a pore of randomly varying diameter, around an average Dp (Figure 6a). Assuming these fluctuations, δ(D), are approximately normally distributed, the probability P(D) of encountering a region of pore diameter D in a pore of nominal diameter Dp is P(D) =

⎛ −(D − D )2 ⎞ 1 p ⎟ exp⎜⎜ ⎟ 2 σ 2π 2σ ⎝ ⎠

∞

⎛ −(D − D )2 ⎞ p ⎟ dD D exp⎜⎜ ⎟ 2 2σ ⎝ ⎠

⟨D⟩ =

1 σ 2π

∫0

≈

1 σ 2π

∫−∞ D exp⎜⎜

∞

⎛ −(D − D )2 ⎞ p ⎟ dD ⎟ 2 σ 2 ⎝ ⎠

= Dp

(11)

if we assume that P(D) decays to zero in small pores rapidly enough such that we can treat the diameter variable D as running from (−∞, ∞) rather than the physically correct interval [0, ∞). As the pores approaching D = 0 are likely to be blocked, and out of the range of the BJH calculation, this will be a serviceable practical approximation. With these assumptions, the probability that a section of pore with a diameter less than 2tm is encountered, therefore potentially blocked, is P(D≤2tm) ⎛ −(D − D )2 ⎞ p ⎟ dD exp⎜⎜ ⎟ 2 σ 2 ⎝ ⎠ ⎡ ⎤ ⎛ 2tm − Dp ⎞ 1 = ⎢1 + erf⎜ ⎟⎥ 2 σ ⎠⎦ 2⎣ ⎝ =

1 σ 2π

2t m

∫−∞

(12)

Pore blocking also requires that two diametrically opposing sides of the pore form a fluoride phase. To assess this probability, we assume the pore length, l, ̧ can be divided into an integer number of potential adsorption sites; there will be l/tm such sites on each wall (assuming the length of the adsorbed phase is also tm). Therefore, considering both walls, there are in total 2l/tm potential adsorption sites. This assumes a 2D-pore as illustrated in Figure 6, which is strictly accurate for slit pores, and since pore blocking occurs in the most constricted regions by adsorbates on diametrically opposing wall faces, this should also be serviceable for narrow cylindrical pores. Assuming n indistinguishable HF molecules enter the pore and are adsorbed, there are W(2l/tm,n) ways of arranging the adsorbate.

(10) F


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Langmuir W (2l /tm , n) =

(2l /tm)! n! ((2l /tm) − n)!

justified assuming diffusion controlled processes, at least to a first approximation. However, while the use of a form such as that of eq 8 appears reasonable, the model is not yet detailed enough for a physical interpretation of the Dthresh term. We must also justify the use of the same cumulative distribution functions between kinetically inaccessible and pore blocking regions, which arise from a fundamentally different physical process. Blocked pores form a family of pores of sufficiently small size such that surface adsorption at the pore entry closes over their topssee Figure 7. To establish a model

(13)

Blocking can occur in any section of the pore, giving l/tm possible blocking sites; for each blocking scenario, there are W(2l/tm−2,n−2) arrangements (i.e., we must arrange the n − 2 nonblocking HF molecules among the 2l/tm − 2 remaining adsorption sites). This is only accurate in the limit where n ≪ 2l/tm; i.e., the probability that opposing sites are occupied is low as we ignore additional pore length attenuation after the first blocking event. However, this should provide a practical first approach to understanding the adsorbate−absorbent interaction. The probability of achieving an arrangement where adsorption has occurred on opposing sides of the pore, as required for pore blocking, is the number of blocking arrangements divided by the total number of ways of distributing n HF molecules among 2l/tm potential adsorption sites:

Figure 7. Modeling pore blocking of fine pores. (a) The adsorbate thickness is not well-defined; we represent this in uncertainty of the size of the second adsorbate, which can only approach within 2tm of the first. (b) The appearance of a pore between the two adsorbates is modeled as a break along the D axis of distance Dp; at extremes, the adsorbates can anchor at a single point at the pore edges. (c) A break in the D-axis indicates that 2tm − Dp is the position of closest approach of two adsorbates.

l W ((2l /tm) − 2, n − 2) Popp((2l /tm), n) = tm W ((2l /tm), n) =

n(n − 1)tm 2(2l − tm)

(14)

The probability of pore blocking, Pblock, requires both that adsorption occurs at two opposing sites of the pore (eq 14) and that this site has a diameter D ≤ 2tm (eq 12). Pblock =

⎛ 2tm − Dp ⎞⎤ n(n − 1)tm ⎡ ⎢1 + erf⎜ ⎟⎥ 2 σ ⎠⎦ 4(2l − tm) ⎣ ⎝

of pore blocking, we assume a two-adsorbate blocking model. (Blocking is probably achievable with a single adsorbate, but only in the smallest pores to which the BJH model is no longer reliable; i.e., D < 1 nm). Using a 2D model again, this should also be equally applicable to slit and small cylindrical pores. We set the left-hand edge of a well-defined adsorbate species at the origin, and refer to the position of a second adsorbate species by its right-hand edge for adsorption at positive D values (and not the left also, to account for extreme cases of adsorption across a poresee Figure 7bwhere such a definition would lead to scenarios in which we must refer to the position of the adsorbate in the center of the pore opening). We assume the second adsorbate has a poorly defined edge to account for the assumed inhomogeneous nature of the adsorbate layer or the diameter of the pore opening. With a normally distributed variation of monolayer thicknesses, the probability that the two adsorbates touch (necessary for pore blocking) when the second adsorbate fixes to the surface at point D is

(15)

Therefore, pore attenuation can be fit to a general curve of the form (assuming lsec = lprimPblock): ⎡ ⎛ Dattenuated − Dp ⎞⎤ lsec = A([HF])lprim⎢1 + erf⎜ ⎟⎥ ⎢⎣ ⎝ 2 σattenuated ⎠⎥⎦

(16)

This modifies the prim-Al2O3 pore length, lprim, to give that of the secondary, lsec, where the term Dattenuated now denotes a general pore diameter threshold parameter to fit; on comparison with eq 15 (which also defines Pblock) one expects that this should correspond to 2tm. The form of the function A([HF]) will likely deviate from that given in eq 15 at higher HF loadings (increased concentrations and/or reaction times), but we expect the general form, that of increasing probability of pore blocking with greater adsorbate exposure, to hold. We now leave pore shortening to justify the use of error functions of the type in eq 8 to partition pores into distinct families in the three-zone model (derived in section 4.2) somewhat more rigorously. We first consider the unrestricted to kinetically restricted pores division, noting that these will only be inaccessible insofar as there is a low probability that the adsorbate will work its way into narrow pores rather than being explicitly denied entry. Taking the underlying statistics to be approximately normal is a fair assumption, based on the Central Limit Theorem, given the large number of adsorbate/adsorbent interactions. However, these are probably fundamentally diffusion-based processes, and the solution to the diffusion equation (with point-release initial conditions) is a Gaussian function. Vestiges of this Gaussian-like form are retained under more sophisticated initial conditions (such as the introduction of fluid flow).33,34 This suggests the use of normal distributions in the delineation of accessible and inaccessible regimes is

Ptouch(D) =

⎡ −(D − 2t )2 ⎤ 1 m ⎥ exp⎢ σ 2π 2σ 2 ⎣ ⎦

(17)

This assumes that the two adsorbates are constrained to be, on average, 2tm apart at closest approach. However, an opening Dp between the two adsorbate species, which we model as a discontinuity in the D-axis, reduces this position of closest approach to 2tm − Dpsee Figure 7c. The probability that pore blocking is achieved is the probability that two adsorbates either side of a pore of diameter Dp touch, summed over all D, the points where the second adsorbate bonds. That is, we require (with Ptouch defined in eq 17) ∞

Pblock =

∫−∞ dD Ptouch(D)

(18)

Because this problem is fundamentally symmetric around the origin, we may evaluate this integral as G


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Figure 8. PSD curves of (a) unimodal and (b) bimodal aluminas after industrial fluorination, and (c) unimodal and (d) bimodal samples after laboratory fluorination with the pore attenuation and three-zone model applied. Fitting parameters are given in Table 3. 0

Pblock(Dp) = 2

∫−∞

dD

⎡ −(D − [2t − D ])2 ⎤ 1 m p ⎥ exp⎢ ⎢⎣ ⎥⎦ σ 2π 2σ 2

again two of those (Dattenuated in eq 16 and Dblock in eq 20) should to be both equal to 2tm on the grounds of physical argument. Therefore, the model is in fact identical to the monolayer/three-zone model, differing only in the treatment of the largest pores. In section 4.2, a poor fitting in the HFaccessible larger pores contaminated the entire model, including regions uninfluenced by adsorption phenomena (see Figure 4). As such, the good fit achieved here is very unlikely to be a consequence of simply overfitting data. In addition to strong physical arguments for the existence of the distinct families of pores, physically reasonable values can be extracted from the fitted data. Fitting parameters are provided in Table 3, and several important inferences may be drawn that both support the validity of this model and inform the underlying chemistry occurring. The terms Dkinetic and Dblocked represent the pore size thresholds that separate the accessible/inaccessible and inaccessible/blocked families of pores, respectively (Figure 5). The Dattenuated term is defined by eq 16. Equation 20 indicates that Dblocked most reasonably corresponds with 2tm, as should Dattenuated on comparison of eq 15 with eq 16. There is clearly very good agreement between both parameters for a given alumina; indeed, the fits attained use values of Dblocked and Dattenuated that differ by only 0.3 nm within a given sample. There is more variability across different aluminas, but all are very consistent at Dblocked = Dattenuated = 2tm ∼ 1.5 nm. The consistency between these two parameters, and across different samples, is strong evidence toward the reliability of our model. This is further demonstrated by the fact that, from these data, we may estimate the layer thickness

(19)

This leads to our operative form: ⎡ ⎛ D block − Dp ⎞⎤ Pblock(Dp) = ⎢1 + erf⎜ ⎟⎥ ⎢⎣ ⎝ 2 σblock ⎠⎥⎦

(20)

Again, the cumulative probability distribution form of eqs 7−9 naturally arises, with 2tm appearing as the Dthresh term. With a full theoretical description of the pore attenuation/ three-zone model in hand, it is possible to revisit the modeling results. Modeled PSDs of the studied aluminas are shown in Figure 8. Extremely good fits in all scenarios are noted under this model. Of particular note is the attenuation of midrange (5−10 nm) pores while maintaining the negligible shift in the leading edge of the PSD. The probability of encountering a region of pore across which blocking can occur is negligibly low in a nominally large pore, and therefore the pore structure changes very little; decreasing pore size increases the chances of encountering regions susceptible to blocking. The good agreement here requires a balance in the interplay between the fitting regions. While it could be argued that the introduction of separate regions could lead to overfitting, and therefore agreement achieved is inevitable, this interplay of regions provides strong evidence against this. The current model possesses only two extra parameters when compared with the monolayer/three-zone model in section 4.2. However, H


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However, modifying the full model by assuming, instead of monolayer formation, blocking occurs somewhere in the interior of the pore (thereby attenuating their length), leads to very good replication of experimental observations across all samples considered (section 4.3). This is unlikely the result of overfitting; the equivalent model based on monolayer formation contained an almost identical number of parameters but resulted in poor fits. Further, simple arguments leading to a mathematical description of the underlying processes reveal that key parameters, particularly monolayer thickness, tm, should appear independently in different parts of the model. This provides eight independently fitted parameters modeling tm, and all are in close agreement with one another, with the variation noted appearing to be no more than measurement error (i.e., there is no correlation with HF exposure). Indeed, invariance of some model parameters, and changes in others, as a function of HF dosing are in line with expectation based on our models. Further, the tm values provide an estimate for adsorbate layer thickness, at 0.74 ± 0.16 nm, which is in excellent agreement with previous estimates of 0.65−0.83 nm.21 Our model therefore confirms the presence of families of pores which have unhindered access, kinetically restricted access, or are capped with little to no adsorption occurring inside the pores. The critical threshold delimiting accessible and kinetically restricted pores in alumina used for HF dry scrubbing purposes is 2.4−2.8 nm. Incidentally, our previous work13 examining fluoride transport within the smelter failed to find a correlation between fluoride uptake and BET SSA; BJH methods did, on the other hand. The strongest correlation was obtained when pores smaller than 3 nm were omitted, consistent with the critical threshold found here; thus, this stands as further independent evidence for the inactivity of readily blocked and inaccessible pores in adsorbent behavior. The extremely good fits achieved, physically reasonable dependence of parameters in model fitting, and agreement with previous monolayer thicknesses all indicate that pore evolution in aluminas used for HF dry scrubbing purposes is dictated almost exclusively by pore blocking processes, and not pore thinning by adsorbate layer deposition. This further indicates that simple BET estimates for adsorption capacity, while perhaps acceptable far from saturation, become an increasingly poor measure when considering higher loadings. The results derived and conclusions drawn here assumed nothing regarding the specific interaction between the adsorbent and adsorbate other than a sufficiently strong interaction such that it is irreversible under the conditions present in the adsorption reaction. Therefore, pore evolution by pore blocking, and the splitting of pores into regions of accessible, kinetically restricted, and quickly blocked pores, is likely a general phenomenon in adsorbate/adsorbent systems. A particularly salient example may lie in sorbent aging in the very closely related system of excess fluoride removal from drinking water by mesoporous and activated aluminas.35−38 This is recognized as a serious health concern worldwide39 and is a system to which we intend to apply the current approach to in future studies; indeed, our preliminary studies with sol−gel synthesized aluminas aged in aqueous NaF solutions indicate very similar PSD changes occur in such systems. This will have ramifications on the efficiency of contaminant removal from drinking water reserves and may even also ultimately allow for the determination of important parameters such as adsorbate binding energies and diffusivities.

Table 3. Fitting Parameters for the Models of Pore Attenuation in the Studied Aluminas unimodal industrial

bimodal industrial

unimodal laboratory

bimodal laboratory

Three Zone Model Parameter (Unrestricted/Kinetically Restricted Zones); cf. Equation 8 Dkinetic 2.8 2.5 2.5 2.4 σkinetic 5.0 1.5 1.9 4.0 Three Zone Model Parameter (Kinetically Restricted/Blocked Pore Zones); Equation 20 Dblocked 1.5 1.3 1.5 1.7 σblocked 0.6 0.6 0.7 1.0 Pore Attenuation Model Parameters; Equation 16 A([HF]) 0.4 0.4 1.5 1.7 Dattenuated 1.2 1.5 1.5 1.6 σattenuated 5.5 3.5 5.6 5.7

to be (tm ± 2σn−1), at 0.74 ± 0.16 nm, in excellent agreement with previous estimates, 0.65−0.83 nm,21 obtained with a different experimental and modeling approach. The terms A([HF]) and Dkinetic also behave in a physically reasonable fashion. Equations 15 and 16 indicate that A([HF]) should increase as a function of increased exposure to HF. And indeed, this term is considerably higher (by a factor of ∼4) in the materials treated to saturation and is very similar within the industrial and laboratory-treated sample sets. On the other hand, Dkinetic is expected to be fundamental to the adsorbate/ adsorbent system; the invariance noted across all four samples strongly supports this.

5. CONCLUSIONS Aluminas of relevance to the dry scrubbing needs of aluminum smelters have been studied in terms of their pore evolution and dry scrubbing capacity. Unimodal PSD materials (aluminas achieved by even calcination across all grains to a common transition alumina) have been compared against bimodal materials (those with a blended undercalcined/high SSA and overcalcined/low SSA mixture of grains) allowing studies of chemically similar systems which possess similar BET surface areas and average pore diameters, but very different PSDs. Materials have been treated with HF in both an industrial dry scrubbing unit (and therefore representative of practical application) and to saturation in a laboratory-based HF dosing system. BET SSAs and PSDs computed through the BJH model reveal that scrubbing lowers surface area and that this primarily influences the smaller pores. The large pore limit of the distribution reveals little change in the PSDs, even once scrubbing is pushed to saturation. These observations are inconsistent with current thinking in the aluminum industry which suggests the smallest pores are kinetically inaccessible and that dry scrubbing occurs through the formation of a monolayer-like build-up confined primarily to the largest pores. Modifying the pore geometries obtained in BJH calculations according to a series of simple models has allowed us to reconstruct PSDs of treated aluminas to contrast with experiment. The simple monolayer model (section 4.1) is found to be inadequate in general. Where reasonable agreement was found, this was due to the low levels of adsorbate loadings producing only minor changes in the postadsorption PSD. Attempts to improve the model by including regions of kinetically inaccessible and blocked pores (the three-zone model) were also inadequate (section 4.2). I


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ASSOCIATED CONTENT

S Supporting Information *

Data rebinning procedures for the monolayer/blocking model, model results for the bimodal laboratory treated material, resulting fitting parameters for all materials fit to this model, and reproduction of all model results employing Halsey statistical thickness equation in the BJH calculation. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.5b00664.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge Hydro Aluminum, ASA, for granting permission for data from a related project to be used in this publication. We are also grateful for invaluable discussions with Dr. Colin Beeby and to Hasini Wijayaratne for ESEM images.

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REFERENCES

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K


Adsorptive Capacity and Evolution of the Pore Structure of Alumina on

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