Simulations of the Structure and Properties of Amorphous Silica

Chemical Engineering Science 56 (2001) 4205–4216 www.elsevier.com/locate/ces

Simulations of the structure andproperties of amorphous silica surfaces Jeanette M. Stallons, Enrique Iglesia ∗

Department of Chemical Engineering, University of California at Berkeley, Berkeley, CA 94720, USA

Received10 December 1999; accepted26 December 2000

Abstract The structure andtransport properties of solidsurfaces have been describedusingmodelsof varying complexity andrigor without systematic comparisons among available methods. Here, we describe the surface of amorphous silica using four techniques: (1) an orderedsurfacecreatedby cutting the structure of a known silica polymorph ( -cristobalite); (2) an unrelaxedamorphous surface obtainedby cutting bulk amorphous silica structures createdby molecular dynamicsmethods;(3) a relaxedamorphous surface createdby relaxing the amorphous surface; and(4) a randomsurface createdby Monte Carlo sphere packing methods. Calculations of the adsorption potential surface and simulation of the surface di4usion of weakly bound adsorbates (N2, Ar, CH4) interacting via Lennard–Jones potentials with these surfaces were used to compare surface models and to judge their ÿdelity by comparisons with available experimental values. Similar heats of adsorption were obtained on the relaxed, unrelaxed, and random surfaces (±0:5kJ=mol), but the relaxed surface showed greater heterogeneity with a wider distribution of adsorption energies. Surface di4usion on the relaxed surface was slower than on the other surfaces, with slightly higher activation energies (0:5–1:0kJ=mol). Rigorous comparisons between simulatedandexperimental surface di4usivitiesare not possible, because of scarce surface di4usion data on well-characterized surfaces. The values obtained from simulations on silica were similar to experimental surface di4usivities reported on borosilicate glasses. ? 2001 Publishedby Elsevier Science Ltd.

Keywords: Porous media; Simulation; Transport processes; Di4usion; Surface di4usion; Adsorption

1. Introduction the adsorption and surface di4usion properties of amorphous silica. Amorphous solids are used because they Simulations are increasingly useful tools in the design pose more formidable simulation challenges than known of porous solids by helping to understand and to pre- andwell-deÿnedcrystallinestructures. Silica was cho- dict the structure and transport properties of these solids sen as an example because of its widespread use as ad- (MacElroy & Raghavan, 1991; Reyes & Iglesia, 1993; sorbents, catalyst supports, andporous membranes. Sev- Drewry & Seaton, 1995; Keil, 1996). Most surface mod- eral previous studies of the bulk and surface structures of els, however, neglect the details of the surface structure amorphous solidshave been carriedout on silica andthe andthey adoptinsteadthe pragmatic approach of de- requiredinteratomic potentials are available from these scribing surfaces using simple lattice models with math- studies (Feuston & Garofalini, 1989, 1990; Athanasopou- ematically convenient distributions of binding sites. A los & Garofalini, 1992; Kohler & Garofalini, 1994; Litton systematic study of how these approaches lead to vary- & Garofalini, 1997). ing degrees of accuracy, ÿdelity, and predictive value Recent studies have introduced heterogeneity into sim- appears to be currently unavailable. Here, we attempt ple lattice models in order to attempt to describe the such a systematic comparison by describing surface struc- chemical non-uniformity of amorphous surfaces. For ex- tures using several methods, di4ering in approach and ample, adsorption energy distributions were assigned to level of complexity, andcomparing their ability to predict various available lattice sites (Nicholson & Silvester, ∗ Corresponding author. Tel.: +1-510-642-9673. fax: +1-510-642- 1977; O’Brien & Myers, 1985). Surface features, such 4778. as steps, terraces, holes, or grooves have been placed E-mail address: [email protected] (E. Iglesia). on square lattices (Haus & Kehr, 1987; Bojan & Steele,

0009-2509/01/$ - see front matter ? 2001 Publishedby Elsevier Science Ltd. PII: S 0009-2509(01)00021-5 4206 J. M. Stallons, E. Iglesia / Chemical Engineering Science 56 (2001) 4205–4216

1988; Gomer, 1990; Rudzinski & Everett, 1992; Steele, radialstructure functions andbondangle distributions). 1997; Zgrablich, 1997). The smaller holes andbarriers are The choice of potential energy function is critical, be- distributed randomly on the surfaces; the larger dimen- cause it determines the positions of atoms at the surface. sions andspacing of the steps andgrooves are approxi- Few studies have included this level of detail in describ- matedfrom experimental microscopy data(Gomer, 1990; ing surfaces. Garofalini et al. have examinedthe structure Steele, 1997). Disorder of the underlying surface struc- of silica surfaces (Feuston & Garofalini, 1989; Athana- ture itself has been introduced using lattices with more sopoulos & Garofalini, 1992; Kohler & Garofalini, 1994) complex topology, for example, by perturbing all vertex andthe surface di4usionof silicon andoxygen lattice positions in a square lattice by a random displacement atoms at high temperatures (Litton & Garofalini, 1997). vector (Benegas, Pereyra, & Zgrablich, 1987). These ap- Bakaev (1999) andBakaev andSteele (1999a, b) have proaches retain the assumption that random, heteroge- recently addressed the incomplete surface annealing in neous surfaces can be described by distributions of sites the Garofalini methods, which leads to pendant oxygens on regular or distorted ordered structures, without in- suggestedto account for OH groups experimentally found cluding the detailed topological connectivity among sur- at the surface of silica. These authors proposedthat hy- face atoms or between surface atoms andthe bulk atoms drophobic silica surfaces require the substantial absence immediately below. Nevertheless, it seems that connec- of OH groups andrequire much higher annealing tem- tivity andinteractions with sub-surface atoms shouldin- peratures in the structural simulations of the surface. Sil- Iuence strongly the transport andbindingproperties of ica surfaces usedas adsorbentsandas catalyst supports, these amorphous surfaces. however, contain signiÿcant OH surface densities; thus, Bakaev andSteele (1992) useda “Bernal surface” the Garofalini methodappears to be more realistic in the (Finney, 1983), createdby randomlypacking spheres, simulation of silica surfaces formedby abrupt termina- representing oxygen atoms, within a three-dimensional tion of particle growth in aqueous media followed by container with periodic boundary conditions using Monte thermal treatments at temperatures well below the glass Carlo methods. The sphere size was set as the diameter transition temperature. MacElroy andRaghavan (1990, of lattice oxygen anions. Riccardo and Steele (1996) 1991) exploredthe di4usionof methane andsilicon hex- usedthis Bernal surface to simulate Ar surface di4u- aIouride within random microporous silicas. Both groups sion on amorphous titania. The interactions of Ar with usedmolecular dynamicsor Monte Carlo simulations to surface oxygens were described by Lennard–Jones place Si andO atoms accurately within bulk silica and potentials parameterizedby comparing experimental and then createda surface by cutting and,in some cases, re- simulatedadsorptionisotherms. Surface “roughness” laxing the bulk structure. was variedby randomremoval of some surface spheres; In this study, various silica surfaces were created and the resulting “roughness” hada markede4ect on surface their adsorption and transport properties were compared transport rates at low temperatures (85–160 K). For ex- with one another andwith available experimental results ample, di4usion activation energies increased from 4.7 in order to assess the level of detail required for the faith- to 7:1kJ=mol as three monolayers of atoms were ran- ful representations of these heterogeneous random sur- domly removed from the initial Bernal surface. Fractal faces. Speciÿcally, surfaces were createdby simple or dimensions (Df) of 2.2–2.5 were calculatedfrom the ef- relaxedcuts of simulatedbulk silica (unrelaxedandre- fects of the diameter of the adsorbates on the maximum laxedsurfaces, respectively) andby Monte Carlo packing achievable coverage (monolayer capacity) on various methods (random surface). The properties of these amor- simulatedsurfaces. The approximate agreement observed phous surfaces were comparedwith those of orderedsys- between these andexperimental fractal dimensionsfor tems, in which uniform adsorption sites are placed in a silica surfaces (Df =2:03–2.30) was taken as evidence lattice corresponding to -cristobalite, a crystalline silica of the ÿdelity of the simulation approach. The activation polymorph (ordered surface). Both the thermodynamics energies andsurface di4usivitiesobtainedin these simu- of adsorption and the dynamics of surface transport of lations on roughenedBernal surfaces were not compared weakly bound adsorbates are simulated. with experimental results. Clearly, atoms on surfaces are not locatedat random positions but at positions of minimum energy, andthese 2. Bulk silica simulation positions are not static but they relax both after the surface is ÿrst formedby fracturing, abrupt termination of 2.1. Method particle growth, or cooling from melts, and during adsorption anddi4usionof molecules. The forces among Bulk silica was described using well-established surface atoms, those in subsurface regions, andthose in molecular dynamics techniques (Woodcock, Angell, & adsorbed molecules can be described accurately by em- Cheeseman, 1976; Soules, 1979; Garofalini & Melman, pirical potential energy functions parametrizedto de- 1986; Feuston & Garofalini, 1988). Simulations were scribe the bulk structural details of amorphous solids (e.g. carried out using methods described by Feuston and J. M. Stallons, E. Iglesia / Chemical Engineering Science 56 (2001) 4205–4216 4207

Garofalini (1988). Atoms were placedin initial conÿgu- approximated by a form of Ewald’s sums (Woodcock, rations of -cristobalite, ÿ-cristobalite, -quartz, or in an Angell, & Cheeseman, 1976; Soules, 1979; Garofalini arbitrary regular structure with four oxygen atoms around & Melman, 1986; Feuston & Garofalini, 1988). Sil- each silicon atom. Initial velocity components were cho- ica, however, is not a purely ionic solidandSi–O sen randomly from a Maxwell–Boltzmann distribution bonds have a clear covalent and directional nature. in a way that ensuredthat the system hadno net initial Hence, three-body angle-bending terms are added to momentum. Newton’s equations of motion were then the potential; they discourage signiÿcant deviations integratedat constant energy using fourth-orderGear from the tetrahedral Si–O–Si and O–Si–O angles predictor–corrector methods with periodic boundary con- (Feuston & Garofalini, 1988; Vessal, Leslie, & Catlow ditions and nearest-neighbor lists. Forces between the 1989). atoms were described using a two-body Born–Mayer– The three-body potential was developed by Feuston Huggins potential anda three-bodyStillinger–Weber andGarofalini (1988) andit is basedon the three-body type (Feuston & Garofalini, 1988). The bulk silica struc- potential developedbyStillinger andWeber (1985) for ture was annealedto its lowest energy form at room silicon. The form of these additional potential terms is temperature by a melt-quench sequence. The initial con- shown below: ÿguration was meltedat 6000 K by periodicvelocity (r ;r ;r )=h(r ;r ;Â )+h(r ;r ;Â ) scaling during the ÿrst 3000 molecular dynamics (MD) 3 i j k ij ik jik jk ji kji time steps followedby a 7 ps constant energy run. The + h(r ;r ;Â ); (3) system was cooledstepwise to 300 K, while ensuring ki kj ikj internal equilibrium at intermediate temperatures (4000, h(r ;r ;Â ) 2000, and1000 K), andthen it was run at constant en- ij ik jik  ergy for 17 ps at 1000 K in order to ensure complete  exp i relaxation before a ÿnal quench to 300 K. After a 7 ps  i c  rij − ri equilibration at 300 K, radial distribution functions and = c 2 c c bondangles were calculatedby collecting the results  (cos Âjik − cos Âjik ) ;rij ¡rij and rik ¡rik ;  of the MD simulations for up to 20; 000 fs. Simulations  c c 0;rij ¿ rij or rik ¿ rik ; were performedusing 100–3600 atoms with a 0 :001 ps time step at a constant density of 2:20 g=cm3, which (4) corresponds to the bulk density of amorphous silica. The where rij;rik ;rjk are the interatomic separation dis- speciÿc number of atoms in each simulation was chosen c c c c c c tances; rij;rik ;rjk ;Âkji, Âjik ;Âikj are cut-o4 constants; so that surface di4usivities varied by less than 5% with and i;j;k , i;j, k are adjustable parameters. The a twofoldincrease in the sample size. parameter values usedare shown in Tables 1 and2; they were obtainedby Feuston andGarofalini (1988) based 2.2. Interatomic potential energy functions on comparisons of the simulatedandexperimental radial distribution functions obtained by X-ray scatter (Mozzi The interactions between atoms in the system were & Warren, 1969) andneutron scattering (Misawa, Price, described using both a two-body Born–Mayer–Huggins & Susuki, 1980). potential (2) anda three-bodyStillinger-Weber-type potential (3): 3. Silica surface simulations V = 2(ri;rj)+ 3(ri;rj;rk ); (1) i¡j i¡j¡k 3.1. Methods where V is the total potential energy and ri, rj, rk are the atomic positions of the atoms in each pair or triplet. The 3.1.1. Unrelaxed surfaces two-body potential consists of two terms: Surfaces were createdby cutting the simulatedbulk 2 silica to the desired geometry. Planar surfaces were gen- −rij ZiZje rij 2(rij)=Aij exp + erfc ; (2) erated by removing the periodic boundary condition in rij ÿij one direction. The dangling oxygen bonds that form when where rij is the separation distance, e is the unit proton siloxane bridges are cleaved become the hydroxyl groups charge, Zi is the formal ionic charge, and Aij and ÿij are present on the surface of silica materials preparedby adjustable parameters. The ÿrst term is a Born–Mayer abrupt interruption of particle growth in gaseous or aque- repulsive term. The secondterm is an approximate ous media. A similar approach was used by MacElroy and description of long-range attractive Coulombic interac- Raghavan (1990, 1991) in order to create silica micro- tions. For amorphous materials, which lack long-range spheres, from which they removedall tri-silanol groups crystallinity, the attractive forces can be accurately de- (Si bonded to three OH groups), which are not present scribedby local, short-range forces (5–6 A),O which are on SiO2 surfaces, by removing the corresponding Si atom 4208 J. M. Stallons, E. Iglesia / Chemical Engineering Science 56 (2001) 4205–4216

Table 1 Born–Mayer–Huggins potential parameters (Feuston & Garofalini, 1988)

ASi-O AO-O ASi-Si ÿSi-O ÿO-O ÿSi-Si (×10−9 ergs) (A)O

2.96 0.72 1.88 2.55 2.53 2.60

Table 2 Parameter values for the three-body Stillinger–Weber-type potential (Feuston & Garofalini, 1988)

−11 −11 Si =18:0 × 10 ergs 8 =0:3 × 10 ergs Si =2:6 AO Si =0:6 AO c O c O rSi =3:0 A rSi =2:60 A c c cos ÂO-Si-O = −1=3 cos ÂSi-O-Si = −1=3 andthe three associatedO atoms. The resulting surfaces were not allowedto relax after cutting (MacElroy & Fig. 1. Generation of a random surface from a Monte Carlo packing Raghavan, 1990), but the authors concluded that this ap- of spheres. proach ledto a realistic surface basedon the similarity between the simulatedconcentration of surface hydroxyl groups (6:3OH=nm2) andthe values obtainedfrom spec- troscopic measurements of silica surfaces (2–7 OH=nm2) them at positions determined by intermolecular potential (Iler, 1979). energy functions. Adsorbates were assumed to interact only with the oxygen atoms in the structure. Hence, only 3.1.2. Relaxed surfaces the positions of the oxygen atoms were speciÿed. Their The unique position of a surface as the interface be- positions were assignedusing Monte Carlo methods,in tween the bulk solidanda gas or liquidphase leads which spheres (oxygen atoms) were dropped randomly to structures andchemical properties that di4erfrom into a large cylindricalcontainer andallowedto fall un- those within the bulk solid. Therefore, simulated sur- til they reacheda stable three-point contact (Jodrey& faces createdby fracturing the solidmust be allowed Tory, 1981). The code was adapted from one reported to relax to their minimum energy if they are to re- previously to describe pore structures in random meso- Iect accurately silica surfaces. Relaxedsurfaces were porous solids (Reyes & Iglesia, 1991). The sphere size createdby allowing atoms in the fracturedsolidto was chosen in order to match the ÿrst peak in the ra- move under the inIuence of the same forces used in dial distribution function with the experimental O–O dis- bulk solidsimulations. Surface relaxation was carried tance (1:61 A)O (Mozzi & Warren, 1969; Misawa, Price, out using a methoddevelopedbyFeuston andGaro- & Susuki, 1980). This packing was then fracturedat a falini (1989). In this method, the bottom section of random position, keeping all atoms located below the atoms (10%) in a fracturedbulk solidsample was fracture line. Only the atoms within the central portion of ÿxedandthe system allowedto relax at high tem- the cylindrical container were included in order to avoid peratures (6000 K). The system was then cooledto wall e4ects on the packing density. A schematic of this 300 K in steps, as in the bulk simulation, at which process to generate this random silica surface is shown point additional atoms were immobilized (in ∼10% in Fig. 1. increments) until about 50% of the total atoms were ÿxed. 3.1.4. Ordered surface 3.1.3. Random surfaces Finally, a simple ordered surface was also considered. The signiÿcant e4ort requiredin orderto perform rig- The crystalline silica polymorph, -crystobalite, was cre- orous MD simulations of silica surfaces ledus to ex- atedby repeating multiple unit cells with atomic positions amine the accuracy of simpler random surface models obtainedfrom reportedcrystal structure data(Hyde& by comparing their predictions with those from the MD Andersson, 1989). An ordered surface was then created simulations. A two-step methodinvolving bulk andsur- by fracturing the -cristobalite structure andkeeping all face generation steps was used. A bulk solid was created of the oxygens boundto the silicon atoms locatednear by placing atoms at random positions instead of locating the surface. J. M. Stallons, E. Iglesia / Chemical Engineering Science 56 (2001) 4205–4216 4209

Surface geometry, by itself, cannot be usedto establish the ÿdelity of these simulated surfaces, because it is not possible to measure such details of the surface structure in amorphous materials. The energetics andconnectiv- ity of such surfaces, however, can be characterizedby their chemical interactions with speciÿc molecules during adsorption–desorption and by the dynamics with which the adsorbed species di4use. In the next section, surface thermodynamic properties were probed by estimating heats of adsorption and comparing them with experimental results andby simulating the dynamicsof surface di4usion.

Fig. 2. Radial distribution functions as a function of distance from 4. Heats of adsorption the silica surface. 4.1. Method

3.2. Surface structure results In order to investigate the interactions of adsorbed species with surfaces, we require a potential energy The geometric properties of the surfaces generatedby function that describes accurately such interactions. With the di4erent methods were examined ÿrst. The concen- the development of quantum mechanical methods, these trations of surface hydroxyl groups (non-bridging oxy- energy functions are increasingly being obtainedfrom gens) obtainedon simulatedsurfaces were comparedto ab initio treatments (Wiesenekker, Kroes, & Baerends, experimental values. The ordered surface, -cristobalite 1996; Sorescu & Yates, 1998). Such potential energy re-scaledto the densityof amorphous silica (2 :2g=cm3), functions, however, are not yet available to describe hada surface hydroxylconcentration of 8 OH per nm 3. interactions between small molecules andsurfaces. For The relaxedandunrelaxedsurfaces hadsurface hydroxyl clean, single-crystal metal surfaces, scattering measure- concentrations of 6–7 OH per nm2 when fractured. No ments with angular distributions and energy transfer corresponding values were calculated for the random sur- data at multiple incidence angles can be used to develop faces, which do not contain the structural silicon por- empirical potential energy functions (Cardillo, 1985; tion needed for the calculation. These values lie at the Barker & Rettner, 1992). This methodcannot be applied high endof the experimental OH surface densityrange to amorphous surfaces. Therefore, we must use instead (4.2–6.2 OH per nm2), which varies depending on ther- an interaction potential with parameters available from mal treatment of the samples after synthesis (Iler, 1979; accessible data, such as atomic size and polarizabil- Zhuravlev, 1987). ity andHenry’s adsorptionconstants. Here, we use a Radial distribution functions were obtained as a func- Lennard–Jones-type potential energy function: tion of distance from the surface. Fracture without re- 12 6 laxation leads to bulk-like radial distribution functions at 2(rij)=4 − ; (5) all distances from the surface. Only the presence of sur- rij rij face dangling oxygen bonds, created by cleaving Si–O–Si bonds, distinguishes the geometric and chemical features where rij is the separation distance between the molecule of the surface from those in the bulk of silica particles. anda surface atom and and are adjustable parame- Radial distribution functions for relaxed surfaces, how- ters. Lennard–Jones parameters are available in the lit- ever, vary with distance from the surface. Relaxation of erature for interactions between silica andvarious small simple planar cuts leads to a shorter Si–O distance (1:5 A)O molecules (MacElroy & Raghavan, 1990; Kohler & near the surface, corresponding to the shorter Si–O dis- Garofalini, 1994). Adsorbed molecules can interact with tance for terminal oxygens, in agreement with previous terminal oxygen atoms in hydroxyl groups, with oxygens work (Feuston & Garofalini, 1989). A more detailed anal- in siloxane bridges, or with silicon atoms. The interaction ysis by Feuston andGarofalini (1989) andthe simulated with silicon atoms is very weak (∼0:1 × 10−14 ergs) radial structure functions shown in Fig. 2 show that the (Kohler & Garofalini, 1994). Interactions with the two relaxed surface also contains a higher density of defects types of oxygens were either assumedto be equivalent, (over andundercoordinatedsiliconandoxygen atoms) or a larger size parameter was usedin the potential to ac- andof small rings containing 2–4 silicon atoms instead count for the larger size of the hydroxyl oxygen. Values of the six silicon rings typical of bulk silica. usedin the calculations are shown in Table 3. 4210 J. M. Stallons, E. Iglesia / Chemical Engineering Science 56 (2001) 4205–4216

Table 3 Lennard–Jones potential parameters for surface oxygen-adsorbate interactions

(A)O (erg × 10−14) Reference

N2 3.2175 2.7813 (Kohler & Garofalini, 1994) Ar 3.070 3.1217 (Kohler & Garofalini, 1994) CH4 3.2 (bridging oxygen) 2.5410 (MacElroy & Raghavan, 1990) 3.35 (non-bridging oxygens) 2.5410 (MacElroy & Raghavan, 1990)

Fig. 3. Potential energy surface for N2 on a relaxedsimulatedsilica surface.

4.2. Results

Fig. 4. Potential energy contour plot for N2 on a relaxedsimulated 4.2.1. Potential energy surfaces silica surface. Interaction energies between molecules andsurfaces were calculatedby placing the molecule above the surface (6–10 AO depending upon the surface topology), calculat- ing the potential energy, and then decreasing the distance The relaxedsilica surface (Figs. 3, 4) has richer fea- between the molecule andthe surface in small increments tures andbetter deÿnedpeaksandvalleys than unrelaxed (0.1–0:001 A)O until the energy reacheda minimum value surfaces (Fig. 5). Random surfaces (Fig. 6) show more (i.e. when energies for three consecutive steps decreased localizedandbetter-deÿnedadsorptionsites than unre- andthen increasedagain). Potential energy surfaces were laxedsurfaces (Fig. 5), but they also show much Iatter obtained by dividing the surface into a grid (0:2 A)O and potential energy surfaces, which lack the high- and repeating this calculation at each gridpoint. The poten- low-energy values observedfor relaxedsilica surfaces (Fig. 3). In contrast, the ordered surface (Fig. 7) possesses tial energy surface for N2 on a relaxedsilica surface is shown in Fig. 3. It shows a non-uniform distribution of only one type of adsorption site with regular connectivity adsorption energies with well-deÿned adsorption sites lo- among identical sites; the potential energy surfaces are catedat the bottom of the energy wells. The correspond- Iatter than those for the other surfaces. The surface ing topological energy contour map is shown in Fig. 4. properties andnon-uniformity of these surfaces can be quantitatively comparedusing the adsorptionenergy The energy contour plot for N2 on an unrelaxedsilica surface is shown in Fig. 5. The calculation process was distributions described in the next section. slightly di4erent for the random surface, because it was not periodic. In this case, minimum energies were cal- 4.2.2. Adsorption energy distributions culatedusing the same procedureas for MD-generated The distribution of adsorption energies on a given surfaces but only for the central regions of larger sur- surface was obtainedfrom the potential energy surface faces. The resulting potential energy surface for N2 on a by determining the energy values at the bottom of each random surface is shown in Fig. 6. For comparison, the potential energy well. These minima were obtainedby potential energy surface for the ordered surface created comparing the energy value of each gridpoint on the from -cristobalite is also shown (Fig. 7). potential energy surface (points spacedever 0 :2 A)O to its J. M. Stallons, E. Iglesia / Chemical Engineering Science 56 (2001) 4205–4216 4211

Fig. 5. Potential energy contour plot for N2 on an unrelaxedsimulated Fig. 7. Potential energy contour plot for N2 on an ordered simulated silica surface. silica surface.

Fig. 8. Adsorption energy distributions for N2; CH4, andAr on a relaxedsimulatedsilica surface.

by ÿtting adsorption energy distributions on ten di4erent relaxedsurfaces to a Gaussian distribution.The resulting adsorptionenergies on relaxedsimulatedsurfaces are

Fig. 6. Potential energy contour plot for N2 on a random simulated comparedto experimental values in Table 4. The aver- silica surface. age value for CH4 adsorption on relaxed silica surfaces is within 0:2kJ=mol of experimental values reported previously (Gangwal, Hudgins, & Silveston, 1979). The eight adjacentgridpoints (2 vertical, 2 horizontal, and4 average values for N2 andAr are within 2 kJ =mol of their diagonal points). If the energy value was lower than the corresponding experimental values. The di4erences values at all eight other points, that gridpoint was denoted among these molecules in the agreement between as a minimum. Minimum energies were calculatedfor ten calculatedandexperimental heats of adsorptionreIect di4erent (25 AO × 25 A)O relaxedandunrelaxedsurfaces the methodusedin orderto obtain the Lennard–Jones andfor one (150 AO × 150 A)O random surface. Adsorp- parameters requiredfor each molecule. CH 4 values were tion energy distributions for N2; CH4, andAr on relaxed obtainedby ÿtting simulation results for unrelaxedsur- silica surfaces are shown in Fig. 8. The average heats faces to experimental Henry law constants; thus, ensuring of adsorption for each type of molecule were obtained agreement of the simulations with experiment (MacElroy 4212 J. M. Stallons, E. Iglesia / Chemical Engineering Science 56 (2001) 4205–4216

Table 4 Average heats of adsorption on silica

SHads (kJ=mol) simulatedS Hads (kJ=mol) experimental Reference

N2 11.9 10.2 (Brunauer, Emmett, & Teller, 1938) Ar 13.1 11.5 (Brunauer, Emmett, & Teller, 1938) CH4 10.5 10.6 (Gangwal, Hudgins, & Silveston, 1979)

evident in the potential energy surface of the unrelaxed surface. The random surface, which also possesses a distribution of sites with random connectivity, has a higher heat of adsorption than experimentally found (10:2kJ=mol) (Brunauer, Emmett, & Teller, 1938).

5. Surface diusion

5.1. Method

Adsorption measurements and simulations probe the distribution of energy binding sites on the surface, but

Fig. 9. Adsorption energy distributions for N2 on various types of they provide no information about the connectivity be- simulatedsilica surfaces. tween these sites. Measurements of the dynamics of molecules di4using on the surface, however, reIect the binding energy and the connectivity of surface sites. As a result, surface di4usivities are more sensitive to & Raghavan, 1990). Therefore, excellent agreement is the chemical andgeometric detailsof the surface than expected. The parameters for Ar and N2, however, were binding energies. Unfortunately, experimental surface extrapolatedfrom those reportedfor zeolite systems. di4usion measurements are scarce. In this paper, the Adsorption energy distributions were also calculated dynamic properties of simulated silica surfaces were for N2 on the various types of simulatedsilica surfaces probed by comparing the di4usional behavior of ad- (Fig. 9). The ordered system has only one type of mini- sorbates on each surface. Speciÿcally, the migration of mum energy site, andhence, a single adsorptionenergy weakly boundmolecules, such as N 2 was followedusing (14:3kJ=mol). The surfaces createdby the molecular molecular dynamics methods that explicitly solve the dynamics and Monte Carlo methods, however, show equations of motion for these migrating species on each sites with a range of binding energies. Similar aver- surface. In these calculations, surface atom positions age adsorption energies were obtained on the relaxed were ÿxedfrom the previous simulations, andonly the (11:9kJ=mol) andunrelaxed(11 :4kJ=mol) surfaces, adsorbedspecieswere allowedto move. This simpliÿ- anda slightly higher value was obtainedon the random cation is appropriate for weakly interacting adsorbates, surface (12:8kJ=mol). Relaxedsurfaces ledto broader which exchange little energy with the surface (Utrera distributionsthan unrelaxedsurfaces. Relaxedsurfaces & Ramirez, 1992). Indeed, the di4usivity estimates ob- have weaker andstronger bindingsites than unrelaxed tainedfrom our static surface simulations were within surfaces, as expectedfrom their more ruggedpotential 5–10% of those in which the top layers of the surface energy surfaces (Figs. 4 and 5). The breadth of the ad- were allowed to relax during migration of adsorbed sorption energy distribution for the unrelaxed surface is species. The adsorbates were placed randomly at a height similar to that obtainedfor randomsurfaces, even though of 3–8 AO above the surface. This distance depended upon they have very di4erent energy topology and markedly the roughness of the surface, andit was chosen to mini- di4erent connectivity among adsorption sites. mize the frequent initial desorption of molecules placed The features of the actual amorphous surface seem to near the surface. Simulations were carriedout using the be representedmost accurately by the relaxedsurface ob- same technique outlinedfor the bulk simulations, except tainedfrom molecular dynamicssimulations. This surface at constant temperature insteadof energy, because of the possesses a distribution of energy sites with non-uniform nature of the dynamic system being examined, in which distribution, in contrast to the single adsorption site potential andkinetic energies can Iuctuate but the tem- energy present on the ordered surface or the channels perature is kept constant. Interactions between adsorbed J. M. Stallons, E. Iglesia / Chemical Engineering Science 56 (2001) 4205–4216 4213

Fig. 11. Arrhenius surface di4usion plots obtained for adsorbed N Fig. 10. Trajectory plot of N2 molecule di4using on a relaxed simu- 2 latedsilica surface (100 fs). on various types of simulatedsilica surfaces. species andsurface atoms were calculatedusing the do not have the highest average adsorption energy, re- Lennard–Jones potential used in the adsorption calcula- laxed surfaces show the broadest distribution of adsorp- tions. The surface di4usivity was calculated using two tion energies andthe sites with the strongest bindingsites methods. In one method, the Einstein di4usion equation among all surfaces examined. Di4usion was fastest on in two-dimensions: unrelaxedsurfaces, which have a similar average adsorp- 2 tion energy, but a narrower range of adsorption energies. R =4Dst; (6) Surface di4usion was also faster on ordered and random was used, where R2 is the mean-squareddisplacement surfaces than on relaxedsurfaces; although both ordered of a molecule, Ds is the surface di4usion coeUcient, and random surfaces have higher average adsorption en- and t is the elapsedtime. Another methodusedthe ergies, they lack the stronger binding sites present on re- velocity-autocorrelation functions: laxedsurfaces. ∞ Surface di4usion rates on the random, unrelaxed, and 1 Ds = 2 (vx(t)vx(0)) + (vy(t)vy(0)) dt; (7) ordered surfaces are more similar to each other than to 0 the values obtainedfor relaxedsurfaces. The di4erence where vx and vy are the velocity components in the x in surface di4usivities between these systems reIects a and y directions at time 0 or time t (Riccardo & Steele, subtle balance between the average adsorption properties 1996). In this method, di4usivities are obtained from the of the surface andthe distributionin both energy and dynamics of the short-term randomization of the initial connectivity among these adsorption sites. The ordered velocity vector for a given set of molecules. Activation system has the highest average binding energy among the energies for surface di4usion were calculated assuming surfaces examined, a feature that leads to slower di4u- that surface di4usion is an activated process that can be sional processes. But the regular arrangement of binding described accurately by an Arrhenius equation: sites prevents the stranding of molecules in unconnected D = D e−Ea=RT ; (8) or poorly connectedregions of the surface. Randomand s 0 unrelaxedsurfaces contain some bindingsites stronger 2 where Ds is the surface di4usivity (cm =s);D0 is the than in the ordered surfaces, but fewer than in relaxed 2 pre-exponential factor (cm =s);Ea is the activation en- surfaces. The density of unconnected regions, however, ergy for surface di4usion (kJ=mol), R is the universal gas is smaller than on the relaxedsurfaces andthe surface dif- constant (kJ=mol K), and T is the temperature (K). fusivities consequently larger. Di4usion is fastest on the unrelaxedsurface which has the smallest average adsorp- 5.2. Results tion energy andwhose potential energy surface exhibited “channels” between energy sites, which can enhance dif- A typical trajectory for N2 migration on a relaxedsil- fusion comparedto the more randomlyconnectedsites ica surface is shown in Fig. 10. Molecules di4use pre- present on random or relaxed surfaces. dominately by following the lowest energy paths across The relaxedsurface showedthe largest di4usionac- the surface. Surface di4usivities obtained for each surface tivation energy (7:8kJ=mol); this value equals 0.65 of at 200–300 K are shown in Fig. 11. The rate of surface the average adsorption energy. Activation energies on the di4usion was slowest on relaxed surfaces. Although they unrelaxed, random, and ordered surfaces were smaller 4214 J. M. Stallons, E. Iglesia / Chemical Engineering Science 56 (2001) 4205–4216

(7.0, 7.0, and6 :9kJ=mol) than on the relaxedsurfaces, deeperenergy wells andto unconnectedor poorly con- reIecting the Iatter potential energy surfaces shown in nectedsurface regions in which molecules can remain Figs. 5–7 andthe balance between the depthandthe con- for long periods of time. Ordered, random, and unrelaxed nectivity of the minima in the potential energy surface. surfaces, although possessing di4erent adsorption prop- Unfortunately, experimental surface di4usivities are erties, hadvery similar surface di4usivitiesanddi4usion scarce for well-characterizedamorphous surfaces. Re- activation energies, in spite of the varying level of com- portedsurface di4usivitiesfor weakly boundmolecules plexity usedin their synthesis. Surface di4usivitieson on silica glasses are in the range of 10−2–10−4 cm2=s simplest (ordered) surface were closest in value to those (Sladek, Gilliland, & Baddour, 1974; Tamon, Okazaki, obtainedon the most complex (relaxed)surfaces, even & Toei, 1985). Experimental surface di4usivities though the adsorption distribution of the relaxed surface for CH4; N2, andAr on Vycor glass are similar resembledmost closely that of the randomsurface. From (∼2 × 10−4 cm2=s) at 300–350 K andthey show similar a geometric andchemical point of view, the relaxedsur- activation energies (6 kJ=mol) (Barrer, Gabor, & Gabor, face seems to be the best representation of the amorphous 1959). These values are within the range obtainedby the silica surface. In light of the meager surface di4usivity simulations of these molecules on silica. The agreement data, it is not possible at this time to reach a deÿnite con- is encouraging, because the dynamic simulations were clusion that the relaxedsurfaces, although more rigorous basedon Lennard–Jonesparameters basedexclusively andappropriate, providethe best representation of silica on thermodynamic data (adsorption constants). Slightly surfaces treatedat low temperatures. higher activation energies were obtainedin the simulations, which could be indicative of the greater disorder of the amorphous silica surfaces comparedwith the surface of Vycor glass. Even though the simulatedsurface di4usivitieslie Notation within the range of experimental values, the selection of one type of theoretical surface description over another Aij adjustable parameter in BMH two-body po- cannot be made rigorously without additional experi- tential, kJ mental data. Slower di4usion (up to a factor of four) D0 pre-exponential factor in Arrhenius expres- 2 was obtainedon relaxedsurfaces than on the others, and sion for surface di4usion, cm =s 2 indeedsuchrelaxedsurfaces seem to be the most appro- Ds surface di4usivity, cm =s priate andrepresentation of actual amorphous surfaces Df fractal dimension formedby the abrupt interruption of particle growth e unit proton charge in gaseous or aqueous media. Di4erences among sur- Ea activation for surface di4usion, kJ=mol face di4usivities obtained on the ordered, random, and h functional term in three-body potential unrelaxedsurfaces were smaller. ri;rj;rk atomic position of atom, i; j; k; in each pair or triplet rij separation distance between two atoms or a molecule andan atom, AO 6. Conclusions rik interatomic separation distance between atoms i and k, AO Increasing the level of detail and of complexity used rjk interatomic separation distance between to describe silica surfaces leads to more faithful descrip- atoms j and k, AO c O tions of the heterogeneous nature of the surfaces andto rij cut-o4 constant in three-body potential, A c O the appearance of adsorption sites with the experimen- rik cut-o4 constant in three-body potential, A c O tally foundadsorptionenergies andwith the expected rjk cut-o4 constant in three-body potential, A random connectivity among adsorption sites. Potential R universal gas constant, kJ=mol K energy contours for relaxedsurfaces showedgreater het- R2 mean squareddisplacement erogeneity than those for unrelaxed, random, or ordered t time surfaces, and a wider range of adsorption site energies. T temperature, K Similar average heats of adsorption were obtained for the vx velocity component in the x direction random,relaxed,andunrelaxedsurfaces. The relaxedand vy velocity component in the y direction random surfaces, however, showed more random adsorp- V total potential energy, kJ tion site connectivity, in contrast with the regular connec- Zi formal ionic charge tivity of ordered surfaces or the more distinct low-energy channels on unrelaxed surfaces. As expected, surface dif- Greek letters fusion on relaxedsurfaces was slower andshoweda higher activation energy than on the other surfaces be- ÿij adjustable parameter in BMH two-body po- cause their more non-uniform site distributions led to tential, AO J. M. Stallons, E. Iglesia / Chemical Engineering Science 56 (2001) 4205–4216 4215

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