Modelling mesoporous materials

Studies in Surface Science and Catalysis 142 R. Aiello, G. Giordano and F. Testa (Editors) 9 2002 Elsevier Science B.V. All rights reserved.

1149

M o d e l l i n g M e s o p o r o u s Materials M.W. Anderson a, C.C. Egger a, G.J.T. Tiddy b, J.L. Casci c* aUMIST Centre for Microporous Materials, Department of Chemistry, UMIST, P.O. Box 88, Manchester M60 1QD, UK bDepartment of Chemical Engineering, UMIST, P.O. Box 88, Manchester M60 1QD, UK CSynetix, P.O. Box 1, Billingham. Cleveland TS23 1LB UK

1. I N T R O D U C T I O N Mesoporous materials of the MCM or SBA variety are by their nature organised amorphous material. Consequently, in order to describe their structure, it is necessary to utilise a model which is able to accommodate both the organisation and the disorder. Such models are useful for a number of reasons. First, as a method of characterisation, if a model can be generated then a variety of experimental data can be simulated e.g. x-ray diffraction, electron microscopy, gas adsorption etc. Second, a structural model allows further properties of a phase to be anticipated. Third, a model allows a visualistion of a structure which aids our understanding of these novel complex materials. Fourth, the details of the structure yield clues to the synthetic mechanism thereby aiding strategies to design and control new structures. Recently it has been shown [1 ] that an electron density map of a mesoporous structure can be directly determined from electron crystallography. That is, a very well ordered sample was examined by electron microscopy, both electron diffraction patterns and images were collected providing, after indexing, both intensity and phase information resulting in a low resolution electron density map (low resoltuion means that only the wall structure is located, as precise atomic coordinates are random for an amorphous structure). This is the ultimate method to directly determine the structure (wall curvature and thickness) of such a material. However, the technique is laborious and is not suitable for screening materials. Furthermore, subtle structural features are not readily extracted from the resulting three-dimensional electron density map. Our approach is to build structures using a certain amount of previous knowledge in a manner which is then easily manipulated to reflect different synthetic conditions and qualities of material. In this paper we concentrate on the structure of SBA-1 [2] a material first synthesised by the Santa Barbara group, Stucky et al. 2. BUILDING S T R U C T U R E S BY HAND SBA-1 is a hydroxylated silica mesoporous material which is synthesised using a suffactant template cetyl-triethylammonium bromide in a highly acidic silica solution. Under the conditions of synthesis the surfactant forms globular micelles which pack together to give a cubic unit cell. The space group of the resulting material appears to be Pm3n and is related to the suffactant mesophase known as the I1 phase[3]. Working on the basis that the resulting This work was funded by Synetix.

1150 silica structure will in some manner wrap around the globular micellar water structure, in order to describe the silica walls in an analytical mathematical form a type of mathematics is required which will easily describe surfaces wrapped around spheres and distorted spheres. Such a mathematics exists based upon the Gauss distribution function: e

_x 2

--C

and is described in detail by Jacob and Andersson[4]. For our purposes we will require spheres for which the x, y, and z coordinates are given by the equation:

e -(x~§247 = C The radius of the sphere is determined by the constant C and the centre of the sphere can be moved to any coordinate h, k, I by the following transformation: e-[(x-hi +(y-kr162 ] = C The sphere can be elongated or squashed in any dimension to produce for instance an oblate ellipsoid by the following transformation: e-[a (~:-hr +b~O,-er +b3(z-l)Z].__C Finally, an object with a different radius can be formed not only by changing C but also by adding a constant, a, within the exponential thus: e "-[~ (~-h)~+b~('v-e)~+b~(z- t)~] = C This provides the tools to build a mesoporous material synthesised from globular micelles by now adding these functions in the exponential scale. Figure 1 shows what happens when two exponential functions are added together, one representing a sphere, and a second oblate ellipsoid displaced to a different coordinate. When the objects are far apart they form perfect spheres or ellipsoids. However, as they approach the surface begins to form a continuous wrapping. Such a construction should be ideal for the description of mesoporous materials based on globular micelles as the silica surface should indeed wrap around the micellar body forming a continuous surface. Pores

G

a

C

b d

Figure 1. a) and b) are oblique and top views of a small sphere and oblate ellipsoid generated with Gauss distribution functions. c) and d) show how the surfaces wrap as the object become larger and approach one another.

1151 will be generated by the excluded zone between the objects. The arrangement of micelles in the Pm3n, 11, structure is given by adding 21 objects together according to the following equation:

s {O'-h)~+Cv-k)~+~:z-')}+ ~ s h,k,t h',~',I' Table 1. h,k,l fractional coordinates of 9 spheres

h 0

k 0

1

0 0

Table 2. h ' k' /' coordinates for 12 ellipsoids with values of bl, b2 and b3 in terms o f f r~ = radius o f sphere

rz

f = radius o f long axis o f ellipsoid radius o f short axis o f ellipsoid

C = e -~

= radius o f short axis o f ellipsoid a = r~ + In(C)

1 0

h' 1

k' 0.25

1' 0.5

bl 1/f2

be 1

0

0

1

0.75

0.5

1/f2

1

1 0

0 1

1

1

0

1

0

1

0

1

1

1

1

1

0.5

0.5

0.5

0 0 0.5 0.5 0.5 0.5 0.25

0.25 0.75 1 1 0 0 0.5

0.5 0.5 0.25 0.75 0.25 0.75 1

1/f2 1/ f 2 l/ f 2 1/ f 2 1/f2 1If 2 1

1 1 l/ f 2 l/ f 2 1/f2 l/ f 2 1/f2

b3 1/f2 1 / f2 1/f2 1/f2 1 1 1 1 1/f2

0.75 0.25 0.75

0.5 0.5 0.5

1 0 0

1 1 1

1/ f 2 1/f2 1 / f2

1/ f 2 1/f2 1 / f2

Figure 2: Pm3n arrangement of 9 spherical micelles, one marked $5, (on a body centre) and 12 oblate ellipsoids, marked El-E4, two on each face o f t h e unit cell. Based on a unit cell of 85A the sphere radius and the radius of the short axis of the ellipsoid are both 10A resulting in the constant, a, equal zero.

1152 The first summation represents 9 spheres with a body centred arrangement and the second summation 12 oblate ellipsoids. The coordinates for these objects are given in tables 1 and 2 as well as the derivation of the constants. The result is the arrangement of micelles shown in figure 2. This figure is just a schematic representation of the relative positions of the spherical and oblate micelles and the sizes are scaled in order to aid the reader to understand this arrangement. In this figure the extension of the oblate ellipsoid is given by the factor f, which in this case has been chosen as 1.3. Although the final structure can be calculated for any unit cell in order for the mathematics to remain robust it is important that the constant C does not become excessively large or small. In order to prevent this from occurring all surface calculations have been based on a unit cell of 6A which is then scaled accordingly. '~

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................ : : , ; G ' ~ : . r

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Figure 3: left picture shows the surface generated with an 85A unit cell, radii rl and r2 equal 19.5A and an oblateness given by f=l.18A; right picture shows the atomistic model of SBA-1 by placing a random army of silicon atoms on the outer side of the surface. The structure ofSBA-1 is then built by increasing the size of the spheres and ellipsoids until they merge and the surface becomes continuously wrapped. However, as the surface will wrap into adjacent unit cells it is important to include in the calculation a further 12 virtual ellipsoids. The coordinates of these twelve ellipsoids can be derived from those in Table 2 by replacing the coordinate 0.25 by -0.25 and 0.75 by 1.25. Consequently a total of 33 objects are required to describe the whole structure ofSBA-1 (9 spheres, 12 ellipsoids and 12 virtual ellipsoids). When this is done the result is the surface shown in figure 3. One unit cell is shown and windows can be seen which are generated as the surface wraps from one object to the next. This surface will represent the periphery of the edge of the wall of SBA-1 which will presumably be in contact with a water sheath around the micelles. In order to generate an atomistic model of SBA-1 it is the necessary to fill the space on the outer side of the surface, not occupied by the surfactant molecules and water layer with a random array of silicon atoms. This method was successfully used before in an atomistic description of

1153 MCM-4815]. The density of silicon atoms is maintained by keeping an average Si...Si separation of 3.5/!t and the atomistic model so formed is shown in the lower part of figure 3. In order to optimise the parameters, radius of the sphere rs, radius of the short axis of the ellipsoid re and degree of oblatenessfa large number of structures were generated in this manner from which the x-ray diffraction pattern and electron micrographs could be generated for comparison between the our model and experimental data. The results of some of these calculations are shown in figure 4.

'

!i

.............

~"

t

. . . . .

t

~

t

.

123456123456123456 "20

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234567

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Figure 4: top 16 x-ray diffraction patterns calculated with sphere radius rl equal to the radius of the short axis of the oblate ellipsoid r2. The value off, the measure of oblateness is given inset in each figure. The 16 atomistic models generated are layed out below in the same order. The x-ray diffractions pattems agree very well with those reported in the literature and with those that we measure. The best agreement between is found for rs=rz=l 9 to 19.5/!t and f=l.18. As soon as rs deviated from r2 extra reflections appeared, most significantly the [ 110] reflection, which rapidly became very strong and is not observed experimentally. The fact that the radius of the sphere and that of the short axis of the oblate ellipsoid are similar is not surprising as both will be governed by the length of the surfactant chain. The models which best fit the x-ray diffractions patterns also show a strong correspondence between the projected electron potential maps (not shown) and the electron micrographs described in the literature[ 1]. As the ellipsoids tend to spheres (when f tends to 1) the size of the two types of micelles has to be significantly different (e.g. 23A and 19A radii to be able to explain the electron micrographs. This was the description for the structure given previously[1 ], however, the x-ray diffraction pattems of such a structure are vastly incorrect, including a large [110] reflection (see figure 5).

1154

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2

3

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Figure 5: X-ray diffraction patterns and atomistic models illustrating presence of low angle [110] reflection, *, when the radii rs and r2 deviate from each other. Top model for r~=18.5A, re=21A, f=l.13; bottom for spherical micelles similar to description in reference 1, rr r2=23.5A, f=l.0. The optimum model for SBA-1 synthesised in our laboratory is with rs=re=19 to 19.5A andf=l.18 for an 85/~ unit cell. Such a surface generates three types of windows between pores all of which can be seen in the [100] projection by transmission electron microscopy[1 ]. The three window types are illustrated in figure 6 and are generated when the micelles are closest, separated by a water layer. The largest windows are created by two oblate ellipsoid micelles with the flatter sides close together. The medium pores are created by an oblate ellipsoid and a spherical micelle. In this case the oblate micelle nearly has the thin end towards the spherical micelle. The third, and smallest window is generated by two oblate ellipsoid micelles, orthogonal to each other, where the thin edges are close together.

1155 It is interesting to conjecture why the windows are of such different sizes. There appears to be a relationship between the window size and the contact angle between the micelles. For instance the largest window is between the two oblate micelles which approach on the flatter sides giving a small contact angle between the micelles. The water layer between the micelles will probably exclude the silicate thereby generating the pore. This is illustrated in figure 7. When the contact angle is small, E2 approaching E4, then a water layer of a given thickness will exclude a relatively large window. When the contact angle is larger, E1 approaching $5, then the same water thickness layer will exclude a smaller window. When E1 approaches E2 at the largest contact angle only a very small region is excluded by the water layer.

Figure 6: four unit cells of the optimum surface of SBA-1 in the [100] projection. Three windows are apparent. The A window were micelles E2 and E4

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approach; the B window where micelles E1 and $5 approach; the narrow window where micelles E2 and E1 approach. Nomenclature from figure 2. 3. CONCLUSIONS The wall structure of mesoporous materials can in general be describe via an analytical expression. Where the mesoporous material is synthesised from a surfactant mesophase based upon a three-dimensional packing of globular micelles then the mathematics based on the exponential scale of a Gaussian distribution works very well. We have successfully described the structure of SBA-1 in this manner and revealed that the details of the micellar structure, including oblate distortions of globular micelles, are retained in the final inorganic structure. A preliminary mechanism for window size in mesoporous materials is discussed.

1156

Figure 7: micelles surrounded by a water layer and then the silica wall. The water layer excludes the silica wall from a region between the micelles thereby generating a window. The window size is governed by the contact angle between the micelles. 4. R E F E R E N C E S

1. Sakamoto, Y; Kaneda, M; Terasaki, O.; Zhao, D.; Kim, J.M.; Stucky, G.; Shin, H.J.; Ryoo, R. Nature 2000, 408, 449. 2. Huo, G; Margolese, D.I.; Ciesla, U.; Demuth, D.G.; Feng, P.; Gier, T.E.; Siegel P.; Firouzi, A.; Chmelka, B.F.; Schtith, F.; Stucky, G.D. Chem. Mater., 1994, 6, 1176. 3. Luzzati, V." Delacroix, H.; Gulik, A. J. Phys. II, 1996, 6, 405. 4. "The Nature of Mathematics and the Mathematics of Nature" by Jacob, M. and Andersson, S., Elsevier 1998. 5. Alfredsson, V; Anderson, M.W. Chem. Mater., 1996, 8, 1141.