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Fractal dimension of zeolite surfaces by calculation
Chaos, Solitons and Fractals 12 (2001) 1145±1155
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Fractal dimension of zeolite surfaces by calculation Melkon Tatlõer, Ays e Erdem-S enatalar * Department of Chemical Engineering, Istanbul Technical University, Maslak, 80626, Istanbul, Turkey Accepted 3 April 2000
Abstract A theoretical method for the estimation of the fractal dimensions of the pore surfaces of zeolites is proposed. The method is an analogy to the commonly employed box-counting method and uses imaginary meshes of various sizes s to trace the pore surfaces determined by the frameworks of crystalline zeolites. The surfaces formed by the geometrical shapes of the secondary building units of zeolites are taken into account for the calculations performed. The characteristics of the framework structures of the zeolites 13X, 5A and silicalite are determined by the help of the solid models of these zeolites and the total numbers of grid boxes intersecting the surfaces are estimated by using equations proposed in this study. As a result, the fractal dimension values of the zeolites 13X, 5A and silicalite are generally observed to vary in signi®cant amounts with the range of mesh size used, especially for the relatively larger mesh sizes that are close to the sizes of real adsorbates. For these relatively larger mesh sizes, the fractal dimension of silicalite falls below 1.60 while the fractal dimension values of zeolite 13X and 5A tend to rise above 2. The fractal dimension values obtained by the proposed method seem to be consistent with those determined by using experimental adsorption data in their relative magnitudes while the absolute magnitudes may dier due to the dierent size ranges employed. The results of this study show that fractal dimension values much dierent from 2 (both higher and lower than 2) may be obtained for crystalline adsorbents, such as zeolites, in ranges of size that are close to those of real adsorbates. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction The observation of scaling behavior in diverse ®elds of science has promoted the studies carried out to relate various phenomena to the concept of fractal geometry. Systems such as adsorbents, aggregates, catalyst supports and electrodes most often exhibit complex behavior, which cannot be easily explained by considering concepts of Euclidean geometry. The determination of the fractal nature of these systems may be of crucial importance for an accurate prediction of the scaling behavior encountered in various processes, such as adsorption [1±4], reaction [5], aggregation [6,7], electrodeposition [8,9], etc. The estimation of the fractal dimension values of adsorbents does not only help to characterize these materials but it also provides valuable information about the adsorption behavior of adsorbates with various sizes. The most commonly employed method to determine the fractal dimension values of adsorbents is based on the fact that the accessible surface area of a solid decreases or increases with increasing dimensions of the adsorbate molecules, for fractal dimension values above and below 2, respectively. The variation of the monolayer capacities for a homologous series of gas molecules used in the process of adsorption with respect to their eective sizes enables the evaluation of the fractal dimension values of porous adsorbents [1,2]. It has been demonstrated that the fractal dimension may also be estimated by employing a modi®ed form of the Frenkel±Halsey±Hill isotherm equation [10] as well as a method taking the Freundlich isotherm as a starting point for the calculations [11]. The former method utilizes a single adsorption isotherm and is valid *
Corresponding author. Tel.: +90-212-285-6896; fax: +90-212-285-2925. E-mail address: [email protected] (A. Erdem-S enatalar).
0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 0 8 5 - 0
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only for multilayer adsorption, while the latter employs a single adsorption isotherm and data representing the characteristics of a reference adsorbent for the same adsorbate. Besides adsorption, various other methods, such as small angle X-ray or neutron scattering (SAXS or SANS) [3,12], electronic energy transfer [3,12] and thermoporosimetry [13], may also be used in the determination of the fractal dimension values of adsorbents. All methods of fractal analysis involve the employment of ``yardsticks'' of various sizes for tracing the surfaces of adsorbents. Depending on the method used, the yardsticks may be adsorbate molecules of dierent eective sizes, electromagnetic waves diracted at dierent angles, energy transfer from a donor molecule to an acceptor molecule at dierent distances and ®lms of variable thickness, etc. It should also be mentioned that the fractal dimension value of any object that does not exhibit self-similar properties may vary with respect to the range of size of the yardsticks employed in the estimation. Thus, it is possible that dierent methods, employing yardsticks of various sizes, may estimate dierent fractal dimension values for a certain adsorbent. Zeolites have microporous crystalline structures and are commonly used in processes involving adsorption, as well as in catalysis and ion-exchange. The fractal dimensions of three of the most commonly employed zeolites, namely 13X, 5A and silicalite have been recently evaluated by using dierent methods based on adsorption. The employment of the Pfeifer±Avnir and point-slope methods resulted in fractal dimensions of about 2.08, 2.13 and 1.68 for the zeolites 5A, 13X and silicalite [11], respectively. The range of size employed in the estimations varied between the sizes of methane and propane for the zeolites 5A and 13X and the sizes of methane and n-butane for silicalite. Due to the uniformity of the pore structures of the zeolites, it is commonly thought that these materials should exhibit smooth surfaces with fractal dimension values of about 2. Thus, as the fractal dimension value of a zeolite gets further from 2, the reason underlying this type of behavior becomes less apparent. In this study, the fractal dimensions of the pore surfaces as determined by the framework structures of the zeolites 13X, 5A and silicalite are evaluated by making an analogy to the commonly used box-counting method. The geometric shapes of the pore surfaces formed by the secondary building units constituting the frameworks of the zeolites 13X, 5A and silicalite are taken as a basis for the calculations. The characteristics of the framework structures of the zeolites 13X, 5A and silicalite are determined by the help of the solid models of these zeolites and equations are proposed for estimating the total numbers of grid boxes intersecting the pore surfaces formed by sections of their frameworks. The variation of the fractal dimension value is investigated with respect to the range of scale utilized in the estimations and the upper limit of the scale used is restricted by the sizes of the secondary building units involved. A discussion of the results obtained in relation to those obtained by various other methods employed to determine the fractal dimensions of adsorbents is also presented. 2. Theoretical The crystalline structures of the zeolites consist of tetrahedra known as primary building units. A T-atom (silicon or aluminum), surrounded by four oxygen atoms, is located at the center of each tetrahedron. A number of these units constitute the secondary building units, which may be of various types, such as, double four-ring (D4R), single six-ring (S6R), single ®ve-ring (S5R), etc. An arrangement of the secondary building units ends up with the formation of the regular crystalline structures of the zeolites with cavities and/or channels of molecular sizes. The openings to the cavities or channels of zeolites are constricted mainly by the oxygens of the tetrahedra forming rings, which may have various free diameter values. The number of oxygen atoms in a ring determines the magnitude of the diameter of the opening to the channel or the cavity. For an ideal planar con®guration, the 6-, 8-, 10- and 12-member ring openings are approximately equal to 2.8, 4.4, 6.0 and respectively [14,15]. The actual opening, however, may slightly deviate from these values. The 7.7 A, cavities or channels formed by the six-member rings are so small that only the very small polar molecules, such as H2 O and NH3 can penetrate. The internal pore surfaces of the zeolites are formed by the linkage of the secondary building units in dierent geometries and the characteristic shapes of these surfaces may be partly dierent than those pertaining to the original secondary building units forming them. The secondary building unit of silicalite is
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referred to as the 5-1 unit shown in Fig. 1, which consists of a regular pentagon with an extension. However, the pentagons in silicalite are folded along one diagonal to create a trapezoid and a triangle adjacent at their base at an angle of about 144°. Thus, the inner surfaces of silicalite [16] are mainly composed of trapezoids and triangles, but some squares and rectangles also exist. Two distinct channel systems exist in silicalite, namely the straight and sinusoidal ones. The framework structure of silicalite may be seen in Fig. 2(a). The secondary building units of the zeolites 13X and 5A [17,18] consist of single or double four- and sixrings and the framework surfaces of these zeolites may be represented by squares and regular hexagons. The framework structures of the zeolites 13X and 5A may be seen in Fig. 2(b) and (c), respectively. The unit cells of zeolites are the smallest units that possess all the characteristics of the crystal structures of these materials. Thus, in order to evaluate the fractal dimension values of zeolite surfaces, it is sucient to consider a single unit cell of a zeolite. The characteristics of the zeolites 13X, 5A and silicalite, as determined by the help of the investigation of the solid framework models are given in Table 1. The number of the dierent geometric shapes in a given structure may be seen in the table. The fractal dimension of a structure can be determined by using various methods. The box-counting method may systematically characterize structures that have scaling properties. This method may commonly be applied to structures of very dierent natures since self-similarity is not a prerequisite for the estimations made. The box-counting fractal dimension can be determined easily by using 2-D meshes for any structure in a plane and may also be adapted for structures in 3-D space. In this method, the structure investigated is covered with a regular mesh with size s and the number of grid boxes (N) that contain some of the structure is determined. The number of boxes counted will depend on the size of the mesh used and the size of the mesh may be gradually varied. As a result, the fractal dimension value may be evaluated by plotting log N vs log (1/s). The slope of the straight line ®tted to the plotted points represents the fractal dimension value of the structure investigated. The box-counting method is adapted in this study for the measurements carried out to evaluate the fractal dimensions of the pore surfaces of the zeolites 13X, 5A and silicalite in 3-D space. The fractal dimension values of the surface units are initially determined in a 2-D plane after which the structure is investigated in 3-D space by taking into account the number of overlaps between the neighboring units. The ratio of the number of overlapping units to the total number of the structural units is an indicator of the extent of the fractality of the pore surfaces, as examined in 3-D space. The 3-D imaginary meshes used to trace the pore surfaces of the zeolites are assumed to be placed on the structural units without an angle of inclination. This approach is used to make an analogy with the adsorptive behaviors of adsorbates tracing the pore surfaces. The 3-D meshes placed on neighboring units may intersect due to the inclinations between the units, but the double counting of the same mesh grid for two neighboring units is dealt with by subtracting the number of mesh boxes corresponding to the overlapping units from the total number of
Fig. 1. The secondary building unit of silicalite.
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Fig. 2. The framework structures of the zeolites (a) silicalite, (b) 13X and (c) 5A. Table 1 The number of the dierent geometric shapes in single unit cells of silicalite and zeolite 13X as well as in a single pseudocell of zeolite 5A Zeolite
Number of units Trapezoid
Triangle
Hexagon
Square
Rectangle
Silicalite 5A 13X
74 ± ±
44 ± ±
± 8 16
4 12 72
10 ± ±
mesh boxes counted. This subtraction also accounts for the fractality of the structure in 3-D space, as mentioned before. The geometric shapes of the framework surfaces formed by the secondary building units of the zeolites 13X, 5A and silicalite are taken as a basis for the measurements made. The ®rst step of the calculations performed involves the application of the box-counting method to the surface structural units, which generally have well-known geometrical shapes, such as squares, pentagons, hexagons, etc. In this study, it is assumed that the relatively small pore openings formed by six (or less than six)-member rings mentioned above behave as full surfaces. It is well known that the sizes of these pore openings are smaller than the molecular sizes of most of the real adsorbates. Once the above assumption is made, in order to better
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demonstrate the tendency of the variation of the fractal dimensions of adsorbent surfaces with the size range, the mesh sizes used in this study are made to vary in a relatively broad range. The Si±O (or Al±O) bond lengths and the angles of the planar geometric structures formed by the arrangement of the secondary building units may generally be assumed to be equal. The Si±O (or Al±O) bond length is taken to be equal to for all the cases investigated. The number of grid boxes that intersect the geometric framework 1.6 A structures, formed by the arrangement of the secondary building units, when various scaling factors are used may be easily determined in a plane. The evaluation of the fractal dimensions of the zeolites 13X, 5A and silicalite for various scales is detailed below. In order to estimate the fractal dimension value, ®rst of all, the number of grid boxes intersecting the structural units of the surface is determined. Accordingly, regular trapezoids, triangles, hexagons, rectangles and squares should be taken into consideration, as mentioned before. The dimensions of a regular trapezoid and a triangle forming the folded pentagons of silicalite are shown in Fig. 3. In the same ®gure, the relative dimensions pertaining to the same structural units are also indicated in parantheses The relative areas in terms of the as a fraction of the largest base length of the trapezoid (a 2b 5:12 A). number of meshes N s of the trapezoid and the triangle vary with respect to the mesh size used (s) and the number of grid boxes counted is inversely proportional to this size. Some examples depicting the relationship between the mesh size and the relative areas of the trapezoids and the triangles pertaining to the folded pentagons of silicalite are given in Table 2. It may be observed that the widths of both the trapezoid a 2b and the triangle 2e are equal to the 1=s value in each case such that as the mesh size is decreased, the number of grid boxes intersecting a structure will increase. It should also be noted that the relative dimension values given in Fig. 3 correspond to the employment of a mesh size, s 1. The number of grid boxes (N) counted for rectangles and squares is simply equal to the product of the two sides of these quadrilaterals, while for less regular structures, such as triangles, dierent relationships should be sought
Fig. 3. The dimensions of a regular trapezoid and a triangle forming the folded pentagons of silicalite. The relative dimensions pertaining to the same structural units are indicated in parantheses as a fraction of the largest base length of the trapezoid (a 2b 5:12 A). Fig. 4. Determination of the number of grid boxes intersecting a right triangle, for which the relative dimensions of b and c are equal to 8 and 6, respectively.
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Table 2 The relationship between the mesh size and the relative dimensions of a regular pentagon 1=s
a
b
c
d
e
1 4 16
0.62 2.5 10
0.19 0.75 3
0.59 2.37 9.5
0.36 1.46 5.87
0.50 2.0 8.0
for. The N value of a trapezoid is determined as the sum of a rectangle and two right triangles by the following equation: Ntrapezoid a c c b c b:
1
The ®rst term corresponds to the number of grid boxes counted for the rectangle, whereas the second one represents the sum of those counted for the two similar triangles. The latter term is obtained by considering a simple approximated relationship existing between the number of boxes counted and the number of meshes in a single row and column of a mesh structure (or the width b and height c of the unit). An example for the calculation of the number of grid boxes intersecting a right triangle, for which the relative dimensions of b and c are equal to 8 and 6 respectively, is illustrated in Fig. 4. Both of the right triangles seen in the ®gure intersect 31 grid boxes, in accordance with the proposed equation c b c b. The number of grid boxes obtained for the triangle constituting the second part of the folded pentagon along with the trapezoid is determined in a similar way by c and b replacing d and e, respectively Ntriangle d e d e:
2
The dimensions of the squares and rectangles found in the structure of silicalite are shown in Fig. 5(a) and (b), respectively. The relative dimensions in terms of the number of meshes are also indicated in parantheses The number of boxes counted for these as a fraction of the largest base length of a trapezoid (5.12 A). quadrilateral units found in the structure of silicalite may be represented by Nrectangle a 2e;
3
Nsquare a a;
4
where a and 2e denote the width and length of the units, respectively. In order to estimate the fractal dimension values of the pore surfaces of the zeolites 5A and 13X, the number of grid boxes intersecting regular and planar hexagons and squares should be determined. The
Fig. 5. The dimensions of the (a) squares and (b) rectangles found in the structure of silicalite. The relative dimensions in terms of the number of meshes are indicated in parantheses as a fraction of the largest base length of a trapezoid.
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Fig. 6. The dimensions of a regular hexagon. The relative dimensions in terms of the number of meshes are indicated in parantheses as a fraction of the length of the diagonal of the hexagon (f 2g 6:4 A). Fig. 7. Determination of the number of meshes counted for one of each two neighboring units sharing an overlap (m2 ).
dimensions of a regular hexagon are shown in Fig. 6 and the relative dimension values in terms of the number of meshes are indicated in parantheses for the case where the mesh size is equal to 1. The relative dimensions are given as a fraction of the length of the diagonal of the hexagon denoted by f 2g which is The number of grid boxes (N) counted for a hexagon, which is constituted by a rectangle equal to 6.4 A. and four right triangles may be obtained by using the following equation: Nhexagon f 2h 2 g h g h:
5
The ®rst term corresponds to the number of grid boxes counted for the rectangle, whereas the second one represents the sum of those counted for the four similar triangles. The squares of the zeolites 13X and 5A may be represented by Fig. 5(a), and the N values may be estimated by employing Eq. (4), provided that a and the relative dimension are replaced by f and 0.5, respectively. As mentioned before, in order to eliminate any inconsistency that may originate from the double counting of the same mesh grid for two neighboring framework units, such as a triangle and a trapezoid for silicalite, as well as for determining the extent of the fractality of the pore surfaces in 3-D space, the number of extra boxes counted, corresponding to the overlaps shared by the neighboring units are subtracted from the total number of boxes determined. For angles of 90 6 h < 180° between two neighboring units, the amount subtracted (Noverlap ) may be taken to be equal to half of the boxes counted for the neighboring rows of two distinct units, while no overlapping is expected to occur between units with h P 180°. The number of overlapping units increase (since units other than the neighboring units are also involved) as h falls below 90°. For all the cases investigated in this study, the angles between the neighboring units are found to be in the range 90 < h 6 180°. Table 3 represents the total number of overlapping units (k1 and/or k2 ) in the unit cells of zeolite 13X and silicalite as well as in a single pseudocell of zeolite 5A. The overlaps in silicalite are respectively, whereas those in 13X and 5A have an of two dierent actual lengths, that is 3.2 and 5.12 A, The number of overlaps that are subtracted from the total number of boxes counted actual length of 3.2 A. are calculated by using the following equation: Noverlap k1 m1 k2 m2 ;
6
m1 and m2 represent the number of meshes counted for one of each two neighboring units sharing an overlap and correspond to relative dimensions of 0.62 and 1, respectively. The values of m1 and m2 are directly proportional to the 1=s values. k1 and k2 denote the number of overlaps that have relative dimensions of 0.62 and 1, respectively. As an example, the two similar rectangles seen in Fig. 7 lead to obtaining an m2 value of 5 1=s 5.
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Table 3 The total number of overlapping units in the unit cells of zeolite 13X and silicalite as well as in a single pseudocell of zeolite 5A Type of zeolite
Number of overlapping units k1
k2
Silicalite 5A 13X
102 24 96
63 ± ±
The characteristics of the framework structures of the zeolites 13X, 5A and silicalite are determined by the help of the solid models of these zeolites and the total numbers of grid boxes intersecting the surfaces are estimated by using the above equations. The fractal dimension values are estimated by plotting log Ntotal vs log 1/s values. The slope of the plot gives the fractal dimension. 3. Results and discussion The fractal dimensions of the pore surfaces of the zeolites 13X, 5A and silicalite are determined by using the method proposed in this study. The fractal dimension values are evaluated by varying the 1=s values in a range. The upper limit of the 1=s values employed is varied (generally between 8 and 256) while the lower limit is kept constant at two distinct values, namely 2 and 16. The 1=s values given in the abscissa of the ®gures represent the values corresponding to the upper limit of the range investigated. In order to check for the reliability of the estimations performed in this study, ®rst of all, the fractal dimensions of two geometric shapes, namely a regular pentagon and a hexagon are determined by using the proposed method and a comparison is made with the results obtained by the commonly employed box-counting method. For the latter method, 2-D square meshes may be easily prepared and applied to the investigated structure. The fractal dimension values of a pentagon and a hexagon as determined by using the proposed and the boxcounting methods and the same range of mesh size for both cases may be seen in Table 4. It may be observed that the variation of the fractal dimension with the range used exhibits a similar form and the values obtained are quite close for both methods. It should also be noted that the fractal dimension of a regular pentagon is smaller than that of a regular hexagon and for both cases, the fractal dimension values vary as the range of the mesh size employed changes. The fractal dimension of silicalite is estimated by using the proposed method and two distinct lower limits for the 1=s values. The imaginary 3-D meshes used are assumed to be placed without an angle of inclination on each of the structural units of silicalite, as mentioned before. The results obtained are depicted in Fig. 8. It may be seen that relatively higher fractal dimension values very close to 2 are obtained for the case, where the lower limit is set at 16 while the fractal dimension varies in considerable amounts when the lower limit is taken to be equal to 2. In case 1/s values between 2 and 16 are employed, the fractal dimension of silicalite falls below 1.8, which may be considered to be a quite low value for a zeolite. This value decreases further and approaches 1.6 as 1/s values between 2 and 8 are utilized. Quite high correlation coecient values r > 0:99 are obtained regardless of the range of 1=s values employed in this study. An interesting point is the proximity of the fractal dimension value estimated for silicalite by the proposed theoretical method to that obtained by using adsorption data [11] (D 1:68) pertaining to the range determined by the sizes of methane±butane. Although the range of size employed in that study most probably Table 4 The fractal dimension values of a pentagon and a hexagon as determined by using the proposed and the box-counting methods and the same range of mesh size for both cases Range of 1=s values
Dpentagon proposed method
Dpentagon box-counting method
Dhexagon proposed method
Dhexagon box-counting method
16±32 16±48 16±64
1.800 1.815 1.842
1.820 1.835 1.860
1.855 1.877 1.901
1.849 1.888 1.903
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Fig. 8. The variation of the fractal dimension of silicalite with respect to the higher limit of the 1=s values when the lower limit of the 1=s values is kept constant at () 2 and (+) 16. Fig. 9. The variation of the fractal dimension of 5A with respect to the higher limit of the 1=s values when the lower limit of the 1=s values is kept constant at () 2 and (+) 16.
corresponds to 1/s values of about 0.9±1.8 when eective values, considering the actual surface that can be employed in adsorption, are assumed for the cylindrical molecules, it should be noted that the sizes of the gas molecules are still comparable with those of the secondary building units. Thus, the geometrical shapes of the framework units may be of considerable signi®cance in the determination of the fractal dimension values of zeolites. The variation of the fractal dimension of zeolite 5A with respect to the 1/s values is given in Fig. 9. The calculations are performed in a manner similar to the estimation of the fractal dimension of silicalite. It may be observed from Fig. 9 that almost constant fractal dimension values of about 2 are obtained for the case where the lower limit is set at 16 while the fractal dimension varies in moderate amounts when the lower limit is taken to be equal to 2. In case 1/s values between 2 and 8 are employed, the fractal dimension of zeolite 5A slightly exceeds 2.1. A comparison of the fractal dimension values obtained in this study with that determined by using experimental adsorption data [11] (2.08) reveals that quite consistent values for zeolite 5A are obtained. However, it should also be reminded that the range of molecular size pertaining to the adsorption data used in the previous study [11] (methane±propane) corresponded to 1/s values between 1.2 and 2, a range lower than the ones investigated in this study. In any case, it is apparent that employing a narrow range of relatively high mesh sizes may lead to fractal dimension values quite dierent from 2. The fractal dimension of zeolite 13X is estimated by using an approach similar to those employed for zeolite 5A and silicalite. Two distinct lower limits for the range of mesh size are employed for this case, too. The variation of the fractal dimension with respect to the 1/s values are given in Fig. 10. It may be observed that the fractal dimension value remains slightly above 2 when a lower limit of 16 is employed for the 1/s values. For the relatively lower 1/s values using a lower limit of 2, quite high fractal dimension values are obtained. The fractal dimension of zeolite 13X approaches 2.45 when 1/s values between 2 and 8 are employed. Although the fractal dimension values obtained in a narrow range of relatively high mesh sizes tend to exceed in considerable amounts, the value estimated in a previous study using experimental adsorption data [11] D 2:13, the results still seem to be meaningful, especially when the relative magnitudes obtained for the zeolites 13X, 5A and silicalite are taken into account. It should also be noted that the size range corresponding to real adsorbates is found at a critical point, where eects originating from the structural geometrical units and the monotonously repeating unit cells may both be signi®cant. In case size ranges in the vicinity of the size of a unit cell are employed in the estimations, a fractal dimension of 2 is very probable to be obtained. Thus, the tendencies of the variation of the fractal dimension values observed in this study when narrow ranges of relatively high mesh sizes are taken into account may change to some degree for the even higher mesh sizes corresponding to the sizes of real adsorbates.
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Fig. 10. The variation of the fractal dimension of 13X with respect to the higher limit of the 1=s values when the lower limit of the 1=s values is kept constant at () 2 and (+) 16.
Zeolites generally have one type of channel or pore. Due to this fact, there exists a general tendency to regard zeolites as adsorbents having smooth surfaces [3]. The results obtained by using the method proposed in this study clearly exhibit that zeolites behave as fractal adsorbents in the vicinity of the range of molecular size pertaining to the most commonly used adsorbates. The geometrical shapes of the framework structures and the relative magnitudes of the gas molecules with respect to the framework units seem to be the most important factors determining the fractal dimension values obtained for the zeolites 13X, 5A and silicalite, by using experimental adsorption data. Although one-to-one correspondence between the range of mesh size used in this study and the range of molecular size valid for adsorption could not be provided, the general tendency of the fractal dimension values to rise above or fall below 2 when narrow ranges of relatively low 1=s values are employed, seems to be sucient for such an explanation. It should be mentioned that the range of molecular size valid for adsorption is still comparable with the secondary building units and thus the range of mesh sizes used in this study. It should also be noted that the methods, such as SAXS and SANS may overlook the signi®cant information presented above, which exhibits the fractal nature of zeolites, since their range of reliable applicability in the evaluation of fractal dimension values is a value nearly comparable with the sizes of the unit cells of most of the zeolites. Since the unit over 10 A, cells are simply replicas of each other, a fractal dimension of 2 may be expected to be obtained for almost all the zeolites in the corresponding range. 4. Conclusions A theoretical method that may be used to evaluate the fractal dimensions of the pore surfaces of crystalline adsorbents is proposed. The geometrical shapes of the framework units are taken into account for the evaluation of the surface fractal dimensions. The fractal dimension value of silicalite varies in signi®cant amounts with respect to the range of mesh size used and falls below 1.70 for 1=s values between 2 and 8. The magnitude of the fractal dimension of zeolite 5A seems to vary less with the range of mesh size used and the fractal dimension value slightly exceeds 2.10 for the 1/s values between 2 and 8. The fractal dimension of zeolite 13X attains quite high values above 2 for the relatively lower 1=s values employed and approaches 2.45 for the 1/s values between 2 and 8. Although the range of mesh size used in this study and the range of molecular size valid for adsorption do not exactly coincide, the general tendency of the fractal dimension values to fall below and rise above 2, most often in signi®cant amounts, when narrow ranges of relatively low 1/s values are utilized seems to be sucient to explain the reason for obtaining fractal dimension values other than 2 employing adsorption data for zeolites. The relative magnitudes of the fractal dimensions of the zeolites 13X, 5A and silicalite are never distorted for the cases investigated in this study. The resemblence of the relative magnitudes of the fractal dimension values obtained by using the method proposed in this study to the ones determined by
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employing experimental adsorption data is striking. The absolute magnitudes, on the other hand, may show some dierences due to the dierent size ranges employed. The claims that zeolites should have fractal dimension values of 2 due to their uniform pore structures seems to hold true only for certain ranges of 1/s values, that is for sizes of yardsticks comparable to the sizes pertaining to unit cells or for very high 1/s values, which are not of any practical signi®cance at the present time and also which do not comply with the ¯at surface assumption for rings having six or less members. The range of molecular size corresponding to the most commonly employed adsorbates is comparable with the sizes of the secondary building units of zeolites, thus obtaining fractal dimension values far from 2, that may be both above and below this value, should not be surprising. Indeed, a challenging task remains for determining the existence of a zeolite having a fractal dimension equal to 2 in the range determined by the molecular sizes of real adsorbates. It should also be mentioned that the methods such as SAXS and SANS may overlook the signi®cant information presented in this study, which exhibits the fractal nature of zeolites as adsorbents and catalysts, a since their range of reliable applicability in the evaluation of the fractal dimension value is over 10 A, value nearly comparable with the sizes of the unit cells of most of the zeolites. References [1] Pfeifer P, Avnir D. Chemistry in noninteger dimensions between two and three. I. Fractal theory of heterogeneous surfaces. J Chem Phys 1983;79:3558±65. [2] Avnir D, Farin D, Pfeifer P. Chemistry in noninteger dimensions between two and three. II. Fractal surfaces of adsorbents. J Chem Phys 1983;79:3566±71. [3] Avnir D, Farin D, Pfeifer P. A discussion of some aspects of surface fractality and of its determination. New J Chem 1992;16: 439±49. [4] Erdem-S enatalar A, Tatlier M. Eects of fractality on accessible surface area values of zeolite adsorbents. Chaos, Solitons & Fractals 2000;16:953±60. [5] Farin D, Avnir D. The reaction dimension in catalysis on dispersed metals. J Am Chem Soc 1988;110:2039±45. [6] Shang CH, Liu BX, Sun JG, Li HD. Observation of two-dimensional fractal growth in solids. Phys Rev B 1991;44:5035±9. [7] Sheintuch M, Brandon S. Deterministic approaches to problems of diusion reaction and adsorption in a fractal porous catalyst. Chem Eng Sci 1989;44:69±79. [8] Iwasaki H, Yoshinobu T. Self-ane growth of copper electrodeposits. Phys Rev B 1993;48:8282±5. [9] Carro P, Marchiano SL, Hernandez Creus A, Gonzalez S, Salvarezza RC, Arvia AJ. Growth of three-dimensional silver fractal electrodeposits under damped free convection. Phys Rev E 1993;48:R2374±2377. [10] Pfeifer P, Wu YJ, Cole MW, Krim J. Multilayer adsorption on a fractally rough surface. Phys Rev Lett 1989;62:1997±2000. [11] Tatlier M, Erdem-S enatalar A. Method to evaluate the fractal dimensions of solid adsorbents. J Phys Chem B 1999;103:4360±5. [12] Drake JM, Levitz P, Klafter J. A comment on the fractal dilemma in porous silica gels. New J Chem 1990;14:77±81. [13] Ehrburger F, Jullien R. Determination of the fractal dimension of aggregates using the intra-aggregate pore-size distribution. In: Unger KK, editors. Characterization of porous solids. Amsterdam: Elsevier, 1988. p. 441±9. [14] Szostak R. Molecular sieves. New York: van Nostrand Reinhold; 1989. [15] Ruthven DM. Principles of adsorption and adsorption processes. New York: Wiley; 1984. [16] Flanigen EM, Bennett JM, Grose RW, Cohen JP, Patton RL, Kirchner RM, Smith JV. Silicalite, a new hydrophobic crystalline silica molecular sieve. Nature 1978;271:512±6. [17] Meier WM, Olson DH, Baerlocher Ch. Atlas of zeolite structure types. Zeolites 1996;17:1±229. [18] Breck DW. Zeolite molecular sieves. New York: Wiley; 1974.
www.elsevier.nl/locate/chaos
Fractal dimension of zeolite surfaces by calculation Melkon Tatlõer, Ays e Erdem-S enatalar * Department of Chemical Engineering, Istanbul Technical University, Maslak, 80626, Istanbul, Turkey Accepted 3 April 2000
Abstract A theoretical method for the estimation of the fractal dimensions of the pore surfaces of zeolites is proposed. The method is an analogy to the commonly employed box-counting method and uses imaginary meshes of various sizes s to trace the pore surfaces determined by the frameworks of crystalline zeolites. The surfaces formed by the geometrical shapes of the secondary building units of zeolites are taken into account for the calculations performed. The characteristics of the framework structures of the zeolites 13X, 5A and silicalite are determined by the help of the solid models of these zeolites and the total numbers of grid boxes intersecting the surfaces are estimated by using equations proposed in this study. As a result, the fractal dimension values of the zeolites 13X, 5A and silicalite are generally observed to vary in signi®cant amounts with the range of mesh size used, especially for the relatively larger mesh sizes that are close to the sizes of real adsorbates. For these relatively larger mesh sizes, the fractal dimension of silicalite falls below 1.60 while the fractal dimension values of zeolite 13X and 5A tend to rise above 2. The fractal dimension values obtained by the proposed method seem to be consistent with those determined by using experimental adsorption data in their relative magnitudes while the absolute magnitudes may dier due to the dierent size ranges employed. The results of this study show that fractal dimension values much dierent from 2 (both higher and lower than 2) may be obtained for crystalline adsorbents, such as zeolites, in ranges of size that are close to those of real adsorbates. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction The observation of scaling behavior in diverse ®elds of science has promoted the studies carried out to relate various phenomena to the concept of fractal geometry. Systems such as adsorbents, aggregates, catalyst supports and electrodes most often exhibit complex behavior, which cannot be easily explained by considering concepts of Euclidean geometry. The determination of the fractal nature of these systems may be of crucial importance for an accurate prediction of the scaling behavior encountered in various processes, such as adsorption [1±4], reaction [5], aggregation [6,7], electrodeposition [8,9], etc. The estimation of the fractal dimension values of adsorbents does not only help to characterize these materials but it also provides valuable information about the adsorption behavior of adsorbates with various sizes. The most commonly employed method to determine the fractal dimension values of adsorbents is based on the fact that the accessible surface area of a solid decreases or increases with increasing dimensions of the adsorbate molecules, for fractal dimension values above and below 2, respectively. The variation of the monolayer capacities for a homologous series of gas molecules used in the process of adsorption with respect to their eective sizes enables the evaluation of the fractal dimension values of porous adsorbents [1,2]. It has been demonstrated that the fractal dimension may also be estimated by employing a modi®ed form of the Frenkel±Halsey±Hill isotherm equation [10] as well as a method taking the Freundlich isotherm as a starting point for the calculations [11]. The former method utilizes a single adsorption isotherm and is valid *
Corresponding author. Tel.: +90-212-285-6896; fax: +90-212-285-2925. E-mail address: [email protected] (A. Erdem-S enatalar).
0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 0 8 5 - 0
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only for multilayer adsorption, while the latter employs a single adsorption isotherm and data representing the characteristics of a reference adsorbent for the same adsorbate. Besides adsorption, various other methods, such as small angle X-ray or neutron scattering (SAXS or SANS) [3,12], electronic energy transfer [3,12] and thermoporosimetry [13], may also be used in the determination of the fractal dimension values of adsorbents. All methods of fractal analysis involve the employment of ``yardsticks'' of various sizes for tracing the surfaces of adsorbents. Depending on the method used, the yardsticks may be adsorbate molecules of dierent eective sizes, electromagnetic waves diracted at dierent angles, energy transfer from a donor molecule to an acceptor molecule at dierent distances and ®lms of variable thickness, etc. It should also be mentioned that the fractal dimension value of any object that does not exhibit self-similar properties may vary with respect to the range of size of the yardsticks employed in the estimation. Thus, it is possible that dierent methods, employing yardsticks of various sizes, may estimate dierent fractal dimension values for a certain adsorbent. Zeolites have microporous crystalline structures and are commonly used in processes involving adsorption, as well as in catalysis and ion-exchange. The fractal dimensions of three of the most commonly employed zeolites, namely 13X, 5A and silicalite have been recently evaluated by using dierent methods based on adsorption. The employment of the Pfeifer±Avnir and point-slope methods resulted in fractal dimensions of about 2.08, 2.13 and 1.68 for the zeolites 5A, 13X and silicalite [11], respectively. The range of size employed in the estimations varied between the sizes of methane and propane for the zeolites 5A and 13X and the sizes of methane and n-butane for silicalite. Due to the uniformity of the pore structures of the zeolites, it is commonly thought that these materials should exhibit smooth surfaces with fractal dimension values of about 2. Thus, as the fractal dimension value of a zeolite gets further from 2, the reason underlying this type of behavior becomes less apparent. In this study, the fractal dimensions of the pore surfaces as determined by the framework structures of the zeolites 13X, 5A and silicalite are evaluated by making an analogy to the commonly used box-counting method. The geometric shapes of the pore surfaces formed by the secondary building units constituting the frameworks of the zeolites 13X, 5A and silicalite are taken as a basis for the calculations. The characteristics of the framework structures of the zeolites 13X, 5A and silicalite are determined by the help of the solid models of these zeolites and equations are proposed for estimating the total numbers of grid boxes intersecting the pore surfaces formed by sections of their frameworks. The variation of the fractal dimension value is investigated with respect to the range of scale utilized in the estimations and the upper limit of the scale used is restricted by the sizes of the secondary building units involved. A discussion of the results obtained in relation to those obtained by various other methods employed to determine the fractal dimensions of adsorbents is also presented. 2. Theoretical The crystalline structures of the zeolites consist of tetrahedra known as primary building units. A T-atom (silicon or aluminum), surrounded by four oxygen atoms, is located at the center of each tetrahedron. A number of these units constitute the secondary building units, which may be of various types, such as, double four-ring (D4R), single six-ring (S6R), single ®ve-ring (S5R), etc. An arrangement of the secondary building units ends up with the formation of the regular crystalline structures of the zeolites with cavities and/or channels of molecular sizes. The openings to the cavities or channels of zeolites are constricted mainly by the oxygens of the tetrahedra forming rings, which may have various free diameter values. The number of oxygen atoms in a ring determines the magnitude of the diameter of the opening to the channel or the cavity. For an ideal planar con®guration, the 6-, 8-, 10- and 12-member ring openings are approximately equal to 2.8, 4.4, 6.0 and respectively [14,15]. The actual opening, however, may slightly deviate from these values. The 7.7 A, cavities or channels formed by the six-member rings are so small that only the very small polar molecules, such as H2 O and NH3 can penetrate. The internal pore surfaces of the zeolites are formed by the linkage of the secondary building units in dierent geometries and the characteristic shapes of these surfaces may be partly dierent than those pertaining to the original secondary building units forming them. The secondary building unit of silicalite is
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referred to as the 5-1 unit shown in Fig. 1, which consists of a regular pentagon with an extension. However, the pentagons in silicalite are folded along one diagonal to create a trapezoid and a triangle adjacent at their base at an angle of about 144°. Thus, the inner surfaces of silicalite [16] are mainly composed of trapezoids and triangles, but some squares and rectangles also exist. Two distinct channel systems exist in silicalite, namely the straight and sinusoidal ones. The framework structure of silicalite may be seen in Fig. 2(a). The secondary building units of the zeolites 13X and 5A [17,18] consist of single or double four- and sixrings and the framework surfaces of these zeolites may be represented by squares and regular hexagons. The framework structures of the zeolites 13X and 5A may be seen in Fig. 2(b) and (c), respectively. The unit cells of zeolites are the smallest units that possess all the characteristics of the crystal structures of these materials. Thus, in order to evaluate the fractal dimension values of zeolite surfaces, it is sucient to consider a single unit cell of a zeolite. The characteristics of the zeolites 13X, 5A and silicalite, as determined by the help of the investigation of the solid framework models are given in Table 1. The number of the dierent geometric shapes in a given structure may be seen in the table. The fractal dimension of a structure can be determined by using various methods. The box-counting method may systematically characterize structures that have scaling properties. This method may commonly be applied to structures of very dierent natures since self-similarity is not a prerequisite for the estimations made. The box-counting fractal dimension can be determined easily by using 2-D meshes for any structure in a plane and may also be adapted for structures in 3-D space. In this method, the structure investigated is covered with a regular mesh with size s and the number of grid boxes (N) that contain some of the structure is determined. The number of boxes counted will depend on the size of the mesh used and the size of the mesh may be gradually varied. As a result, the fractal dimension value may be evaluated by plotting log N vs log (1/s). The slope of the straight line ®tted to the plotted points represents the fractal dimension value of the structure investigated. The box-counting method is adapted in this study for the measurements carried out to evaluate the fractal dimensions of the pore surfaces of the zeolites 13X, 5A and silicalite in 3-D space. The fractal dimension values of the surface units are initially determined in a 2-D plane after which the structure is investigated in 3-D space by taking into account the number of overlaps between the neighboring units. The ratio of the number of overlapping units to the total number of the structural units is an indicator of the extent of the fractality of the pore surfaces, as examined in 3-D space. The 3-D imaginary meshes used to trace the pore surfaces of the zeolites are assumed to be placed on the structural units without an angle of inclination. This approach is used to make an analogy with the adsorptive behaviors of adsorbates tracing the pore surfaces. The 3-D meshes placed on neighboring units may intersect due to the inclinations between the units, but the double counting of the same mesh grid for two neighboring units is dealt with by subtracting the number of mesh boxes corresponding to the overlapping units from the total number of
Fig. 1. The secondary building unit of silicalite.
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Fig. 2. The framework structures of the zeolites (a) silicalite, (b) 13X and (c) 5A. Table 1 The number of the dierent geometric shapes in single unit cells of silicalite and zeolite 13X as well as in a single pseudocell of zeolite 5A Zeolite
Number of units Trapezoid
Triangle
Hexagon
Square
Rectangle
Silicalite 5A 13X
74 ± ±
44 ± ±
± 8 16
4 12 72
10 ± ±
mesh boxes counted. This subtraction also accounts for the fractality of the structure in 3-D space, as mentioned before. The geometric shapes of the framework surfaces formed by the secondary building units of the zeolites 13X, 5A and silicalite are taken as a basis for the measurements made. The ®rst step of the calculations performed involves the application of the box-counting method to the surface structural units, which generally have well-known geometrical shapes, such as squares, pentagons, hexagons, etc. In this study, it is assumed that the relatively small pore openings formed by six (or less than six)-member rings mentioned above behave as full surfaces. It is well known that the sizes of these pore openings are smaller than the molecular sizes of most of the real adsorbates. Once the above assumption is made, in order to better
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demonstrate the tendency of the variation of the fractal dimensions of adsorbent surfaces with the size range, the mesh sizes used in this study are made to vary in a relatively broad range. The Si±O (or Al±O) bond lengths and the angles of the planar geometric structures formed by the arrangement of the secondary building units may generally be assumed to be equal. The Si±O (or Al±O) bond length is taken to be equal to for all the cases investigated. The number of grid boxes that intersect the geometric framework 1.6 A structures, formed by the arrangement of the secondary building units, when various scaling factors are used may be easily determined in a plane. The evaluation of the fractal dimensions of the zeolites 13X, 5A and silicalite for various scales is detailed below. In order to estimate the fractal dimension value, ®rst of all, the number of grid boxes intersecting the structural units of the surface is determined. Accordingly, regular trapezoids, triangles, hexagons, rectangles and squares should be taken into consideration, as mentioned before. The dimensions of a regular trapezoid and a triangle forming the folded pentagons of silicalite are shown in Fig. 3. In the same ®gure, the relative dimensions pertaining to the same structural units are also indicated in parantheses The relative areas in terms of the as a fraction of the largest base length of the trapezoid (a 2b 5:12 A). number of meshes N s of the trapezoid and the triangle vary with respect to the mesh size used (s) and the number of grid boxes counted is inversely proportional to this size. Some examples depicting the relationship between the mesh size and the relative areas of the trapezoids and the triangles pertaining to the folded pentagons of silicalite are given in Table 2. It may be observed that the widths of both the trapezoid a 2b and the triangle 2e are equal to the 1=s value in each case such that as the mesh size is decreased, the number of grid boxes intersecting a structure will increase. It should also be noted that the relative dimension values given in Fig. 3 correspond to the employment of a mesh size, s 1. The number of grid boxes (N) counted for rectangles and squares is simply equal to the product of the two sides of these quadrilaterals, while for less regular structures, such as triangles, dierent relationships should be sought
Fig. 3. The dimensions of a regular trapezoid and a triangle forming the folded pentagons of silicalite. The relative dimensions pertaining to the same structural units are indicated in parantheses as a fraction of the largest base length of the trapezoid (a 2b 5:12 A). Fig. 4. Determination of the number of grid boxes intersecting a right triangle, for which the relative dimensions of b and c are equal to 8 and 6, respectively.
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Table 2 The relationship between the mesh size and the relative dimensions of a regular pentagon 1=s
a
b
c
d
e
1 4 16
0.62 2.5 10
0.19 0.75 3
0.59 2.37 9.5
0.36 1.46 5.87
0.50 2.0 8.0
for. The N value of a trapezoid is determined as the sum of a rectangle and two right triangles by the following equation: Ntrapezoid a c c b c b:
1
The ®rst term corresponds to the number of grid boxes counted for the rectangle, whereas the second one represents the sum of those counted for the two similar triangles. The latter term is obtained by considering a simple approximated relationship existing between the number of boxes counted and the number of meshes in a single row and column of a mesh structure (or the width b and height c of the unit). An example for the calculation of the number of grid boxes intersecting a right triangle, for which the relative dimensions of b and c are equal to 8 and 6 respectively, is illustrated in Fig. 4. Both of the right triangles seen in the ®gure intersect 31 grid boxes, in accordance with the proposed equation c b c b. The number of grid boxes obtained for the triangle constituting the second part of the folded pentagon along with the trapezoid is determined in a similar way by c and b replacing d and e, respectively Ntriangle d e d e:
2
The dimensions of the squares and rectangles found in the structure of silicalite are shown in Fig. 5(a) and (b), respectively. The relative dimensions in terms of the number of meshes are also indicated in parantheses The number of boxes counted for these as a fraction of the largest base length of a trapezoid (5.12 A). quadrilateral units found in the structure of silicalite may be represented by Nrectangle a 2e;
3
Nsquare a a;
4
where a and 2e denote the width and length of the units, respectively. In order to estimate the fractal dimension values of the pore surfaces of the zeolites 5A and 13X, the number of grid boxes intersecting regular and planar hexagons and squares should be determined. The
Fig. 5. The dimensions of the (a) squares and (b) rectangles found in the structure of silicalite. The relative dimensions in terms of the number of meshes are indicated in parantheses as a fraction of the largest base length of a trapezoid.
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Fig. 6. The dimensions of a regular hexagon. The relative dimensions in terms of the number of meshes are indicated in parantheses as a fraction of the length of the diagonal of the hexagon (f 2g 6:4 A). Fig. 7. Determination of the number of meshes counted for one of each two neighboring units sharing an overlap (m2 ).
dimensions of a regular hexagon are shown in Fig. 6 and the relative dimension values in terms of the number of meshes are indicated in parantheses for the case where the mesh size is equal to 1. The relative dimensions are given as a fraction of the length of the diagonal of the hexagon denoted by f 2g which is The number of grid boxes (N) counted for a hexagon, which is constituted by a rectangle equal to 6.4 A. and four right triangles may be obtained by using the following equation: Nhexagon f 2h 2 g h g h:
5
The ®rst term corresponds to the number of grid boxes counted for the rectangle, whereas the second one represents the sum of those counted for the four similar triangles. The squares of the zeolites 13X and 5A may be represented by Fig. 5(a), and the N values may be estimated by employing Eq. (4), provided that a and the relative dimension are replaced by f and 0.5, respectively. As mentioned before, in order to eliminate any inconsistency that may originate from the double counting of the same mesh grid for two neighboring framework units, such as a triangle and a trapezoid for silicalite, as well as for determining the extent of the fractality of the pore surfaces in 3-D space, the number of extra boxes counted, corresponding to the overlaps shared by the neighboring units are subtracted from the total number of boxes determined. For angles of 90 6 h < 180° between two neighboring units, the amount subtracted (Noverlap ) may be taken to be equal to half of the boxes counted for the neighboring rows of two distinct units, while no overlapping is expected to occur between units with h P 180°. The number of overlapping units increase (since units other than the neighboring units are also involved) as h falls below 90°. For all the cases investigated in this study, the angles between the neighboring units are found to be in the range 90 < h 6 180°. Table 3 represents the total number of overlapping units (k1 and/or k2 ) in the unit cells of zeolite 13X and silicalite as well as in a single pseudocell of zeolite 5A. The overlaps in silicalite are respectively, whereas those in 13X and 5A have an of two dierent actual lengths, that is 3.2 and 5.12 A, The number of overlaps that are subtracted from the total number of boxes counted actual length of 3.2 A. are calculated by using the following equation: Noverlap k1 m1 k2 m2 ;
6
m1 and m2 represent the number of meshes counted for one of each two neighboring units sharing an overlap and correspond to relative dimensions of 0.62 and 1, respectively. The values of m1 and m2 are directly proportional to the 1=s values. k1 and k2 denote the number of overlaps that have relative dimensions of 0.62 and 1, respectively. As an example, the two similar rectangles seen in Fig. 7 lead to obtaining an m2 value of 5 1=s 5.
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Table 3 The total number of overlapping units in the unit cells of zeolite 13X and silicalite as well as in a single pseudocell of zeolite 5A Type of zeolite
Number of overlapping units k1
k2
Silicalite 5A 13X
102 24 96
63 ± ±
The characteristics of the framework structures of the zeolites 13X, 5A and silicalite are determined by the help of the solid models of these zeolites and the total numbers of grid boxes intersecting the surfaces are estimated by using the above equations. The fractal dimension values are estimated by plotting log Ntotal vs log 1/s values. The slope of the plot gives the fractal dimension. 3. Results and discussion The fractal dimensions of the pore surfaces of the zeolites 13X, 5A and silicalite are determined by using the method proposed in this study. The fractal dimension values are evaluated by varying the 1=s values in a range. The upper limit of the 1=s values employed is varied (generally between 8 and 256) while the lower limit is kept constant at two distinct values, namely 2 and 16. The 1=s values given in the abscissa of the ®gures represent the values corresponding to the upper limit of the range investigated. In order to check for the reliability of the estimations performed in this study, ®rst of all, the fractal dimensions of two geometric shapes, namely a regular pentagon and a hexagon are determined by using the proposed method and a comparison is made with the results obtained by the commonly employed box-counting method. For the latter method, 2-D square meshes may be easily prepared and applied to the investigated structure. The fractal dimension values of a pentagon and a hexagon as determined by using the proposed and the boxcounting methods and the same range of mesh size for both cases may be seen in Table 4. It may be observed that the variation of the fractal dimension with the range used exhibits a similar form and the values obtained are quite close for both methods. It should also be noted that the fractal dimension of a regular pentagon is smaller than that of a regular hexagon and for both cases, the fractal dimension values vary as the range of the mesh size employed changes. The fractal dimension of silicalite is estimated by using the proposed method and two distinct lower limits for the 1=s values. The imaginary 3-D meshes used are assumed to be placed without an angle of inclination on each of the structural units of silicalite, as mentioned before. The results obtained are depicted in Fig. 8. It may be seen that relatively higher fractal dimension values very close to 2 are obtained for the case, where the lower limit is set at 16 while the fractal dimension varies in considerable amounts when the lower limit is taken to be equal to 2. In case 1/s values between 2 and 16 are employed, the fractal dimension of silicalite falls below 1.8, which may be considered to be a quite low value for a zeolite. This value decreases further and approaches 1.6 as 1/s values between 2 and 8 are utilized. Quite high correlation coecient values r > 0:99 are obtained regardless of the range of 1=s values employed in this study. An interesting point is the proximity of the fractal dimension value estimated for silicalite by the proposed theoretical method to that obtained by using adsorption data [11] (D 1:68) pertaining to the range determined by the sizes of methane±butane. Although the range of size employed in that study most probably Table 4 The fractal dimension values of a pentagon and a hexagon as determined by using the proposed and the box-counting methods and the same range of mesh size for both cases Range of 1=s values
Dpentagon proposed method
Dpentagon box-counting method
Dhexagon proposed method
Dhexagon box-counting method
16±32 16±48 16±64
1.800 1.815 1.842
1.820 1.835 1.860
1.855 1.877 1.901
1.849 1.888 1.903
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Fig. 8. The variation of the fractal dimension of silicalite with respect to the higher limit of the 1=s values when the lower limit of the 1=s values is kept constant at () 2 and (+) 16. Fig. 9. The variation of the fractal dimension of 5A with respect to the higher limit of the 1=s values when the lower limit of the 1=s values is kept constant at () 2 and (+) 16.
corresponds to 1/s values of about 0.9±1.8 when eective values, considering the actual surface that can be employed in adsorption, are assumed for the cylindrical molecules, it should be noted that the sizes of the gas molecules are still comparable with those of the secondary building units. Thus, the geometrical shapes of the framework units may be of considerable signi®cance in the determination of the fractal dimension values of zeolites. The variation of the fractal dimension of zeolite 5A with respect to the 1/s values is given in Fig. 9. The calculations are performed in a manner similar to the estimation of the fractal dimension of silicalite. It may be observed from Fig. 9 that almost constant fractal dimension values of about 2 are obtained for the case where the lower limit is set at 16 while the fractal dimension varies in moderate amounts when the lower limit is taken to be equal to 2. In case 1/s values between 2 and 8 are employed, the fractal dimension of zeolite 5A slightly exceeds 2.1. A comparison of the fractal dimension values obtained in this study with that determined by using experimental adsorption data [11] (2.08) reveals that quite consistent values for zeolite 5A are obtained. However, it should also be reminded that the range of molecular size pertaining to the adsorption data used in the previous study [11] (methane±propane) corresponded to 1/s values between 1.2 and 2, a range lower than the ones investigated in this study. In any case, it is apparent that employing a narrow range of relatively high mesh sizes may lead to fractal dimension values quite dierent from 2. The fractal dimension of zeolite 13X is estimated by using an approach similar to those employed for zeolite 5A and silicalite. Two distinct lower limits for the range of mesh size are employed for this case, too. The variation of the fractal dimension with respect to the 1/s values are given in Fig. 10. It may be observed that the fractal dimension value remains slightly above 2 when a lower limit of 16 is employed for the 1/s values. For the relatively lower 1/s values using a lower limit of 2, quite high fractal dimension values are obtained. The fractal dimension of zeolite 13X approaches 2.45 when 1/s values between 2 and 8 are employed. Although the fractal dimension values obtained in a narrow range of relatively high mesh sizes tend to exceed in considerable amounts, the value estimated in a previous study using experimental adsorption data [11] D 2:13, the results still seem to be meaningful, especially when the relative magnitudes obtained for the zeolites 13X, 5A and silicalite are taken into account. It should also be noted that the size range corresponding to real adsorbates is found at a critical point, where eects originating from the structural geometrical units and the monotonously repeating unit cells may both be signi®cant. In case size ranges in the vicinity of the size of a unit cell are employed in the estimations, a fractal dimension of 2 is very probable to be obtained. Thus, the tendencies of the variation of the fractal dimension values observed in this study when narrow ranges of relatively high mesh sizes are taken into account may change to some degree for the even higher mesh sizes corresponding to the sizes of real adsorbates.
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Fig. 10. The variation of the fractal dimension of 13X with respect to the higher limit of the 1=s values when the lower limit of the 1=s values is kept constant at () 2 and (+) 16.
Zeolites generally have one type of channel or pore. Due to this fact, there exists a general tendency to regard zeolites as adsorbents having smooth surfaces [3]. The results obtained by using the method proposed in this study clearly exhibit that zeolites behave as fractal adsorbents in the vicinity of the range of molecular size pertaining to the most commonly used adsorbates. The geometrical shapes of the framework structures and the relative magnitudes of the gas molecules with respect to the framework units seem to be the most important factors determining the fractal dimension values obtained for the zeolites 13X, 5A and silicalite, by using experimental adsorption data. Although one-to-one correspondence between the range of mesh size used in this study and the range of molecular size valid for adsorption could not be provided, the general tendency of the fractal dimension values to rise above or fall below 2 when narrow ranges of relatively low 1=s values are employed, seems to be sucient for such an explanation. It should be mentioned that the range of molecular size valid for adsorption is still comparable with the secondary building units and thus the range of mesh sizes used in this study. It should also be noted that the methods, such as SAXS and SANS may overlook the signi®cant information presented above, which exhibits the fractal nature of zeolites, since their range of reliable applicability in the evaluation of fractal dimension values is a value nearly comparable with the sizes of the unit cells of most of the zeolites. Since the unit over 10 A, cells are simply replicas of each other, a fractal dimension of 2 may be expected to be obtained for almost all the zeolites in the corresponding range. 4. Conclusions A theoretical method that may be used to evaluate the fractal dimensions of the pore surfaces of crystalline adsorbents is proposed. The geometrical shapes of the framework units are taken into account for the evaluation of the surface fractal dimensions. The fractal dimension value of silicalite varies in signi®cant amounts with respect to the range of mesh size used and falls below 1.70 for 1=s values between 2 and 8. The magnitude of the fractal dimension of zeolite 5A seems to vary less with the range of mesh size used and the fractal dimension value slightly exceeds 2.10 for the 1/s values between 2 and 8. The fractal dimension of zeolite 13X attains quite high values above 2 for the relatively lower 1=s values employed and approaches 2.45 for the 1/s values between 2 and 8. Although the range of mesh size used in this study and the range of molecular size valid for adsorption do not exactly coincide, the general tendency of the fractal dimension values to fall below and rise above 2, most often in signi®cant amounts, when narrow ranges of relatively low 1/s values are utilized seems to be sucient to explain the reason for obtaining fractal dimension values other than 2 employing adsorption data for zeolites. The relative magnitudes of the fractal dimensions of the zeolites 13X, 5A and silicalite are never distorted for the cases investigated in this study. The resemblence of the relative magnitudes of the fractal dimension values obtained by using the method proposed in this study to the ones determined by
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employing experimental adsorption data is striking. The absolute magnitudes, on the other hand, may show some dierences due to the dierent size ranges employed. The claims that zeolites should have fractal dimension values of 2 due to their uniform pore structures seems to hold true only for certain ranges of 1/s values, that is for sizes of yardsticks comparable to the sizes pertaining to unit cells or for very high 1/s values, which are not of any practical signi®cance at the present time and also which do not comply with the ¯at surface assumption for rings having six or less members. The range of molecular size corresponding to the most commonly employed adsorbates is comparable with the sizes of the secondary building units of zeolites, thus obtaining fractal dimension values far from 2, that may be both above and below this value, should not be surprising. Indeed, a challenging task remains for determining the existence of a zeolite having a fractal dimension equal to 2 in the range determined by the molecular sizes of real adsorbates. It should also be mentioned that the methods such as SAXS and SANS may overlook the signi®cant information presented in this study, which exhibits the fractal nature of zeolites as adsorbents and catalysts, a since their range of reliable applicability in the evaluation of the fractal dimension value is over 10 A, value nearly comparable with the sizes of the unit cells of most of the zeolites. References [1] Pfeifer P, Avnir D. Chemistry in noninteger dimensions between two and three. I. Fractal theory of heterogeneous surfaces. J Chem Phys 1983;79:3558±65. [2] Avnir D, Farin D, Pfeifer P. Chemistry in noninteger dimensions between two and three. II. Fractal surfaces of adsorbents. J Chem Phys 1983;79:3566±71. [3] Avnir D, Farin D, Pfeifer P. A discussion of some aspects of surface fractality and of its determination. New J Chem 1992;16: 439±49. [4] Erdem-S enatalar A, Tatlier M. Eects of fractality on accessible surface area values of zeolite adsorbents. Chaos, Solitons & Fractals 2000;16:953±60. [5] Farin D, Avnir D. The reaction dimension in catalysis on dispersed metals. J Am Chem Soc 1988;110:2039±45. [6] Shang CH, Liu BX, Sun JG, Li HD. Observation of two-dimensional fractal growth in solids. Phys Rev B 1991;44:5035±9. [7] Sheintuch M, Brandon S. Deterministic approaches to problems of diusion reaction and adsorption in a fractal porous catalyst. Chem Eng Sci 1989;44:69±79. [8] Iwasaki H, Yoshinobu T. Self-ane growth of copper electrodeposits. Phys Rev B 1993;48:8282±5. [9] Carro P, Marchiano SL, Hernandez Creus A, Gonzalez S, Salvarezza RC, Arvia AJ. Growth of three-dimensional silver fractal electrodeposits under damped free convection. Phys Rev E 1993;48:R2374±2377. [10] Pfeifer P, Wu YJ, Cole MW, Krim J. Multilayer adsorption on a fractally rough surface. Phys Rev Lett 1989;62:1997±2000. [11] Tatlier M, Erdem-S enatalar A. Method to evaluate the fractal dimensions of solid adsorbents. J Phys Chem B 1999;103:4360±5. [12] Drake JM, Levitz P, Klafter J. A comment on the fractal dilemma in porous silica gels. New J Chem 1990;14:77±81. [13] Ehrburger F, Jullien R. Determination of the fractal dimension of aggregates using the intra-aggregate pore-size distribution. In: Unger KK, editors. Characterization of porous solids. Amsterdam: Elsevier, 1988. p. 441±9. [14] Szostak R. Molecular sieves. New York: van Nostrand Reinhold; 1989. [15] Ruthven DM. Principles of adsorption and adsorption processes. New York: Wiley; 1984. [16] Flanigen EM, Bennett JM, Grose RW, Cohen JP, Patton RL, Kirchner RM, Smith JV. Silicalite, a new hydrophobic crystalline silica molecular sieve. Nature 1978;271:512±6. [17] Meier WM, Olson DH, Baerlocher Ch. Atlas of zeolite structure types. Zeolites 1996;17:1±229. [18] Breck DW. Zeolite molecular sieves. New York: Wiley; 1974.