340
Langmuir 1987, 3, 340-345
which are presented in Table V, where the jump distance has been calculated assuming a molecule is located in the middle of a channel segment or intersection. NMR self-diffusion studies29of propane in silicalite-1 found the self-diffusivityto decrease with coverage through a decrease in jump rate as opposed to a decreased jump distance for 10 m/uc the jump distance was found to be -1.17 nm and activated jumps between channel intersections was suggested. For >10 m/uc the jump distance decreased to 0.80 nm at 12 m/uc and 0.73 nm at 13.2 m/uc (saturation loading). This behavior was concluded to agree with the Jacobs et al. model, and a random walk model applied to a simulation of molecular transport in a ZSM-5 grid.30 This latter study30 concluded that intracrystalline mobility decreased with increased loading because of mutual interactions. The jump length was considered to remain constant at -1 nm. The calculated jump lengths presented in Table V indicate, for a random distribution of molecules over all the adsorption sites a, b, and c in the pore network, that the average jump length (for a triple sequence of a-c-a, etc.) would be 1.12 f 0.08 nm. If the sorption model of Jacobs et al. is invoked and an equal probability of activated jumps along sinusoidal or linear channel segments is assumed, then the average jump length would be 1.15 f 0.07 nm. For the model presented here for loadings of 8 m/uc, since each segment would contain a sorbate molecule, jump sequences of type a-c, b-c, etc. would increase in probability. For the model presented here (i.e., for C2-C4 n-alkanes) the molecules sorbed in excess of 8 m/uc are located in intersections. Therefore, an increased number of sorbate cooperative movements would be required for a-c-a, etc., type jump sequences which would result in a decreased jump rate. At a loading of 10 m/uc two intersections are occupied according to the n-alkane/ZSM-5/silicalite-l model. Therefore, the degree of sorbate interdependence for diffusional jumps would increase dramatically (i.e., the jump rate would decrease). The jump distance also would have to decrease when all intersections are occupied. Jumps of type a-c, c-a would become more predominant (it is important to realize that intersections can hold >2 molecules). In summary the following contributions toward a better understanding of zeolitic (ZSM-5/silicalite-l) sorptiondiffusion behavior have been made in this study: (i) The sorption and molecular packing of n-alkanes in ZSM-5/ silicalite-1 and the coverage dependence of Qst have been outlined. (ii) The n-alkane adsorption model proposed seems to be consistent with the NMR diffusional behavior of these systems.29 Acknowledgment. R.E.R. thanks SERC for a CASE studentship under the sponsorship of British Petroleum, Sunbury-on-Thames, England. Registry No. Ethane, 74-84-0;propane, 74-98-6;n-butane, 106-97-8;n-hexane, 110-54-3;p-xylene, 106-42-3.
Fractal Description of the Surface Structure of Coke Particlest Siauw H. Ng and Craig Fairbridge* CANMET, Energy Research Laboratories, Energy, Mines and Resources, Ottawa, Ontario, K I A OGl Canada
Brian H. Kaye Physics Department, Laurentian University, Sudbury, Ontario, P3E 2C6 Canada Received October 24, 1986. In Final Form: January 22,1987 Several particle size ranges of an oil sand coke were studied by nitrogen and carbon dioxide adsorption. Apparent surface area, A , increased with decreased particle radius, R , according to the proportionality A 0: RS3, where D is the surface fractal dimension. The fractal dimension is an intrinsic, quantitative measurement of surface irregularity as measured by gas adsorption. The same dimension is derived from adsorption data by using either capillary-condensation or micropore volume filling equations. The value for Syncrude coke was determined to be 2.48, and this value was compared with the fractal dimension calculated from the rate of oxygen adsorption and mercury intrusion data. The application of fractal dimension as measured by gas adsorption may provide a convenient and size-independentparameter for the comparison of surface physical properties. Introduction
A description of a porous solid must consider the bulk solid, the pore volume, and the interface or boundary between solid and pore space or solid and adsorbate. The
* To whom correspondence should be addressed. Presented at the “Kiselev Memorial Symposium”, 60th Colloid and Surface Science Symposium, Atlanta, GA, June 15-18,1986; K. S. W. Sing and R. A. Pierotti, Chairmen.
view that a porous solid has a finite volume and mass and that the specific surface area is absolute and determinable may be incorrect. Mandelbrot has outlined mathematical strategies for describing rugged or indeterminate boundaries.l It was pointed out that calculus mathematics of continuous curves was inadequate for describing rugged (1) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: San Francisco, 1982.
This article not subject to U.S. Copyright. Published 1987 by the American Chemical Society
Surface Structure of Coke Particles boundaries and, further, mathematical curves of infinite length existed for which the boundaries had no differential function. Kaye has extended the discussion into the area of fine particle science where the operative properties of an interface are determined by a significant interplay of surface and mass forces.2 The surface structure of a porous solid is the description of a boundary-gas/solid, liquid/solid, or solid/solid. Recently, it has been demonstrated that the characterization of irregular surfaces and the analysis of molecular interactions with such surfaces may be described by fractal ge~metry.~ The classical concept of dimensionality has been extended in the description of boundaries to include a fractional index which is added to the classical integer dimension to derive the fractal dimension as a measure of the space-filling ability of the rugged structure. The original mathematical theory of fractal dimensions has therefore been extended to physical properties such as specific surface, tortuosity, and poro~ity.~ The fractal dimension of a surface, D, varies between 2 and 3 and is a relative measure of how well an irregular surface fills three-dimensional space. Just as the classical concept of dimension varies with operational perspective, so the surface fractal dimension may change with resolution.2 In nature, the forces which operate to form a boundary at a certain resolution are related to the dominant forces operational at that magnitude of scale.5 At the molecular level of scale, most surfaces have been described by fractal geometry.6 A recent review summarizes the fractal dimensions of several surfaces and briefly describes eight methods that have been used in determining fractal dimen~ion.~ As Stanlef has indicated, there are several uses of the word "fractal" in the literature and this has produced some confusion. This discussion is concerned with an interface or surface which has indeterminate area as measured by adsorption techniques in the same manner as the examination of a wiggly line of indeterminate length as measured by various yardsticks. A surface may appear two dimensional in Euclidean geometry or under very low magnification and an irregular surface can be envisioned as one which approaches three dimensions under high magnifications or at a range of scale of interest to the study of molecular interactions. At the same time, we must consider the measurement of specific pore space or volume and the relationship between this parameter and specific surface area. In the domain of adsorption chemistry there are several techniques available for estimating some degree of surface irregularity. Two important methods are the surface area/molecular probe size and surface area/particle size relationships. It has been demonstrated that a material has a fractal surface if the monolayer capacity, n,, is related to adsorbate cross-sectional area a, by
n,
a,-D/2
where the range of scale is typically of the order 0.1-1.0 pm2 for adsorbate size. Alternatively, the monolayer capacity for a single adsorbate may be related to adsorbent particle radius, R, by (2) Kaye, B.H.Part. Charact. 1985, 2, 91. (3) Avnir, D.; Farin, D.; Pfeifer, P. Nature (London) 1984, 308, 261. (4) Pape, H.;Riepe, L.; Schopper, J. R. Part. Charact. 1984, 1, 66. (5) Kaye, B.H.; Leblanc, J. E.; Abbot, P. Part. Charact. 1985,2, 56. (6) Avnir, D.;Farin, D.; Pfeifer, P. J. Colloid Interface Sci. 1985,103,
.*o
lib.
(7) Avnir, D. In Better Ceramic Through Chemistry; Brinker, L. J., Clark, D. E., Ulrich, D. R., Eds.; Material Res. SOC.Symp. Ser. 73, 1986. (8) Stanley, E. In On Growth and Form; Stanley, E., Ostrovsky, N., Eds.; Martinus Nijhoff Dordrecht, 1986.
Langmuir, Vol. 3, No. 3, 1987 341
n,
a
RD-3
The assumption of course is that the number of particles per gram of material is proportional to R-3 and that the powders are mechanically stable. The possible range of fractality explored by this procedure may extend to 2 or 3 orders of magnitude. For many surface reactions the rate of reaction has been related to the surface subfractal dimensions, D, the fractal dimension experienced by the species reacting with or at the ~ u r f a c e .In ~ this case the rate is related to particle radius by rate a reactive surface area Rb-3 (3) Porous solids have also been extensively characterized with data generated by mercury porosimetry. The use of this technique in describing the fractal geometry of surfaces has not been quantified. If surface heterogeneity arises from pores, then the following equation was derived9 -dV/dr r2-0 (4) where V is the total volume of pores of diameter >2r. For any pore of diameter 2r, there are 2O similar pores of diameter r or less. Mercury intrusion data are normally presented as an intrusion volume/pressure curve and transformed into a pore size distribution plot. Friesen and Mikulal" have recently interpreted mercury intrusion data for coal and coal char by the equation dV,/dP 0: p4 (5) where V, is the volume of mercury forced into the pores by external pressure, P. This equation may be derived directly from eq 4 by substitution for r by P from the Washburn equation. Kaye has begun to consider an analogy between mercury porosimetry data and the use of fractal analysis to describe boundaries." The application of increasing pressure results in the intrusion of liquid mercury into small pores. When sufficiently high pressure to force mercury into the smallest pores of a porous solid cannot be applied, one encounters the same paradox as in other fractal analysis methods-the specific volume of pores appears to increase indefinitely with increased pressure. It is interesting that in the area of coal science, the linear increase in intrusion volume with pressure for mercury porosimetry has been attributed entirely to coal compressibility.12J3 Image analysis techniques have been reported for the determination of flactal dimensions of particle profiles.14J5 This of course becomes the fractal dimension of a boundary line and a description of two-dimensional textures. Texture analysis techniques are currently being developed by analysis of grey levels to the extent that a surface fractal dimension may be calculated by computerized image analysis.16 At CANMET, the reactivities of Canadian coals have been examined and compared with the chemical and physical properties of the original coal. Similarly, the gasification reactivity of oil sand coke particles has been (9) Pfeifer, P.; Avnir, D.; Farin, D. J. Stat. Phys. 1984, 36, 699. (10) Friesen, W.;Mikula, R. 3. CANMET Div. Rep. ERP/CRL 86-128. Available from Energy, Mines and Resources Canada, CANMET Technology Information Division, Technical Enquiries, Ottawa, Ontario, K1A OG1 Canada. (11) Kaye, B. H.Presented at Nuremberg Conference on the Characterization of Fineparticles, Nuremberg, May 9-11, 1984. Kaye, B. H. Ann. Isr. Phys. SOC.1986, 8. (12) Zwietering, P.; van Krevelen, D. Fuel 1954, 33, 331. (13) Spitzer, Z. Powder Technol. 1981, 29, 177. (14) Farin, D.;Peleg, S.; Yavin, D.; Avnir, D. Langmuir 1985, I , 399. (15) Kaye, B. H. Part. Charact. 1984, 1, 14. (16) Peleg, S.;Naor, J.; Hartley, R.; Avnir, D. IEEE Trans. Pattern Anal. Mach. Intelligence 1984, 6, 518.
Ng et al.
342 Langmuir, Vol. 3, No. 3, 1987
-
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Figure 2. Effect of particle size on the volume of C02 (gas) adsorbed at 40 kPa (M) and the D-R micropore volume (A).
10
~
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,
, 0.03
,
I
0.04
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RELATIVE PRESSURE
Figure 3. C02 adsorption isotherm expressed by the D-A equation with m = 1 for coke particles of diameter 69 (a) and 113 km (0).
(17) Furimsky, E. Fuel Proc. Technol. 1985,11, 167. (18) Furimsky, D.; Palmer, A. Appl. Catal. 1986,23, 355. (19) Fairbridge, C.; Ng, S. H.; Palmer, A. Fuel 1986, 65, 1759. (20) Fairbridge, C.; Palmer, A.; Ng, S. H.; Furimsky, E. Fuel, in press. (21) Kaganer, M. G . Zhur. Fiz. Khim. 1959, 33, 2202. (22) Dubinin, M. M.; Radushkevich, L. V. Roc. Acad. Sci. USSR 1947, 55,331. Dubinin, M. M. RUSS.J.Phys. Chem. (Engl. Transl.) 1965,39, 697.
R-l. In fact, the latter case is derived from eq 2 when D = 2.00 for a surface containing relatively little irregularity or for the special case where fractal and Euclidean geometries overlap. The same fractal dimension was derived for the adsorption of N2 and COz by using either BET or DRK theories for the C02 data. Similarly, the same fractal dimension can be derived from the C02adsorption data by plotting the apparent micropore volume as determined from eq 7 or the amount of gas adsorbed at 40 kPa for each particle size range as in Figure 2. This is one of the corollaries of fractal analysis in that it does not matter whether one plots an absolute value for surface area or micropore volume; all that matters is that an equivalent measurement be obtained for each sample. This of course obviates any discussion of the nature of pore-filling models since the same fractal dimension would be calculated regardless of the theory chosen as long as the mechanism of adsorption is invariant of particle size. In the analogous fractal analysis then, one is concerned with the increased amount of gas adsorbed with decreasing particle size. Examination of a typical C02adsorption isotherm indicates a similar effect in that the volume of gas adsorbed increases with increased pressure. The isotherm can therefore be shown in a log-log plot of volume adsorbed against relative pressure as in Figure 3. In this figure, the volume adsorbed (Vag)corresponds to the volume of gas adsorbed and the relative pressure (pP/Pog)corresponds
Langmuir, Vol. 3, No. 3, 1987 343
Surface Structure of Coke Particles Table 1. Particle Size Effects av dim, cLm
12988 446 274 225 189 163 137 113 99 84 69
intrusion vol, mL/g
%$g
ZgL
1.690 1.646 1.630 1.666 1.628 1.613 1.680 1.580 1.667 1.602 1.595
1.392 1.481 1.451 1.387 1.352 1.365 1.351 1.255 1.332 1.275 1.405
413MPa 0.270 0.240 0.224 0.413 0.398 0.451 0.446 0.471 0.453 0.473 0.484
413 kPa 0.183 0.173 0.178 0.343 0.351 0.388 0.403 0.401 0.390 0.410 0.425
to that of the adsorbed gas. This again can be interpreted from a fractal viewpoint where a dimension is given by 2 plus the absolute value of the slope of the log-log plot. For Syncrude coke particles of particle size range where eq 2 was operable, the fractal dimension calculated from the slope of Figure 3 was 2.39. The amount of C 0 2 adsorbed has been calculated from the difference in pressure before and after adsorption and as such simply represents the number of moles of gas which no longer contribute to the pressure of the gas (Pg). From the context of a fractal discussion, this experiment involves increasing the pressure of C02 over a porous sample, making a pressure measurement after l h, and repeating. There is no reason for expecting adsorption equilibria to be attained or for having the fractal dimension measured by this procedure be equivalent to the surface fractal dimension determined by eq 2. However, the data in Figure 3 correspond to an equation proposed by Dubinin and A ~ h t a k h o vwhich ~ ~ can be written as log W = log Wo - D* logm(Po/P) (8) where m is an integer and m = 1 for the C 0 2 data. The D* value in eq 6-8 is related to a structural parameter. The introduction of parameter m was an attempt to introduce a more general equation for the degree of micropore filling by considering other than a Gaussian pore size distribution. Rand24considered the D-A equation with values of m between 1 and 2 and the equation has been further modified.25 A recent paper by Wojsz and Rozwadowski%examines the distribution function of adsorption energy for heterogeneous microporous adsorbents. Values for the characteristic free energy of adsorption may be calculated from eq 7 or 8. These calculated parameters scale with particle size in the same manner as apparent surface area. Kadlec2' has recently described an equation for the pressure P'of the adsorbed phase for the treatment of micropore volume filling (MVF) and indicates three classifications of MVF. Thus far, the application of fractal geometry of gas adsorption has been from a phenomenological point of view to describe surface ruggedness at molecular levels. The description of adsorption on microporous solids and the adsorption potential in micropores usually only considers pores of equivalent cylindrical diameter 60-nm diameter for a mercuryfcarbon interface. In this event, it would not be possible to determine the surface fractal dimension using mercury porosimetry at pressures greater than 30 MPa since the pore volumes are no longer filled in order of decreasing diameter at increased pressure.30 Nonetheless, several interesting results were obtained from the porosimetry data. The interparticle void volume may be subtracted from the intrusion volume at 413 MPa to yield a specific pore volume, V,. This pore volume, for example, also scales with decreased particle size to derive a fractal dimension of 2.45, Table 11. If the intrusion volume at 413 kPa is subtracted from the intrusion volume at every pressure step, plots of low (corrected intrusion volume, V,) against log (pressure) were linear (Figure 6) with slopes from 0.6 to 0.7 for each particle size range. Volume V,,, should correspond to intraparticle pore volume. In this case, a volume fractal (30) Kadlec, 0. Adsorpt. Sci. Technol. 1984, I, 177.
Langmuir, Vol. 3, No. 3, 1987 345
Surface Structure of Coke Particles
dimension, D,, can be described by 2 plus the absolute value of slope or approximately 2.65, in an analogous fashion to the C02adsorption volume/pressure curves of Figure 3. From the integral of eq 4, one could derive an expression for the fractal dimension in terms of (3 - slope) from a log V/log r plot. Similarly, from eq 5 an expression for fractal dimension in terms of (3 slope) is derived for a plot of log V against log P. The values of V must correspond only to true pore volume. When the uncorrected intrusion data are plotted according to eq 5 as log (AVIAP) vs log (Pa,),then the various particle size ranges fit the same approximate linear correlation to produce D = 2.31 over a limited range of pressure. It must be stressed that the mercury porosimetry data may measure throat accessibility rather than actual pore volumes a t specific pressures and this distorts the pore volume data.
+
Image Analysis Scanning electron microscopy of particle sections revealed that each particle consisted essentially of concentric rings which appeared to have grown outward by some iterative process. The oil sand derived coke was obviously created in just such a process where successive layers of coke were formed in a fluid bed. Liquid bitumen was adsorbed onto a particle and then heated to form the coke and this process was repeated. The outer surface or particle profile, however, appeared to be relatively smooth and definitely not that of a rugged perimeter. The outermost layer in fact looks like a smooth barrier or crust and may be formed as a result of the way the particles are cooled and removed from the coking reactor system. Microscopic profilimetry would therefore reveal that the profile is not rugged and may not be fractal in nature at this level of magnification. Conclusions The results of gas adsorption studies on Syncrude coke reveal that the surface may be described by fractal descriptors. Both N2and C02 adsorbates yielded the same surface fractal dimension regardless of mechanistic adsorption theory. I t is apparent that the two gases are adsorbed by very different methods-capillary forces and surface adsorption potentials. The use of apparent surface area as a surface characteristic leads to unnecessary confusion for the special case of fractal surfaces. The use of a surface fractal dimension determined from adsorption data yields a more useful parameter which from eq 1can be used to calculate equivalent monolayer capacity (micropore volume) or effective cross-sectional area for each adsorbate. In the realm of catalysis, for example, the apparent surface area of a fractal surface as experienced by each reactant would be related to each molecule’s cross-sectional area and not the nitrogen specific surface area. It is not necessary to complete isotherms to derive a surface fractal dimension. Equations 1or 2 may be used where the surface saturation volume or some other easily
measured parameter may be used to calculate D,. Surface subfractal dimension may be determined by the measurement of reaction rates, particularly for reactions which are surface structure sensitive. The use of mercury porosimetry to determine surface or volume fractal dimensions is beginning to show promise. Values of fractal dimension for Syncrude coke are summarized in Table I1 and indicate that several different techniques produce equivalent values for surface fractal dimension. A fractal surface, or a solid with pores of molecular dimensions, interacts with an adsorbate in some manner similar to that associated with the volume filling of micropores. The classification of micropores is primarily based upon nitrogen adsorption and one should bear in mind that the interaction of molecules with an irregular surface or with pores of molecular dimensions may be equivalent to the interaction of nitrogen with micropores. This micropore-like behavior may extend to very large pores interacting with large adsorbates. It is apparent that the use of a fractal model to describe pore distribution is no less applicable than other models. While pore size distributions are normally described in terms of Gaussian distributions, it is evident that Gaussian distributions are insensitive at the tails.31 In addition there is a subtle difference between a surface fractal dimension, D,, and a void fractal dimension, D,. There is some evidence from the Syncrude coke studies to suggest that there is a continuous void structure ressembling some sort of continuous pore network. It is apparent that a smaller particle of Syncrude coke has a higher probability of containing continuities in its pore space as measured by gas adsorption or mercury intrusion. This can be envisioned as void space which is not open to the external environment. This in turn is a result of the iterative building up of the particles from smaller to larger diameter, as in agglomeration, and not the result of any grinding procedure. Recent discussions of coal structure indicate that small-angle neutron scattering and X-ray small-angle scattering indicate that coals may well possess a fractal void structure.32 The use of mathematical relationships which begin with the assumptions of well-defined pore and surface geometries and Gaussian pore size distribuions to describe these same features may be a liability in the description of some systems. For some materials then, fractal dimension is a convenient parameter for characterizing surface interactions.
Acknowledgment. We acknowledge the assistance of
Dr.P. Mainwaring for scanning electron microscopic examination. Registry No. N2,7727-37-9; C02,124-38-9; 02,7782-44-7; Hg, 7439-97-6. (31) Stauffer, D. In On Growth and Form; Stanley, E., Ostrovsky, N., Eds.; Martinus Nijhoff Dordrecht, 1986. (32) Gorbaty, M. L.; Mraw, S. C.; Gethner, J. S.; Brenner, D. Fuel h o c . Technol. 1986,12,31. Bale, H. D.; Schmidt, P. W. Phys. Reu. Lett. 1984, 53, 596.
