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NMR and fractal modelling studies of transport in porous media
Pergamon
Chemical Engineering Science, Vol. 51, No. 10, PP. 2263-2272, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved
S0009-2509(96)00083-8
0009-2509/96 $1S.00 + 0.00
NMR AND FRACTAL MODELLING STUDIES OF TRANSPORT IN POROUS MEDIA S.P. RIGBY AND L.F. GLADDEN University of Cambridge, Department of Chemical Engineering, Pembroke Street, Cambridge, CB2 3RA, U.K. Abstract - Magnetic Resonance Imaging (MRI) and PGSE NMR studies have shown that it is necessary to consider heterogeneities in the porous structure over different lengthscales in order to be able to understand the relationship between structure and transport in porous solids. Moreover, the spatial distribution in pore structure is seen to influence strongly mass transfer processes occurring within the porous medium. The MRI studies described here have suggested that a fractal representation might be appropriate in describing heterogeneous porous systems. Numerical simulations have been performed of transport within ClusterCluster Aggregate (CCA) structures. A comparison of methods of simulating the diffusion process is presented. The results of the simulations are compared with MRI and PGSE NMR measurements of the tortuosity of commercial catalyst pellets. In the light of complementary nitrogen desorption and mercury porosimetry data, a multifractal description of porous media is also proposed in the form of a Composite CCA structure in which both the macroscopic heterogeneity associated with the pore-size distribution and the fractal characteristics of the microscopic pore structure itself are represented.
INTRODUCTION The use of porous materials in chemical and process engineering is widespread. However, our ability to characterise fully the structure of porous materials and to describe the transport of gas or liquid phases within these porous structures is extremely limited. The initial aim of research in this area is to be able to measure one, or more, of the characteristic properties ( e.g. voidage fraction, pore connectivity, pore-size distribution ) of a porous structure and then predict the transport properties, namely diffusivity and tortuosity, of that structure for a particular system of gases and/or liquids. Previous workers have attempted to describe the pore space to varying degrees of complexity. Early work, such as the dusty gas model of Evans et al. (1961) avoided any assumption about the internal structure of the solid. Wakao and Smith (1962) proposed a method to correlate and predict diffusion rates in terms of voidage fraction and poresize distribution data for a bi-disperse porous system. Of particular significance was the simple result that in a random distribution of pores the diffusion coefficient is proportional to the square of the voidage. Later, Johnson and Stewart (1965) considered the catalyst as a slab penetrated by straight pores with a distribution of pore sizes and inclinations. In the case of the orientations of the pores being random, the tortuosity becomes a simple geometric factor and takes the value of three. Foster and Butt (1966) modelled the void volume within a solid as a structure composed of two major arrays of pores, centrally convergent and centrally divergent respectively, inter-connected at specified intervals within the arrays. The exact shape of these arrays is determined uniquely from the volume-area distribution of the pore structure. Reyes and Jensen (1985) used a Bethe lattice representation of the porous medium. Whilst the Bethe lattice can be applied to unimodal or bimodal pore-size distributions the disadvantage of this type of model is that the voidage appears to increase from the centre, the connectivity and distribution of pores in space are not truly random and there are no closed loops. An alternative approach to modelling porous solids is based on capillary networks (Gavalas and Kim, 1981; Petropoulos et al., 1991) or stochastic pore networks (Sharratt and Mann, 1987; Mann and Sharratt, 1988). A stochastic pore network is a network of pores; each pore segment having a radius assigned at random from a given pore-size distribution. This type of model represents the voidage as an interconnected space, with many paths linking discrete nodes. The tortuosities obtained from this model cover the range of experimental observations for monodisperse pore-size distributions. Simulations on three-dimensional networks demonstrated the importance of characterising the connectivity of such pore networks and representing the extent of randomness characteristic of the porous solid in the model representation (Hollewand and Gladden, 1992a, b). Early work had suggested that in the case of a bimodal distribution in which the size of the pores throughout the model is assigned in an uncorrelated manner, network models predict values of tortuosity far in excess of those measured experimentally. In real catalyst materials macropore and micropore networks are most likely, to a certain extent, to exist as distinct regions within the pellet and it was observed (Hollewand and Gladden 1992b) that if random pore networks were constructed assuming distinct but inter-connecting regions of microporosity and macroporosity, anomalously high values of tortuosity were indeed avoided. In response to the observation that, under a microscope, many porous solids appear as a random packing of spherical particles, workers have modelled porous catalysts as random packings of randomly sized and placed spheres or rods (Reyes and Iglesia, 1991; Mace and Wei, 1991; Smith and Wei, 1991; Drewry and Seaton, 1995). Although these models bear a stronger resemblance to the microstructure of real materials, the continuously changing width and connectivity of the pore space makes a unique representation of the pore space of a real material difficult and the numerical solution of transport phenomena within these systems is a subject of ongoing study. 2263
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S.P. RIGBYand L. E GLADDEN
In summary, despite the representation and simulation of transport of porous materials receiving considerable attention over the past forty years, a numerical approach for predicting the inter-relationship of structure and transport within porous solids is yet to be developed. The motivation for the work presented in this paper, is the identification of our inability to represent heterogeneities in real porous solids as a major limiting factor in our ability to obtain a quantitative understanding of transport processes within these systems. In particular, when we construct a representation of a porous solid on the computer, we have to assume that the given model is representative of the porous solid of interest over macroscopic lengthscales. In reality, many porous solids are described by heterogeneous voidage and pore structure over lengthscales larger than the typical pore dimension but smaller than the dimension of the porous pellet itself. Such heterogeneities are clearly observed in magnetic resonance imaging (MRI) studies. Our earlier studies have already demonstrated that spatial variation in mean pore size plays an important role in catalyst deactivation (Cheah et al., 1994) and transport processes (Hollewand and Gladden, 1993). In the course of this work, it was observed that the fractal dimension calculated from spin-lattice relaxation time images (which probe the spatial variation of locally-averaged mean pore size within the material) recorded for six pellets drawn from the same batch was constant (Gladden et al., 1995). In this paper we have extended this study and demonstrate that while the fractal dimension appears constant within a batch it varies between batches. The physical interpretation of this result is that the manufacturing process imposes a characteristic clustering of similar pore size regions within the porous pellet. The observation of a characteristic fractal dimension associated with a given batch suggests that we are now able to measure the extent of structural heterogeneity within these porous solids. This paper outlines our first attempts at applying a fractal representation to describe porous catalyst pellets. In the context of this work, the single Cluster-Cluster Aggregate (CCA) models we employ are representing the heterogeneity in structure within the pellet, and do not represent the structure at the level of the single pore. In this first study, we have used and extended the CCA fractal representation proposed by Elias-Kohav et al. (1991). It has also been necessary to investigate the sensitivity of the results of transport simulations to the numerical methods used. The concept of the CCA is then developed further and Composite CCA structures have been constructed to represent fractal behaviour characterising different structural features of a porous solid; for example, macroscopic variations in voidage and pore size, and fractal properties of the microscopic pore structure itself. The structure of the paper is as follows. First, the relevant background to the PGSE NMR and MRI experimental studies is given, and their application to the study of structure-transport relationships is illustrated. Second, the CCA fractal representation used in this work and the methods used to predict transport phenomena within these structures are presented. The concept of a Composite CCA is introduced and its associated transport characteristics are reported. Finally, the ability of the fractal models presented here to describe transport processes in the porous catalyst samples studied is discussed.
PGSE NMR AND MRI STUDIES OF POROUS CATALYST PELLETS The samples studied in this work are described in Table 1. Prior to the NMR experiments, samples were prepared by impregnating the pellets with pure water for at least 12 hours under ambient conditions. Excess water was removed from the external surface of a pellet by contacting it with pre-soaked filter paper. The paper was wetted in order to prevent any liquid being removed from within the pores of the pellet. It has been shown that this method results in the filling of all pores detectable using mercury porosimetry and gives the same results as samples impregnated under vacuum conditions (Hollewand and Gladden, 1995a).
PGSE NMR Pulsed Gradient Spin Echo (PGSE) NMR allows the measurement of self-diffusion coefficients without the need for isotopic tracers. The relevant theoretical background to PGSE NMR and experimental details relevant to the data presented here can be found elsewhere (Hollewand and Gladden, 1995a); a stimulated echo pulse sequence was used. In this earlier work we have shown that for the case of macroscopically heterogeneous materials the average diffusion coefficient, Dav, of a liquid imbibed within the porous pellet can be approximated by:
Dav= ~piDi i
(1)
where Di is the diffusion coefficient associated with phase i, and Pi is the proportion of the spin population in phase i. Constant relaxation properties in each phase or weakly correlated diffusion and relaxation is assumed. The fitted average diffusivity and width of the diffusivity distribution have been shown to be independent of the number of components assumed in the fit. In the earlier work we have shown that pellets characterised by a wider distribution of self-diffusivities within the pore structure are described by higher values of tortuosity. PGSE experiments were performed using a Bruker Z32FHP-DIFF 200WB 1H 7.5 probe and a Bruker B-Z 18B gradient unit. Samples consisted of 2-6 pellets placed in quartz tubes of internal diameter 7.2mm. The maximum sample length was
NMR and fractal modellingstudies of transport in porous media
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Sample
Material
Voidage
Nominal Diameter
Pellet Form
Fractal Dimension
E1 E2 E3 GI G2 I1 12 P1 S1 $2
Alumina Alumina Alumina Silica Silica Alumina CaO/Alumina Silica Silica Silica
0.49 0.65 0.59 0.66 0.69 0.39 0.43 0.70 0.60 0.61
3 3 3 3 3 10 10 1
Tablet Extrudate Tablet Gel sphere Gel sphere Tablet Tablet Extrudate Gel sphere Gel sphere
1.64:L-0.03 1.625:L-0.049 1.567i-0.043 1.68:k-0.02 1.64!'0.02 1.60"~.03 n/d n/d rdd 1.58!-0.03
2.2
Table 1 The porouscatalystsamplesstudied.The fractaldimensiondeterminedfromthe spin-latticerelaxationtimeimagesrecorded for these pelletsis also given(n/d= not determined). constrained by the region over which uniform linear gradients could be produced and was 15 + 1 mm. The gradients were driven by a steady d.c. power supply provided by two 12 V lead-acid accumulators connected in series. The maximum attainable gradient was 190 G cm"1. PGSE NMR data recorded for samples listed in Table 1 are shown in Fig. 4. MRI One of the advantages of MRI techniques over other imaging methods, is that the measurement can be made sensitive to various physical and chemical parameters such as pore size, voidage, diffusion coefficient and chemical species using so-called contrast techniques. The fractal dimensions referred to in this paper have been obtained from spin-lattice relaxation (T1) images; the value of TI correlating with the locally-averaged surface area-to-volume ratio (S/V) of the pore structure within that image pixel. The correlation between T 1 and S/V is understood as follows. Following perturbation of the nuclear spin system by a radiofrequency (rf) pulse, the magnetisation vector associated with the sample under study while placed in the static magnetic field Bo is rotated away from its equilibrium alignment with Bo, thereby giving rise to components of both transverse and longitudinal magnetisation relative to the equilibrium vector Mo. The exponential recovery of the longitudinal component of the magnetisation to its equilibrium value is determined by a process known as spin-lattice relaxation and occurs via the transfer of excess energy from the spin population to its surroundings (commonly referred to as the "lattice"). The time constant of this relaxation process is therefore known as the spin-lattice, or longitudinal, relaxation time, T 1. The T 1 relaxation behaviour of a fluid confined within a pore is sensitive to both the pore geometry and size; in particular, the rate of TI relaxation is enhanced due to interactions atthe solid-liquid interface (Woessner, 1962). The simplest physical model that can be used to interpret T 1 data is the two-fraction, fast exchange model (Brownstein and Tarr, 1977) which, assuming that diffusion of the fluid to the surface is much faster than the relaxation process, leads to an observed relaxation rate that can be related to properties of the pore structure: 1
T1
1 XS 1 -~ T1B V T1S
(2)
where T1B is the relaxation time characteristic of the bulk fluid, T1S is the relaxation time characteristic of a surface layer thickness, X, and S/V ( =2/r, for cylindrical pores of radius r ) is the surface area-to-volume ratio of the pore. However, in general, an unambiguous physical interpretation of T1 data requires knowledge of the relative rates of both TI relaxation and diffusion within the pore space. Thus by pre-conditioning an MRI pulse sequence with a spin-lattice relaxation measurement, the intensity of each pixel within the image represents a quantitative measurement of T1 within that pixel (and hence volume element within the material) and, by equation 2, also corresponds to a typical S/V ratio for that region of the material. MRI contrast techniques allow us to gain insights into structural variations within porous solids which are impossible to obtain using conventional probes of porosity such as nitrogen adsorption and mercury porosimetry analyses which yield only a one-dimensional description of the pore space. 1H NMR imaging experiments were performed using a Bruker DMX 200 NMR spectrometer with a static field strength of 4.7 T yielding a proton resonance frequency of 200.13 MHz. Spin density and TI images were obtained using a spin-echo pulse sequence employing 90° selective and 180° non-selective pulses, the imaging sequence was pre-conditioned using a saturation recovery pulse sequence and an echo time of 2.6 ms was used. All images used in the fractal analysis were of dimension 128 x 128 pixels, acquired using a read gradient strength of 33.09 G cm "1
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S.P. RIGBYand L. E GLADDEN
Figure 1 NMR images of porous catalyst pellets. (a) Spin density, (b) spin-lattice relaxation time (scale 150-400 ms) and (c) diffusivity (scale 0 1.5x10-9 m2s"I) images for a 2.2 mm diametersilica sphere (Sample$2); lightershades indicate larger valuesof the parameter of interest. A slice of 0.3 mm has been imaged with in-plane resolution of 45 ~tmand an array size of 64 x 64. Althoughthe lowestvalue recorded on the diffusivity scale is 0, it is noted that the lowest accessiblediffusivityin this experimentalconfigurationis approximately 1.0 x 10"10 m2s"1. yielding an in-plane resolution of 40 p.m. A 90 ° Gaussian slice selection pulse truncated at 10% of maximum intensity was employed. Image slice thicknesses in the range 0.3-1mm were used. Spin-density and T 1 images were obtained from eight T 1-weighted spin-echo images and experimental times were typically 12-16 hours. The variable T 1 contrast was obtained by varying the delay time between the non-selective 90 ° pulse of the saturation recovery pre-conditioning and the selective 90 ° pulse of the imaging pulse sequence; in this work images were acquired for eight variable delay times. To obtain a TI image the signal intensity I, from the same pixel, i, in each of the eight images is fitted to the expression describing the magnetisation recovery appropriate to the saturation recovery experiment ( Fukushima and Roeder, 1981 ): I(i,VD) = 9s (i)[1 - e x p ( - V D / T 1(i))]
(3)
where VD is the variable delay time in the saturation-recovery pre-conditioning pulse sequence, thereby obtaining a value of ps(i) and T1 (i) for pixel i. By repeating this procedure for each pixel in the image a complete image of spin density (corresponding to water concentration and hence a measure of voidage) and spin-lattice relaxation time within the sample is produced. Each pixel contains many pores and thus the signal intensity in each pixel is the resultant average over all the pores in the volume of the material associated with that pixel. T 1 and Ps weighted images of water phantoms were obtained to establish that all heterogeneities observed in the image are associated with the sample and do not arise from a spatially varying response of the imaging probehead. A spatially resolved image of the self-diffusivity of the imbibed water within the pore space is obtained by pre-conditioning the imaging sequence with a PGSE pulse sequence and performing the analogous analysis; each pixel then provides a measurement of the self-diffusivity of fluid imbibed within that region of the solid (Hollewand and Gladden, 1995b). Figure 1 shows images of (a) spin density, (b) spin-lattice relaxation time constant and (c) self-diffusivity of water imbibed within a porous catalyst pellet. All three images were obtained from the same slice of the same 2.2 mm diameter silica pellet ($2). The spin density and spin-lattice relaxation time images were obtained using a T 1preconditioned imaging pulse sequence, while the diffusion image was obtained by preconditioning the imaging sequence with a PGSE pulse sequence. In particular, note that the lengthscale of structural heterogeneities (typically 0.05 - 1 mm) as revealed by the TI image is similar to the dimension of inhomogeneities in the self-diffusion image. In general it was observed that there was a stronger correlation between heterogeneities in pore size and diffusivity, than there was between spin density and diffusivity. Having established the importance of spatial variations in pore structure in determining transport processes the further development of the work requires that we identify a method of representing the spatial variations which appear to determine transport characteristics. A fractal approach seems particularly well-suited to the challenge as discussed below.
NMR and fractal modelling studies of transport in porous media
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Characterisation of Fractal Properties of Porous Pellets using MRI Data Fractal physics provides a method for characterising strongly disordered systems (Mandelbrot, 1982). In the case of the T 1 images used in this study, fractal analysis is used to classify the shape of homogeneous clusters of pixel intensity by estimating the fractal dimension of the boundary or perimeter of the clusters. The algorithm used is described elsewhere (Gladden et al., 1995). If the clusters obeyed planar Euclidean-type geometry then the perimeter, P, would be expected to scale with the size of the cluster, R, as P ~ R and the area, A, to scale as A R 2. For a fractal boundary the length of the perimeter will depend on the resolution with which the perimeter is measured and will scale with cluster size, R, as P ~: R d where d is the fractal dimension of the boundary. The fractal dimension then characterises the degree of disorder and must satisfy dtop < d < dem where dtop is the topological dimension of the structure under analysis (dtop=l for the boundaries in the images considered here) and dem is the embedding dimension i.e. 2. From the preceding discussion we obtain an area-perimeter relation for homogeneous clusters of the form A ,,: p2/d. The fractal dimension is therefore estimated from the gradient of the log-area versus log-perimeter plot using a form of the boundary extraction method (Wang et al., 1991). Fractal dimensions determined from the appropriate spin-lattice relaxation time images are given in Table 1. MRI Studies of Transient Diffusion MRI is also able to probe transient transport processes. Transient diffusion processes are followed by initially saturating a pellet with 1H20 and then immersing it in 2H20. Since the MRI method is only sensitive to IH, the transient diffusion of 2H20 into the pellet can be followed. The transient diffusion profiles so-obtained can then be fitted against solutions of the appropriate diffusion equation, assuming a constant diffusivity; hence a tortuosity factor is evaluated (Hollewand and Gladden, 1995b). Values of tortuosity determined using the MRI method are also shown in Fig. 4 for comparison with the PGSE data discussed earlier. In all cases, for any given batch of pellets, the tortuosity obtained from the MRI experiment is greater than that determined by PGSE NMR.
FRACTAL REPRESENTATIONS OF POROUS CATALYST PELLETS Fractal objects are self-similar structures in which increasing magnifications reveal similar features at different length scales; they show a power-law relation between properties such as mass, void volume or surface area and length scale which are termed mass, pore and surface fractals respectively (Pfeifer, 1989). A large number of natural and processed materials have been shown to have either fractal surfaces or fractal pore distributions; Elias-Kohav et al. (1991) list silicas, aluminas, carbon blacks and rocks amongst others. Experimental determinations of fractal dimension have been obtained using scanning electron microscopy (Krohn and Thompson, 1986), small-angle x-ray scattering (Schaefer et al., 1984; Schmidt, 1988), adsorption (Pfeifer and Avnir, 1983; Rolle-Kampczyk et al., 1993), mercury porosimetry (Pfeifer et al., 1984; Friesen and Laidlaw, 1993), 29Si Magic Angle Spinning (MAS) NMR (Devereux et al, 1990) (though only for silicas doped with paramagnetic impurities) and nuclear magnetic relaxation of a fluid in a porous medium (Mendelson, 1986). In ongoing studies we are comparing measurements of fractal properties of the porous pellets listed in Table i using adsorption, desorption, mercury porosimetry and NMR techniques. Simulating Diffusion in 3D CCA Structures In this work, the approach of Elias-Kohav et al. (1991) has been extended to three-dimensional (3D) and composite two-dimensional (2D) objects. The objects studied were Cluster-Cluster Aggregates (CCA), Negative ClusterCluster Aggregates (NCCA), Random Clusters (RC) and a Menger sponge. Only the results obtained for the CCA models are presented here. The 3D CCAs were grown in an analogous way to those in 2D, reported by Elias-Kohav et al. (1991), on a cubic lattice of side length 64 units; successively larger clusters are formed by the diffusion and aggregation of smaller units (Meakin, 1983). This process resembles gelation in the sol-gel synthesis of glasses and ceramics (Sakka, 1985; Kaufman et al., 1987). The mass fractal dimension of the structures generated was calculated using the 'box-counting' method (Mandelbrot, 1982; Pfeifer, 1989). A typical 2D CCA structure is shown in Fig. 2. Three different numerical approaches to calculating tortuosity were used and the results compared. The three methods are described below in the context of 3D CCA structures. Method I - Whole Object Averaging Method In this approach the local tortuosity is approximated by the number of sideways diversions that a molecule needs to make in order to proceed in the void. If M is the locally averaged number of blocked pixels adjacent to an empty site, then the probability of a one pixel diversion is M/6. After such a move there is a similar probability of a further diversion and when M does not vary with every diversion the local tortuosity after n steps is:
1 x = I + M + ( M ' ~ 2+., .+ ( M ) n --) 6 \ 6) -61-(M/6) where the limit holds for large n. For a completely random arrangement, M/6= 1-E and "~=1/E.
(4)
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S. R RIOBY and L. F. GLADDEN
M e t h o d 2 - Diffusion Flux M e t h o d
This method simulates diffusion occurring under a concentration gradient, typical of a Wicke-Kallenbach experiment. In this case the lattice occupied by the CCA is divided up into 512 small boxes of 8 x 8 x 8 sites, and the voidage fraction and local tortuosity (using equation 4 above) is found in each box giving rise to a value of diffusivity in each box. The three dimensional steady-state diffusion equation:
ri:,
'
aX_l
? [I:,(x L '
aY
aYj
az L
'
(5)
aZ_l
is then solved using a finite difference algorithm. A value for the effective diffusivity for the whole structure is obtained from the flux predicted by the simulation and thus a value for the tortuosity is deduced.
Figure 2 A 2D CCA of void fraction 0.65. Black and white pixels correspond to solid phase and void space respectively.
A
B
A
B
A
B
A
B
A
Figure 3 A Composite CCA. In each of the regions labelled A and B, a single CCA structure is present; A and B have different void fractions. Transport in this composite CCA has been studied for a range of overall void fraction up to 0.675. Void fractions in the A regions of 0.7, 0.65 and 0.6 have been used together with appropriate void fractions in the B regions to obtain the overall desired void fraction.
M e t h o d 3 - R a n d o m Walk Method
A Monte-Carlo study was made of classical diffusion using random walks in the void surrounding the clusters; the method used is that reported by Pandey et al. (1984) in a similar study of transport on 2D and 3D random clusters. The general procedure is such that one unoccupied lattice site is selected randomly as the local origin; from here the particle starts its random motion. One of the six nearest neighbour sites is selected randomly and the particle is moved to this site if it is unoccupied, otherwise the particle stays at its previous position. In both cases the time (corresponding to number of steps taken) is increased by a unit step, whether the attempt to move was successful or not. The process of randomly choosing a neighbour of the current particle position and attempting to move to it is repeated again and again for a preset number of steps, the maximum time. This is equivalent to each step of the random walk being said to take the same time interval. The straight line, or root mean square, distance between the starting and finishing points of the walk is then found. Each walk of a particular length is independent of all other walks of the same or differing lengths as different random number seeds were chosen in each case. Each walk of a particular length is repeated as many times as gives rise to a reasonable degree of random error (typically 104 random walking particles per path length) and was repeated for five to ten clusters with the same void fraction. In the case of classical diffusion the mean square displacement is related to the time, t, of the random walk and the total length of the random walk, !, by: = 6Dt = 6DKI .
(6)
If DAS is defined as the 'self-diffusion' of a random walker on an empty lattice then the effective diffusivity, Deft, of a random walker on a lattice partly occupied by a cluster is given by: Deft -- ~ D A s
(7)
2269
NMR and fractal modelling studies of transport in porous media
where e is the fraction of empty sites and x is the tortuosity factor for that structure. Thus the ratio of the slopes of the lines of the plots of against 1 for an occupied and unoccupied lattice would be equal to the ratio rdx. If the fraction of empty sites is known then the tortuosity can be calculated. Figure 4 shows the results of the simulations of diffusion, using the three different methods described above, for a range of 3D CCA structures. The first two simulation methods give a tortuosity more evenly weighted over the whole of the model structure than does the Random Walk Method. In a random walk it is unlikely, except at the limit of a walk of infinite length, that a walker would visit every accessible voidage site on the lattice and it can never visit those empty sites completely enclosed by sites occupied by the cluster. Thus the voidage seen by a random walker tends to be significantly less than actually exists and the tortuosities predicted by this method are greater than those obtained by the Whole Object Averaging and Diffusion Flux methods; this point has been discussed by Elias-Kohav et al. (1991).
Transport within 2D Composite CCA Structures A typical 2D Composite CCA structure is shown in Fig. 3. The composite structure consisted of nine single CCA clusters, each grown on a lattice of size 128 x 128, arranged in a 3 x 3 square arrangement. With reference to Fig. 3, in the simulations presented here the composite structure consisted of a checkerboard arrangement of A and B type single CCA structures. A particular voidage fraction was assigned to the A-squares and the voidage of the B-squares was then varied to give composite structures of a range of overall voidages. This type of structure represents a first step in modelling systems of greater heterogeneity than can be described by a single CCA, particularly where large variations in voidage occur in specific regions of the sample; hierarchical structures typical of those proposed for some silicas (Rolle-Kampczyk et al., 1993) might also be represented using this approach. In the latter case, regions A and B could be constructed with different fractal dimensions which would relate to the internal structure of the secondary particles and the inter-particle space between secondary particles respectively. The possibility of using this approach to describe macropore and micropore distributions within a bidisperse pore structure is also being considered. Simulations of diffusion on the Composite CCA structures were performed using the Whole Object Averaging Method and Diffusion Flux Method described earlier. The results of simulations on such structures are given in Fig. 5. It can be seen that for a given overall average voidage fraction, 0.55 say, as the difference between the voidage fractions in the A and B squares required to achieve this mean is increased the tortuosity of the whole structure increases. It is also seen that the tortuosities associated with these more heterogeneous structures can be significantly greater than those predicted for the single CCA models. Further, differences between the tortuosities predicted by the Whole Object Averaging Method and Diffusion Flux Method are much more significant than in the case of simulating transport in a single CCA structure; the Diffusion Flux results are consistently greater than those obtained using the Whole Object Averaging Method. A possible explanation of this is that within the composite structure, even in models of overall high voidage, transport will still occur in or around regions of lower voidage. Both phenomena would act so as to decrease the overall
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Voidage Fraction Figure 4 Simulations of tortuosity on 3D CCA models. Results are shown for the Whole Object Averaging( ), Diffusionflux ( . . . . . ) and Random Walk (. . . . ) methods. MRI and PGSE NMR measurements of tortuosity are also shown for the commercial catalyst pellets listed in Table 1: o silica/PGSE; • silica/MRI; C3 alumina/PGSE;• alumina/MRI.
0.45
, 0.5
0.55
0.6
0.65
0.7
Average Voidage Fraction Figure 5 Simulations of Diffusion on a Composite CCA Model. The void fraction (E) in the A-type regions and the method of determining the tortuosity were as follows: c3 ( ) ~=0.7, DiffusionFlux; • ( ) ~--0.7, Whole Object Averaging;o ( . . . . ) ~=0.6. DiffusionFlux; • (. . . . ) e=0.6, WholeObject Averaging.
2270
S. E RIGIBYand L. E GLADDEN
diffusivity leading to increased tortuosity. The Whole Object Averaging Method is evenly weighted to the voidage that exists over the whole grid, whereas in the Diffusion Flux method a molecule will see in addition a tortuosity imposed by the macroscopic transport through the heterogeneous structure. DISCUSSION A number of observations can be drawn from the NMR data presented here. First, for all seven batches of commercial catalyst pellets studied so far it has been found that the fractal dimension, as determined from the T1 images, for each batch is constant to within experimental error. The physical interpretation of this fractal parameter is as a characteristic measure of the disorder of a porous solid comprising macroscopic regions (0.05 - lmm) of similar pore size. The spatial variation in pore size has previously been shown to play an important role in transport and reaction processes. In general, decreasing voidage fraction is associated with increasing tortuosity. In the case of isotropic porous pellets, such as silica spheres, estimates of tortuosity determined from MRI studies of transient diffusion are consistently larger than values of tortuosity determined by PGSE NMR. Comparing the values of tortuosity determined experimentally by MRI and PGSE NMR with the results of the transport simulations performed on 3D CCA structures (Fig. 4) it is seen that all experimentally determined values of the tortuosity lie within the bounds of the three transport simulations applied in this work, and follow the same trends. Comparison of the predictions of the Diffusion Flux and Whole Object Averaging Methods with the values determined for the real pellets using PGSE NMR, which is a measure of steady-state self-diffusion of water within the pore structure, shows that the CCA representation successfully predicts the tortuosity of pellets S 1, $2 and P1; these silica pellets are characterised by narrow pore-size distributions with no evidence of heirarchical pore structures as determined by nitrogen desorption and mercury porosimetry analyses following the approach of RolleKampczyk et al. (1993). The good agreement with the experimental data obtained for the silica gel spheres, S1 and $2, is pleasing since the CCA structural representation had been proposed as a realistic representation of sol-gel silica materials. The difference between the results obtained using the Diffusion Flux and Whole Object Averaging Methods compared to the Random Walk Method has already been discussed and attributed to the ability of a given diffusing molecule to sample the complete void space during the simulation; a similar argument might explain the difference between tortuosities measured by MRI and PGSE NMR methods. For the case of an experiment which measures directly the self-diffusion coefficient (for example PGSE) the results will be sensitive to molecules randomly exploring their local volume and in this case the molecules will tend to follow the path of least resistance. When the experiment involves a measurement under an applied concentration gradient (as in the MRI experiments) then the measured diffusive flux will be dependent on the net transfer of diffusing species within a generally macroscopic heterogeneous region. At a molecular level the diffusion may still be dominated by motion along the path of least resistance, but now additional tortuosity is introduced via the heterogeneity on the larger scale as the molecular diffusion process avoids the higher tortuosity regions thereby leading to a lower experimental measurement of the diffusivity and hence higher value of the tortuosity. An alternative explanation, but one which is also dependent on macroscopic heterogeneity, has been proposed by Hollewand and Gladden (1995b) where it was suggested that the difference arises because tortuosity is sampled over different length and time scales in the two experimental methods. The PGSE NMR technique observes the ensemble average of molecular displacements occurring in a time typically 0.01 to 0.1 s corresponding to displacements of order 10 I.tm. In this case each molecule only samples a local environment defined by its displacement and the measured value of diffusivity is therefore an average over the sample volume of each of the microscopic displacements, In the MRI experiments molecules sample environments corresponding to the pellet length or radius which are in the range 1-10 mm. In a real heterogeneous system we expect both of these effects to be in operation and the relative importance of incomplete sampling of the void space and averaging during experimental measurements of tortusoity is the subject of ongoing study. Finally, we consider the Composite CCA structure. The immediate observation from this study is that the increased heterogeneity introduced by this approach leads to significant increases in tortuosity compared with the single CCA representation. It is of particular interest to note that for all samples other than S 1, $2 and P1 (discussed above) the nitrogen desorption and mercury porosimetry analyses showed evidence of multiple fractal structures. All these samples yielded PGSE NMR measurements of tortuosity larger than that predicted by the single CCA representation. In particular, the data recorded for samples G1 and G2 are noted. Our experimental studies would suggest that for these two specific silica gel materials, a fractal representation of structural disorder on both the pore-scale and the macroscopic scale is necessary, thereby explaining why the single CCA representation does not adequately describe transport processes in these sol-gel silicas. Comparison of the results of the Diffusion Flux and Whole Object Averaging Methods for calculating tortuosity also suggest an explanation for discrepancies between the PGSE NMR and MRI determinations of tortuosity. As discussed above, the PGSE NMR experiment measures an average of local molecular displacements over a fixed timescale, this process is physically very similar to finding an average value of M in the Whole Object Averaging Method. In the MRI method molecules sample environments corresponding to the pellet length or radius which is better described by the Diffusion Flux simulation. As seen in Fig. 5, the Diffusion Flux approach predicts consistently higher values of tortuosity compared to the Overall Averaging Method.
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CONCLUSIONS PGSE NMR and MRI have been used to demonstrate that the range of local diffusivities and spatial variation in pore size and diffusion coefficient play a significant role in determining the transport processes occurring within the pore space of commercial porous catalyst pellets. Spin-lattice relaxation time imaging experiments have shown that the heterogeneity in structure occurring over lengthscales of 0.05-1 mm in a given batch of pellets is characterised by a constant fractal dimension. These observations have been the motivation to simulating transport within such pore systems using fractal models. The 2D CCA structures and associated transport simulations of Elias-Kohav et al. (1993) have been extended to three dimensions. The tortuosities predicted using Whole Object Averaging, Diffusion Flux and Random Walk algorithms have been compared and the results discussed. Experimental measurements of tortuosity for ten commercial catalyst pellets all fall within the limits of the simulated values. Following the observation that a fractal parameter might be used to characterise disorder both at the microscopic, pore level in addition to the macroscopic level as discussed previously, a Composite CCA structure has been proposed. These structures predict higher tortuosities for a given voidage than do the single CCA structures. It is also suggested that different methods of simulating the transport process might be more appropriate in predicting transport under the varying boundary conditions presented by the PGSE NMR and MRI techniques used here. ACKNOWLEDGEMENTS We acknowledge the financial support and interest shown in this work by ICI, ZENECA and the EPSRC. S.P. Rigby wishes to thank EPSRC for an EPSRC studentship and ICI for a Research Scholarship.
NOTATION cluster area, dimensionless A concentration, dimensionless C fractai dimension d embedding dimension dem topological dimension dtop D diffusivity, dimensionless self-diffusivity of random walks on an empty lattice, dimensionless DAS effective diffusivity, dimensionless Deft pixel number i K proportionality constant between total path length and time of a random walk, dimensionless 1 total length of random walk, dimensionless M locally averaged number of blocked pixels adjacent to an empty site, dimensionless P cluster perimeter, dimensionless r pore radius, m mean square displacement in time t, lattice units cluster size, dimensionless R surface area of pore, m 2 S I MRI signal intensity, dimensionless t time, dimensionless spin-lattice relaxation time, s T1 spin-lattice relaxation time in bulk liquid, s T1B spin-lattice relaxation time in surface layer, s T1S volume of pore, m 3 V VD variable delay time, s X Cartesian co-ordinate, dimensionless Y Cartesian co-ordinate, dimensionless Z Cartesian co-ordinate, dimensionless Greek Letters e voidage fraction, dimensionless X surface layer thickness, m Ps spin-density in pixel, dimensionless x tortuosity, dimensionless
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REFERENCES
Brownstein, K.R. and Tarr, C.E., 1977, J.Magn. Reson., 26, 17-24. Cheah, K.Y., Chiaranussati, P., Hollewand, M.P. and Gladden, L.F., 1994, Appl. Catal.,A115, 147-155. Devereux, F., Boilot, J.P.,Chaput, F. and Sapoval, B., 1990, Phys. Rev. Left.,65, 614-617. Drewry, H. and Seaton, N.A., 1995, A.I.Ch.E.J., 41,880-893. Elias-Kohav, T., Sheintuch, M. and Avnir, D., 1991, Chem. Engng Sci., 46, 2787-2798. Evans, R.B., Watson, G.M. and Mason, E.A., 1961, J. Chem. Phys., 35, 2076-2083. Foster, R.N. and Butt, J.B., 1966, A.I.Ch..E. £, 12, 180-185. Friesen, W.I. and Laidlaw, W.G., 1993, £ Colloid Interface Sci., 160, 226-235. Fukushima, E. and Roeder, S.B.W., 1981, Experimental Pulse NMR: A Nuts and Bolts Approach, Addison-Wesley, Reading, MA. Gavalas, G.R. and Kim, S., 1981, Chem. Engng Sci., 36, 1111-1122. Gladden, L.F., Hollewand, M.P. and Alexander, P., 1995, A.I.Ch.E. £, 41, 1-13. Hollewand, M.P. and Gladden, L.F., 1992a, Chem. Engng Sci., 47, 1761-1770. Hollewand, M.P. and Gladden, L.F., 1992b, Chem. Engng Sci., 47, 2757-2762. Hollewand, M.P. and Gladden, L.F., 1993, J. Catal., 144, 254-272. Hollewand, M.P. and Gladden, L.F., 1995a, Chem. Engng Sci., 50, 309-326. Hollewand, M.P. and Gladden, L.F., 1995b, Chem. Engng Sci., 50, 327-344. Johnson, M.F.L. and Stewart, W.E., 1965, J. Catal., 4, 248-252. Kaufman, V.R., Avnir, D., Pines-Rojanski, D. and Huppert, D., 1988, J. Non-Cryst. Solids, 99, 379-386. Krohn, C.E. and Thompson, A.H., 1986, Phys. Rev. B, 33, 6366-6374. Mace, O. and Wei, J, 1991, Ind. Engng Chem. Res., 30, 909-918. Mandelbrot, B.B., 1982, The Fractal Geometry Of Nature, Freeman, New York. Mann, R. and Sharratt, P.N., 1988, Chem. Engng Sci., 41, 711-718. Mendelson, K.S., 1986, Phys. Rev. B, 34, 6503-6505. Meakin, P., 1983, Phys. Rev. Lett., 51, 1119-1122. Pandey, R.B., Stauffer, D., Margolina, A. and Zabolitzky, J.G., 1984, J. Stat. Phys., 34, 427-450. Petropoulos, J.H., Petrou, J.K. and Liapis, A.I., 1991, Ind. Engng Chem. Res., 30, 1281-1289. Pfeifer, P., 1989, Characterization Of Surface Irregularity In Preparative Chemistry Using Supported Reagents (Ed. by P. Laszlo), Academic, New York. Pfeifer, P. and Avnir, D., 1983, J. Chem. Phys., 79, 3558-3565. Pfeifer, P., Avnir, D. and Farin, D., 1984, J. Stat. Phys., 36, 699-716. Reyes, S.C. and Iglesia, E., 1991, J. Catal., 129, 457-472. Reyes, S.C. and Jensen, K.F., 1985, Chem. Engng Sci., 40, 1723-1734. Rolle-Kampczyk, U., Karger, J., Caro, J., Noack, M., Klobes, P. and Rohl-Kuhn, B., 1993, J. Colloid Interface Sci. 159, 366-371. Sakka, S., 1985, Am. Ceram. Soc. Bull., 64, 1463-1466. Schaefer, D.W., Martin, E.J., Wiitzius, P. and Cannel, D.S., 1984, Phys. Rev. Lett., 52, 2371-2374. Schmidt, P.W., 1988, Macromol. Chem., Macromol. Syrup., 15, 153-166. Sharratt, P.N. and Mann, R., 1987, Chem. Engng Sci., 42, 1565-1567. Smith, B.J. and Wei, J., 1991, J. Catal., 132, 41-57. Wakao, N. and Smith, J.M., 1962, Chem. Engng Sci., 17, 825-834. Wang, Z.-Y., Konno, M. and Saito, S., 1991, J. Chem. Eng. Japan, 24, 256-258. Woessner, D.E., 1962, J. Chem. Phys., 36, 1-4.
Chemical Engineering Science, Vol. 51, No. 10, PP. 2263-2272, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved
S0009-2509(96)00083-8
0009-2509/96 $1S.00 + 0.00
NMR AND FRACTAL MODELLING STUDIES OF TRANSPORT IN POROUS MEDIA S.P. RIGBY AND L.F. GLADDEN University of Cambridge, Department of Chemical Engineering, Pembroke Street, Cambridge, CB2 3RA, U.K. Abstract - Magnetic Resonance Imaging (MRI) and PGSE NMR studies have shown that it is necessary to consider heterogeneities in the porous structure over different lengthscales in order to be able to understand the relationship between structure and transport in porous solids. Moreover, the spatial distribution in pore structure is seen to influence strongly mass transfer processes occurring within the porous medium. The MRI studies described here have suggested that a fractal representation might be appropriate in describing heterogeneous porous systems. Numerical simulations have been performed of transport within ClusterCluster Aggregate (CCA) structures. A comparison of methods of simulating the diffusion process is presented. The results of the simulations are compared with MRI and PGSE NMR measurements of the tortuosity of commercial catalyst pellets. In the light of complementary nitrogen desorption and mercury porosimetry data, a multifractal description of porous media is also proposed in the form of a Composite CCA structure in which both the macroscopic heterogeneity associated with the pore-size distribution and the fractal characteristics of the microscopic pore structure itself are represented.
INTRODUCTION The use of porous materials in chemical and process engineering is widespread. However, our ability to characterise fully the structure of porous materials and to describe the transport of gas or liquid phases within these porous structures is extremely limited. The initial aim of research in this area is to be able to measure one, or more, of the characteristic properties ( e.g. voidage fraction, pore connectivity, pore-size distribution ) of a porous structure and then predict the transport properties, namely diffusivity and tortuosity, of that structure for a particular system of gases and/or liquids. Previous workers have attempted to describe the pore space to varying degrees of complexity. Early work, such as the dusty gas model of Evans et al. (1961) avoided any assumption about the internal structure of the solid. Wakao and Smith (1962) proposed a method to correlate and predict diffusion rates in terms of voidage fraction and poresize distribution data for a bi-disperse porous system. Of particular significance was the simple result that in a random distribution of pores the diffusion coefficient is proportional to the square of the voidage. Later, Johnson and Stewart (1965) considered the catalyst as a slab penetrated by straight pores with a distribution of pore sizes and inclinations. In the case of the orientations of the pores being random, the tortuosity becomes a simple geometric factor and takes the value of three. Foster and Butt (1966) modelled the void volume within a solid as a structure composed of two major arrays of pores, centrally convergent and centrally divergent respectively, inter-connected at specified intervals within the arrays. The exact shape of these arrays is determined uniquely from the volume-area distribution of the pore structure. Reyes and Jensen (1985) used a Bethe lattice representation of the porous medium. Whilst the Bethe lattice can be applied to unimodal or bimodal pore-size distributions the disadvantage of this type of model is that the voidage appears to increase from the centre, the connectivity and distribution of pores in space are not truly random and there are no closed loops. An alternative approach to modelling porous solids is based on capillary networks (Gavalas and Kim, 1981; Petropoulos et al., 1991) or stochastic pore networks (Sharratt and Mann, 1987; Mann and Sharratt, 1988). A stochastic pore network is a network of pores; each pore segment having a radius assigned at random from a given pore-size distribution. This type of model represents the voidage as an interconnected space, with many paths linking discrete nodes. The tortuosities obtained from this model cover the range of experimental observations for monodisperse pore-size distributions. Simulations on three-dimensional networks demonstrated the importance of characterising the connectivity of such pore networks and representing the extent of randomness characteristic of the porous solid in the model representation (Hollewand and Gladden, 1992a, b). Early work had suggested that in the case of a bimodal distribution in which the size of the pores throughout the model is assigned in an uncorrelated manner, network models predict values of tortuosity far in excess of those measured experimentally. In real catalyst materials macropore and micropore networks are most likely, to a certain extent, to exist as distinct regions within the pellet and it was observed (Hollewand and Gladden 1992b) that if random pore networks were constructed assuming distinct but inter-connecting regions of microporosity and macroporosity, anomalously high values of tortuosity were indeed avoided. In response to the observation that, under a microscope, many porous solids appear as a random packing of spherical particles, workers have modelled porous catalysts as random packings of randomly sized and placed spheres or rods (Reyes and Iglesia, 1991; Mace and Wei, 1991; Smith and Wei, 1991; Drewry and Seaton, 1995). Although these models bear a stronger resemblance to the microstructure of real materials, the continuously changing width and connectivity of the pore space makes a unique representation of the pore space of a real material difficult and the numerical solution of transport phenomena within these systems is a subject of ongoing study. 2263
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In summary, despite the representation and simulation of transport of porous materials receiving considerable attention over the past forty years, a numerical approach for predicting the inter-relationship of structure and transport within porous solids is yet to be developed. The motivation for the work presented in this paper, is the identification of our inability to represent heterogeneities in real porous solids as a major limiting factor in our ability to obtain a quantitative understanding of transport processes within these systems. In particular, when we construct a representation of a porous solid on the computer, we have to assume that the given model is representative of the porous solid of interest over macroscopic lengthscales. In reality, many porous solids are described by heterogeneous voidage and pore structure over lengthscales larger than the typical pore dimension but smaller than the dimension of the porous pellet itself. Such heterogeneities are clearly observed in magnetic resonance imaging (MRI) studies. Our earlier studies have already demonstrated that spatial variation in mean pore size plays an important role in catalyst deactivation (Cheah et al., 1994) and transport processes (Hollewand and Gladden, 1993). In the course of this work, it was observed that the fractal dimension calculated from spin-lattice relaxation time images (which probe the spatial variation of locally-averaged mean pore size within the material) recorded for six pellets drawn from the same batch was constant (Gladden et al., 1995). In this paper we have extended this study and demonstrate that while the fractal dimension appears constant within a batch it varies between batches. The physical interpretation of this result is that the manufacturing process imposes a characteristic clustering of similar pore size regions within the porous pellet. The observation of a characteristic fractal dimension associated with a given batch suggests that we are now able to measure the extent of structural heterogeneity within these porous solids. This paper outlines our first attempts at applying a fractal representation to describe porous catalyst pellets. In the context of this work, the single Cluster-Cluster Aggregate (CCA) models we employ are representing the heterogeneity in structure within the pellet, and do not represent the structure at the level of the single pore. In this first study, we have used and extended the CCA fractal representation proposed by Elias-Kohav et al. (1991). It has also been necessary to investigate the sensitivity of the results of transport simulations to the numerical methods used. The concept of the CCA is then developed further and Composite CCA structures have been constructed to represent fractal behaviour characterising different structural features of a porous solid; for example, macroscopic variations in voidage and pore size, and fractal properties of the microscopic pore structure itself. The structure of the paper is as follows. First, the relevant background to the PGSE NMR and MRI experimental studies is given, and their application to the study of structure-transport relationships is illustrated. Second, the CCA fractal representation used in this work and the methods used to predict transport phenomena within these structures are presented. The concept of a Composite CCA is introduced and its associated transport characteristics are reported. Finally, the ability of the fractal models presented here to describe transport processes in the porous catalyst samples studied is discussed.
PGSE NMR AND MRI STUDIES OF POROUS CATALYST PELLETS The samples studied in this work are described in Table 1. Prior to the NMR experiments, samples were prepared by impregnating the pellets with pure water for at least 12 hours under ambient conditions. Excess water was removed from the external surface of a pellet by contacting it with pre-soaked filter paper. The paper was wetted in order to prevent any liquid being removed from within the pores of the pellet. It has been shown that this method results in the filling of all pores detectable using mercury porosimetry and gives the same results as samples impregnated under vacuum conditions (Hollewand and Gladden, 1995a).
PGSE NMR Pulsed Gradient Spin Echo (PGSE) NMR allows the measurement of self-diffusion coefficients without the need for isotopic tracers. The relevant theoretical background to PGSE NMR and experimental details relevant to the data presented here can be found elsewhere (Hollewand and Gladden, 1995a); a stimulated echo pulse sequence was used. In this earlier work we have shown that for the case of macroscopically heterogeneous materials the average diffusion coefficient, Dav, of a liquid imbibed within the porous pellet can be approximated by:
Dav= ~piDi i
(1)
where Di is the diffusion coefficient associated with phase i, and Pi is the proportion of the spin population in phase i. Constant relaxation properties in each phase or weakly correlated diffusion and relaxation is assumed. The fitted average diffusivity and width of the diffusivity distribution have been shown to be independent of the number of components assumed in the fit. In the earlier work we have shown that pellets characterised by a wider distribution of self-diffusivities within the pore structure are described by higher values of tortuosity. PGSE experiments were performed using a Bruker Z32FHP-DIFF 200WB 1H 7.5 probe and a Bruker B-Z 18B gradient unit. Samples consisted of 2-6 pellets placed in quartz tubes of internal diameter 7.2mm. The maximum sample length was
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Sample
Material
Voidage
Nominal Diameter
Pellet Form
Fractal Dimension
E1 E2 E3 GI G2 I1 12 P1 S1 $2
Alumina Alumina Alumina Silica Silica Alumina CaO/Alumina Silica Silica Silica
0.49 0.65 0.59 0.66 0.69 0.39 0.43 0.70 0.60 0.61
3 3 3 3 3 10 10 1
Tablet Extrudate Tablet Gel sphere Gel sphere Tablet Tablet Extrudate Gel sphere Gel sphere
1.64:L-0.03 1.625:L-0.049 1.567i-0.043 1.68:k-0.02 1.64!'0.02 1.60"~.03 n/d n/d rdd 1.58!-0.03
2.2
Table 1 The porouscatalystsamplesstudied.The fractaldimensiondeterminedfromthe spin-latticerelaxationtimeimagesrecorded for these pelletsis also given(n/d= not determined). constrained by the region over which uniform linear gradients could be produced and was 15 + 1 mm. The gradients were driven by a steady d.c. power supply provided by two 12 V lead-acid accumulators connected in series. The maximum attainable gradient was 190 G cm"1. PGSE NMR data recorded for samples listed in Table 1 are shown in Fig. 4. MRI One of the advantages of MRI techniques over other imaging methods, is that the measurement can be made sensitive to various physical and chemical parameters such as pore size, voidage, diffusion coefficient and chemical species using so-called contrast techniques. The fractal dimensions referred to in this paper have been obtained from spin-lattice relaxation (T1) images; the value of TI correlating with the locally-averaged surface area-to-volume ratio (S/V) of the pore structure within that image pixel. The correlation between T 1 and S/V is understood as follows. Following perturbation of the nuclear spin system by a radiofrequency (rf) pulse, the magnetisation vector associated with the sample under study while placed in the static magnetic field Bo is rotated away from its equilibrium alignment with Bo, thereby giving rise to components of both transverse and longitudinal magnetisation relative to the equilibrium vector Mo. The exponential recovery of the longitudinal component of the magnetisation to its equilibrium value is determined by a process known as spin-lattice relaxation and occurs via the transfer of excess energy from the spin population to its surroundings (commonly referred to as the "lattice"). The time constant of this relaxation process is therefore known as the spin-lattice, or longitudinal, relaxation time, T 1. The T 1 relaxation behaviour of a fluid confined within a pore is sensitive to both the pore geometry and size; in particular, the rate of TI relaxation is enhanced due to interactions atthe solid-liquid interface (Woessner, 1962). The simplest physical model that can be used to interpret T 1 data is the two-fraction, fast exchange model (Brownstein and Tarr, 1977) which, assuming that diffusion of the fluid to the surface is much faster than the relaxation process, leads to an observed relaxation rate that can be related to properties of the pore structure: 1
T1
1 XS 1 -~ T1B V T1S
(2)
where T1B is the relaxation time characteristic of the bulk fluid, T1S is the relaxation time characteristic of a surface layer thickness, X, and S/V ( =2/r, for cylindrical pores of radius r ) is the surface area-to-volume ratio of the pore. However, in general, an unambiguous physical interpretation of T1 data requires knowledge of the relative rates of both TI relaxation and diffusion within the pore space. Thus by pre-conditioning an MRI pulse sequence with a spin-lattice relaxation measurement, the intensity of each pixel within the image represents a quantitative measurement of T1 within that pixel (and hence volume element within the material) and, by equation 2, also corresponds to a typical S/V ratio for that region of the material. MRI contrast techniques allow us to gain insights into structural variations within porous solids which are impossible to obtain using conventional probes of porosity such as nitrogen adsorption and mercury porosimetry analyses which yield only a one-dimensional description of the pore space. 1H NMR imaging experiments were performed using a Bruker DMX 200 NMR spectrometer with a static field strength of 4.7 T yielding a proton resonance frequency of 200.13 MHz. Spin density and TI images were obtained using a spin-echo pulse sequence employing 90° selective and 180° non-selective pulses, the imaging sequence was pre-conditioned using a saturation recovery pulse sequence and an echo time of 2.6 ms was used. All images used in the fractal analysis were of dimension 128 x 128 pixels, acquired using a read gradient strength of 33.09 G cm "1
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Figure 1 NMR images of porous catalyst pellets. (a) Spin density, (b) spin-lattice relaxation time (scale 150-400 ms) and (c) diffusivity (scale 0 1.5x10-9 m2s"I) images for a 2.2 mm diametersilica sphere (Sample$2); lightershades indicate larger valuesof the parameter of interest. A slice of 0.3 mm has been imaged with in-plane resolution of 45 ~tmand an array size of 64 x 64. Althoughthe lowestvalue recorded on the diffusivity scale is 0, it is noted that the lowest accessiblediffusivityin this experimentalconfigurationis approximately 1.0 x 10"10 m2s"1. yielding an in-plane resolution of 40 p.m. A 90 ° Gaussian slice selection pulse truncated at 10% of maximum intensity was employed. Image slice thicknesses in the range 0.3-1mm were used. Spin-density and T 1 images were obtained from eight T 1-weighted spin-echo images and experimental times were typically 12-16 hours. The variable T 1 contrast was obtained by varying the delay time between the non-selective 90 ° pulse of the saturation recovery pre-conditioning and the selective 90 ° pulse of the imaging pulse sequence; in this work images were acquired for eight variable delay times. To obtain a TI image the signal intensity I, from the same pixel, i, in each of the eight images is fitted to the expression describing the magnetisation recovery appropriate to the saturation recovery experiment ( Fukushima and Roeder, 1981 ): I(i,VD) = 9s (i)[1 - e x p ( - V D / T 1(i))]
(3)
where VD is the variable delay time in the saturation-recovery pre-conditioning pulse sequence, thereby obtaining a value of ps(i) and T1 (i) for pixel i. By repeating this procedure for each pixel in the image a complete image of spin density (corresponding to water concentration and hence a measure of voidage) and spin-lattice relaxation time within the sample is produced. Each pixel contains many pores and thus the signal intensity in each pixel is the resultant average over all the pores in the volume of the material associated with that pixel. T 1 and Ps weighted images of water phantoms were obtained to establish that all heterogeneities observed in the image are associated with the sample and do not arise from a spatially varying response of the imaging probehead. A spatially resolved image of the self-diffusivity of the imbibed water within the pore space is obtained by pre-conditioning the imaging sequence with a PGSE pulse sequence and performing the analogous analysis; each pixel then provides a measurement of the self-diffusivity of fluid imbibed within that region of the solid (Hollewand and Gladden, 1995b). Figure 1 shows images of (a) spin density, (b) spin-lattice relaxation time constant and (c) self-diffusivity of water imbibed within a porous catalyst pellet. All three images were obtained from the same slice of the same 2.2 mm diameter silica pellet ($2). The spin density and spin-lattice relaxation time images were obtained using a T 1preconditioned imaging pulse sequence, while the diffusion image was obtained by preconditioning the imaging sequence with a PGSE pulse sequence. In particular, note that the lengthscale of structural heterogeneities (typically 0.05 - 1 mm) as revealed by the TI image is similar to the dimension of inhomogeneities in the self-diffusion image. In general it was observed that there was a stronger correlation between heterogeneities in pore size and diffusivity, than there was between spin density and diffusivity. Having established the importance of spatial variations in pore structure in determining transport processes the further development of the work requires that we identify a method of representing the spatial variations which appear to determine transport characteristics. A fractal approach seems particularly well-suited to the challenge as discussed below.
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Characterisation of Fractal Properties of Porous Pellets using MRI Data Fractal physics provides a method for characterising strongly disordered systems (Mandelbrot, 1982). In the case of the T 1 images used in this study, fractal analysis is used to classify the shape of homogeneous clusters of pixel intensity by estimating the fractal dimension of the boundary or perimeter of the clusters. The algorithm used is described elsewhere (Gladden et al., 1995). If the clusters obeyed planar Euclidean-type geometry then the perimeter, P, would be expected to scale with the size of the cluster, R, as P ~ R and the area, A, to scale as A R 2. For a fractal boundary the length of the perimeter will depend on the resolution with which the perimeter is measured and will scale with cluster size, R, as P ~: R d where d is the fractal dimension of the boundary. The fractal dimension then characterises the degree of disorder and must satisfy dtop < d < dem where dtop is the topological dimension of the structure under analysis (dtop=l for the boundaries in the images considered here) and dem is the embedding dimension i.e. 2. From the preceding discussion we obtain an area-perimeter relation for homogeneous clusters of the form A ,,: p2/d. The fractal dimension is therefore estimated from the gradient of the log-area versus log-perimeter plot using a form of the boundary extraction method (Wang et al., 1991). Fractal dimensions determined from the appropriate spin-lattice relaxation time images are given in Table 1. MRI Studies of Transient Diffusion MRI is also able to probe transient transport processes. Transient diffusion processes are followed by initially saturating a pellet with 1H20 and then immersing it in 2H20. Since the MRI method is only sensitive to IH, the transient diffusion of 2H20 into the pellet can be followed. The transient diffusion profiles so-obtained can then be fitted against solutions of the appropriate diffusion equation, assuming a constant diffusivity; hence a tortuosity factor is evaluated (Hollewand and Gladden, 1995b). Values of tortuosity determined using the MRI method are also shown in Fig. 4 for comparison with the PGSE data discussed earlier. In all cases, for any given batch of pellets, the tortuosity obtained from the MRI experiment is greater than that determined by PGSE NMR.
FRACTAL REPRESENTATIONS OF POROUS CATALYST PELLETS Fractal objects are self-similar structures in which increasing magnifications reveal similar features at different length scales; they show a power-law relation between properties such as mass, void volume or surface area and length scale which are termed mass, pore and surface fractals respectively (Pfeifer, 1989). A large number of natural and processed materials have been shown to have either fractal surfaces or fractal pore distributions; Elias-Kohav et al. (1991) list silicas, aluminas, carbon blacks and rocks amongst others. Experimental determinations of fractal dimension have been obtained using scanning electron microscopy (Krohn and Thompson, 1986), small-angle x-ray scattering (Schaefer et al., 1984; Schmidt, 1988), adsorption (Pfeifer and Avnir, 1983; Rolle-Kampczyk et al., 1993), mercury porosimetry (Pfeifer et al., 1984; Friesen and Laidlaw, 1993), 29Si Magic Angle Spinning (MAS) NMR (Devereux et al, 1990) (though only for silicas doped with paramagnetic impurities) and nuclear magnetic relaxation of a fluid in a porous medium (Mendelson, 1986). In ongoing studies we are comparing measurements of fractal properties of the porous pellets listed in Table i using adsorption, desorption, mercury porosimetry and NMR techniques. Simulating Diffusion in 3D CCA Structures In this work, the approach of Elias-Kohav et al. (1991) has been extended to three-dimensional (3D) and composite two-dimensional (2D) objects. The objects studied were Cluster-Cluster Aggregates (CCA), Negative ClusterCluster Aggregates (NCCA), Random Clusters (RC) and a Menger sponge. Only the results obtained for the CCA models are presented here. The 3D CCAs were grown in an analogous way to those in 2D, reported by Elias-Kohav et al. (1991), on a cubic lattice of side length 64 units; successively larger clusters are formed by the diffusion and aggregation of smaller units (Meakin, 1983). This process resembles gelation in the sol-gel synthesis of glasses and ceramics (Sakka, 1985; Kaufman et al., 1987). The mass fractal dimension of the structures generated was calculated using the 'box-counting' method (Mandelbrot, 1982; Pfeifer, 1989). A typical 2D CCA structure is shown in Fig. 2. Three different numerical approaches to calculating tortuosity were used and the results compared. The three methods are described below in the context of 3D CCA structures. Method I - Whole Object Averaging Method In this approach the local tortuosity is approximated by the number of sideways diversions that a molecule needs to make in order to proceed in the void. If M is the locally averaged number of blocked pixels adjacent to an empty site, then the probability of a one pixel diversion is M/6. After such a move there is a similar probability of a further diversion and when M does not vary with every diversion the local tortuosity after n steps is:
1 x = I + M + ( M ' ~ 2+., .+ ( M ) n --) 6 \ 6) -61-(M/6) where the limit holds for large n. For a completely random arrangement, M/6= 1-E and "~=1/E.
(4)
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S. R RIOBY and L. F. GLADDEN
M e t h o d 2 - Diffusion Flux M e t h o d
This method simulates diffusion occurring under a concentration gradient, typical of a Wicke-Kallenbach experiment. In this case the lattice occupied by the CCA is divided up into 512 small boxes of 8 x 8 x 8 sites, and the voidage fraction and local tortuosity (using equation 4 above) is found in each box giving rise to a value of diffusivity in each box. The three dimensional steady-state diffusion equation:
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az L
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aZ_l
is then solved using a finite difference algorithm. A value for the effective diffusivity for the whole structure is obtained from the flux predicted by the simulation and thus a value for the tortuosity is deduced.
Figure 2 A 2D CCA of void fraction 0.65. Black and white pixels correspond to solid phase and void space respectively.
A
B
A
B
A
B
A
B
A
Figure 3 A Composite CCA. In each of the regions labelled A and B, a single CCA structure is present; A and B have different void fractions. Transport in this composite CCA has been studied for a range of overall void fraction up to 0.675. Void fractions in the A regions of 0.7, 0.65 and 0.6 have been used together with appropriate void fractions in the B regions to obtain the overall desired void fraction.
M e t h o d 3 - R a n d o m Walk Method
A Monte-Carlo study was made of classical diffusion using random walks in the void surrounding the clusters; the method used is that reported by Pandey et al. (1984) in a similar study of transport on 2D and 3D random clusters. The general procedure is such that one unoccupied lattice site is selected randomly as the local origin; from here the particle starts its random motion. One of the six nearest neighbour sites is selected randomly and the particle is moved to this site if it is unoccupied, otherwise the particle stays at its previous position. In both cases the time (corresponding to number of steps taken) is increased by a unit step, whether the attempt to move was successful or not. The process of randomly choosing a neighbour of the current particle position and attempting to move to it is repeated again and again for a preset number of steps, the maximum time. This is equivalent to each step of the random walk being said to take the same time interval. The straight line, or root mean square, distance between the starting and finishing points of the walk is then found. Each walk of a particular length is independent of all other walks of the same or differing lengths as different random number seeds were chosen in each case. Each walk of a particular length is repeated as many times as gives rise to a reasonable degree of random error (typically 104 random walking particles per path length) and was repeated for five to ten clusters with the same void fraction. In the case of classical diffusion the mean square displacement is related to the time, t, of the random walk and the total length of the random walk, !, by:
(6)
If DAS is defined as the 'self-diffusion' of a random walker on an empty lattice then the effective diffusivity, Deft, of a random walker on a lattice partly occupied by a cluster is given by: Deft -- ~ D A s
(7)
2269
NMR and fractal modelling studies of transport in porous media
where e is the fraction of empty sites and x is the tortuosity factor for that structure. Thus the ratio of the slopes of the lines of the plots of
Transport within 2D Composite CCA Structures A typical 2D Composite CCA structure is shown in Fig. 3. The composite structure consisted of nine single CCA clusters, each grown on a lattice of size 128 x 128, arranged in a 3 x 3 square arrangement. With reference to Fig. 3, in the simulations presented here the composite structure consisted of a checkerboard arrangement of A and B type single CCA structures. A particular voidage fraction was assigned to the A-squares and the voidage of the B-squares was then varied to give composite structures of a range of overall voidages. This type of structure represents a first step in modelling systems of greater heterogeneity than can be described by a single CCA, particularly where large variations in voidage occur in specific regions of the sample; hierarchical structures typical of those proposed for some silicas (Rolle-Kampczyk et al., 1993) might also be represented using this approach. In the latter case, regions A and B could be constructed with different fractal dimensions which would relate to the internal structure of the secondary particles and the inter-particle space between secondary particles respectively. The possibility of using this approach to describe macropore and micropore distributions within a bidisperse pore structure is also being considered. Simulations of diffusion on the Composite CCA structures were performed using the Whole Object Averaging Method and Diffusion Flux Method described earlier. The results of simulations on such structures are given in Fig. 5. It can be seen that for a given overall average voidage fraction, 0.55 say, as the difference between the voidage fractions in the A and B squares required to achieve this mean is increased the tortuosity of the whole structure increases. It is also seen that the tortuosities associated with these more heterogeneous structures can be significantly greater than those predicted for the single CCA models. Further, differences between the tortuosities predicted by the Whole Object Averaging Method and Diffusion Flux Method are much more significant than in the case of simulating transport in a single CCA structure; the Diffusion Flux results are consistently greater than those obtained using the Whole Object Averaging Method. A possible explanation of this is that within the composite structure, even in models of overall high voidage, transport will still occur in or around regions of lower voidage. Both phenomena would act so as to decrease the overall
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Voidage Fraction Figure 4 Simulations of tortuosity on 3D CCA models. Results are shown for the Whole Object Averaging( ), Diffusionflux ( . . . . . ) and Random Walk (. . . . ) methods. MRI and PGSE NMR measurements of tortuosity are also shown for the commercial catalyst pellets listed in Table 1: o silica/PGSE; • silica/MRI; C3 alumina/PGSE;• alumina/MRI.
0.45
, 0.5
0.55
0.6
0.65
0.7
Average Voidage Fraction Figure 5 Simulations of Diffusion on a Composite CCA Model. The void fraction (E) in the A-type regions and the method of determining the tortuosity were as follows: c3 ( ) ~=0.7, DiffusionFlux; • ( ) ~--0.7, Whole Object Averaging;o ( . . . . ) ~=0.6. DiffusionFlux; • (. . . . ) e=0.6, WholeObject Averaging.
2270
S. E RIGIBYand L. E GLADDEN
diffusivity leading to increased tortuosity. The Whole Object Averaging Method is evenly weighted to the voidage that exists over the whole grid, whereas in the Diffusion Flux method a molecule will see in addition a tortuosity imposed by the macroscopic transport through the heterogeneous structure. DISCUSSION A number of observations can be drawn from the NMR data presented here. First, for all seven batches of commercial catalyst pellets studied so far it has been found that the fractal dimension, as determined from the T1 images, for each batch is constant to within experimental error. The physical interpretation of this fractal parameter is as a characteristic measure of the disorder of a porous solid comprising macroscopic regions (0.05 - lmm) of similar pore size. The spatial variation in pore size has previously been shown to play an important role in transport and reaction processes. In general, decreasing voidage fraction is associated with increasing tortuosity. In the case of isotropic porous pellets, such as silica spheres, estimates of tortuosity determined from MRI studies of transient diffusion are consistently larger than values of tortuosity determined by PGSE NMR. Comparing the values of tortuosity determined experimentally by MRI and PGSE NMR with the results of the transport simulations performed on 3D CCA structures (Fig. 4) it is seen that all experimentally determined values of the tortuosity lie within the bounds of the three transport simulations applied in this work, and follow the same trends. Comparison of the predictions of the Diffusion Flux and Whole Object Averaging Methods with the values determined for the real pellets using PGSE NMR, which is a measure of steady-state self-diffusion of water within the pore structure, shows that the CCA representation successfully predicts the tortuosity of pellets S 1, $2 and P1; these silica pellets are characterised by narrow pore-size distributions with no evidence of heirarchical pore structures as determined by nitrogen desorption and mercury porosimetry analyses following the approach of RolleKampczyk et al. (1993). The good agreement with the experimental data obtained for the silica gel spheres, S1 and $2, is pleasing since the CCA structural representation had been proposed as a realistic representation of sol-gel silica materials. The difference between the results obtained using the Diffusion Flux and Whole Object Averaging Methods compared to the Random Walk Method has already been discussed and attributed to the ability of a given diffusing molecule to sample the complete void space during the simulation; a similar argument might explain the difference between tortuosities measured by MRI and PGSE NMR methods. For the case of an experiment which measures directly the self-diffusion coefficient (for example PGSE) the results will be sensitive to molecules randomly exploring their local volume and in this case the molecules will tend to follow the path of least resistance. When the experiment involves a measurement under an applied concentration gradient (as in the MRI experiments) then the measured diffusive flux will be dependent on the net transfer of diffusing species within a generally macroscopic heterogeneous region. At a molecular level the diffusion may still be dominated by motion along the path of least resistance, but now additional tortuosity is introduced via the heterogeneity on the larger scale as the molecular diffusion process avoids the higher tortuosity regions thereby leading to a lower experimental measurement of the diffusivity and hence higher value of the tortuosity. An alternative explanation, but one which is also dependent on macroscopic heterogeneity, has been proposed by Hollewand and Gladden (1995b) where it was suggested that the difference arises because tortuosity is sampled over different length and time scales in the two experimental methods. The PGSE NMR technique observes the ensemble average of molecular displacements occurring in a time typically 0.01 to 0.1 s corresponding to displacements of order 10 I.tm. In this case each molecule only samples a local environment defined by its displacement and the measured value of diffusivity is therefore an average over the sample volume of each of the microscopic displacements, In the MRI experiments molecules sample environments corresponding to the pellet length or radius which are in the range 1-10 mm. In a real heterogeneous system we expect both of these effects to be in operation and the relative importance of incomplete sampling of the void space and averaging during experimental measurements of tortusoity is the subject of ongoing study. Finally, we consider the Composite CCA structure. The immediate observation from this study is that the increased heterogeneity introduced by this approach leads to significant increases in tortuosity compared with the single CCA representation. It is of particular interest to note that for all samples other than S 1, $2 and P1 (discussed above) the nitrogen desorption and mercury porosimetry analyses showed evidence of multiple fractal structures. All these samples yielded PGSE NMR measurements of tortuosity larger than that predicted by the single CCA representation. In particular, the data recorded for samples G1 and G2 are noted. Our experimental studies would suggest that for these two specific silica gel materials, a fractal representation of structural disorder on both the pore-scale and the macroscopic scale is necessary, thereby explaining why the single CCA representation does not adequately describe transport processes in these sol-gel silicas. Comparison of the results of the Diffusion Flux and Whole Object Averaging Methods for calculating tortuosity also suggest an explanation for discrepancies between the PGSE NMR and MRI determinations of tortuosity. As discussed above, the PGSE NMR experiment measures an average of local molecular displacements over a fixed timescale, this process is physically very similar to finding an average value of M in the Whole Object Averaging Method. In the MRI method molecules sample environments corresponding to the pellet length or radius which is better described by the Diffusion Flux simulation. As seen in Fig. 5, the Diffusion Flux approach predicts consistently higher values of tortuosity compared to the Overall Averaging Method.
NMR and fractal modelling studies of transport in porous media
2271
CONCLUSIONS PGSE NMR and MRI have been used to demonstrate that the range of local diffusivities and spatial variation in pore size and diffusion coefficient play a significant role in determining the transport processes occurring within the pore space of commercial porous catalyst pellets. Spin-lattice relaxation time imaging experiments have shown that the heterogeneity in structure occurring over lengthscales of 0.05-1 mm in a given batch of pellets is characterised by a constant fractal dimension. These observations have been the motivation to simulating transport within such pore systems using fractal models. The 2D CCA structures and associated transport simulations of Elias-Kohav et al. (1993) have been extended to three dimensions. The tortuosities predicted using Whole Object Averaging, Diffusion Flux and Random Walk algorithms have been compared and the results discussed. Experimental measurements of tortuosity for ten commercial catalyst pellets all fall within the limits of the simulated values. Following the observation that a fractal parameter might be used to characterise disorder both at the microscopic, pore level in addition to the macroscopic level as discussed previously, a Composite CCA structure has been proposed. These structures predict higher tortuosities for a given voidage than do the single CCA structures. It is also suggested that different methods of simulating the transport process might be more appropriate in predicting transport under the varying boundary conditions presented by the PGSE NMR and MRI techniques used here. ACKNOWLEDGEMENTS We acknowledge the financial support and interest shown in this work by ICI, ZENECA and the EPSRC. S.P. Rigby wishes to thank EPSRC for an EPSRC studentship and ICI for a Research Scholarship.
NOTATION cluster area, dimensionless A concentration, dimensionless C fractai dimension d embedding dimension dem topological dimension dtop D diffusivity, dimensionless self-diffusivity of random walks on an empty lattice, dimensionless DAS effective diffusivity, dimensionless Deft pixel number i K proportionality constant between total path length and time of a random walk, dimensionless 1 total length of random walk, dimensionless M locally averaged number of blocked pixels adjacent to an empty site, dimensionless P cluster perimeter, dimensionless r pore radius, m
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S.P. RIGBYand L. E GLADDEN
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