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Characterization of macropores using quantitative microscopy
J. Rouquerol, F. Rodrigucz-Rcinoso, K.S.W. Sing and K.K. Unger (Eds.) Characterization of Porous Solids 111 Studies in Surhcc Science and Cat~lysis,Vol. 87 0 1994 Etscvicr Scicncc B.V. All rights rcscrvcd.
327
Characterization of macropores using quantitative microscopy
Brian McEnaney and Timothy J. Mays School of Materials Science, University of Bath, BATH, Avon BA2 7AY, United Kingdom
Abstract The application of quantitative microscopy, or image analysis, to the characterization of macropores is reviewed using as examples different types of porous carbon materials. The principal advantages of the technique are the ability to measure both open and closed porosity and the shape, location and orientation of pores. It is also an advantage to be able to measure these parameters for different classes of macropores in a given porous body. The stereological problems of relating two-dimensional measurements to structural parameters of the threedimensional material are a disadvantage, particularly when dealing with anisotropic and heterogeneous substances.
1.
INTRODUCTION
The performance of porous materials in many technical applications results from an interplay between properties that depend to different extents upon three main pore types: micropores (width, w < 2 nm); mesopores (2 Iw 1.50 nm) and macropores (w > 50 nm) [ 11. For example, the oxidation in air of anode carbons used in aluminium smelting depends upon the specific surface area and therefore mainly upon the extent of meso- and microporosity [2]. However, reactivity also depends upon the temperature of the anode surface, which is related to its thermal conductivity. This property is dominated by the nature and extent of macropores. Examples of other important properties that are influenced by pore structure include adsorption, electrical conductivity, Young's modulus and strength. Consequently, it is often necessary to characterize several pore types, using appropriate methods, in order to assess the overall performance of a material. Here, we focus attention on the characterization of macropores. There are three main methods presently is use for characterizing macropores: mercury porosimetry, fluid flow (i. e., diffusivity and permeability of gases and liquids) and microscopy. The relative merits of these techniques have been considered previously [3]. Porosimetry and fluid flow only probe open pores, i. e., pores that are accessible to fluids penetrating from external surfaces. This is a limitation where both open and closed pores are relevant, e. g., when electrical, thermal and mechanical properties are being considered, but one that does not apply with microscopical methods. A further limitation of porosimetry and fluid flow methods is that simple, model pore structures are usually assumed, e. g., bundles
328
of non-intersecting capillaries, whereas in microscopy pores are viewed directly and therefore an u priori assumption about pore structure is not required. This paper is a review of some of the capabilities of quantitative microscopy for characterizing macropores using examples taken mainly from work in the authors' group on porous carbons and graphites. These examples include measurements made using quantitative microscopy which can also be made using porosimetry and fluid flow, e. g., pore volume fractions, pore size distributions and fractal dimensions. In addition, examples are presented of measurements made using quantitative microscopy, such as pore shape, orientation and location, that are difficult or impossible to make using the other two methods. 2.
EXPERIMENTAL TECHNIQUES
The preparation of a sample for microscopy involves the usual methods of grinding and polishing a flat specimen (often in a resin mount) to yield a planar section through the porous body. Samples can be viewed in an optical microscope (resolution 2 1 pm) or a scanning electron microscope, SEM, for greater resolution. The development of an effective polishing technique is of critical importance to ensure a faithful definition of pore edges and to avoid artifacts that can cause pore enlargement or closure. It is also necessary to ensure effective contrast between pores and the solid matrix to facilitate 'segmentation' (see below). Open pores can be distinguished from closed pores by impregnation with a fluid of high optical or electron contrast, e. g., a fluorescent resin or a liquid metal respectively [4]. While manual techniques can be used to measure the geometry of pore cross-sections, 6'. g., using calibrated graticules either in the microscope system or on micrographs, they are laborious and error-prone. Instead, the most-widely used methods for quantitative microscopy of porosity involve computer-based image analysis. A block diagram of a typical image analysis system using optical microscopy is shown in Figure 1 . I I I I I I I I I I I I I I I
Figure 1. A block diagram of a typical image analysis system for quantitative microscopy. The general mode of operation of the image analysis system, Figure l., is as follows. A monochrome television camera captures the image from the microscope. This image is then
329 digitised by the computer into an array of square picture elements, 'pixels', whose locations in the array are stored in the computer. Each pixel is assigned a grey level corresponding to the average brightness of the image at its particular location. Typically, in modem, commercial image analysers, there are lo5 - lo6 pixels and -lo2 - lo3 grey levels in an image array. The grey image is next converted to a binary image by a process called 'segmentation', in which the operator selects a range of grey levels that corresponds to the objects of interest, i. e., macropores in the present case. In the binary image, the pixels within the segmented grey level range are 'on' while the rest are 'off'. Segmentation is a critical step in the process since it requires the subjective judgement of the operator. It is also for this reason that specimen preparation is so important, since it is necessary to ensure that all of the objects of interest have grey levels that fall within the range selected for segmentation. Modem image analysis systems are equipped with overlay facilities which allow the grey and binary images to be compared and a suite of image refining algorithms which the operator can use to ensure that the binary image is a faithful representation of the objects of interest. The final step in the image analysis process is the computation of geometrical parameters of objects in the binary image (see below). The image analysis system can easily make the large numbers of measurements that are required to ensure results are statistically significant. Typically many thousands of objects in a microscopical field can be measured, and many fields can be measured for a given sample using automatic stepping stages. Thus, once the parameters required for effective segmentation have been selected, a high degree of automation can be achieved. With objects where there is a wide range of grey levels, segmentation may be difficult. In such cases, the objects of interest can be selected and outlined manually, e. g., using a light pen. This circumvents the problem of segmentation, but the advantages of automation are lost. A cross-section through a macropore in a plane, polished surface is sketched in Figure 2. In a binary image, the continuous curve which is the perimeter of the pore section is represented by an irregular polygon whose vertices are the co-ordinates of the boundary pixels. The image analysis system then computes geometric parameters of this polygon. Some examples of such parameters are shown in Figure 2. In addition to simple parameters such as area, A, and perimeter, P, other, more complex parameters may also be measured, e. g., maximum and minimum Feret diameters, d, and dmi,, Figure 2. Derived parameters may also be computed, such as aspect ratio = d,,, / dmin,equivalent circle diameter = d(41cA) and roundness = P2I 4aA. If a reference axis is defined, then orientation parameters can also be determined, e. g., the angle 8 in Figure 2.
-
3.
EXAMPLES OF MACROPORE CHARACTERIZATIONS QUANTITATIVE MICROSCOPY
USING
Presented here are some examples of macropore characterizations using quantitative image analysis. While these examples are taken mainly from the authors' work on carbons and graphites many of the points that they illustrate are applicable to a wide range of macroporous solids.
330
'
boundary of macropore cross-section
approximating polygon whose vertices are the centres of boundary pixels smallest superscribed circle, diameter = d (maximum Feret diameter)
; I
perimeter, P
Figure 2. A sketch of a macropore cross-section in a plane, polished surface, and some examples of simple image analysis parameters.
3 . 1 , Distributions of pore areas and pore volumes A basic function of quantitative microscopy is the estimation of pore area distributions. For example, image analysis was used [5] to determine macropore size distributions in carbons made from phenolic resins, which are of interest as porous catalyst supports. The macropore sizes were correlated with fabrication parameters, e. g., pressing conditions and carbonization rates, and mechanical properties such as Young's modulus, flexural strength and toughness. These carbons provide simple images for analysis with sharp contrast between pores and matrix as shown in Figure 3. A typical pore area distribution for such carbons obtained using computer-based image analysis, Figure 4., shows a unimodal pore area distribution with a modal value of 8 pm 2.
-
331
20 l m Figure 3. Macropores in a resin-based carbon.
m
!i
2
1
0
0
1 log ( pore area, A / pm2 )
2
10
Figure 4. A pore area distribution for a resin-based carbon obtained from quantitative optical microscopy. To equate the pore area distribution to a pore volume distribution requires an assumption about the stereological relationship between the two-dimensional images and the threedimensional objects from which they are derived. This is Delesse's theorem and, in the present
332 context, it implies that the macropores are spherical and uniformly distributed throughout the sample. This assumption is often made in image analysis studies, including those on materials for which the theorem is clearly inappropriate, e. g., anisotropic materials or on objects which are far from spherical. In the present case a number of initial measurements were made including distributions of aspect ratio of pore sections cut in different directions in the sample which indicated that pores were approximately equiaxed. In addition, observations on a number of fields in different sections indicated that, subject to careful fabrication, pores were distributed uniformly throughout pressed carbon beams. This evidence for isotropy and homogeneity of the macropores in the resin-based carbons suggested that it was reasonable to apply Dellesse's theorem to these materials, i. e., the area fraction of pores, typically around 0.40, was a good approximation to the volume fraction. However, in general, the stereological problem of relating data from a two-dimensional image to structural parameters of the threedimensional body from which it is derived is a serious limitation of quantitative microscopy.
3 . 2 Pore shape and orientation The above example illustrates the application of quantitative microscopy to a class of macroporous materials which are particularly simple, i. e., they contain a single class of uniformly-distributed, near-spherical pores. However, in many solids pore structures are more complex, e. g., they comprise non-uniform distributions of pores of different shapes which originate at different stages in the manufacturing process. Quantitative microscopy has proved very useful in characterising the size, shape and orientation of different classes of pores in complex engineering materials. One class of pores of particular interest are cracks, i. e., pores of high aspect ratio. The orientation of cracks can play an important role in determining a number of properties of porous materials, e. g., strength, electrical conductivity and thermal expansion coefficent. The capabilities of image analysis in this area is illustrated with two examples: (i) grain orientations in an electrode graphite and (ii) cracking around fibres in a carbon-carbon composite. Synthetic graphites, used as electrodes in the smelting of metals, are made by extruding and subsequently heat-treating a mix of liquid pitch and solid, needle-coke filler grains. Figure 5. shows the general microstructure of an electrode graphite, including the macroporous calcination cracks that run parallel to the long axis of a filler grain. Quantitative microscopy was used to determine the angle, 9,between the maximum Feret diameter (see Figure 2.) of a single calcination crack - and hence the long axis of the filler grain - and the extrusion direction in each of a large number of grains across the radius of a graphite electrode [61. The extent of orientation of grains, as measured by the width of the distribution of 8 for localized regions, was greatest at the edge of the electrode log and least at its centre, see Figure 6. This was as expected from consideration of shear forces in the liquid mix during extrusion. The different extents of grain orientation, Figure 6., have implications for the radial variation of electrical conductivity of electrodes and hence their performance in service. This shows how quantitative microscopy of macropores can provide structural information that would be difficult or impossible to obtain using other methods.
333
1 rnm
Figure 5 . Microstructure in an electrode graphite. b - calcination cracks in the needle-coke filler, a-a'. d - globular rnacropores in binder phase, c. Extrusion direction parallel to base of micrograph.
20
.
15 10
0
> .-
c1
m
3
5
0 -90
-60
-30
orientation angle, 0 /
O,
0
30
60
90
with respect to extrusion direction
Figure 6. Distributions of orientation angle, 0, of axial, rnacroporous calcination cracks in the needle-coke filler grains of an electrode graphite, with respect to the extrusion direction of a cylindrical log.
334
Another example that illustrates this point concerns the nature of cracking around fibres in carbon-carbon composites, CC, i. e., composites consisting of carbon fibres in a carbon matrix. CC have high strength, stiffness and toughness which they retain to very high temperatures; they also have excellent thermal shock resistance. For this reason they have been used extensively in components for rocket engines and re-entry vehicles in aerospace engineering and they are also candidate first-wall materials in the next generation of fusion reactors [7]. However, little is known about the relationships between structure and properties in these materials under irradiation conditions. As part of a programme assessing the prospects of CC in fusion reactors, image analysis studies have been undertaken recently to characterize the nature of cracking around fibres in CC [7]. Such cracks may be important in determining the fracture and thermal expansion of the composites. Distributions of macroporous, fibrematrix cracks in a three-directionally reinforced CC have been measured on images provided by a scanning electron microscope. Figure 7. shows the model for quantifying these cracks in terms of a characteristic angle 8, and it illustrates the frequency of cracks as a function 8. There is preferential cracking in the arc 30 < 8 c 90 ', Figure 7., probably due differential stresses arising from the anisotropic reinforcement architecture of the composite. O
a
count
1"
90.120
21Q240
270-300
Figure 7. Analysis of macroporous fibre-matrix interface cracks in a three-directionally reinforced carbon-carbon composite. a. definition of angle 8; b. distribution function of 8 (from [7]).
3 . 3 Fractal analyses of macropore surfaces The examples of quantitative microscopy above have all involved the use of simple parameters of planar, Euclidian geometry to characterize macropore cross-sections, e. g., ma, Feret diameter, and orientation angle. However, image analysis can also be used to explore the fractal nature of objects, including pores. A simple technique is the divider stepping method derived from Richardson's classical work on the length of coastlines (see [S]). In the present context, if the perimeter of a pore viewed in the image analyser is fractal then the estimate of its length, P(n), using a yardstick of length n is related to the fractal dimension D by
335
where B is a constant. Thus, the value of D may be estimated by measuring the perimeter of the pore using chords (yardsticks) of different lengths. This can be done manually with a light pen or, alternatively, by using pixel size as a yardstick. Figure 2. shows that the image analyser represents the pore perimeter as an irregular polygon whose vertices are the coordinates of the boundary pixels. If the pixel size is varied by viewing the pore image at different magnifications, then the fractal dimension can be estimated. An example of a Richardson plot for a pore in a resin carbon obtained using this method is shown in Figure 8. The value of D = 1.33 obtained from this plot is in the range 1 < D < 2, as expected for a fractal line. However, the fractal nature of the pore perimeter is not clearly established because the range of yardstick size (n = 0.2 - 2.0 pm) is limited. Another limitation of this approach is that microscopical estimations of pore sizes are influenced by the magnification used if the pore edges are rounded as a result of polishing [9]. The estimation of fractal dimensions using image analysis has been reviewed by Kaye [lo].
2.9 h
E
2.8
3.
2.7 2.6 2.5 -0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
log 1o (pixel width / pm) Figure 8. A Richardson plot for a macropore in a resin-based carbon.
3 . 4 Comparison of quantitative microscopy and other methods No systematic studies appear to have been made yet to compare results from quantitative
microscopy, porosimetry and fluid flow for characterizing macropores in a particular material. However, cumulative open macropore size distributions for various graphites obtained from image analysis, using metallic and fluorescent impregnants for contrast enhancement, were compared with those obtained form mercury porosimetry [4].This work was part of a study of the influence of pore structure on the oxidation of graphite moderators in nuclear reactors. As Figure 9. shows, porosimetry underestimates mean pore sizes compared with those obtained from image analysis due, it was presumed, to the sensitivity of porosimetry to constrictions in the pore network.
336
OPV 15.2 */a
-
Characlcrtsl~c~ Q T O dimCMOn ( p m l
Figure 9. Comparison of cumulative open pore size distributions for a nuclear graphite obtained from quantitative optical microscopy and mercury porosimeay (from [4]). One implication of the different macropore size distributions in Figure 9. is that, if oxidation is not influenced by pore constrictions, then image analysis is more useful in studies of moderator corrosion than porosimetry. It may be noted that, in a separate study [ 111, it appeared that fluid flow, like image analysis, is not sensitive to pore constrictions. This suggests that macropore size distributions from image analysis and fluid flow might be similar. Further work is required to c o n f i i this expectation. 4 . CONCLUDING REMARKS The principal advantages of quantitative microscopy as a tool for characterising macropores in porous solids are the ability to measure both open and closed porosity and the shape, location and orientation of pores. It is also an advantage to be able to measure these parameters for different classes of macropores in a given porous body. The stereological problems of relating two-dimensional measurements to structural parameters of the threedimensional material are a disadvantage of the technique, particularly when delaing with anisotropic and heterogeneous substances. Acknowledgements
We thank: Dr T.D. Burchell of Oak Ridge National Laboratory, USA, and Dr A.J. Wickham of Nuclear Electric plc, UK, for permission to use Figures 7. and 9. respectively.
337 REFERENCES 1.
2. 3.
4. 5. 6. 7. 8. 9. 10.
11.
K.S.W. Sing, D.H. Everett, R.A.W. Haul, L. Moscou, R.A. Pierotti, J. Rouquerol and T. Siemieniewsaka, Pure Appl. Chem., 57 (1985) 603. K. Grjotheim and B.J. Welch, Aluminium Smelter Technology, 2nd edn., Aluminium Verlag, Dusseldorf, 1988. B. McEnaney and T.J. Mays, in: H. Marsh (ed.), Introduction to Carbon Science, Buttenvorths, London, 1989, pp. 153-196. J.V. Best, W.J. Stephen and A.J. Wickham, Prog. Nucl. Energy, 16 (1985) 127. B. McEnaney, I.M. Pickup and L. Bodsworth, Catal. Today, 7 (1990) 299. Y. Yin, B. McEnaney and T.J. Mays, Carbon, 27 (1988) 113. Y.Q. Fei, B. McEnaney, F.J. Derbyshire, and T.D. Burchell, In: Extended Abstracts 21st American Carbon Conference, American Carbon Society, Buffalo, 1993, pp. 66-67. B.B. Mandelbrot, Fractals: Form, Chance and Dimension, W.H. Freeman and Co., New York, 1983. J. Piekarczyk and R. Pampuch, Ceramurgia Int., 2 (1976) 177. B.H. Kaye, In: D. Avnir (ed.), The Fractal Approach to Heterogeneous Chemistry, Wiley, New York, 1989, pp. 55-66. T.J. Mays, PhD Thesis, University of Bath (1988).
327
Characterization of macropores using quantitative microscopy
Brian McEnaney and Timothy J. Mays School of Materials Science, University of Bath, BATH, Avon BA2 7AY, United Kingdom
Abstract The application of quantitative microscopy, or image analysis, to the characterization of macropores is reviewed using as examples different types of porous carbon materials. The principal advantages of the technique are the ability to measure both open and closed porosity and the shape, location and orientation of pores. It is also an advantage to be able to measure these parameters for different classes of macropores in a given porous body. The stereological problems of relating two-dimensional measurements to structural parameters of the threedimensional material are a disadvantage, particularly when dealing with anisotropic and heterogeneous substances.
1.
INTRODUCTION
The performance of porous materials in many technical applications results from an interplay between properties that depend to different extents upon three main pore types: micropores (width, w < 2 nm); mesopores (2 Iw 1.50 nm) and macropores (w > 50 nm) [ 11. For example, the oxidation in air of anode carbons used in aluminium smelting depends upon the specific surface area and therefore mainly upon the extent of meso- and microporosity [2]. However, reactivity also depends upon the temperature of the anode surface, which is related to its thermal conductivity. This property is dominated by the nature and extent of macropores. Examples of other important properties that are influenced by pore structure include adsorption, electrical conductivity, Young's modulus and strength. Consequently, it is often necessary to characterize several pore types, using appropriate methods, in order to assess the overall performance of a material. Here, we focus attention on the characterization of macropores. There are three main methods presently is use for characterizing macropores: mercury porosimetry, fluid flow (i. e., diffusivity and permeability of gases and liquids) and microscopy. The relative merits of these techniques have been considered previously [3]. Porosimetry and fluid flow only probe open pores, i. e., pores that are accessible to fluids penetrating from external surfaces. This is a limitation where both open and closed pores are relevant, e. g., when electrical, thermal and mechanical properties are being considered, but one that does not apply with microscopical methods. A further limitation of porosimetry and fluid flow methods is that simple, model pore structures are usually assumed, e. g., bundles
328
of non-intersecting capillaries, whereas in microscopy pores are viewed directly and therefore an u priori assumption about pore structure is not required. This paper is a review of some of the capabilities of quantitative microscopy for characterizing macropores using examples taken mainly from work in the authors' group on porous carbons and graphites. These examples include measurements made using quantitative microscopy which can also be made using porosimetry and fluid flow, e. g., pore volume fractions, pore size distributions and fractal dimensions. In addition, examples are presented of measurements made using quantitative microscopy, such as pore shape, orientation and location, that are difficult or impossible to make using the other two methods. 2.
EXPERIMENTAL TECHNIQUES
The preparation of a sample for microscopy involves the usual methods of grinding and polishing a flat specimen (often in a resin mount) to yield a planar section through the porous body. Samples can be viewed in an optical microscope (resolution 2 1 pm) or a scanning electron microscope, SEM, for greater resolution. The development of an effective polishing technique is of critical importance to ensure a faithful definition of pore edges and to avoid artifacts that can cause pore enlargement or closure. It is also necessary to ensure effective contrast between pores and the solid matrix to facilitate 'segmentation' (see below). Open pores can be distinguished from closed pores by impregnation with a fluid of high optical or electron contrast, e. g., a fluorescent resin or a liquid metal respectively [4]. While manual techniques can be used to measure the geometry of pore cross-sections, 6'. g., using calibrated graticules either in the microscope system or on micrographs, they are laborious and error-prone. Instead, the most-widely used methods for quantitative microscopy of porosity involve computer-based image analysis. A block diagram of a typical image analysis system using optical microscopy is shown in Figure 1 . I I I I I I I I I I I I I I I
Figure 1. A block diagram of a typical image analysis system for quantitative microscopy. The general mode of operation of the image analysis system, Figure l., is as follows. A monochrome television camera captures the image from the microscope. This image is then
329 digitised by the computer into an array of square picture elements, 'pixels', whose locations in the array are stored in the computer. Each pixel is assigned a grey level corresponding to the average brightness of the image at its particular location. Typically, in modem, commercial image analysers, there are lo5 - lo6 pixels and -lo2 - lo3 grey levels in an image array. The grey image is next converted to a binary image by a process called 'segmentation', in which the operator selects a range of grey levels that corresponds to the objects of interest, i. e., macropores in the present case. In the binary image, the pixels within the segmented grey level range are 'on' while the rest are 'off'. Segmentation is a critical step in the process since it requires the subjective judgement of the operator. It is also for this reason that specimen preparation is so important, since it is necessary to ensure that all of the objects of interest have grey levels that fall within the range selected for segmentation. Modem image analysis systems are equipped with overlay facilities which allow the grey and binary images to be compared and a suite of image refining algorithms which the operator can use to ensure that the binary image is a faithful representation of the objects of interest. The final step in the image analysis process is the computation of geometrical parameters of objects in the binary image (see below). The image analysis system can easily make the large numbers of measurements that are required to ensure results are statistically significant. Typically many thousands of objects in a microscopical field can be measured, and many fields can be measured for a given sample using automatic stepping stages. Thus, once the parameters required for effective segmentation have been selected, a high degree of automation can be achieved. With objects where there is a wide range of grey levels, segmentation may be difficult. In such cases, the objects of interest can be selected and outlined manually, e. g., using a light pen. This circumvents the problem of segmentation, but the advantages of automation are lost. A cross-section through a macropore in a plane, polished surface is sketched in Figure 2. In a binary image, the continuous curve which is the perimeter of the pore section is represented by an irregular polygon whose vertices are the co-ordinates of the boundary pixels. The image analysis system then computes geometric parameters of this polygon. Some examples of such parameters are shown in Figure 2. In addition to simple parameters such as area, A, and perimeter, P, other, more complex parameters may also be measured, e. g., maximum and minimum Feret diameters, d, and dmi,, Figure 2. Derived parameters may also be computed, such as aspect ratio = d,,, / dmin,equivalent circle diameter = d(41cA) and roundness = P2I 4aA. If a reference axis is defined, then orientation parameters can also be determined, e. g., the angle 8 in Figure 2.
-
3.
EXAMPLES OF MACROPORE CHARACTERIZATIONS QUANTITATIVE MICROSCOPY
USING
Presented here are some examples of macropore characterizations using quantitative image analysis. While these examples are taken mainly from the authors' work on carbons and graphites many of the points that they illustrate are applicable to a wide range of macroporous solids.
330
'
boundary of macropore cross-section
approximating polygon whose vertices are the centres of boundary pixels smallest superscribed circle, diameter = d (maximum Feret diameter)
; I
perimeter, P
Figure 2. A sketch of a macropore cross-section in a plane, polished surface, and some examples of simple image analysis parameters.
3 . 1 , Distributions of pore areas and pore volumes A basic function of quantitative microscopy is the estimation of pore area distributions. For example, image analysis was used [5] to determine macropore size distributions in carbons made from phenolic resins, which are of interest as porous catalyst supports. The macropore sizes were correlated with fabrication parameters, e. g., pressing conditions and carbonization rates, and mechanical properties such as Young's modulus, flexural strength and toughness. These carbons provide simple images for analysis with sharp contrast between pores and matrix as shown in Figure 3. A typical pore area distribution for such carbons obtained using computer-based image analysis, Figure 4., shows a unimodal pore area distribution with a modal value of 8 pm 2.
-
331
20 l m Figure 3. Macropores in a resin-based carbon.
m
!i
2
1
0
0
1 log ( pore area, A / pm2 )
2
10
Figure 4. A pore area distribution for a resin-based carbon obtained from quantitative optical microscopy. To equate the pore area distribution to a pore volume distribution requires an assumption about the stereological relationship between the two-dimensional images and the threedimensional objects from which they are derived. This is Delesse's theorem and, in the present
332 context, it implies that the macropores are spherical and uniformly distributed throughout the sample. This assumption is often made in image analysis studies, including those on materials for which the theorem is clearly inappropriate, e. g., anisotropic materials or on objects which are far from spherical. In the present case a number of initial measurements were made including distributions of aspect ratio of pore sections cut in different directions in the sample which indicated that pores were approximately equiaxed. In addition, observations on a number of fields in different sections indicated that, subject to careful fabrication, pores were distributed uniformly throughout pressed carbon beams. This evidence for isotropy and homogeneity of the macropores in the resin-based carbons suggested that it was reasonable to apply Dellesse's theorem to these materials, i. e., the area fraction of pores, typically around 0.40, was a good approximation to the volume fraction. However, in general, the stereological problem of relating data from a two-dimensional image to structural parameters of the threedimensional body from which it is derived is a serious limitation of quantitative microscopy.
3 . 2 Pore shape and orientation The above example illustrates the application of quantitative microscopy to a class of macroporous materials which are particularly simple, i. e., they contain a single class of uniformly-distributed, near-spherical pores. However, in many solids pore structures are more complex, e. g., they comprise non-uniform distributions of pores of different shapes which originate at different stages in the manufacturing process. Quantitative microscopy has proved very useful in characterising the size, shape and orientation of different classes of pores in complex engineering materials. One class of pores of particular interest are cracks, i. e., pores of high aspect ratio. The orientation of cracks can play an important role in determining a number of properties of porous materials, e. g., strength, electrical conductivity and thermal expansion coefficent. The capabilities of image analysis in this area is illustrated with two examples: (i) grain orientations in an electrode graphite and (ii) cracking around fibres in a carbon-carbon composite. Synthetic graphites, used as electrodes in the smelting of metals, are made by extruding and subsequently heat-treating a mix of liquid pitch and solid, needle-coke filler grains. Figure 5. shows the general microstructure of an electrode graphite, including the macroporous calcination cracks that run parallel to the long axis of a filler grain. Quantitative microscopy was used to determine the angle, 9,between the maximum Feret diameter (see Figure 2.) of a single calcination crack - and hence the long axis of the filler grain - and the extrusion direction in each of a large number of grains across the radius of a graphite electrode [61. The extent of orientation of grains, as measured by the width of the distribution of 8 for localized regions, was greatest at the edge of the electrode log and least at its centre, see Figure 6. This was as expected from consideration of shear forces in the liquid mix during extrusion. The different extents of grain orientation, Figure 6., have implications for the radial variation of electrical conductivity of electrodes and hence their performance in service. This shows how quantitative microscopy of macropores can provide structural information that would be difficult or impossible to obtain using other methods.
333
1 rnm
Figure 5 . Microstructure in an electrode graphite. b - calcination cracks in the needle-coke filler, a-a'. d - globular rnacropores in binder phase, c. Extrusion direction parallel to base of micrograph.
20
.
15 10
0
> .-
c1
m
3
5
0 -90
-60
-30
orientation angle, 0 /
O,
0
30
60
90
with respect to extrusion direction
Figure 6. Distributions of orientation angle, 0, of axial, rnacroporous calcination cracks in the needle-coke filler grains of an electrode graphite, with respect to the extrusion direction of a cylindrical log.
334
Another example that illustrates this point concerns the nature of cracking around fibres in carbon-carbon composites, CC, i. e., composites consisting of carbon fibres in a carbon matrix. CC have high strength, stiffness and toughness which they retain to very high temperatures; they also have excellent thermal shock resistance. For this reason they have been used extensively in components for rocket engines and re-entry vehicles in aerospace engineering and they are also candidate first-wall materials in the next generation of fusion reactors [7]. However, little is known about the relationships between structure and properties in these materials under irradiation conditions. As part of a programme assessing the prospects of CC in fusion reactors, image analysis studies have been undertaken recently to characterize the nature of cracking around fibres in CC [7]. Such cracks may be important in determining the fracture and thermal expansion of the composites. Distributions of macroporous, fibrematrix cracks in a three-directionally reinforced CC have been measured on images provided by a scanning electron microscope. Figure 7. shows the model for quantifying these cracks in terms of a characteristic angle 8, and it illustrates the frequency of cracks as a function 8. There is preferential cracking in the arc 30 < 8 c 90 ', Figure 7., probably due differential stresses arising from the anisotropic reinforcement architecture of the composite. O
a
count
1"
90.120
21Q240
270-300
Figure 7. Analysis of macroporous fibre-matrix interface cracks in a three-directionally reinforced carbon-carbon composite. a. definition of angle 8; b. distribution function of 8 (from [7]).
3 . 3 Fractal analyses of macropore surfaces The examples of quantitative microscopy above have all involved the use of simple parameters of planar, Euclidian geometry to characterize macropore cross-sections, e. g., ma, Feret diameter, and orientation angle. However, image analysis can also be used to explore the fractal nature of objects, including pores. A simple technique is the divider stepping method derived from Richardson's classical work on the length of coastlines (see [S]). In the present context, if the perimeter of a pore viewed in the image analyser is fractal then the estimate of its length, P(n), using a yardstick of length n is related to the fractal dimension D by
335
where B is a constant. Thus, the value of D may be estimated by measuring the perimeter of the pore using chords (yardsticks) of different lengths. This can be done manually with a light pen or, alternatively, by using pixel size as a yardstick. Figure 2. shows that the image analyser represents the pore perimeter as an irregular polygon whose vertices are the coordinates of the boundary pixels. If the pixel size is varied by viewing the pore image at different magnifications, then the fractal dimension can be estimated. An example of a Richardson plot for a pore in a resin carbon obtained using this method is shown in Figure 8. The value of D = 1.33 obtained from this plot is in the range 1 < D < 2, as expected for a fractal line. However, the fractal nature of the pore perimeter is not clearly established because the range of yardstick size (n = 0.2 - 2.0 pm) is limited. Another limitation of this approach is that microscopical estimations of pore sizes are influenced by the magnification used if the pore edges are rounded as a result of polishing [9]. The estimation of fractal dimensions using image analysis has been reviewed by Kaye [lo].
2.9 h
E
2.8
3.
2.7 2.6 2.5 -0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
log 1o (pixel width / pm) Figure 8. A Richardson plot for a macropore in a resin-based carbon.
3 . 4 Comparison of quantitative microscopy and other methods No systematic studies appear to have been made yet to compare results from quantitative
microscopy, porosimetry and fluid flow for characterizing macropores in a particular material. However, cumulative open macropore size distributions for various graphites obtained from image analysis, using metallic and fluorescent impregnants for contrast enhancement, were compared with those obtained form mercury porosimetry [4].This work was part of a study of the influence of pore structure on the oxidation of graphite moderators in nuclear reactors. As Figure 9. shows, porosimetry underestimates mean pore sizes compared with those obtained from image analysis due, it was presumed, to the sensitivity of porosimetry to constrictions in the pore network.
336
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-
Characlcrtsl~c~ Q T O dimCMOn ( p m l
Figure 9. Comparison of cumulative open pore size distributions for a nuclear graphite obtained from quantitative optical microscopy and mercury porosimeay (from [4]). One implication of the different macropore size distributions in Figure 9. is that, if oxidation is not influenced by pore constrictions, then image analysis is more useful in studies of moderator corrosion than porosimetry. It may be noted that, in a separate study [ 111, it appeared that fluid flow, like image analysis, is not sensitive to pore constrictions. This suggests that macropore size distributions from image analysis and fluid flow might be similar. Further work is required to c o n f i i this expectation. 4 . CONCLUDING REMARKS The principal advantages of quantitative microscopy as a tool for characterising macropores in porous solids are the ability to measure both open and closed porosity and the shape, location and orientation of pores. It is also an advantage to be able to measure these parameters for different classes of macropores in a given porous body. The stereological problems of relating two-dimensional measurements to structural parameters of the threedimensional material are a disadvantage of the technique, particularly when delaing with anisotropic and heterogeneous substances. Acknowledgements
We thank: Dr T.D. Burchell of Oak Ridge National Laboratory, USA, and Dr A.J. Wickham of Nuclear Electric plc, UK, for permission to use Figures 7. and 9. respectively.
337 REFERENCES 1.
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K.S.W. Sing, D.H. Everett, R.A.W. Haul, L. Moscou, R.A. Pierotti, J. Rouquerol and T. Siemieniewsaka, Pure Appl. Chem., 57 (1985) 603. K. Grjotheim and B.J. Welch, Aluminium Smelter Technology, 2nd edn., Aluminium Verlag, Dusseldorf, 1988. B. McEnaney and T.J. Mays, in: H. Marsh (ed.), Introduction to Carbon Science, Buttenvorths, London, 1989, pp. 153-196. J.V. Best, W.J. Stephen and A.J. Wickham, Prog. Nucl. Energy, 16 (1985) 127. B. McEnaney, I.M. Pickup and L. Bodsworth, Catal. Today, 7 (1990) 299. Y. Yin, B. McEnaney and T.J. Mays, Carbon, 27 (1988) 113. Y.Q. Fei, B. McEnaney, F.J. Derbyshire, and T.D. Burchell, In: Extended Abstracts 21st American Carbon Conference, American Carbon Society, Buffalo, 1993, pp. 66-67. B.B. Mandelbrot, Fractals: Form, Chance and Dimension, W.H. Freeman and Co., New York, 1983. J. Piekarczyk and R. Pampuch, Ceramurgia Int., 2 (1976) 177. B.H. Kaye, In: D. Avnir (ed.), The Fractal Approach to Heterogeneous Chemistry, Wiley, New York, 1989, pp. 55-66. T.J. Mays, PhD Thesis, University of Bath (1988).