- Email: [email protected]
Pore size determination from charged particle energy loss measurement
NUCLEAR INSTRUMENTS AND METHODS 143 (1977) 595-600 ; ©
NORTH-HOLLAND PUBLISHING CO.
P O R E SIZE D E T E R M I N A T I O N F R O M CHARGED PARTICLE ENERGY LOSS M E A S U R E M E N T F. P. BRADY* and B. H. ARMITAGE
U.K.A.E.A., Harwell, Didcot, Oxon., OXll ORA, England
Received 14 February 1977 A new method aimed at measuring porosity and mean pore size in materials has b~n developed at Harwell. The energy width or variance of a transmitted or backscattered charged particle beam is measured and related to the mean pore size via the assumption, that the variance in total path length in the porous material is given by (dx~= na 2, where n is the mean number of pores and a the mean pore size. It is shown on the basis of a general and rigorous theory of total path length distribution that this approximation can give rise to large errors in the mean pore size determination particularly in the case of large porosities (e>0.5). In practice it is found that it is not easy to utilize fully the general theory because accurate measurements of the first four moments are required to determine the means and variances of the pore and inter-pore path length distributions. Several models for these distributions are proposed. When these are incorporated in the general theory the determinations of mean pore size from experimental measurements on powder samples are in good agreement with values determined by other methods. 1. I n t r o d u c t i o n
During the past few years a new m e t h o d aimed at measuring porosity and pore size in materials has been u n d e r d e v e l o p m e n t at Harwell Laboratories t h r o u g h the use of charged particle beams. Originally it was s h o w n 1) that by m e a s u r i n g the average energy loss and the energy width of a proton b e a m after passage t h r o u g h a porous sample target (of u n i f o r m geometrical thickness) it is possible to d e t e r m i n e the porosity of the material and, u n d e r certain a s s u m p t i o n s , the average pore size. H o w e v e r a g r e e m e n t b e t w e e n these d e t e r m i n a t i o n s and m e a s u r e m e n t s of pore size based on other techniques has been on the whole not completely satisfactory and for large porosities the discrepancies a m o u n t to differences of several h u n d r e d percent. Following a very brief review of the b a c k g r o u n d and experimental technique 1,2) this report focuses on the theoretical model and calculational procedure used to d e t e r m i n e the m e a n pore size from the energy width. It is found that the simple model which has been used to calculate the m e a n pore size can give rise to considerable error particularly in the case of high porosity (>0.5). Model dependent calculations, based on a general theory due to Clement3), are proposed for d e t e r m i n i n g pore size more accurately. W h e n the experimental data are analysed on the basis of these calculatfons the results are in m u c h better a g r e e m e n t with pore sizes obtained from gas adsorption m e a s u r e m e n t s . * On leave from the University of California Davis, CA 95616, U.S.A.
M e a s u r e m e n t s of porosity and m e a n pore size are of importance in m a n y areas of science and engineering, and in technology. Closely related problems are those of particle size m e a s u r e m e n t and the packing of particulate solids. For example porosity and pore size are relevant in oil and water production, and the controlled packing of particle solids, over a wide range of particulate sizes, is important in powder metallurgy, the loading of nuclear fuel e l e m e n t s , the production of c e m e n t s and ceramics, and the structure of catalysts and molecular separation columns. Current techniques of porosity m e a s u r e m e n t are based mainly on the mercury porosimeter or gas adsorption. I n f o r m a t i o n on the size distribution of pores accessible from the sample surface is obtainable by the penetration of a non-wetting liquid such as mercury. In the mercury porosimeter pressure is applied to the mercury such that as the pressure is increased open pores are penetrated of progressively decreasing entrance diameter. T h e gas adsorption technique provides information on the specific surface area of a sample u n d e r investigation. In addition pore size information is obtained s o m e w h a t less directly from an analysis of capillary condensation. In the charged particle m e t h o d on the other hand, one a t t e m p t s to relate the m e a s u r e m e n t s of the energy s p e c t r u m to the m e a n path length across the pores and thus derive a basic physical quantity which is different from those arising from the a b o v e - m e n t i o n e d methods. T h e energy width of a charged particle b e a m
596
F.
P.
BRADY
AND
transmitted through a porous sample is enhanced over and above the energy width (due mainly to straggling) for a non-porous sample of the same material and areal density. The explanation, as given in ref. 1 is that ir~dividual protons of the beam, whose cross-sectional area is assumed to be much greater than the pore size, have paths which pass through different numbers of pores and hence intercept different net total thicknesses of material. Thus they undergo different energy losses as they traverse the sample. The model assumed 1) is that of a random distribution of pores whose length along the direction of the beam is uniform and equal to a. Then the variance in total path length through solid material is na 2, where n is the mean number of pores traversed and a is the (mean) path length across the pores. Thus, both the effects of path length dispersion across a given pore, and also pore size dispersion are neglected. The connection with the measured energy distribution of the transmitted beam is made via ( A E 2 ) ~ - n a 2 ( d E / d x ) 2, where (AE 2) is the variance about the mean energy loss of the energy distribution and d E / d x is the mean differential energy loss in the sample. Usually Gaussian distributions are assumed 1,2) so that the experimental (AE2)--(fwhm/2.35) 2, where fwhm is the full width at half-maximum of the energy distribution. The mean energy loss of the beam provides a measure of the effective thickness, te, of material (in equivalent length of matrix material or in areal density), and thus a measure of na via na = t o - t e , where to is the geometrical thickness of the sample. It is then possible to calculate n and a and to show that the mean pore size, a, is proportional to the energy variance ( A E 2 ) , the latter having been corrected for the straggling variance as measured with an equivalent non-porous sample. The porosity, e, can also be calculated : e = ( t o - re)/to. More recently this method of pore size measurement has been extended by using resonant backscattering from nuclei in the porous materail2). In partiizular the enhancement of the resonance width in the elastic backscattering of protons and alpha particles from 160 at 2.66 and 3.05 MeV lab energy respectively is measured at 170° lab. The width of the resonance in a porous oxide is enhanced over that measured with the equivalent non-porous oxide. Details concerning the experimental and calculational methods are given in ref. 2. The calculation of mean pore size, a, is made by assuming,
B.
tt.
ARMITAGE
in analogy to the case of transmission geometry, that the variance in path length is 2 n a 2, where n is the mean number of pores encountered on the way in, prior to the scattering. This number is assumed to be equal to, but independent of, the mean number encountered on the way out. 2. Theoretical calculations The distribution of total path lengths of a beam of particles through a porous material has been considered by Clement3). The assumption is made that there is no correlation between the distances through pores and those in between the pores, i.e. that the path lengths, x;, between pores and the path lengths, y;, through pores are independent random variables with means and variances (~1, ~ ) and (g2, ~ ) r e s p e c t i v e l y . A requirement is that the mean number of pores, n, traversed in the mean total path length, x, be large, where a total path length through the material is x = ~,x;. Clement 3) was not able to obtain an explicit form for the total path length distribution, but was able to calculate the moments. The first x moment is E l ( x ) = n~l and the higher moments about El(X) (leading terms) are E2(x ) = n ( ~ 2 + 0 - 2 ) ;
E3(x) = - 6 / / 3 n 2 _].. 30-2/tn ;
E 4 ( x ) = 12n3/~ 4 + 3/'/202(0 2 - 10//2) ;
Em(x)
=
(-1)
m
Em(y),
(m ~
2),
where ]/
~-
2 - - 0"2 0"2~ 2 2 0"~ +0-2
~//2
2 ,
0"
--"
2 2 0" 1 0"2 2
2,
0- +0-2
and n is the mean number of pores. Unfortunately the basic parameters sensitive to the higher moments" n = 3E3/4E~,
are very
It = - 2 E3/3 E] , etc.
These higher moments are difficult to determine accurately from the experimental data, which are usually in the form of energy spectra whose accuracies are limited by counting statistics and experimental effects such as low-energy tails due to slit scattering and detector characteristics. The first and second moments can in general be determined quite well from the data. It is instructive to compare the predictions of the simple model, in which pore size variances are neglected, and which gives the second moment as E 2 ( a ) = na 2, with the predictions according to the theory of
PORE
I
I
I
I
I
I
SIZE
DETERMINATION
length distributions across the pores and in between the pores and to calculate pj and ~(j = 1, 2) for these distributions. For example exponential distributions of paths in and between pores correspond to
I
b l = b 2 =1
,.6
.......
bl = ~ ,
X
___
b,=~s.b2-'
\
_._
b2=1
bl~1OqgDgiQ
~
-
.
.\.
~1o,
fx~'~X~x "" \ _. ..
Y"..
!\
\
5- 0 . 8 - - - \ ~~-1"1 _ _
o
\ \
'....
\
X\
\:
"..
.
X \ "\",
\
\\\ x,...\ \
_
\
0.6 ~
\ \ -...\.\. \ .,. \
\
,.
",
o,-
0.2 -
o,o
\ ~
!
1
I
0.2
0.3
0./.
"\
1
I
1
1
0.5
0.6
0.7
0.8
POROSITY
Fig. 1. The ratio of mean pore size, a, calculated from the simple model to the mean pore size, P2, calculated from more rigorous theory is plotted against porosity using various assumptions for the pore and particle path length distributions.
Clement 3)
where
the
second moment is To be quantitative, further assumptions need to be made. Models for the distributions of path lengths, x~, between pores and of path lengths, y~, through the pores can be assumed and for each distribution a mean, pj, and a variance, ~ , with ~ = b j p f ( / ' = 1 , 2 ) c a n be calculated. The ratio pl/P2 is related to the porosity, :/1 ~/P2 = ~- ~- 1. (In the transmission geometry can be determined using the mean energy loss, but in the backscattering case it must be otherwise determined.) One can write
E20; ) -- E2(P2 ) -- r/(p 2 + 02).
E2(P2)
=nP
El(p2) =
2" F(e, b, b2), npl = np2(e-'--1).
(1)
Given bl and b 2 and ~ one can calculate P2 from E2(P2) and E~(p2). The simple model, where tr2 = 0, gives E 2 ( a ) - n a 2 and E 1 - - h a ( e - l - - I ) . The mean pore size in each case is calculated from E2/E1, hnd the ratio for the two cases is
a/P2 =
597
F(,~, b l , b2).
To calculate F as a function of porosity one needs to assume specific models for the path
b 2 - 1 ( ~ - p~, ~ - ~2).
For this case the ratio a / p 2 - F is plotted against porosity as the continuous curve in fig. 1. It can be seen that the simple theory can be seriously in error particularly for large porosities where it gives much too small a value for mean pore size. (Here mean "size" means the average path length across the pores.) A uniform distribution of pore path lengths extending from zero to some maxim u m path length has b 2 - ] . F ( e , 1, ]), the case for b ~ - 1 and b 2 - ] , is shown in fig. 1 as the dash-dot curve. For spherical pores of uniform size, radius R, the path length frequency distribution through the pores can be shown to increase linearly up to a m a x i m u m path lenght of 2R and to have p 2 - 4 R / 3 and ~ - p 2 / 8 . Many, if not most, porous materials are in fact packings or compacts of solid particles and often one can justifiably speak of p! and 0 1 as arising from the distributions of path lengths through the individual particles. A case of particular interest here, in that energy spectra have been measured, is that of a powder or packing of very small and nearly spherical particles whose radii have a small distribution of values about R. In this case P l - 4 R / 3 and ~ - p ~ / 8 so b l - 1/8. In addition to b 1 a value of b2 is needed in order to calculate F(e, bl, b2) as a function of porosity. Once the porosity is specified the ratio a/p: and p: can be calculated from F and from measurements which provide the first and second moments of the total path length distribution. These are directly related via energy loss data to the corresponding moments of the energy spectrum. The measurements have been made 2) on powders of alumina and silica with porosities between 0.50 and 0.75. We still need to characterize the path length distribution through the pores of the packings of spherical or nearly spherical particles and so determine b2. We shall assume for these high porosity samples that a good approximation to the pore path length distribution is that of an exponential, so b: can be taken equal to unity. In justification consider a random distribution of spheres of crosssectional area C - ~ R 2. Suppose for the m o m e n t that the thickness of the spheres in the direction
598
v.
P. BRADY
AND
of the beam of particles is negligible so that one has essentially a random distribution of discs with their axes parallel to the beam. Then the path distribution in the Z direction is e x p ( - N C Z ) , where N is the mean number of discs per unit volume, N = 3(1-e)/(4zr R3). The mean path length, P2, is (NC) - ~ - ] R / ( 1 - e ) ( - 4 R for e - ~). For a distribution of spheres one must correct for their finite thickness and for their shape. If one subtracts from (NC) -1 the mean path length, P l, through a sphere ( P l - J R ) t h e n the resulting mean pore path length, /z2, is that which satisfies p l / ~ 2 2 = e - 1 - 1. Several geometrically systematic arrangements or regular packings 6f spheres were considered as another possible way of calculating the pore path length distribution. However for the case of the relevant high porosity packings such as the cubic and tetrahedral, very long path lengths are allowed, and the path length distribution is unbounded as the path length increases indefinitely. A cut-off can be assumed on the basis that the sample is finite or that in practice a distribution of sphere sizes is present, but the path length cut-off is somewhat arbitrary. In any case it seems likely that the distribution of spheres is essentially random aside from the effects imposed by finite size. Thus the pair distribution function should be similar to that of a liquid. Regular geometric packing, where it does occur, is expected to be one of short range order. BernaP) considered the random assembly of hard spheres, and obtained the general shape of the two-molecule pair distribution function, g(r), of a liquid from geometrical packing conditions. Using his g(r) we derived from a combination of analytical and graphical calculations the approximate distribution of paths, N(y), between spheres. This distribution is more linear than an exponential for short paths but on the whole the exponential provides a reasonable approximation. It turns out that the factor F(e, bl, b2) is not very sensitive to b2 for the large porosities of interest here. If b2 is increased from 1.0 to 1.5, F increases by only 4% at e - ~ and by 15% at e - 1. The dependence of F on porosity for the case of b 2 - 1 and b l - ~, is shown by the dotted curve in fig. 1. Also plotted, (dashed curve) are the values of F for b 2 - 1 and b l - ~. The latter b~ allows for the distribution of sphere sizes in the silica and alumina powders and compacts of experimental interest here. A question of interest is why the simple model,
B. H. A R M I T A G E
t,0
AI203 35
{~10)
~" 30 E
w
25
N
w 20
g
Si 02 (150)
-
AI203 Si02 (~5) (/.,5) ~ / 1l / * /, ///i
A,203 (150)
1
{3_
z 15 IE ,,,"
#2 o'" . . . I
GA...,,"
0.5
~
l
0-6 ' POROSITY
-
-
(j.7
Fig. 2. The pore size for six samples determined from experimental measurements or the second moments as calculated using in one case the approximation (Ax2)= na 2 (solid circles) and in the second case model dependent general theory (the open circles) is compared to gas adsorption measurements (squares connected by the dashed lines). The lines connecting the points are solely to guide the eye. The number in parentheses just below the chemical formula or the the sample is the compaction pressure in MPa.
which assumes (Ax2) = na 2, is such a poor approximation for large porosities. This model corresponds to neglecting the variance in path length through the pores, i.e. setting a 2 = 0 (and /~2=a). With a2 = 0 the dependence of the second moment, E2, on dl vanishes. In the case of large porosity (>0.5) the mean pore path length/12 is larger than/~l, and the mean path length variance. ~ , is expected in general to be relatively large. In this case it would seem to be more appropriate to assume (Ax 2) - nl~ - n~z~(a- 1 - 1 ) 2 - nc2(e - 1 - 1)2 (putting/~2 = c) which neglects the variance, ~ , in /~1. In this case a/c -- ( e - ~ - 1)2. This dependence on e is shown in fig. 1 as the dash-two-dot curve which is similar to the curves for those cases where an exponential pore path distribution is assumed. As expected this dependence is very close to the F(e, 81, 1) and F(e, 15, 1) curves for porosities
~>~.
The approximation (Ax2) - ritzY, might be called model-independent in the sense that no model ex-
PORE
SIZE
DETERMINATION
pressions for the pore or particle path length distributions are required. In any case this approximation represents for large porosities an improvement over that of (zlx 2) = na 2 which has been used earlier~,2). For small porosities, e <0.5, (Ax 2 ) - n/u~ is a poor approximation. Lacking further information concerning the sample or accurate measurements of the higher moments one should use (dx2)= na 2, where the porosity is less than 0.5. The ultimate goal is not to rely on models for the path-length distributions but to determine from experiment the parameters which characterize the pore and particle path length distributions, This goal, [after Clement3)] can be achieved by accurate experimental determinations of the moments up to and including the fourth. However an attempt to determine such sets of moments from measured spectra 7) of a proton beam transmitted through porous samples did not produce self-consistent or reasonable sets of the parameters ~Zl, al, ~z2, a2, and n. The spectra when measured were not intended to be used for more than first and second moment determinations and very likely the inconsistencies are due to limited statistical accuracy of the distribution tails and to slit scattering effects. Nevertheless it should in principle be possible in the transmission case to obtain data of sufficient quality. In the backscattering case this will be much more difficult as the "background" spectra are distorted by various other effects such as those due to other resonanceg, A slightly different and more direct approach would be to work with the total path length distribution rather than with the moments. The calculated distribution could be transformed into an energy-loss distribution (including the effects of straggling) and then fit to the experimentally measured energy distribution by varying the path length parameters so as to minimize a Z 2 in the usual way. This procedure will also require very good data. In addition the usual ambiguities of seeking m i n i m u m Z 2 in multiparameter space will arise, 3. Comparison with experimental results The technique of resonance backscattering from nuclei has been used by Mackenzie and Armitage 2) to determine the mean pore size, a, of porous powders of silica and alumina. The powders, consisting of small spherical or nearly spherical particles with a small size distribution, were made at Harwell by Avery and Ramsay as described in
599
ref. 5. Some samples were compacted at pressures of 45 and 150 MPa. As described in ref. 2 the data were analysed and the values for the mean pore size, a, calculated assuming ( d x 2) = na 2. These mean values of a for 4 alumina (A1203) and 2 silica (SIO2) samples are plotted in fig. 2 as the solid circles. The actual values plotted are slightly different from these obtained in ref. 2. The data were reanalysed using stopping power and range-energy curves constructed from those of ref. 6 which are deemed more appropriate for oxygen in a solid. The silica values are about 20% less and the alumina values about 10% less than those of ref. 2. The experimental errors in the values of a are estimated to be about 20%. Also plotted in fig. 2 are values of/~2 obtained from the values of a by dividing by the factor F(e,b~,b2), taken here to be F(e, 15, 1) for the reasons discussed in section 2. The resulting values of the mean pore size are larger and are in considerably better agreement with the values measured by Avery and Ramsay s) using the method of gas adsorption. These are shown by the squares. The agreement is not perfect. However one needs to keep in mind that the two techniques measure different physical quantities. The charged particle method measures essentially the mean path length across the pores averaged over the cross sections of the pores. In gas adsorption measurements it is usually assumed that one is measuring a mean radius which corresponds to that of the pore throat. For example, in a cubic packing, e =0.47, the throat radius 5) is rv--0.41 R, where R is the particle radius. For tetragonal e --0.66 and rw = 0.73 R. Therefore allowing for these physical differences and for experimental errors of about 20% on both sides one can conclude that the results of the two methods are consistent with each other. There are a number of measurements of mean pore size in graphite ~,7) using the energy width of a transmitted proton beam. The porosities of the samples ranged from 0.18 to 0.25 and the mean pore sizes from about 10 to 50 lzm. The data were analysed and mean pore size, a, computed via (dx2) - na 2. The results were compared to those values, aM, found using a mercury porosimeter (MP). The mean of the ratio a/aM varied from 1.1 to 2.2 over the six samples with a mean of 1.65. This agreement is of modest quality, however again it must be kept in mind that there exists evidence that the MP can produce mean values on the small side in that some larger pores are acces-
600
v.p.
BRADY AND B. H. ARMITAGE
sible only through a smaller throat. It can be seen from fig. 1 that the agreement can be improved if the exponential path length distributions are assumed. However only in about half the cases could the MP distributions be approximated by exponentials, Calculations based on the simple assumption that (Ax 2>= n a 2 thus appear to provide a tolerable approximation for e<0.5, but become progressively more in error as e increases above 0.5. As described earlier a reasonable approximation for the latter case is < A x 2 > - n c 2 ( e - l - 1 ) 2. Accordingly the high porosity data of ref. 2 were analysed using this method and agreement with gas absorption values was nearly as good as that obtained with model dependent values (fig. 2). The overall agreement between mean pore sizes calculated using these simple approximations with those obtained from other techniques (mercury porosimeter and gas adsorption) was within 40%. These results were obtained from 6 low porosity and 6 high porosity measurements, and covered a pore size range of a factor of 104 (5 nm to 50~m). The charged particle values, however, were nearly always larger than those obtained from mercury porosimetry and gas adsorption, so that part of the difference can be assumed to be due to basic physical differences in the techniques, 4. Conclusions The method of measuring pore size in materials via the use of MeV charged particle beams is one which provides essentially different physical information concerning pore size and structure. When using this technique, which measures the mean path length across the pores, some care must be taken in the calculation of mean pore size from the experimental energy distributions. Calculations based on the simple assumption that the variance in path length is given by ( A x 2> = n a 2 will be a fairly reliable approximation for e <0.5 but will become seriously in error for e >0.5. For the latter a fairly good and simple approximation is - n/u~(e - ~ - 1)2. Agreement to within ~-40% with mercury porosimeter and gas adsorption measurements has been obtained using these simple approximations •
Ideally the general theory of Clement 3) should be used to calculate the means and variances of
the path length distributions inside and between the pores. In practice we have found that it is difficult to utilize fully the general theory because rather accurate measurements of the first four moments are required in order to characterize the two path length distributions. To achieve this the experimental data would have to be of high statistical accuracy and essentially free from instrumental effects such as slit scattering. It has been shown that the theory of Clement can be applied to determine the mean pore size from the first two moments if one specifies a model for the path length distributions through and between pores. The model is needed to determine c?/~2(=b) for both the path length across pores and between pores. A case of immediate interest here is that of a series of fairly high porosity (0.51
References 1) B. H. Armitage, T. W. Conlon and B. Rose, unpublished; J . A . Cookson, B. H. Armitage and A. T. G. Ferguson, Non-Destructive Testing 5 (1972) 225. 2) C. D. McKenzie and B. H. Armitage, Nucl. Instr. and Meth. 133 (1976) 489 and unpublished data. 3) C. F. Clements, J. Phys. D, Appl. Phys. 5 (1972) 793. 4) j. D. Bernal, Nature 185 (1960) 68. 5) R. G. Avery and J. D. F. Ramsay, J. Colloid Interface Sci. 42 (1973) 597 and private communication. 6) C. Williamson, J. P. Boujot and J. Picard, CEAR-3042. 7) B. H. Armitage and D. R. Porter, unpublished report.
NORTH-HOLLAND PUBLISHING CO.
P O R E SIZE D E T E R M I N A T I O N F R O M CHARGED PARTICLE ENERGY LOSS M E A S U R E M E N T F. P. BRADY* and B. H. ARMITAGE
U.K.A.E.A., Harwell, Didcot, Oxon., OXll ORA, England
Received 14 February 1977 A new method aimed at measuring porosity and mean pore size in materials has b~n developed at Harwell. The energy width or variance of a transmitted or backscattered charged particle beam is measured and related to the mean pore size via the assumption, that the variance in total path length in the porous material is given by (dx~= na 2, where n is the mean number of pores and a the mean pore size. It is shown on the basis of a general and rigorous theory of total path length distribution that this approximation can give rise to large errors in the mean pore size determination particularly in the case of large porosities (e>0.5). In practice it is found that it is not easy to utilize fully the general theory because accurate measurements of the first four moments are required to determine the means and variances of the pore and inter-pore path length distributions. Several models for these distributions are proposed. When these are incorporated in the general theory the determinations of mean pore size from experimental measurements on powder samples are in good agreement with values determined by other methods. 1. I n t r o d u c t i o n
During the past few years a new m e t h o d aimed at measuring porosity and pore size in materials has been u n d e r d e v e l o p m e n t at Harwell Laboratories t h r o u g h the use of charged particle beams. Originally it was s h o w n 1) that by m e a s u r i n g the average energy loss and the energy width of a proton b e a m after passage t h r o u g h a porous sample target (of u n i f o r m geometrical thickness) it is possible to d e t e r m i n e the porosity of the material and, u n d e r certain a s s u m p t i o n s , the average pore size. H o w e v e r a g r e e m e n t b e t w e e n these d e t e r m i n a t i o n s and m e a s u r e m e n t s of pore size based on other techniques has been on the whole not completely satisfactory and for large porosities the discrepancies a m o u n t to differences of several h u n d r e d percent. Following a very brief review of the b a c k g r o u n d and experimental technique 1,2) this report focuses on the theoretical model and calculational procedure used to d e t e r m i n e the m e a n pore size from the energy width. It is found that the simple model which has been used to calculate the m e a n pore size can give rise to considerable error particularly in the case of high porosity (>0.5). Model dependent calculations, based on a general theory due to Clement3), are proposed for d e t e r m i n i n g pore size more accurately. W h e n the experimental data are analysed on the basis of these calculatfons the results are in m u c h better a g r e e m e n t with pore sizes obtained from gas adsorption m e a s u r e m e n t s . * On leave from the University of California Davis, CA 95616, U.S.A.
M e a s u r e m e n t s of porosity and m e a n pore size are of importance in m a n y areas of science and engineering, and in technology. Closely related problems are those of particle size m e a s u r e m e n t and the packing of particulate solids. For example porosity and pore size are relevant in oil and water production, and the controlled packing of particle solids, over a wide range of particulate sizes, is important in powder metallurgy, the loading of nuclear fuel e l e m e n t s , the production of c e m e n t s and ceramics, and the structure of catalysts and molecular separation columns. Current techniques of porosity m e a s u r e m e n t are based mainly on the mercury porosimeter or gas adsorption. I n f o r m a t i o n on the size distribution of pores accessible from the sample surface is obtainable by the penetration of a non-wetting liquid such as mercury. In the mercury porosimeter pressure is applied to the mercury such that as the pressure is increased open pores are penetrated of progressively decreasing entrance diameter. T h e gas adsorption technique provides information on the specific surface area of a sample u n d e r investigation. In addition pore size information is obtained s o m e w h a t less directly from an analysis of capillary condensation. In the charged particle m e t h o d on the other hand, one a t t e m p t s to relate the m e a s u r e m e n t s of the energy s p e c t r u m to the m e a n path length across the pores and thus derive a basic physical quantity which is different from those arising from the a b o v e - m e n t i o n e d methods. T h e energy width of a charged particle b e a m
596
F.
P.
BRADY
AND
transmitted through a porous sample is enhanced over and above the energy width (due mainly to straggling) for a non-porous sample of the same material and areal density. The explanation, as given in ref. 1 is that ir~dividual protons of the beam, whose cross-sectional area is assumed to be much greater than the pore size, have paths which pass through different numbers of pores and hence intercept different net total thicknesses of material. Thus they undergo different energy losses as they traverse the sample. The model assumed 1) is that of a random distribution of pores whose length along the direction of the beam is uniform and equal to a. Then the variance in total path length through solid material is na 2, where n is the mean number of pores traversed and a is the (mean) path length across the pores. Thus, both the effects of path length dispersion across a given pore, and also pore size dispersion are neglected. The connection with the measured energy distribution of the transmitted beam is made via ( A E 2 ) ~ - n a 2 ( d E / d x ) 2, where (AE 2) is the variance about the mean energy loss of the energy distribution and d E / d x is the mean differential energy loss in the sample. Usually Gaussian distributions are assumed 1,2) so that the experimental (AE2)--(fwhm/2.35) 2, where fwhm is the full width at half-maximum of the energy distribution. The mean energy loss of the beam provides a measure of the effective thickness, te, of material (in equivalent length of matrix material or in areal density), and thus a measure of na via na = t o - t e , where to is the geometrical thickness of the sample. It is then possible to calculate n and a and to show that the mean pore size, a, is proportional to the energy variance ( A E 2 ) , the latter having been corrected for the straggling variance as measured with an equivalent non-porous sample. The porosity, e, can also be calculated : e = ( t o - re)/to. More recently this method of pore size measurement has been extended by using resonant backscattering from nuclei in the porous materail2). In partiizular the enhancement of the resonance width in the elastic backscattering of protons and alpha particles from 160 at 2.66 and 3.05 MeV lab energy respectively is measured at 170° lab. The width of the resonance in a porous oxide is enhanced over that measured with the equivalent non-porous oxide. Details concerning the experimental and calculational methods are given in ref. 2. The calculation of mean pore size, a, is made by assuming,
B.
tt.
ARMITAGE
in analogy to the case of transmission geometry, that the variance in path length is 2 n a 2, where n is the mean number of pores encountered on the way in, prior to the scattering. This number is assumed to be equal to, but independent of, the mean number encountered on the way out. 2. Theoretical calculations The distribution of total path lengths of a beam of particles through a porous material has been considered by Clement3). The assumption is made that there is no correlation between the distances through pores and those in between the pores, i.e. that the path lengths, x;, between pores and the path lengths, y;, through pores are independent random variables with means and variances (~1, ~ ) and (g2, ~ ) r e s p e c t i v e l y . A requirement is that the mean number of pores, n, traversed in the mean total path length, x, be large, where a total path length through the material is x = ~,x;. Clement 3) was not able to obtain an explicit form for the total path length distribution, but was able to calculate the moments. The first x moment is E l ( x ) = n~l and the higher moments about El(X) (leading terms) are E2(x ) = n ( ~ 2 + 0 - 2 ) ;
E3(x) = - 6 / / 3 n 2 _].. 30-2/tn ;
E 4 ( x ) = 12n3/~ 4 + 3/'/202(0 2 - 10//2) ;
Em(x)
=
(-1)
m
Em(y),
(m ~
2),
where ]/
~-
2 - - 0"2 0"2~ 2 2 0"~ +0-2
~//2
2 ,
0"
--"
2 2 0" 1 0"2 2
2,
0- +0-2
and n is the mean number of pores. Unfortunately the basic parameters sensitive to the higher moments" n = 3E3/4E~,
are very
It = - 2 E3/3 E] , etc.
These higher moments are difficult to determine accurately from the experimental data, which are usually in the form of energy spectra whose accuracies are limited by counting statistics and experimental effects such as low-energy tails due to slit scattering and detector characteristics. The first and second moments can in general be determined quite well from the data. It is instructive to compare the predictions of the simple model, in which pore size variances are neglected, and which gives the second moment as E 2 ( a ) = na 2, with the predictions according to the theory of
PORE
I
I
I
I
I
I
SIZE
DETERMINATION
length distributions across the pores and in between the pores and to calculate pj and ~(j = 1, 2) for these distributions. For example exponential distributions of paths in and between pores correspond to
I
b l = b 2 =1
,.6
.......
bl = ~ ,
X
___
b,=~s.b2-'
\
_._
b2=1
bl~1OqgDgiQ
~
-
.
.\.
~1o,
fx~'~X~x "" \ _. ..
Y"..
!\
\
5- 0 . 8 - - - \ ~~-1"1 _ _
o
\ \
'....
\
X\
\:
"..
.
X \ "\",
\
\\\ x,...\ \
_
\
0.6 ~
\ \ -...\.\. \ .,. \
\
,.
",
o,-
0.2 -
o,o
\ ~
!
1
I
0.2
0.3
0./.
"\
1
I
1
1
0.5
0.6
0.7
0.8
POROSITY
Fig. 1. The ratio of mean pore size, a, calculated from the simple model to the mean pore size, P2, calculated from more rigorous theory is plotted against porosity using various assumptions for the pore and particle path length distributions.
Clement 3)
where
the
second moment is To be quantitative, further assumptions need to be made. Models for the distributions of path lengths, x~, between pores and of path lengths, y~, through the pores can be assumed and for each distribution a mean, pj, and a variance, ~ , with ~ = b j p f ( / ' = 1 , 2 ) c a n be calculated. The ratio pl/P2 is related to the porosity, :/1 ~/P2 = ~- ~- 1. (In the transmission geometry can be determined using the mean energy loss, but in the backscattering case it must be otherwise determined.) One can write
E20; ) -- E2(P2 ) -- r/(p 2 + 02).
E2(P2)
=nP
El(p2) =
2" F(e, b, b2), npl = np2(e-'--1).
(1)
Given bl and b 2 and ~ one can calculate P2 from E2(P2) and E~(p2). The simple model, where tr2 = 0, gives E 2 ( a ) - n a 2 and E 1 - - h a ( e - l - - I ) . The mean pore size in each case is calculated from E2/E1, hnd the ratio for the two cases is
a/P2 =
597
F(,~, b l , b2).
To calculate F as a function of porosity one needs to assume specific models for the path
b 2 - 1 ( ~ - p~, ~ - ~2).
For this case the ratio a / p 2 - F is plotted against porosity as the continuous curve in fig. 1. It can be seen that the simple theory can be seriously in error particularly for large porosities where it gives much too small a value for mean pore size. (Here mean "size" means the average path length across the pores.) A uniform distribution of pore path lengths extending from zero to some maxim u m path length has b 2 - ] . F ( e , 1, ]), the case for b ~ - 1 and b 2 - ] , is shown in fig. 1 as the dash-dot curve. For spherical pores of uniform size, radius R, the path length frequency distribution through the pores can be shown to increase linearly up to a m a x i m u m path lenght of 2R and to have p 2 - 4 R / 3 and ~ - p 2 / 8 . Many, if not most, porous materials are in fact packings or compacts of solid particles and often one can justifiably speak of p! and 0 1 as arising from the distributions of path lengths through the individual particles. A case of particular interest here, in that energy spectra have been measured, is that of a powder or packing of very small and nearly spherical particles whose radii have a small distribution of values about R. In this case P l - 4 R / 3 and ~ - p ~ / 8 so b l - 1/8. In addition to b 1 a value of b2 is needed in order to calculate F(e, bl, b2) as a function of porosity. Once the porosity is specified the ratio a/p: and p: can be calculated from F and from measurements which provide the first and second moments of the total path length distribution. These are directly related via energy loss data to the corresponding moments of the energy spectrum. The measurements have been made 2) on powders of alumina and silica with porosities between 0.50 and 0.75. We still need to characterize the path length distribution through the pores of the packings of spherical or nearly spherical particles and so determine b2. We shall assume for these high porosity samples that a good approximation to the pore path length distribution is that of an exponential, so b: can be taken equal to unity. In justification consider a random distribution of spheres of crosssectional area C - ~ R 2. Suppose for the m o m e n t that the thickness of the spheres in the direction
598
v.
P. BRADY
AND
of the beam of particles is negligible so that one has essentially a random distribution of discs with their axes parallel to the beam. Then the path distribution in the Z direction is e x p ( - N C Z ) , where N is the mean number of discs per unit volume, N = 3(1-e)/(4zr R3). The mean path length, P2, is (NC) - ~ - ] R / ( 1 - e ) ( - 4 R for e - ~). For a distribution of spheres one must correct for their finite thickness and for their shape. If one subtracts from (NC) -1 the mean path length, P l, through a sphere ( P l - J R ) t h e n the resulting mean pore path length, /z2, is that which satisfies p l / ~ 2 2 = e - 1 - 1. Several geometrically systematic arrangements or regular packings 6f spheres were considered as another possible way of calculating the pore path length distribution. However for the case of the relevant high porosity packings such as the cubic and tetrahedral, very long path lengths are allowed, and the path length distribution is unbounded as the path length increases indefinitely. A cut-off can be assumed on the basis that the sample is finite or that in practice a distribution of sphere sizes is present, but the path length cut-off is somewhat arbitrary. In any case it seems likely that the distribution of spheres is essentially random aside from the effects imposed by finite size. Thus the pair distribution function should be similar to that of a liquid. Regular geometric packing, where it does occur, is expected to be one of short range order. BernaP) considered the random assembly of hard spheres, and obtained the general shape of the two-molecule pair distribution function, g(r), of a liquid from geometrical packing conditions. Using his g(r) we derived from a combination of analytical and graphical calculations the approximate distribution of paths, N(y), between spheres. This distribution is more linear than an exponential for short paths but on the whole the exponential provides a reasonable approximation. It turns out that the factor F(e, bl, b2) is not very sensitive to b2 for the large porosities of interest here. If b2 is increased from 1.0 to 1.5, F increases by only 4% at e - ~ and by 15% at e - 1. The dependence of F on porosity for the case of b 2 - 1 and b l - ~, is shown by the dotted curve in fig. 1. Also plotted, (dashed curve) are the values of F for b 2 - 1 and b l - ~. The latter b~ allows for the distribution of sphere sizes in the silica and alumina powders and compacts of experimental interest here. A question of interest is why the simple model,
B. H. A R M I T A G E
t,0
AI203 35
{~10)
~" 30 E
w
25
N
w 20
g
Si 02 (150)
-
AI203 Si02 (~5) (/.,5) ~ / 1l / * /, ///i
A,203 (150)
1
{3_
z 15 IE ,,,"
#2 o'" . . . I
GA...,,"
0.5
~
l
0-6 ' POROSITY
-
-
(j.7
Fig. 2. The pore size for six samples determined from experimental measurements or the second moments as calculated using in one case the approximation (Ax2)= na 2 (solid circles) and in the second case model dependent general theory (the open circles) is compared to gas adsorption measurements (squares connected by the dashed lines). The lines connecting the points are solely to guide the eye. The number in parentheses just below the chemical formula or the the sample is the compaction pressure in MPa.
which assumes (Ax2) = na 2, is such a poor approximation for large porosities. This model corresponds to neglecting the variance in path length through the pores, i.e. setting a 2 = 0 (and /~2=a). With a2 = 0 the dependence of the second moment, E2, on dl vanishes. In the case of large porosity (>0.5) the mean pore path length/12 is larger than/~l, and the mean path length variance. ~ , is expected in general to be relatively large. In this case it would seem to be more appropriate to assume (Ax 2) - nl~ - n~z~(a- 1 - 1 ) 2 - nc2(e - 1 - 1)2 (putting/~2 = c) which neglects the variance, ~ , in /~1. In this case a/c -- ( e - ~ - 1)2. This dependence on e is shown in fig. 1 as the dash-two-dot curve which is similar to the curves for those cases where an exponential pore path distribution is assumed. As expected this dependence is very close to the F(e, 81, 1) and F(e, 15, 1) curves for porosities
~>~.
The approximation (Ax2) - ritzY, might be called model-independent in the sense that no model ex-
PORE
SIZE
DETERMINATION
pressions for the pore or particle path length distributions are required. In any case this approximation represents for large porosities an improvement over that of (zlx 2) = na 2 which has been used earlier~,2). For small porosities, e <0.5, (Ax 2 ) - n/u~ is a poor approximation. Lacking further information concerning the sample or accurate measurements of the higher moments one should use (dx2)= na 2, where the porosity is less than 0.5. The ultimate goal is not to rely on models for the path-length distributions but to determine from experiment the parameters which characterize the pore and particle path length distributions, This goal, [after Clement3)] can be achieved by accurate experimental determinations of the moments up to and including the fourth. However an attempt to determine such sets of moments from measured spectra 7) of a proton beam transmitted through porous samples did not produce self-consistent or reasonable sets of the parameters ~Zl, al, ~z2, a2, and n. The spectra when measured were not intended to be used for more than first and second moment determinations and very likely the inconsistencies are due to limited statistical accuracy of the distribution tails and to slit scattering effects. Nevertheless it should in principle be possible in the transmission case to obtain data of sufficient quality. In the backscattering case this will be much more difficult as the "background" spectra are distorted by various other effects such as those due to other resonanceg, A slightly different and more direct approach would be to work with the total path length distribution rather than with the moments. The calculated distribution could be transformed into an energy-loss distribution (including the effects of straggling) and then fit to the experimentally measured energy distribution by varying the path length parameters so as to minimize a Z 2 in the usual way. This procedure will also require very good data. In addition the usual ambiguities of seeking m i n i m u m Z 2 in multiparameter space will arise, 3. Comparison with experimental results The technique of resonance backscattering from nuclei has been used by Mackenzie and Armitage 2) to determine the mean pore size, a, of porous powders of silica and alumina. The powders, consisting of small spherical or nearly spherical particles with a small size distribution, were made at Harwell by Avery and Ramsay as described in
599
ref. 5. Some samples were compacted at pressures of 45 and 150 MPa. As described in ref. 2 the data were analysed and the values for the mean pore size, a, calculated assuming ( d x 2) = na 2. These mean values of a for 4 alumina (A1203) and 2 silica (SIO2) samples are plotted in fig. 2 as the solid circles. The actual values plotted are slightly different from these obtained in ref. 2. The data were reanalysed using stopping power and range-energy curves constructed from those of ref. 6 which are deemed more appropriate for oxygen in a solid. The silica values are about 20% less and the alumina values about 10% less than those of ref. 2. The experimental errors in the values of a are estimated to be about 20%. Also plotted in fig. 2 are values of/~2 obtained from the values of a by dividing by the factor F(e,b~,b2), taken here to be F(e, 15, 1) for the reasons discussed in section 2. The resulting values of the mean pore size are larger and are in considerably better agreement with the values measured by Avery and Ramsay s) using the method of gas adsorption. These are shown by the squares. The agreement is not perfect. However one needs to keep in mind that the two techniques measure different physical quantities. The charged particle method measures essentially the mean path length across the pores averaged over the cross sections of the pores. In gas adsorption measurements it is usually assumed that one is measuring a mean radius which corresponds to that of the pore throat. For example, in a cubic packing, e =0.47, the throat radius 5) is rv--0.41 R, where R is the particle radius. For tetragonal e --0.66 and rw = 0.73 R. Therefore allowing for these physical differences and for experimental errors of about 20% on both sides one can conclude that the results of the two methods are consistent with each other. There are a number of measurements of mean pore size in graphite ~,7) using the energy width of a transmitted proton beam. The porosities of the samples ranged from 0.18 to 0.25 and the mean pore sizes from about 10 to 50 lzm. The data were analysed and mean pore size, a, computed via (dx2) - na 2. The results were compared to those values, aM, found using a mercury porosimeter (MP). The mean of the ratio a/aM varied from 1.1 to 2.2 over the six samples with a mean of 1.65. This agreement is of modest quality, however again it must be kept in mind that there exists evidence that the MP can produce mean values on the small side in that some larger pores are acces-
600
v.p.
BRADY AND B. H. ARMITAGE
sible only through a smaller throat. It can be seen from fig. 1 that the agreement can be improved if the exponential path length distributions are assumed. However only in about half the cases could the MP distributions be approximated by exponentials, Calculations based on the simple assumption that (Ax 2>= n a 2 thus appear to provide a tolerable approximation for e<0.5, but become progressively more in error as e increases above 0.5. As described earlier a reasonable approximation for the latter case is < A x 2 > - n c 2 ( e - l - 1 ) 2. Accordingly the high porosity data of ref. 2 were analysed using this method and agreement with gas absorption values was nearly as good as that obtained with model dependent values (fig. 2). The overall agreement between mean pore sizes calculated using these simple approximations with those obtained from other techniques (mercury porosimeter and gas adsorption) was within 40%. These results were obtained from 6 low porosity and 6 high porosity measurements, and covered a pore size range of a factor of 104 (5 nm to 50~m). The charged particle values, however, were nearly always larger than those obtained from mercury porosimetry and gas adsorption, so that part of the difference can be assumed to be due to basic physical differences in the techniques, 4. Conclusions The method of measuring pore size in materials via the use of MeV charged particle beams is one which provides essentially different physical information concerning pore size and structure. When using this technique, which measures the mean path length across the pores, some care must be taken in the calculation of mean pore size from the experimental energy distributions. Calculations based on the simple assumption that the variance in path length is given by ( A x 2> = n a 2 will be a fairly reliable approximation for e <0.5 but will become seriously in error for e >0.5. For the latter a fairly good and simple approximation is
Ideally the general theory of Clement 3) should be used to calculate the means and variances of
the path length distributions inside and between the pores. In practice we have found that it is difficult to utilize fully the general theory because rather accurate measurements of the first four moments are required in order to characterize the two path length distributions. To achieve this the experimental data would have to be of high statistical accuracy and essentially free from instrumental effects such as slit scattering. It has been shown that the theory of Clement can be applied to determine the mean pore size from the first two moments if one specifies a model for the path length distributions through and between pores. The model is needed to determine c?/~2(=b) for both the path length across pores and between pores. A case of immediate interest here is that of a series of fairly high porosity (0.51
References 1) B. H. Armitage, T. W. Conlon and B. Rose, unpublished; J . A . Cookson, B. H. Armitage and A. T. G. Ferguson, Non-Destructive Testing 5 (1972) 225. 2) C. D. McKenzie and B. H. Armitage, Nucl. Instr. and Meth. 133 (1976) 489 and unpublished data. 3) C. F. Clements, J. Phys. D, Appl. Phys. 5 (1972) 793. 4) j. D. Bernal, Nature 185 (1960) 68. 5) R. G. Avery and J. D. F. Ramsay, J. Colloid Interface Sci. 42 (1973) 597 and private communication. 6) C. Williamson, J. P. Boujot and J. Picard, CEAR-3042. 7) B. H. Armitage and D. R. Porter, unpublished report.