Pore Size Distribution in Porous Materials
4
Interpretation of Small-Angle X-Ray Scattering Patterns
The theory of sniall-angle x-ray scattering as developed b) Guinier is applied simply and approximately to porous aggregates and the results are reduced to continuous distributions of pore size. 4 method is presented for the iniersiori of the Guinier integral under certain simplifqing assumptions. A simple semiempirical method of making the correction for a finite collimating system is developed and applied. Results are correlated with data from adsorption measurements and from the mercury porosimeter.
T
made. The latter two methods are tedious for numerical work and are perhaps too laborious to justify application to masses that' obey only very a,pprosimately the assumptions made in the development of thc (iuinier Theory.
HE: relation between the state of subdivision of a inas8 of matt,er and the angular distribution of x-ray intensity scattered by the inass has been discussed analytically by Guiiiier ?). Several methods of using this theory for interpreting t,hv scattering patterns in terms of particle or pore size have hei'ii advanced.
,.1 his paper presents a nietliod for aiialyziny small angle s - r a i
vattering data in terms of a continuous distribution of pore (or particle) sizw. hlso presented are some typical results together with coinp:irisons Tvith the results of other physical methods of elucidating pure geometry. I n March 1947, before t,he present paper was submitted for piiblication, the work of Shull and Roms (1.5) appeared; this tiad liccn presented in suhstance tlefure (libson Island Conferrwces in 1944 and 1945. There is a strong similarity in the two treatments, and there is no doubt that Shull and Itoess h a w established priority. The present paper is an independent work and places emphasis on the alternative pore size interpretation. The experimental arrangement is familiar. Figure 1 shows a diagram of that used by the authors.
Biscoe and Warren (3) evaluated the average particle size of carbon black on the assumption of only small departures from the average size. Jellinek and Fankuchen (9) did the same for certain inorganic gels. I n a later paper, Jellinek, Solomon, and Fankuchen ( I O ) amplified their method to the case where the size, distribution may be considerd as a collection of a small nunibrr of discrete constant sizes. Shull and R o e s (5, 13-15) developed a workable method for deterniining a continuous distribution based on the determination of parameters in a postulated flexible distribution. More recently Bauer ( 2 ) and Itoess (12) devised methods for the determination of continuous distributions wherein no a priori assumption concrming the kind of distribution is 1 Present addresa, Department Madison. W i a .
L/,X-RAY
(if
C;heiii:stry, rniver.;it,-
SOURCE
of Wi*consixi.
VACUUM
B E A M STOP,
/ C o L L ' MATOR
FILM1
PENTAERYTWRITOL -I CRYSTAL (001)
CLEAVAGE FACE
HOLDER
2 7
0.6MM.
DETAIL A
Figure 1.
DETAIL
Small-Angle X-Ray Camera
665
0
rCAP
ANALYTICAL CHEMISTRY
666 The camera tube is a length of 10-cni. (4inch) aluminum pipe, in one end of which is fitted a collimating tube made vacuumtight by a piece of cellophane cemented over the entrance slit. The other end is capped with a gasketed disk. The film holder is attached to a piston which slides in the tube and may be placed a t any desired position up to 60 cm. from the sample. . Provision is made to evacuate the camera (to about 15 mm. of mercurv) during exposure in order to eliminate air-scattering. The main beam is caught by a beam-stop 0.6 mm. wide to prevent overexposure a t the center of the film. Intensity readings are therefore possible to within about 0.3 mm. of zero scattering angle a t sample-to-film distances up to about 60 em. The collimator is 15 cm. long. I t s entrance slit is a pinhole 0.15 mm. in diameter punched in a lead sheet 0.6 mm. thick. The exit slit is a narrow rectangle 4 mm. long and 0.1 mm. aide. The main beam is therefore collimated to a very thin and narrox fan shape. The edges of the exit slit are inlaid with 1-mm. platinum wire Figure 2. Sample to prevent transmission through sharp Cap and Anvil slit edges. which affects the film in a mann& indistinguishable from smallangle scattering. The x-ray beam from the target is monochromated by Bragg reflection from pentaerythritol (6). llthough the life of this crystal under the x-ray beam is short (about 100 hours),,the intensity of its 002 reflection is high enough to overweigh the bother of frequent renewal. The sample is ground, mixed with very dilute collodion solution, and placed in the rectangular window of the sample holder by smearing it over a small anvil (Figure 2). The holder is then placed like a cap over the end of the collimating tube for the exposure. Exposures ranged from 10 to 80 hours, depending on sample, sample-to-film distance, kind of radiation, and part of pattern required. Development was according to a standardized procedure on calibrated film, and intensities were determined by a Leeds & Northrup microphotometer. Intensities higher than density 1.1 were discarded, because the film began to depart from linearity above this density; and densities below 0.02 were disrarded because of the relatively high .‘noise” shown by the microphotometer trace. The experimental curve of obseived intensity, l o b s . , versus the scattering chord, z, was obtained by averaging the microphotometer readings from the left and right sides of the theoretically svmmetrical pattern and smoothing the data. Several exposures of one sample were normally made with long and short mposure times. This procedure permits piecing togetha several patterns to obtain a single intensity curve covering a range of intensities impossible to reach with one exposure because of the relatively narrow film latitude. Intensity ranges of at least 1000 to 1 w x e normally obtained, but the lower end of this i ange was frequently discarded as unreliable. The used intensity iange was always a t least several hundred to one. Fitting two patterns together a t the region of overlap offers no difficulty, 8s only relative intensities are of interest here. Figure 3 shows a typical obseived intensity cuive (I, A ) and the derived logarithmic curve (11, A ) . Use of the logarithmic curve greatly facilitates both the smoothing and the fitting processes. Guinier ( 7 ) has shown that for a single spherical particle of radius r , immersed in a finely collimated beam of monochromatic \-lays of wave length and intensity lo,the fraction of incident intensity scattered to a rhoidal distance z from the main btwin IS given approximately by
I r;!
a r6
exp (-uZrZz2)
The very fine beam required by this theory is not practicable because the scattered radiation is too weak. To increase the intensity of the main beam, and hence the scattered radiation, the beam is enlarged into a second dimension by opening up the pinholes to narrow rectangular slits. When the angular spread of the main beam is comparable to the scattering angle, however, it is necessary to employ a slit correction t o reduce the experrmentally obtained intensity back to what would have been obtained had pinholes been used. The complexity of this correction increases with departure of the collimating system (including the detector slit) from infinitesimal pinholes. The pinholeslit combination described above makes the correction particularly simple, and is due to Shull. In both methods of correction described below, the detector slit is taken as small. PORE GEOMETRY
The geometry involved in this system is diagramed in Figure The plane of the fan-shaped beam is the plane of the paper, while the plane of the film is perpendicular to it. We seek the intensity distribution along the equator (z-axis) of the film. 4.
Consider an element of the fan beam, dy, a t Q on the film meridian, distant y from the center. The scattered intensity a t P . distant z from Q, is given by Guinier’s relation d
(i,) a
Inasmuch as evidently z 2
r’
+
y2
1.6
1.2
3 .-
0.8
!
e
.->
2
0.4
e
-+
i
>.
c .-
z
0.0
-
1.8
.-$
2
--z
1.2
0.6
0.0
-0.6
4 8 Scatrering Chord, Mm.
Figur’e 3.
s = sample-to-film distance
=
is the contribution to the intensity at Q due to the pinhole beam, OQ. The total for the entire fan beam is then
0
and
r‘i esp ( -u*r*zZ) dy
12
Observed and Corrected Intensity Curves
A . Observed B . Corrected 0. Observed computed
V O L U M E 20, NO. 7, J U L Y 1 9 4 8
667
I=L7
re exp ( -a2r2z2) exp - aWy2) d y
The integral over y may be evaluated by means of the error function,
where kr3 is t h e volume of one such pore. The scattering for these pores is then dl
a r2
D ( r ) erf
(1/2 a Y r ) exp ( -uzzW) dr
and the total intensity a t any point due to the entire distribution of sizes, providing the pores are randomly and .widely separated spheres which produce no coherent interference, is
to give I
3:
r* erj
( 4 5a
~ r eup )
( - 0 2 r ~ )
r2 D ( r ) e r j
as the scattered intensity a t any point on the film equator due to a single spherical particle of radius r. The intensity due to d.\such particles is just d,\' times this. [The value of Y , half the length of the slit projected on the film, is a function of s and may be determined as follows: Short exposures are made, with no sample in place, with the film a t a Beries of measured distances from the sample cap. rlfter development the length of each slit image, 2Y, is measured and a plot drawn of Y us. s. The proper Y t o use for any s is then either read directly from the graph or computed from the linear function, Y(s).] Let the volume distribution of pores be given by D ( r ) where the total volume, dV, of all pores having radii between r and r dr is d V = D ( r ) d r . The number of pores within this range is
( d ia
I7r) esp (-a2&*)
dr
(1)
I
n r . A
133
B
I96
27 22
-
+
h MERIDIAN
20
0
40
r,
60
80
A.
Distribution Function of SilicaAlumina Gel
Figure 6. 0
Two methods of slit correction
P1
This integral is not easily inverted-that is, there is no simple general method for computing D ( r ) , given I ( z ) . It happens, however, that the erj term is nearly constant for r Z Figure 4.
XS p.
Using
B.,
Cu-K, radiation of X = 1.54 a sample-to-film distance of 15 cm. or less, and Y,of about 5 mm., the n j term is practically constant for r >I 40 A. and changes only slowly for r 2 15 A. As a first approximation, then,
Geometry of Slit Correction
I(z) 2.0
aLrn 1.2
D ( r ) exp ( -a2z2rz)dr
(2)
We now assume that D(r) is of a modified llaxwellian form ( 8 ) , expressible analytically as
D(.)
.*
=
A ($)"exp
(- 2)
(3)
and attempt to find the values of n and ro which best fit the experimental data-that is, we seek that member of the double infinity of curves, Equation 3, which most closely approximates in x-ray scattering behavior the sample whose behavior is given by I ( z ) . The resulting expression for I ( r ) can now be integrated in terms of the gamma function to give
1.6
0
-2
+ 3 Y
It is convenient to write this logarithmically, dropping the constant, to give
1.2
which equation is linear in log I and log
0.8 -3.2 log ( a W
Figure 5.
-2.7 I/r%)
+
Selection of ro for D ( r )
-2.2
I n this last equation, I , a! and z are known and n and ro are to be evaluated. I n application, one chooses various values of T, . That value of TO which and plots log I against log makes the plot most nearly linear is the best ro; and from the slope of this line one may determine n.
. ANALYTICAL CHEMISTRY
668 Figure 5 s1ion.s this procea.; for the material of Figure 3.
+-
(
Plots of log I vs. log a2x2
T i )
are given for 1'0
=
I,(x) =
40,30,27, 24,
and 20 8. The definitc curve to the right for TO too low and to the left for ro too high is evident, and T~ = 27, giving the most nearly linear curve, was taken as the correct value. The slope of this line was found to be -2.83, which gives n = 1.33 accnrding to Equation 1. D ( r ) is therefore given by
Curve h of Figure 6 s h o w the form of this distribution. As plotted here the constant A in Equation 3 has been chosen to normalize the curve to make J d V from 1' = 0 to r = m equal t o V Othe , measured pore volume. Having determined the parameters n and TO it is possible to waluate I(x) numerically, now including the erf term, according to Equation 1. If the computed I deviates too widely from the observed I , D(r) may be distorted to give better agreement. If, however, there is a large contribution of r smaller than about 20 .&., it is usually more convenient to make the slit correction independently than to include the correction in the analysisthat is, i t is simpler to reduce l o b s . to Icor. before applying the Guinier theory. Further, it is necessary to make the slit correction before applying other methods of inverting the integral such as those of Bauer and of Roess. A method for making an independent .lit correction is described belon-. Referring again to Figure 4,and to the elemellt, dy, of the fan beam which strikes the film at &, the scatterlng due to thls :lementary beam alone is radially symmetrical about &, depending only upon the distance from &. In particular, the scattered intensity a t the equatorial point, P , depends only on the distance, QP = z . This dependence is that given by the Guinier theory and is what we seek experimentally. Denote this relation between intensity I and radial distance, z, by I = I ( z ) . The intensity a t P due to the beam, d y , is then dI = I ( d x 2 y2)dy. The intensity most conveniently measured is I,(r), the intensity along the equator, as a function of distance from the center. Along this line d I and dI, are equal. Replacing d I by d I c and forming the integral for all dy's,
b,e-''s2 1
where
Solving for ci, one obtains fiiiallj-
Thus it appea's that if the e\perinientally obtained equatorial intensity I,(z) be expanded empirically into the finite series of Gauss functions (Equation 7 ) , the required slit-corrected intensity will be a new similar seriea of Gams functions (Equation 6) diffeiing only in that the new coefficients c, have been changed from the old ones b, according to Equation 8, and the argument, x, replaced by 2. Experience has shown that from three to five (usually four or five) tei inb in the expansion will repraduce the experimental data !vel1 within the error of reading the microphotometer traces. Curves I B and I I B of Figure 3 show the slit-corrected values corresponding to thv uncorrected curves, A. The necessity for applying the collection is obvious. Having obtained I ( z ) , the corrected intenbity function, it is possible to check the accuracy of the approximation by placing I ( z ) in Equation 5 and numerically evaluating the integral for I,(%) at selected values of r. The circled points in Figure 3 show the results of such a calculation a t several values of the scattering chord. The negligible deviation from the observed intensity iinplies that the series (Equation 6) is an adequate approximation. Lsing this new intensity function we may now employ Guinier's theory dirertlp in the form
+
I n this equation I e ( x ) is the esperimentally determined equatorial intensity, and the required function is I ( z ) . Although there appears t o exist no exact method tor the inversion of this integral, the follouing approximate procedure has been found satisfactory : Assume that it is possible to expand the unknown function Z(z) in a finite series of Gauss functions:
I(Z)=
c
cle-u'z2
(6)
Then the integral equation above becomes
where the error function has again been introduced and distributive constants dropped. Combining everything independent of x we may write
Figure 7.
Selection of r~ in D ( r )
V O L U M E 20, N O . 7, J U L Y 1 9 4 8 0 (12
I ,
I
-
20
0
I
I
I
I
I
I
2n
0
40
Porr KaciiiiF,
Figure 8.
669 1
40
I
t i y is ~ a rctlat$ely sinall contribution of pores brlow about 15 A. Figure 8 shows the plotted distributions as determined by this second method for four silicaaluniina gels of differrnt characteristics. Each has been normalized to T’o, the pore volume, as measured by the usual density determinations. Figure 9 shows the distribution determined by the same method for a fuller’s earth having a very much wider size range than that of the gels. Table I lists some other pertinent date. for all five materials.
-
00
A.
PORE VOLIJME ASl) W R F A C E AREA
Pore Size l)istributionn for Some Silioa-.ilumina ( k l s
Other data g i v e n in Table I. r,,----
-
. 2
~
(9)
a-z O
the correct TO
2
m-ill be linear for
Figure 7 shows a set of these plots for the material of Figure 3, using ro = 60,40, 30,25, 22, 20, and 15 1. Again there is a clear tendency to curve in opposite directions for ro too low and too high. The best ro was taken as TO = 22. The slope of this line is -3.96, giving n = 1.96 according to Equation 8. The distribution function is then given hy
D(r) = A (&>’‘’‘e
-
’l’ahle I.
Pore nata for T’? pica1 P o r o u s >laterials I d w r u t i o n Data Siirfare .AT. area, Imre So, radiur, CC./A. G. T ~ A. ,
c__._____
( and the slope of this line affords a determination of n.
Thus a plot of log I against log
that it is particularly si~uplt:to coinpute F when n and r 0 arc’ known. The values, 7 from aniall-angle x-ray scattering, ancl rc froin density arid adsorption data, are comparable valu?.. and Table I shows some typiral examples of the kind of agi’c’cbmerit, found. The values of 7 arid ye a r also ~ superposrd on til(. plottrd distributions of Figurr 8.
so
Thus for the whole range of r’e from zero to infinity,
(at
where A may be chosen by normalizing to VO. Curve B of Figure 6 compares the distribution obtained using this method of independent slit correction with t h a t obtained with the earlier method. The fact that the two curves differ only slightly, even having their peaks at about the same value of r, shows that the less rigorous first method of slit correction is useful when, as here,
Pore wlunie.
Va CC./G. I
>laterial Silica-alumina gel A 0.286 0,0440 13 B 0.439 0.0466 20 C 0.747 0,0409 37 D 0,733 0,0365 39 lziiller’s earth 0.787 0.0129 122’’ (1 The averam pore radiris from porosinirter data
-~ X-Hay Paramcters n ro
D a_t a .
.1v.
port’
radiii.. 7.
A.
1.00 0.87 0.58 1.96
15 17 20 21 30 28 22 33 0.154 230 164” is 142 Angstriirnn.
and rc, are not st,rictl>. Ailthough the average pow radii, representations of the same quantity, it is not profitable to appl!. any of the obvious refinements. The unfortunate fact is that the a hole theory of Guinier is ba’sed 011 the assumption that thc scattering particles are spherical and sufficiently well and raildomly separated t,o avoid coherent interference betlveen particles. The circumstance of nonspheric.ity may he allo\ved for [as har: been done by Guinier ( 7 j ; see also Roess ( I g j ] ; but failure to meet the other essential rcquiremcnt in most common samplw
ANALYTICAL CHEMISTRY
670
.
cannot be circumvented. The intensity of scattered radiation depends on a difference in scattering power between the tlvo phases (solid and air in the case of porous solids), and the resultant size det’ermination (neglecting interference) is a measure of the distance one can travel in the scattering volume wit,hout passing a phase boundary in either direction. I n the case of a dispersed system of solid and air, when the particles are widely separated, the scattering due t,o their distances apart will be suppressed and one will obtain a particle size dhribution. K h e n the particles are squeezed so closely toget,her that, the interstices are small compared with particle size, the small-angle scattering will be indicative of pore size. There is no difference (except in over-all intensit’y) in scatfering pattern b e t w e n that obtained from a collection of particles with interstices of air, and that obtained from the reverse of this: a sponge in Tvhich the former particles are replaced spatially by air bubbles and the former interstices by a solid matrix. The situation when pore and particle size are about equal is the worst case. Here one must necessarily ohtain a combined distribution of both pore and particle sizw, In addition, the situation is complicated further by t,he fact that interparticlo interference now enters importantly. Thr reason that results on pore size distributions in inorganic grls-which are typical of the worst case-seem to corroborate other measurements so nicely is probably that the pore and particle size distributions are about the same and therefore t,hat the sum of the two is proportional t o either one alone. -1discugsion of this whole problem by Bauer is in preparation ( 1 ) . The several assumptions made in applying geometrical significance to the quantities T~ and ?vitiate from the start any rigorous comparison between the two. Thus r, represents an “average pore size” only in the sense that a uniform open-ended cylinder of t,his radius has the same ratio of volume t o surfacc as t’he porous sample. Similarly, :r has significance as an average pore size only in so far as the Guinier theory is applicable-that i3, only if the pores are actually spheres among which there is no diffractive interference, Inasmuch as the actual situation lies far from these assumed situations, correlation between r0 and F cannot be expected to be closer than merely qualitative. Thus Table I shows t,hat all four silica-alumina gels have approximately the same order of pore size; and that bok11 adsorption mcasurements and x-ray measurements arrange them ih the same sequence. It has been pointed out. by Shull (14) that the ratio of pore size (from adsorption measurements) t o discontinuity size (from x-ray scattering) shown by the materials of Table I shows not only the inconstancy expected from the argument above, but’also a regular trend with the “porosity factor” for these materials. (Porosity factor is the ratio of empty volume to solid volume in any porous material.) Shull has found the same trend in a larger series of silica gels during the course of his own work and concludes t,hat x-ray scattering for such materials is more indicative of particle than of pore size. Table I1 li volume, T.”, the porosity factor, V0!T7,, the “average particle size,” r, = 2T’JS0, and the ratios r C , 7and r,/T for the eilica-alumina gels of Table I. Shull points out that the porosity factor should be proportional to the ratio of pore size to particle size; that such a proportionality, for gels A, B, C, and D, approximately rxists between the porosit,y factor and the ratio re/?, and that’ thercBfore P probably represents particle rather than pore size. On the other hand, the agreement betm-een F and T , is no better than that between 7 and rc, and there is also an approximate proportionality between ?;r8 and porosity factor. Additionally, the agreement among re, 7, and the porosimeter data for the fuller’s earth is strikingly better than with r8. Clearly the exact geometrical significance of size determinations from x-ray scattering is still uncertain, and additional correlative data are urgently needed, particularly unamhiguous methods of determining entire distributions of sizes.
Comparisons of entire distributions as obtained by x-ray scattering with entire distributions (rather than simply average sizes) as determined by other methods are difficult to make. The mercury porosimeter desciibed in an earlier communication from this laboratory ( 1 1 ) affords one method of obtaining complete distributions but is limited to larger pore sizes. Figure 9 compareq the porosimeter distribution for the fuller’s earth (16) Kith the corresponding x-ray data. Considering the limitations of both methods the agreement is surprisingly good. If one proceeds to distributions having average radii of the order of a thousand times the wave length used, the method loses sensitivity. Thus, in an attempt to deduce by x-ray scattering the distribution found by the mercury porosimeter for Coors was as a2x2 + ( 3 , and on the criterion of
porous plate (11), the plot of log I us. log
nearly linear for TO = 500 as for T o = m lincaritv of this plot, anv ro > 500 might have been chosen. Table 11.
Comparison of Pore and Particle Size Data
JIaterial 1-8, c ~ . / G 0.42i Gel .1 0.421 Gel B 0.416 Gel C Gel D 0.421 Fuller’s earrh 0 376
v~’v$
rciF
r/ra
ra, A.
0.668 1.108 1.797 1.748 2 . OB
0.76 0.95 1.32 1.77 0.72
0.89 1.19 1.44 1.47 ?.9
19 18 20
23 58
For convenience the several steps in converting the observed intenqity curve into a size diqtribution are summarized. 1. Expand empirically the observed equatorial intensity function, I , (2)into a closed series of Gauss functions, I , (2) = Zb,exp ( - a&).
The slit-corrected intensity_function*ll then be I,,, ( 2 ) = where c1 = b, d a , l e r j ( J ~ Y c)Y . , 3. Plot log I,,, (2) against log (a29 ro-2) for a series of reasonable values of T ~ . That curve that is most nearly linear gives the best value of T O . 4. Determine the slope of this “best” curve, and find n from the relation, slope = - ( n 2). 5 . If desired, find A in the relation D ( r ) = A 2.
Zc, exp ( -a&,
+
+
by normalizing J d V = J D ( r ) d r to the measured”pore volume,
VO. Obviously, the same method applies, with thesame limitations, to a determination of particle size dist,ribution. ACKNOWLEDGMENT
The authors wish to express their appeciation to R. C. Hansford and F. P. Hochgesang for their valuable criticism and counsel during t,he performance of this work. Thanks are liken-ise due L. C. Drake for his help with the adsorption and porosimeter data rcbportcd ahove, and Miss C. J. Delaney for help with the calculations. LITERATURE CITED (1 ~ Rauer. - -S. H.. Cornel1 Cniversitv. Drivate communication (2) BaueI: S. H., J . Chem. Phys., l“3,450 (1945). (3) Biscoe, J., and Warren, B. E., J . Applied Phys., 13,364 (1942) (4) Brunauer, S., Emmett, P. H., and Teller. E.. J . Am. Chem. Soc., 60, 309 (1938). \-,
( 5 ) Elkin, P. B., Shull, C. G., and Roess, L. C., Ind. Eng. Chem., 37, 327 (1945).
(6) Fankuchen, I., S a t u r e , 139, 193 (1937). (7) Guinier, A , , Ann. phys., 12, 161 (1939). (8) Hosemann, R., 2. Physik.. 113,751 (1939); 114, 133 (1939). (9) Jellinek, AI. H., and Fankuchen, I., Ind. Eng. Chem., 37, 159 fl94.5). ~ ---_
Jellinek, M. H., Solomon, E., and Fankuchen, I., IND.EXQ. CHEM.,ANAL.ED.,18, 172 (1946). (11) Ritter, H. L., and Drake, L. C., Ibid., 17,782 (1945). (12) Roess, L. C., J . Chem. Phus., 14, 695 11946). (13) Shull, C. G., Gibson Island Conference, Aug. 23, 1944. (14) Shull, C. G., private communication. (15) Shull. C. G.. and Roess. L. C., J . AppZiedPhys., 18, 295 (1947). (16) Socony-Vacuum Laboratories, unpublished data. (10)
9. Gibson RECEIVED July 2 5 , 1947. Presented in part before the 8.8.4 Island Conference on Catalysis, June 26. 1946. The first paper in this series appears in ( I f ) .
