P O R E ARE-4 AND
ADSORPTION HYSTERESIS FOR
1395
PACKED SPHERES
Pore Area and Adsorption Hysteresis for Packed Spheres
by Raymond Venable and William H. Wade Department of Chemistry, The UniTersity of Texas, A u s t i n , Texas
(Receiaed November 19, 1964)
The desorption isotherms for water from packed spheres and a commercial gel were obtained and analyzed for their pore volume distributions. In addition, nitrogen film areas were obtained during the desorption process. From these latter measurements pore area distributions were obtained. The calculated distributions provide some insight into the geometry of the packed spheres.
Several recent papers’, have delineated the importance of capillary condensation for certain powdered adsorbent samples which can be treated as an assemblage of spheres. In addition, Wade3 has provided several techniques where the packing coordination number of the spheres could be estimated for a single sample of A1203 whose coordination nuniber was increased by compression of loosely packed powders. For such systems Kiselev, et ul.,l have predicted the shapes of K2 isotherms for both the adsorption and desorption branches. The adsorption branch is assumed to obey the B.E.T. equation-which is not usually found to be true-up to a critical pressure where the porous medium conipletely saturates. (The authors point out the existence of special cases where this is not strictly true.) The existence of such a singular pressure requires yniforni packing geometry of identical spheres. Such powder packs have never been obtained, and Wade3 could find no such behavior for his samples. Although it may be possible to prepare samples of more and more uniform dimensions, it is hard to envision obtaining a simple geometric packing for an entire assemblage by a conipressional process. Kiselev, et al., in treating the desorption process empty the pores once again at a pressure unique for given radii spheres packed in a single geometric pattern. They state that the pores open when the pendular rings around the spheres forming the necks appear as an unbroken circle. illthough this is a pleasing, simple picture in that the liquid meniscus at the instant of pore emptying would be hemispherical with two identical radii of curvature, it must be incorrect. The inscribed circle condition is correctly applied to the porefilling process. Prior to pore opening, the filled windows must additionally dilate and the description of
the resulting meniscus is quite complex and to the present time not analytically described. The spherical radii, r s p h , used by Kiselev, el ul., eq. 5 and 6, must be too small and a inore correct representation will be offered later. Furthermore, in describing pore structure is has been common practice to assume cylindrical symmetry and nieke use of the geometrical relationship
where Vp is the true pore volume, 2, is the pore area, and r s p h is the spherical pore radius using the Kelvin equation in the form
y being the liquid surface tension, P is the molar volume, RG is the gas constant, T is the absolute temperature, and r s p h is the radius of the spherical meniscus. I n general, VP/rBphis summed over the pore volume distribution and the resulting area is compared with the B.E.T. area obtained from a nitrogen isotherm
(3) where t is the thickness of film remaining subsequent to pore opening. Quite good agreement is commonly found between the measured and calculated areas and de Boer4 has discussed the applicability to several sys(1) B. G. Aristov, A . P. Karanaukhov, and A. 1’. Kiselev, RZLSS. J. P h y 8 . Chem., 36, 1159 (1962). (2) W. H. Wade, J . P h y s . Chem., 68, 1029 (1964). (3) W. H. Wade, ibid., 69, 322 (1965).
V o l u m e 69. ,Vumber 4
A p r i l 1966
RAYMOND VENABLE AND WILLIAM H. WADE
1396
tems. Everett5 has tabulated quantities similar to the constant, l / 2 , in eq. 1 for other pore structures. For instance, a similar quantity, V,/B,R, has the value of 0.304 for pores formed by cubically packed spheres and 0.117 for hexagonally close-packed spheres where R is now the adsorbent sphere radius. This coupled with a discussion of the validity of eq. 3 for a variety of more complex pore structures by de Boer4 should dictate against the indiscriminate application of eq. 3 without some knowledge of pore geometry. Wade2 has demonstrated the feasibility of measuring areas of frozen adsorbate films on adsorbent spheres. Such areas taken from the desorption branch of an isotherm should yield an analogous pore area distribution and a combination of the two distributions will yield an experimental measurement of Everett's parameter, CYR = V,/Z,R.
Experimental Two high surface area alumina specimens were investigated, both having been the subject of previous investigation~.23~-~ One is a commercial alumina gel produced by Alcoa, Inc., and designated by them as XF-20. In this study it will be referred to as sample A. The other material is a high-purity yA1203 produced by Whittaker, Clark, Daniels, Inc., and is designated by them as Alucer MA. It was compressed in a hydraulic press to produce samples of greatly decreased porosity. Three such samples are discussed here and bear the designation B, C, and D as in the previous The water desorption isotherm for sample A was obtained on a commercial quartz beam vacuum microbalance manufactured by Worden Laboratories, Houston, Texas. Weighing was always done with the balance a t null, the restoring torque on the beam being applied by a Helmholtz coil interacting with an Alnico V magnet rigidly mounted on the beam. The null position was detected by a 50X microscope and the coil current read as a voltage drop across a standard lO-ohm resistor with a Rubicon Model B potentiometer. Pressures were read from an accompanying oil manometer. 'The desorption isotherms for samples B-D were obtained by standard volumetric techniques. These four isotherms are displayed in Figure 1 where the ordinates have been converted to liquid cubic centimeters per gram for subsequent calculations. The surface areas of samples A and C were measured for samples pre-equilibrated with various amounts of water preadsorbed as previously detailed.2 The film areas corresponding to the water adsorption branch have been p ~ b l i s h e d , ~ but , ~ since experimental film area hysteresis has not been noted previously they are The Journal of Physical Chemistry
f
01
0.2
0.4
0.6
0.8
P / P.
Figure 1. Desorption isotherms for samples A-D.
reported in Figure 2 along with the measured film areas taken from the water desorption branches. It will be noted that there is indeed hysteresis in the film areas and the loops occur in the same relative pressure region as observed in the adsorption-desorption isotherms themselves.
Results and Discussion The desorption isotherms for samples B-D in Figure 1 when combined with the adsorption branches3 show that as the porosity decreases in proceeding from B to C to D, the width of the hysteresis loop increases. Pore emptying is likewise accomplished only at lower pressures when proceeding from sample B to D. Previous calculations3of the average coordination number, n, of B, C, and D yielded approximate values of 6, 8, and 10. The coordination number of packed (4) 3. H. de Boer, Pwc. Colston Res. Soc., 10, 68 (1958). (5) D.H.Everett, ibid., 10, 95 (1958). (6) W.H.Wade and N. Hackerman, J . P h y s . Chem., 68, 1592 (1964).
POREAREAAND ADSORPTION HYSTERESIS FOR PACKED SPHERES
00
1397
ness of the uniform film on samples B-D could be represent'ed by the Frenkel-Halsey-Hill equation as
so
60
CC , m'
.T-l 40
20
0
Figure 2. Adsorption and desorption film areas for samples A and C.
spheres is not singularly related to a single packing geometry but, commonly, two or more different pack porosities can have the same coordination number.' Thus considering only the packing models of Graton and Fraser,* the coordination numbers of 6, 8, and 10 correspond to packs with square, triangular and square, and triangular and rectangular pore windows. (The windows are, of course, made up of circular arcs but for abbreviation will be designated as above.) A given pore volume will always empty through the largest pore and thus Graton and Fraser's n = 6 and 8 packs should empty at the same pressure and the n = 10 sample a t a somewhat lower pressure. Using the model of Kiselev, et d . , l packed spheres with coordination numbers of 6 or 8 empty through the square windows a t a relative pressure when the pendular rings no longer overlap. The window opens a t the point where it is just no longer circular in cross section. For a square array of packed spheres, the radius of the inscribed circle is equal to r B p h in eq. 3, and from geometric considerations is = (d2 - 1)h!
-t
(4) where t is the thickness of the residual adsorbate uniform film and once again R is the radius of the adsorbent spheres. Previously, it was found6 that the thickrsph
over the relative pressure region 0.3 < p / p o < 0.98. Calculations show R to have the value of 75.4 b. By utilizing eq. 2, 4, and 5 , one can predict that samples B and C, if they conform to the Graton and Fraser and Kiselev models, should empty at a pressure of 15.8 mm., whereas experimentally they are found to empty at different pressures, both of which are higher than those predicted. It should be immediately obvious that detailed agreement would not be expected, for the isotherms do not show a discontinuous drop as the simple model predicts. This is due to t,wo causes: a distribution of sphere radii and a packing geometry which is not simple but retains much random character during formation. Sample D also shows the same behavior but on the average the pore necks are smaller and thus the pores empty at lower pressures. Once again using Kiselev's model, pore emptying occurs when the pendular rings a t the ends of the rectangular windows separate. The next step in the analysis is not clear to this author. Apparently a hemispherical meniscus is once again assumed at the ends of the rectangles and, for sample D, eq. 2, 5 , and one similar to (4)would be applied. The menisci separate at (@/5 - l ) R - t and thus
Clearly, this approach is incorrect for the liquid meniscus in these rectangular pores cannot approach a hemispherical shape with two identical radii of curvature. The condition of equilibrium over any point on a l/ez) is constant where liquid meniscus is that (l/el el and u2 are the principal radii of curvature. At the present time the conversion of this to an analytical description of complex meniscus shapes as discussed here is not available. Thus it is proposed to treat the complex radii of curvature as a single equivalent radius of curvature, req, which is the radius of a circle whose area is equal to the actual pore area when it opens
+
where A , is the window area. This still nonrigorous approach lacks simplicity because of some inconvenience (7) S. Kruyer, Trans. Faradau Sac., 54, 1758 (1958). (8) L. C . Graton and H. J. Fraser, J . Geol., 43, 785 (1935).
Volume 69,Number
4 April 1966
RAYMOND VENABLE
1398
A N D wILLIAb.1
H. W A D E
in evaluating A,. If spheres forming the window are of uniform radius, R, then an angle 4 may be defined (Figure 3) as one-half the angle formed at the intersection of the two tangents drawn from the points of contact of any one sphere with the two adjacent spheres. The shaded area of the construction in Figure 3 is the segment of the window pertinent to this sphere. I t is the segment of the pore area diminished by the uniform film and pendular rings. Geometrical considerations yield the expression
A,'
=
R2 cot 4
4 + r C 2 180 + 2 B d A 2 - R2 -
adr,2 -
~2
-
r,2
R sin-' - -
Figure 3. A,, (shaded area) for pore window diminished by a uniform film and the two segments of pendular rings.
Tm
+ +
+
whereA = R t r,, B = [R(R t)/Al - R , C = R t , and r , is the lesser radius of curvature of the pendular rings. If the complete window is a regular polygon, then
+
360 A , = - A,' 29
If not, then A,' must be summed for each value of 4 and overlapping triangular areas eliminated. A ,and reg have been evaluated using the C.D.S. 1604 computer at The University of Texas for a variety of sphere radii packed in regular triangular, square, pentagonal, and hexagonal arrays. res is plotted in Figure 4 for water adsorbed on Alucer MA as a function of pressure. The thickness of the uniform film, t , is given by eq. 5 and the size of the pendular rings related by
(9)
ns before.2 The ring-hand side of each of the four window arrays terminates at the maximum relative pressure at which the pores can remain open, as given by the condition in (4),(4'), and their analogs. Each region is calculated as centered about R = 75.0 A. and is arbitrarily extended to limits on R of 60 and 90 A. for it was felt that this should include 90% of the adsorbent spheres. The rectangular inset framing represents the limits of occurrence of the hysteresis loop for sample C, the pressure limits from the isotherm, and the reg limits from the application of eq. 2 assuming res = r s p h . The line set in the rectangle is a plot of eq. 2 in the region of interest. For a given sphere radius and packing geometry, reg = Tsph at the The Journal of Physical Chemistry
0.2
-
0.4
0:s
66
'
I :o
P / Po
Figure 4. Equivalent radius of triangular, square, pentagonal, and hexagonal windows as a function of relative pressure for 60-, 75-, and 90-A. radius spheres.
pressure corresponding to the pore emptying. For this pressure is 16.7 cubic packing with R = 75 i., mm., which is within the region of the hysteresis loop which, when coupled with the measured coordination
POREAREAAND ADSORPTION HYSTERESIS FOR PACKED SPHERES
number of n = 8 would indicate a packing such as cases I1 and IV discussed by Graton and Fraser. However, a t this pressure the desorption branch is far from vertical and the evident distribution of pore openings poses an insoluble problem. This distribution can arise from a distribution in sphere radii for a single given packing geometry, a distribution of packing geometries for a single sphere size, or a random distribution in both packing geometry and sphere size. Undoubtedly the latter is the more realistic. The most sensible view is that pore volumes communicate through a wide distribution of pore windows with a given pore volume emptying through its largert window. For sample B at p = 20 mm. ( p / p o = 0.84) such windows should need to be regular pentagons made of the larger sphere sizes, regular hexagons of smaller sphere sizes, flattened hexagons of larger sphere size, or some appropriate combination of sphere sizes in pentagonal or hexagonal arrays. For sample C at p = 18 mm. ( p / p o = 0.76) they could be both square and pentagonal arrays of a variety of sphere sizes. For sample D at p = 15.5 mm. ( p / p o = 0.65) the largest windows are probably regular square arrays and even for this sample it is still improbable that triangular arrays of even the largest spheres are the largest windows. Furthermore, for sample D there are probably very few pentagonal or hexagonal windows unless they are so flattened that they differ little from square (or diamond) or triangular arrays. The desorption isotherms in combination with eq. 2 were used to obtain uncorrected pore volume distributions. These derivations of I",,,= f(T,ph) are shown in Figure 5 along with analogous pore area distributions obtained from the measured desorption pore areas. It is immediately apparent that the maxima for the two types of distributions for samples A and C occur at the same pressures and the shape of the distribution curves are quite similar. Barrett, Joyner, and Halendas have discussed the detailed analysis of desorption data to yield true pore volume distribution. They note that corrections must be made to account for the thinning of residual adsorbate in previously opened pores and the uniform film retained in the pores opened in a given desorption increment. Unfortunately, the detailed treatment is limited to cylindrical capillaries and will not be employed here. With an eye toward the calculation of Everett's geometry constant, the quantity ( V , - V,d./&R was calculated for Graton and Fraser's four geometries with n = 6, 8, 10, and 12. V,d,, the volume adsorbed in empty pores a t any coverage, and Zr, the adsorbate film area in the pore, were evaluated by eq. 1 and 2 in ref. 3 connecting these quantities through t and rm to eq. 5 and 9 in this paper
1399
-1
1
Figure 5. Uncorrected pore volumes and area distributions.
relating t and rm to the pressure. The constant aR defined as
is plotted in Figure 6 as a function of the relative pressure. The curves are terminated a t p / p o = 5 mm., for below this point eq. 5 is no longer applicable. aR is found to be insensitive to the adsorbed state a t low relative pressures having the values calculated by Everett. At high pressures aR decreases rapidly ; however, on the desorption branch the pores will be closed at these pressures. The pressures corresponding to the maxima in Figure 5 for samples B, C, and D are noted in Figure 6 on the n = 6 , 8 , and 10 lines since they correspond to the measured values of n. They are located on the knees of the curves where aR is not appreciably lower than for outgassed spheres. Thus for present purposes it should be sufficient to correct AVme,,/Arsph and A&,eaa/Araphonly for the thinning of adsorbate film and pendular rings of the previously opened pores. This was done in the following way. Increments in rSph (2 A.) were chosen and thus were Assuming established increments in AV,,,, and AI;,,,,. n = 6, 8, and 10 and once again using eq. 1 and 2 in ref. 3, AVads/ArSphand azf/b,ph could be evaluated as a function of p / p o and are given in Figure 7 . These two quantities must then be weighted by the fraction of total ~~
(9) E. P. Barrett, L. G. Joyner, and P. P. Halenda, J. Am. Chem. SOC.,7 3 , 373 (1951).
Volume 69, Number 4
April 1966
RAYMOND VENABLEAND WILLIAMH. WADE
1400
pore space opened prior to the present increment. This was done by graphical integrations of initially corrected pore volume and area distributions (the primed quantities below) and the quantities
O?
____x\
0;
N=IO
U
0.
.
.
.
P/ Po
Figure 6 . The variation of the geometry constant, a h , as a function of relative pressure for spheres of radius 75 A. packed with coordination numbers of 6, 8, 10, and 12.
become the corrections for thinning of the open pores. The initially corrected distributions are the uncorrected distributions terminated a t their extremities by a sloping straight line such as those drawn in Figure 5 and thus are just first approximations of (11) and (12) corresponding to linear desorption isotherms. The resulting pore area and volume distributions are drawn in Figure 8 for sample C. The values of cyR taken from
3 1.4
3.2
I.3
0.4
2
AX A
dv..I
*aph
A'SPh
($1 I
XIQQ
0.6
12 A Vnl raph
1
x IO'
rsph
0 0
1.1
0.8
20 riph
40
60
Figure 8. Corrected pore volume and area distributions for samples A and C. I .o
1.0
/
0.9 i
/ O.?
0.6
\\\ 0.8
P/P*
Figure 7. The slope of Vada = f(rm) and 2, = f(rm) us. relative pressures in the region of hysteresis.
The Journal of Physical Chemistry
the ratios of these two distributions for R = 75.4 A. are traced in Figure 6. Little significance should be placed on the variation of aR with p/pO since it is the resulting ratio of two graphical differentiations both of which needed to be smoothed. The average experimental value of aR is 0.27 lying between the values for n = 6 and 8 for Graton and Fraser's models. If the true, fully corrected pore volume and area distributions are
POREAREAAND ADSORPTION HYSTERESIS FOR PACKED SPHERES
1401
desired, the adsorbate remaining in the empty pores could be calculated to a good approximation by aR
=
(16) (%)'E
(13)
(14)
for any given segment of the distribution. These quantities are the fraction of total volume adsorbed and film area in pores of a given TBph and for a given increment in ATEDh and must be added to the appropriate values in Figure 8. This second correction shifts the distributions to slightly larger values of Tsph, but aR remains 0.27 as would be expected from Figure 6. Film area hysteresis was also noted for the commercial alumina gel, sample A. It is probably not valid to treat the pore volume as the interstices between packed spheres as can be done for many silica gels, but without more information it certainly cannot be treated as an assembly of capillaries. The uncorrected pore area and volume distributions for this same are shown in Figure 5. A rather arbitrary correction for the thinning of adsorbate in open pores can be made by assuming that the lower break in the desorption isotherm at p / p o = 0.30 corresponds to complete depopulation of the pore structure and curves of analytic form similar to those for sample C applied as correction for open pore thinning. The resulting distributions are also given in Figure 6. These distributions can then be handled in two ways. Treating the pores as capillaries the quantity
a, =
($)'
(E)'
(15)
rwhsv
was evaluated over the two distributions. The average value obtained is a, = 0.55 compared to the value of 0.50 commonly used. It is difficult to assess the expected agreement but the 10% difference is probably within experimental error. If rather than as capillaries, the pore spaces are considered as formed from packed spheres, the quantity
can be evaluated over the distributions. The average value of aR is found to be 0.26 using R = 28.3 A. as calculated from eq. 5 in ref. 3 for n = 6. This value of aR is quite similar to that for sample C. Without more information it is not possible to delineate further this sample's pore structure. I n Table I are listed values of a, and aR for spheres packed in Graton and Fraser's four arrays. Table I Packing
Cubic Orthorhombic Tetragonal-spheroidal Hexagonal close
QR
0,
0.303 0.218 0.144 0.117
0.579 0.417 0.300 0.518
aR is the geometrical pore volume to pore area ratio divided by the sphere radius for completely uncovered spheres and consists of the values approached p / p o = 0 in Figure 6. It would appear to be a very definitive parameter. Unfortunately, this is not so. If rather than the sphere radius, reg is divided into the volume to area ratio to yield a,, a much smaller spread is observed. In calculating reg for the middle two packings where two different window areas occur, the large window was chosen in each case for it will control the pore opening. The average value of a, in Table I is 0.45 which differs only by 10% from the value of 0.5 for capillary geometry. Perhaps this explains why the condition expressed in (3) is so often observed. If one has spheres packed randomly over the four distributions in Table I or even if they are packed uniformly in cubic or hexagonal close-packed arrays, they exhibit the same capillaries. To illustrate this, evaluating a, for sample C yields a value of 0.55 and shows that (3) would be a suitable if misleading test for this sample. I n conclusion, the experimental pore area distributions give some insight into the common success of treating complex pore shapes as capillaries. Acknowledgment. The authors gratefully acknowledge the Robert A. Welch Foundation and the American Petroleum Institute for their financial support and continued interest. Miss Josephine Barto wrote the necessary computer programs and did many of the auxiliary calculations and Mr. Arnold C. Falk performed many of the film area measurements.
Volume 69,Number 4 April 1966
