2723
J . Phys. Chem. 1986, 90, 2723-2728 the valence band can thus diffuse into the surface and form the surface-trapped holes, resulting in a strong emission of the 800-nm band. A similar mechanism was reported32for the explanation of temperature dependences of the intensities of two bulk luminescence bands from CdS crystals. In conclusion, the in situ photoluminescence study for the ,,-Gap electrode has confirmed the previously proposed mechanism for the photoanodic electron-transfer reactions, clearly indicating that (32) Vuylsteke, A. A.; Sihvonen, Y. T. Phys. Reu. 1959, 113, 40-42.
it is a powerful method for the investigation of the molecular mechanism of surface reactions at semiconductor electrodes.
Acknowledgment. We express our thanks to Professors Y . Hamakawa and T. Nishino and Mr. Y . Fujiwara and Mr. A. Kojima in Osaka University for kindly measuring the photoluminescence ‘pectra at the liquid Registry No. Gap, 12063-98-8; S, 7704-34-9;Te, 13494-80-9;[Fe(EDTA)I2-, 15651-72-6; [Fe(EDTA)]-, 15275-07-7; [Fe(CN)6I4-, 13408-63-4;[Fe(CN)6] 13408-62-3;[Fe(C204),] 15321-61-6; [ Fe(C204)J4-,30948-48-2;ferrocene, 102-54-5.
’-,
’-,
Adsorption and Condensation In Random Microsphere Packings Douglas M. Smith UNM Powders and Granular Materials Laboratory, Department of Chemical and Nuclear Engineering, University of New Mexico, Albuquerque, New Mexico 87131 (Received: November 22, 1985; In Final Form: February 3, 1986)
A model has been developed to describe the coupled phenomena of adsorption and capillary condensation in the toroidal region of sphere contacts in a random packing of microspheres (Le., e 0.36-0.4). The exact meniscus shape for toroidal condensate has been calculated by the solution of the Young-Laplace equation. The volume and surface area of toroidal condensate are found as a function of three dimensionless groups: the relative pressure, A (adsorbate/adsorbent size ratio), and p (a function of adsorbate physical properties). By using the known distribution of sphere contacts and near-neighbors in a random packing, we are able to predict adsorption isotherms and the reduction in film surface areas as a function of relative pressure. It is shown that curvature effects on the accuracy of the BET method will not be significant for A less than 0.001. For smaller sphere sizes, and thus higher A, toroidal condensation effects become significant and will result in an incorrect interpretation of surface area and pore volume distribution data as obtained from adsorption isotherms. The reduction of film surface area as a function of relative pressure can be a very important tool for studying particle contacts and near-contacts in porous media. By use of our model, the variation in surface area has been accurately calculated as compared to previously published experimental results. The fact that we are able to accurately model this process without the use of adjustable parameters implies that this theory may be used for comparison to experiment to obtain additional structural information.
Introduction The phenomena of adsorption and capillary condensation are widely applied tools for the analysis of surface area and pore structure in porous materials. The translation of adsorption/ condensation data into information concerning pore structure typically requires an assumption about the pore shape. In general, the pore shape is assumed to be cylindrical. However, for many porous materials which have been produced via sol agglomeration, this cylindrical pore assumption results in a misleading interpretation of pore structure. Many materials, such as silica and aluminum catalyst supports and certain ceramics, can be better described by a “random assemblage of uniform spheres” pore model. Therefore, considerable interest exists for obtaining a more accurate description of adsorption, capillary condensation in the toroidal region surrounding contacts, and condensation in the internal pore cavities. If a realistic model of these phenomena were available, more detailed information could be extracted from adsorption/condensation data concerning the number of contacts between spheres, pore shape, and pore size distribution. Past attempts to model adsorption and condensation in beds of uniform spheres which are randomly packed assume that toroidal condensation occurs only at points where two spheres are in direct contact. For an actual random packing of spheres, many “near-neighbor” contacts occur for which two spheres are not actually touching, but when the layer of adsorbate reaches a certain thickness, the films will overlap and a toroidal-shaped meniscus will form around the spheresphere axis of symmetry. In addition to ignoring the effect of these “near-neighbor” spheres, the shape of the meniscus has been described by the rotation of either a circle or an ellipse around the contact instead of the correct 0022-3654/86/2090-2723$01.50/0
nodoid shape. In this work, we attempt to formulate a more accurate description of the phenomena of adsorption and toroidal capillary condensation in random packings of spheres. The validity of the model will be assessed by comparing experimental and theoretical results for the reduction in measured BET surface area when a liquid is preadsorbed on a porous solid. If an accurate description of the reduction in these film surface areas can be obtained, a valuable new tool for examining the structure of particle-particle contacts in porous solids will be obtained.
Background Early studies of adsorption and capillary condensation in both random and ordered packings of spheres addressed questions concerning the accuracy of the BET surface area measurement method for these types of porous materials. Karnaukhov and Kiselev’ used a greatly simplified model to study adsorption at the contact between two spheres. Effects due to capillary condensation in the vicinity of the contact point were not considered. The relationship between the average number of sphere contacts, the particle size, and the accuracy of BET measurements was obtained. In an extension of that work, Aristov et a1.* included toroidal condensation by assuming that the meniscus shape surrounding a sphere-sphere contact is given by a circle rotated around the axis of symmetry between the spheres. The presence of a film of adsorbate was shown to have a significant effect on (1) Karnaukhov, A. P.; Kiselev, A. V. Russ. J . Phys. Chem. (Engl. Trawl. 1960, 34, 1019.
(2) Aristov, B. G.; Karnaukhov, A. P.; Kiselev, A. V. Russ.J . Phys. Chem. (Engl. Tronsl.) 1962, 36, 1 159.
0 1986 American Chemical Society
2724
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986
condensation and, hence, the calculated pore volume distribution. In the above work, the average number of sphere contacts is required. Wade3 undertook measurements of the reduction in measured BET surface area when water was preadsorbed on packings of alumina spheres in an attempt to measure the coordination number. The measured surface area was found to decrease monotonically with increasing water content. For powders compressed at higher pressures with a resulting lower porosity, this reduction in surface area was more pronounced. This reduction in surface area is the result of condensate in the toroidal region blocking access to the sphere surface. A theory based upon a combined adsorption/condensation model to describe the observed reduction in surface area has been proposed by Wade.4 As with the work by Aristov et al.,2 the assumption of a circularshaped meniscus was employed. Coordination numbers were determined by matching the reduction in surface area to tfieory. However, calculated coordination numbers were substantially greater than those determined via other methods. This may be attributed to errors associated with the circle meniscus assumption and ignoring the role that spheres which are near-neighbors but not touching play in the adsorption/condensation process. Melrose5 has exactly determined the shape of a meniscus between two uniform spheres in contact via the solution of the Young-Laplace equation. The correct nodoid shape was ascertained, and the relationship between condensate volume in the toroidal region and vapor pressure was determined. This work did not address the effects of adsorption, multiple sphere contacts, and spheres not in contact. Dollimore and Heal6 analyzed the processes of adsorption, toroidal condensation, and condensation in internal pore cavities for ordered sphere packings. In that work, the shape of the toroidal meniscus was assumed to be represented by an ellipse of revolution around the sphere-sphere contact. Although this represents a more realistic approximation than the circle assumption, calculated volumes will not equal those calculated for the correct nodoid curve. Although Dollimore and Heal conclude that little change is observed between calculating pore size distributions from either a cylindrical pore model or their spherical model, this is primarily the result of assuming that the pore size is the pore cavity mouth radius. The technique of analyzing adsorption/condensation in a random packing of spheres by using an ordered sphere packing model has been repeatedly q~estioned.~A rich body of literature exists for the similar problem of mercury porosimetry in random sphere packings which has been reviewed by H a y n e ~ Ordered .~ packing models have been unable to describe mercury intrusion experiments. However, Smith and Stermer* have been able to describe mercury intrusion experiments on random packings of silica spheres in the size range of 102-452 nm using a model which accounts for the actual distribution of sphere spacings. An exact solution of the Young-Laplace equation for both spheres in contact and for near-neighbor spheres was used to relate the meniscus shape to the applied pressure.
Theory In order to accurately calculate the relationship between relative pressure, the adsorbed layer, and toroidal condensate, an expression for describing the quantity of adsorbate as a function of PIP, must be assumed. A number of isotherms exist for relating the thickness of the adsorbed layer, t , to the relative pressure and adsorbate size. The Halsey equation' has been widely applied in pore size analysis via physical adsorption/capillary condensation. The Halsey equation is commonly writtenlo t = 6(5/ln ( P O / P ) ) ' I 3
(1)
(3) Wade, W. H. J . Phys. Chem. 1964, 68, 1029. (4) Wade, W. H. J . Phys. Chem. 1965,69, 322.
( 5 ) Melrose, J. C. AIChE J . 1966, 12, 987. (6) Dollimore, D.; Heal, G . R. J . Colloid Interface Sci. 1973, 42, 233. (7) Haynes, J. M. Spec. Period. Rep.: Colloid Sci. 1975, 2, 101-129. (8) Smith, D. M.; Stermer, D. L. J . Colloid Interface Sci., in press. (9) Halsey, G . J . Chem. Phys. 1948, 16, 931. (IO) Lowell, S. Introduction to Powder Surface Area; Wiley: New York, 1979; p 92.
Smith
I
// / // // / ////
///
I
Figure 1. Capillary condensate in the half-region of sphere near-neighbors.
where 6 is the thickness of one monolayer. Rewriting eq 1 in dimensionless form, we obtain T = A(5/ln ( P 0 / P ) ) 1 / 3
(2)
where T = t / R and A = 6/R. Thus, A represents a scale parameter which describes the relative sizes of the adsorbate and adsorbent. The total uptake will include contributions due to adsorption, toroidal condensation, and condensation in the inner cavities. In order to calculate the shape of the meniscus in the toroidal region, we assume that no interactions occur between the toroidal condensate and the adsorbed layer. In other words, changes in the adsorbed film thickness result only in different effective sphere size ( R + t ) and effective sphere spacing (d - t ) where d is the half-distance between spheres (see Figure 1). As a result of the presence of an adsorbed film, calculations will be conducted for a contact angle, 8, equal to 0 (Le,, perfect wetting). The shape of the meniscus at equilibrium in the absence of gravatational effects may be described by the Young-Laplace equation:' I
2H = A P / y =
Y" (1
+ y'2)3/2
Y'
+
x(l
+ y'2)'/2
(3)
Equation 3 relates the mean curvature, 2H, and the mean pressure differential across the meniscus, AP/y, to meniscus shape as described by the spatial coordinates x and y . If we consider half of the toroidal void space between two spheres (Le., apply symmetry), as depicted in Figure 1 for the case of no adsorbate, eq 3 may be simplified. We define the dimensionless variables 9 = x/Rcr{ = y/&, and p = s i n (a),as suggested by M e l r o ~ ewhere ,~ a is the angle that a line perpendicular to the meniscus makes with they axis and & is the effective sphere size, R t . Applying these transformations, we obtain
+
dP I.L 2HR, = APR,/y = -- - d9 9 Boundary conditions for eq 4 are given by pi =
-sin ($),
tl =
1
+ De - cos ($),
p2
(4)
=
c2 = 0
(5)
where D = d / R (dimensionless sphere spacing), De = D - T (effective sphere spacing), and t+b is the filling angle. Following the work of Orr and co-workers,12multiple solutions of eq 4 may be obtained in terms of elliptic integrals. Since the meniscus must be restricted to a negative mean curvature (2HR, < 0), for the problem which we address, the meniscus shape will always remain ( 1 1) Adamson, A. W. Physical Chemistry of Surfaces; 3rd ed.; Wiley: New York, 1976. (12) Orr, F. M.; Scriven, L. E.; Rivas, A. P. J . Fluid Mech. 1975, 67,723.
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986 2125
Adsorption and Condensation in Microsphere Packings a nodoid and the dimensionless mean curvature, 2HR,, is given by -COS ($) E(a,X)/X (A2 - l)F(a,X)/A 2HRe = (6) De + 1 - COS ($)
+
+
where
x
= (1
+ p)-'/2
cr = a / 2 - $
and p is the solution of p
E/& =
= (2HRe)2sin2 ($) - 4HR, sin ($) sin (4)
(7)
The functions F(a,X) and E(a,X) are elliptic integrals of the first and second kinds. For a given value of the filling angle, $, eq 6 and 7 must be solved simultaneously to obtain the mean curvature. The mean curvature may be related to the relative pressure by the Kelvin equation. The possible sources of error related to the use of the Kelvin equation for this type of problem are minor, as addressed by M e l r ~ s e .Writing ~ the Kelvin equation in our notation, we obtain 2HRe = 6 ( 1 T)/A In (P/P,) (8)
+
where 0 = pLRgTL8/y. The magnitude of 0 is independent of the adsorbent and is a function only of the adsorbate and adsorption temperature. The dimensionless volume of toroidal condensate corresponding to a given filling angle is given by
Vt/R: = *(A1 - Xp(p
+ 4)F(a,X)/3 + X(p + 8 ) X (P
+ 1)E(.,X)/3)(2HR)3 - Az (9)
where AI =
COS
($)(-(/I
+ 4 ) ) + 4 COS' =
where N, is the total number of spheres, N(D=O) is the coordination number, and dN(D)/dD is the sphere spatial distribution. The three terms on the right side of eq 13 represent adsorption on a plane surface, adsorption/condensation at spheres in contact, and adsorption/condensation on near-neighbor spheres which are not touching, respectively. Dcritis the maximum half-distance between spheres for which the adsorbed films will overlap and a toroidal meniscus will form. The reduction in the total film area (i.e., the ratio of the area measured via the BET method to the geometric area) is
($)
+
4 sin ($)[(sin2 (I))+ p ) ] l I 2 / 3
~ ( -7COS~($) + cos3 ($))
The surface area of the condensate in the toroidal region is obtained from
A/R> = a(HRe)[(2/(De+ 1 - COS
($1)) - ~ x F ( a , x ) / ( 2 H R e ) I(10)
For given values of A, 6, D, and PIP,, the value of Tis calculated from eq 2. The mean curvature, 2HR,, is given by eq 8 , and the filling angle corresponding to the starting parameters is found by repeated solution of eq 6 and 7. With the filling angle known, the meniscus volume is calculated from eq 9 and the surface area is determined via eq 10. When De is less than 0, the volume of the adsorbed film at a particular relative pressure is reduced due to sphere-sphere interactions regardless of the meniscus shape. This reduction in volume is
V J R 3 = 7rD>(3
+ De)3/3
(11)
The total increase, or decrease, in the adosrbed volume due to toroidal condensation and adsorbed film exclusion is Vc/R3= ( 1
+ n3Vt/R: - V a / R 3
(12)
The observed adsorption/condensation of a gas into the toroidal void space of a random packing of uniform spheres will be the sum of contributions from all sphere-sphere interactions. All of the near-neighbors of a particular sphere are not necessarily in contact with the sphere (Le., D = 0), as assumed by previous workers, but are distributed over a range of distances. The total uptake a t a particular PIP, value is obtained by summing over all sphere-sphere interactions
V ( P / P o ) / R 3= Ns[ 4 4 3 R
+ 3R2 +
L
R 3 ) / 3 +N(D=O) Vc(D=O)/R3+ J D m t V c"d/ R 3 7dD
Experimental measurements of the radial distribution of spheres in a random packing have been reported by ScottI3 for 0 < D < 3 and by Mason and Clark14 for 0 < D < 0.3. From these measurements, the average number of sphere contacts per sphere (Le., N(D=O)) is estimated to be 5.85. The distribution function, dN(D)/dD, is estimated by fitting a fourth-order polynomial to the histogram of Mason and Clark.14 The total number of spheres, N,, may be related to the packing porosity and total pore volume, VP' by 1-€ N, = Vp/R3e4a/3 The preceding model includes the effects of adsorption and toroidal condensation, but the role of condensation in the internal pore cavities has not been included. Uncertainties concerning the shape of a meniscus formed between three spheres, which may or may not be touching, preclude the accurate calculation of the relationship between relative pressure and characteristic pore size.'
Results and Discussion For two spheres in direct contact, condensation will occur in the vicinity of the contact for all relative pressure values. However, for near-neighbor spheres pairs, a meniscus will not form until the adsorbed film thickness is sufficiently great such that the films overlap. The critical sphere half-distance, Dcrit,is the maximum value of D for which this film overlap occurs. The value of Dcrit is a function of A and PIPo only. Values of this critical spacing have been calculated as a function of PIP, and A from eq 2 and are presented in Figure 2. As expected, the value of Dcritis a strong function of the adsorbate/adsorbent size ratio. The values of L and H in Figure 2 correspond to the lower and upper values of PIP, when condensation in the internal pore cavities is expected. The implications of Figure 2 are that, for A values greater than 0.001, the effect of near-neighbor interactions must be included with direct contact sphere pair interactions when describing adsorption/condensation phenomena. For particular values of A, /3, and P / P o , the thickness of the adsorbed film and the meniscus shape, if one forms, may be calculated via the solution of eq 2-8. Figure 3 represents the calculated film thickness and meniscus shape for two sphere pairs for A equal to 0.05 and /3 equal to 0.5. For the sphere pair in direct contact, a meniscus forms at both relative pressures. However, for the sphere pair with half-spacing equal to 0.1, no toroidal condensation occurs at the lower relative pressure. As the pressure is increased, the thickness of the adsorbed film increases until film overlap occurs. It is readily apparent from Figure 3 that the measured surface area of the two pairs will have decidedly different behavior with changing relative pressure. The 0 value of 0.5 used for Figure 3 is approximately equal to that for water adsorption at 273 K. Adsorption o f N2 at 77 K would result in /3 N 0.7. A A value of 0.05 corresponds to a sphere radius (13) Scott, G . D.Nature (London) 1962, 194, 956. (14) Mason, G.; Clark, W. Nature (London) 1965, 207, 512.
2726
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986 1.0
Smith
I
1
t
t
A
0.I
t b i t
0.0I
0.001
1 1
1
n-o.ool
-
D= 0.005
n-o.ool 0.3
0.4
0.5
0.6
P -
t
0.7
0.8
0.9
1.0
Po
Figure 4. Volume correction factors for sphere-sphere interactions with A = 0.001. 0.000 I 1 0.3
0.4
,
0.5
C6
1
-
0.7
0.8
,
0.9
1.0
0
PO
Figure 2. Critical sphere half-distance for toroidal meniscus formation. D=O
D=O.l
P
C=005
p=05
Figure 3. Adsorption film and toroidal condensate for touching and
near-neighbor sphere pairs. in the range of 5-8 nm. This size is often encountered in high surface area materials. The net effect of a single sphere pair interaction, either direct contact or near-neighbor, is an increase in the observed adsorbate uptake. Although the presence of a second sphere results in lower adsorption if D is less than D,,,t, this is offset by the quantity of adsorbate which condenses in the toroidal region. A correction factor for a single sphere pair interaction, denoted as V,/R3,may be calculated which is added to the volume adsorbed on a single sphere with no interactions. The magnitude of Vc/R3is a function of A, @, D, and relative pressure. Figure 4 contains plots of Vc/R3 vs. relative pressure for a range of @ values and A equal to 0.001. Findings are presented for a sphere pair in direct contact (D = 0) and for a pair which are near-neighbors. As expected, the value of V, continually increases with increasing PIP, for the sphere pair in direct contact. The magnitude of the correction factor is seen to be a strong function of the @ value. @ is a measure of the relationship between meniscus curvature and reduced vapor pressure. As @ decreases, the relative pressure that the adsorbate will condense at in a given pore size will decrease. For the near-neighbor sphere pair, V, is equal to zero for relative pressure . to Figure 2, the value of D,,, is 0.005 less than ~ 0 . 9 5 Referring for A equal to 0.001. When the relative pressure is increased to a value such that D for a particular sphere pair is less than Dent, the correction factor, Vc/R3,increases to the value calculated for a direct contact pair. Figure 5 is another plot of volume correction factors for individual sphere pairs. For this case, the value of A is increased to 0.01, which corresponds to a larger sphere radius of 25-40 nm,
0.0001 1
0.3
0.4
I
0.5
0.6
0.7
0.8
0.9
P Po
Figure 5. Volume correction factors for sphere-sphereinteractions when A = 0.01.
and @ is equal to 0.5. As compared to Figure 4,the correction factor has increased by approximately an order of magnitude. The role of the sudden transition between nonoverlapping adsorbate films and meniscus formation as D increases to Dcritis illustrated via calculations for different sphere spacings. Similar results for the area correction factor, A,/R2,have been calculated as a function of @, A, D, and PIP,. For sphere pairs in direct contact, a continuous reduction in surface area occurs with increasing relative pressure. For near-neighbor sphere pairs, a sudden reduction in surface area occurs when a meniscus forms. Theoretical adsorption isotherms may be calculated for random sphere packings via application of eq 13. This involves summing the contribution of all sphere pairs with D less than Dcrit. Figure 6 represents adsorption isotherms for A equal to O.ooO1 and various @ values. @ equal to 03 corresponds to adsorption on a single sphere with no sphere pair interactions. For @ equal to 0.5 (i.e., the range expected for H,O and N2 experiments), very little effect of sphere-sphere interactions is noted. As a result of the large ratio of sphere size to adsorbate size, capillary condensation occurs in a much smaller fraction of near-neighbor sphere pairs. Also, the size of the meniscus which does form is small as compared to the sphere size. For A equal to 0.0001, the sphere radius would be in the size range of 3-5 pm which would result in a very low surface area material. Figure 7 is an analogous plot to Figure 6 except that A is equal to 0.01. For this case, one would expect a much larger uptake of adsorbate and a greater effect of toroidal
Adsorption and Condensation in Microsphere Packings
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986 2121
--rm i llli
0.3
,
,
I
n= 0.0001
0.012
0.0I O
v* 0.008
v R'
0.0I
4
a=0.00I
0.006
0.004
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
P -
0.002
Po
Figure 8. Fractional pore volume filled by adsorption and toroidal con01 0.3
,
0.4
0.5
,
0.7
0.6
0.8
0.9
P -
1 1.0
densation.
Po
Figure 6. Adsorption isotherm in a random packing of spheres when A = 0.0001
0.9
tl
a=0.01 I
0.6
-
A-0.01
0.5
0.4
0.6
-P
1
p=0.5
V
I
0.3
0.2
0' 0.3
Figure 9. Reduction in film surface area as a function of A.
0.4
0.5
I
I
0.6
P pb
0.7
0.8
0.9
Figure 7. Adsorption isotherm in a random packing of spheres when A = 0.01.
condensation. Calculations for small 6 values are terminated at large relative pressures since condensation would occur in the internal pore cavities under these conditions. As stated earlier, condensation in the internal pore cavities is not included in this work as a result of the uncertainty concerning meniscus shape. For adsorption and capillary condensation in the toroidal region to be significant in comparison to condensation in the internal pore cavity, it is instructive to calculate values of I/* which we define as V / Vp. v* is the fraction of the total pore volume which is filled for particular values of /3, A, and PIP,. Figure 8 contains these plots for a range of parameters. Truncated lines indicate that condensation in the internal pore cavity will result. As discussed previously, for A equal to 0.0001, effects due to sphere-sphere interactions will be negligible. However, for A equal to 0.001, up to 5% of the pore volume will be filled before the onset of condensation in the internal cavity. As A is increased to 0.01, 10-20% of the total pore volume will be filled before the onset of cavity condensation.
The measurement of surface area after preadsorption of a fluid is a potentially useful tool for extracting additional information related to pore structure and particle interactions from adsorption data. Figure 9 is a plot of the predicted reduction in this film surface area as a function of relative pressure. Significant reductions in the surface area are predicted for A greater than 0.001. The circles correspond to values of the reduced surface area when internal pore cavity condensation will onset. One of the historical motivations for studying this problem is to address the accuracy of the BET method for the measurement of surface area for very small particles. Since BET measurements are typically made for PIP, less than 0.3,1° it is apparent from Figure 9 that the maximum error is the BET area will be less than 1% for A less than or equal to 0.001. However, significant errors will result for smaller particle sizes (Le., greater A values). As discussed previously, measurements of the reduction in surface area as water is preadsorbed on a compressed alumina powder have been reported by Wade.3 In an attempt to fit his experimental data, the coordination number was treated as an adjustable parameter. Only direct sphere-sphere contacts were considered, and an approximate toroidal meniscus shape was assumed. For a pellet compressed to porosity equal to 0.41, Wade4 estimated a coordination number of 10, and for a pellet compressed at higher pressure (e = 0.31), a coordination number of 12 was reported. Both of these values are unrealisitically high for packings of uniform spheres.I5 By use of Wade's average alumina sphere (15) Haughey, 130.
D.P.;Beveridge, G.S. G.Can.J . Chem. Eng. 1969, 47,
2728 The Journal of Physical Chemistry, Vol. 90, No. 12, 1986 1.0
0.9
0.8
z zo 0.7
0 0 €=41% U
0 €=31X
0.6
Smith physical properties). By using the known distribution of sphere contacts and near-neighbors in a random packing, we are able to predict adsorption isotherms and the reduction in film surface areas as a function of relative pressure. It is shown that curvature effects on the accuracy of the BET method will not be significant for A less than 0.001. For smaller sphere sizes, and thus higher A, capillary condensation effects become significant and will result in an incorrect interpretation of surface area and pore volume distribution data as obtained from adsorption isotherms. By use of our model, the variation in film surface area has been accurately calculated as compared to previously published experimental results. The fact that we are able to accurately model this process without the use of adjustable parameters implies that this theory may be used for comparison to experiment to obtain additional structural information.
Nomenclature
0.5 0.I
0.2
0.3
0.4
0.5
P Po
Figure 10. A comparison of calculated reduced film surface area with experimental measurements by Wade.3
radius of 7.54 nm and for water adsorption at 273 K, /3 is approximately 0.5 and A is approximately 0.04. We have calculated the reduction in surface area for /3 of 0.5 and A equal to 0.05. A comparison of Wade’s experimental results and our predicted values is presented in Figure 10. Our predicted value, which corresponds to a porosity of 0.37, falls between Wade’s data for porosity of 0.31 and 0.41 at low relative pressure. For data at PIP, equal to 0.47, the observed decrease in film surface area is much greater than that observed. This is a result of condensation in the internal pore cavities which has begun due to the small size of the alumina particles. Smaller internal pore cavities have been filled with adsorbate which results in blockage of some surface area, a mechanism which is not included in our model. Although a comparison between theory and only two data points cannot be considered a rigorous test, the fact that we are able to so closely match the observed reduction in surface area by using a theory which contains only variables determined from physical properties of the adsorbent and adsorbate and without the use of adjustable parameters is significant.
Summary The volume and surface area of toroidal condensate are found as a function of three dimensionless groups: the relative pressure, A (adsorbate/adsorbent size ratio), and /3 (a function of adsorbate
A = area d = half-distance between sphere surfaces D = dimensionless sphere spacing ( d / R ) De = effective sphere spacing ( D - T ) H = mean curvature N = distribution of sphere center spacing N , = number of spheres P = pressure Po = saturation pressure R = sphere radius Re = effective sphere radius ( R + t ) t = adsorbed layer thickness T = dimensionless adsorbed layer thickness ( t / R ) V = volume adsorbed v* = v/v, V, = volume decrease due to adsorbed film overlap at a contact V, = volume correction for a single sphere-sphere contact Vp = total pore volume V, = toroidal condensate volume x = spatial variable y = spatial variable a = angle of the perpendicular to meniscus and y axis P = pLRgTLS/Y y = surface tension 8 = thickness of a monolayer A = dimensionless monolayer thickness ( 6 / R ) AP = pressure drop c = porosity [ = dimensionless spatial variable @ / R e ) 7 = dimensionless spatial variable ( x / R , ) 0 = contact angle p = spatial variable ( p = -sin ( a ) ) $ = filling angle
