Oct., 1954
CUMULATIVE PORE
VOLUME,S'AND PORE
SIZE DISTRIBUTIONS
891
AN ANALYTICAL EXPRESSION FOR CUMULATIVE PORE VOLUMES AND PORE SIZE DISTRIBUTIONS1 BY W. 0. MILLIGANAND C. R. ADAM$ The Rice Institute, Houston, Texas Received March 6, 1064
A new equation for representing pore size distributions has been applied to desorption isotherms and electron microscope data previously obtained for heat-treated ( 2 hr., 500') dual oxide gels in the system BeO-InzOa. The new equation (conT O ) ] , have obvious advanstants defined later) P v0 = h tanh [k(R - T O ) ] and its derivative, dV/dR = hk sech [ tages over equations commonly used to express pore size distributions. Although t;e%%ssian distribution is highly flexible, it suffers from two disadvantages. First, it cannot be integrated into a closed form of elementary functions, and it thus becomes tedious and difficult to compare calculated cumulative pore volumes with the experimental values obtained by the customary methods of determining pore size distributions. Furthermore, the Gaussian distribution is usually too wide a t the top and too narrow a t the bottom for many experimental distributions. The new equation, while still as flexible ns the Gaussian distribution, possesses very simple closed ex ressions for both the differential and integral forms. Furthcrmore, it is narrower a t the top and wider at the bottom t i a n the Gaussian distribution, thus representing more closely many experimentally determined distributions which we have studied. Expressions similar to the Maxwellian distribution, while easily integrable, lack versatility in that the spread of the distribution cannot be controlled independently of the most frequent pore size. The new expression not only possesses a more simple integrated form but also may have its spread adjusted independently of the other parameters. The new function is easily fitted to the experimental data by numerical and/or graphical methods. It has been successfully applied to eleven gels in the system BeO-InzOa, which possess distributions having most frequent pore sizes ranging from 133 to 26 d. and a variation in spread of a factor of 10.
-
-
Introduction Most of the commonly occurring distribution functions have been used to represent pore size distributions. Many of them, however, are restricted in either of two respects: (a) they are insufficiently versatile to cover both sharp and broad distributions of different size ranges, or (b) they are cumbersome and tedious to handle either in the integrated or differentiated form. The Gaussian distribution, widely used to represent pore size distributions, cannot be integrated into a closed form, thus making it difficult to compare calculated cumulative pore volumes with the experimentally measured values. Furthermore, it is usually too wide a t the top and too narrow a t the bottom to fit well most experimental distributions. The Maxwellian distribution, along with the Gaussian distribution, has been used by Shul13 to represent pore size distributions, but thjs equation lacks versatility in that the spread cannot be controlled independently of the most frequent pore size. The equation proposed here dV/dR = hk secha [ k ( R
calculated cumulative pore volumes can be easily compared with the experimental values. I n Fig. 1 the distribution, dV/dR = hk sech2 [k(R- r o ) ] ,is shown with a Gaussian distribution (dashed line) having the same height and area. It is seen that the new expression is closely similar to the Gaussian distribution. The main difference is that the new expression is a little sharper at the top and a little broader a t the bottom, thus approximating more closely many experimental distributions.
- TO)]
where V = cumulative pore volume of all pores having radii equal to or less than R 2h = total pore volume excluding adsorption k = factor determining the sharpness of the distribution TO = most frequent pore radius vo = cumulative pore volume of all pores having radii equal to or less than TO
possesses wide versatility in that the spread, amplitude and position of the distribution can be varied independently. Furthermore, the integrated expression V
- vo
=
h tanh [k(R
- rO)]
possesses a very simple closed analytical form, and (1) Presented before the twenty-eighth National Colloid Symposium which was held under the auspices of the Division of Colloid Chemistry of the American Chemical Society in Troy, New York, June 24-26, 1954. (2) Ethyl Corporation fellow in chemistry a t The Rice Institute, 1953-1954.
(3) C. C.Shull, J . Am. Chem. SOL,TO, 1405 (1948).
Fig. 1.-The distribution dV/dR = hk sech2 [Jc(R- T O ) ] , (solid line), compared with a Gaussian distribution (dashed line) having the same height and area.
Since cumulative pore volumes are usually the only quantities accessible by methods commonly employed t o determine pore sizes, it is desirable to work with the integrated form of the distribution function instead of the distribution function itself. The proposed equation is particularly adapted to such ends. The integrated expression may be fitted t o t l e experimental values and the distribu-
W. 0. MILLIGAN'AND C. R. ADAMS
892 TABLE I
Results and Conclusions As an example of the agreement obtained with the new equation calculated and observed values for the cumulative pore volumes for the electron microscope results on the gel composed of 10 mole % Be0 and 90 mole % ' In203, heat-treated at 500" for two hours are given in Table I. I n Table I1 are given the most frequent pore size (To), the amplitude of the distribution ( h k ) , the width of the distribution a t half-height (1.7628/k), and the standard deviation of the calculated cumulative pore volumes as compared with the observed ones. Most of the deviation occurs a t the extreme ends
10 MOLE% Be0-90 MOLE% InzOs, HEATED 500" Eleotron micrpSCOP?C
pore size
R , A.
18 31 43 55 68 80
Pore volume, cc./g. Obsd. Calcd.
v,
0.0452 .0606 .0801 ,1010 .E19 .1459
0.0500 .0624 .0785 .0980 .1209 .1456
4Sample heated 2 hr., 500° Mole % Mole % Be0 In201
0 10 20 30 40 50 60 70 80 90 100 a
100 90 80 70 60 50 40 30 20 10 0
Electron micrpscop1.c pore aize
Most frequent pore radius ro, A. desa EM b
133 83 78 78 86 96 100 85 86 70 32
des = desorption isotherm.
Pore volume,
v,oc./g.
R,A. Obsd. 92 0.1731 105 .2003 117 ,2251 129 .2491 142 ,2700 154 .2832 = 0.002
120 98 78 84 84 82 101 96 77 66 26 b
Vol. 58
Calcd.
0.1739 .2010 .2265 .2492 .2687 ,2844
TABLE 11 des
Amiplitude hk
0.00262 ,00303 .00317 .00467 ,00892 .00968 .00569 ,00689 ,00566 .01103 .01347
EM
0.00230 * 00222 .0029G .00388 .00452 .00684 .00376 .00540 .00436 .00800 .00360
EM = electron microscope.
Width 1.7028/k des EM
104 88 88 84 44 44 66 73 63 44 12
129 129 73 99 101 75 105 96 85 63 59
Standard dev. des
0.006 .010 .011 .009 .011 ,014 .010 .014 .010 .014 .007
-
av. = 0.011
0.006 .002 .008
,007 .010 .019 .007 .013 .012 .017 .008
0,010
tion function obtained analytically from the fitted of the distributions. In Fig. 2 are shown in graphiequation. cal form the results of the 10% Be0 gel. The open points correspond to cumulative pore volumes obCalculations The integrated expression may be fitted by nu- tained from two desorption isotherms a t two difmerical and/or graphical methods. In the present 1.0 case two parameters, k and 'r0,were found graphically and the other two, h and vOlwere determined by the method of least squares. Cumulative pore 0.8 volumes obtained for eleven gels in the system BeOInz03 by gas adsorption and electron microscope techniques already have been p r e ~ e n t e d . ~The ,~ methods used in determining the parameters were 0.6 as follows. v. Approximate values of vo, k and ro were picked and the quantity tanh [k(R - r o ) ]was plotted us. 0.4 V - v0, using only absolute values of the two quantities so that the points below ro were "folded" back on the same graph paper as the points above To. If the curves have a curvature upward (toward 0.2 the P - vo axis), the value of k must be decreased. If the slope of the negative values is higher than the slope of the positive values, the value of ro must 0.C be decreased. Both the slope and curvature are R a d i u s , 1. very sensitive to the chosen values of k and ro so Fig. 2.-Calculated and observed cumulative pore volthat correct values of k and ro are obtained in only umes and pore size dis9butions for 10 mole % Be0-90 mole about four graphings. In the authors' case special % InrOs, heated 500 . 0 en points are cumulative pore hyperbolic tangent graph paper was used, eliminat- volumes from desorption &tal closed points are pore volumes from electron microscope data. The points in the ing the necessity of obtaining from tables the values inset represented counted distributions from electron microof the hyperbolic tangent. When the best values graphs. of k and rohad been obtained by the above methods the values of h and vo were obtained by the method ferent temperature^.^ The closed points represent cumulative pore volumes obtained from electron of least squares. micrographs.b The curves are the calculated values (4) W. 0. Milligan and C. R. Adams, THZE JOURNAL, 67, 885 using the new equation. The radius of the experi(1953). mental points corresponds to the standard devia( 5 ) C. R. Adsms and W.0. Milligan, ibid., 158, 219 (1954).
Oct., 1954
CUMULATIVE POREVOLUMES AND PORESIZEDISTRIBUTIONS
tion from the calculated values. I n the inset are given the distribution functions obtained by various methods. The solid curve represents the analytical derivative of the new equation fitted to the desorption pore volumes by the method described. The points represent counted distributions obtained from high magnification electron micrographs. The dashed curve represents the new function fitted to the electron microscope pore volumes.
893
The over-all agreement is satisfactory. The largest deviations occur a t the extreme ends of the distributions, where it would be expected that secondary effects, such as aggregation or clumping, would have a pronounced effect. I n summary, the flexibility, ease of applicability, and the simple analytical forms of the new equation appear to justify its addition to the existing distribution functions.
