SURFACE AREA OF MICROPOROUS SOLIDS
18
(23) ROEPKEAND MASON:J . Biol. Chem. 133, 103 (1940). AND PALMER: Cold Spring Harbor Symposia Quant. Biol. 8, 94 (1940). (24) SCHMITT J . Phya. Chem. 36, 1401, 1672, 2955 (1932). (25) SMITH: The Chemistry o j the Sterids. The Williams & Wilkins Company, Baltimore (26) SOBOTKA: (1938). (27) TARTAR A N D WRIGHT: J . Am. Chem. SOC. 61, 539 (1939). (28) YABHOFF AND WHITE: Ind. Eng. Chem. 32, 950 (1940).
T H E CALCULATIO?; OF THE SURFACE AREA OF 3IICROPOROUS SOLIDS FROM SIE.1SURESlESTS OF HEAT COSDUCTIVITY' S. S . KISTLER N o r t o n Company, IVorcestcr, Massachusetts
Received J u l y 14, 1941
In 1934 it was demonstrated ( 5 ) that the avcragc diameter of the pore spaces in a microporous solid, such as an aerogel, can he characterized by a number that represents the mean free path of a gas in the pores at such a low pressure that every impact of each molecule is with the pore walls. It was also shown that this mcan free path characteristic of the particular porous solid can be readily calculated from measurements of the heat conductivity of the solid a t thrce suitably chosen gas pressures. In reality, the method uses the normal mean free path of the gas as a measuring stick for th'e pore spaces. Sincc a t atmospheric pressure this length is very near 10-5 cm. for air, it provides a means for measurement in a region well below the resolving power of the conventional microscope. By using moderate pressures, pore diameters can be measured with accuracy down to lo-' em., provided the heat conductivity of the solid structure is not so large as to mask that due to the gas in the pore spaces. This method of measurement is particularly applicable t o the aerogels and xerogels. THE MEAN FREE PhTII OF THE PORE SPACES, I S D ITS SIGSIFICASCE
Consider that the air pressure within the pores is so low that each molecule bounces from wall to wall with a negligible number of intermolecular impacts in the free space. Now, if one should observe n molecules as each traverses a distance dz, it would be discovered that dn of these would impact a wall, and from kinetic theory these quantities are related by the equation :
in which 1 is the mean free path. 1 Presented a t the Eighteenth Colloid Symposium, which was held at Cornell University, Ithaca, New York, June 19-21, 1941.
20
S. 8. IUSTLER
Now, if these pore spaces are enclosed by the solid structure so that they form tubules, bubbles, lenses, etc., the quantity I will have a definite relationship to the diameters of these structures and to the internal surface area. This relationship has not yet been investigated and is, therefore, beyond the scope of this paper. If, as is almost certainly the case with gels and jellies, the solid structure is composed of an assemblage of rods, plates, ribbons, spheres, or other such units, and the pore spaces are the irregularly shaped spaces between, it is more difficult to think of a pore diameter, and the mean free path of such a structure is probably the most significant measure of openness. When a swarm of molecules a t low pressure tries to pass through such a structure, each will move in a straight line until it strikes a solid surface, and the fre-
i
FIG.1. Reduction in free path by a cylinder
FIQ.2
quency of such impacts is determined by the projected area of the structural elements, according to the equation
in which L is the average distance a molecule moves between impacts, A , is the projected area per cubic centimeter of porous solid (the sum of the projected areaa of all of the elements of structure projected on the same plane), and C is a correction factor for the thickness of the elements of structure. The magnitude of C varies slightly with the character of the structural element, but within sufficientlynarrow limits so that the value for long cylinders is a close enough approximation for the present purpose. All three paths terminating on the surface of a cylinder the axis of which is a t right angles to the path will be shortened, owing to the thicknesa of the cylinder.
SURFACE AREA OF MICROPOROUS SOLIDS
21
In figure 1, since all paths to the surface of the cylinder are equally probable, the average reduction in path will be
-
Area of semicircle Diameter
-
wr2 ... 2 2T
-
T
5'
(3)
where r = the radius of the cylinder. Now all orientations of the cylinder with respect to the parallel paths are possible and equally probable, so that the extension of the axis would uniformly describe the surface of a sphere, and the probability that the axis will fall between the angles a and a da (see figure 2) will be
+
r:da e 2 n - r sin a = sin a da 2m-l
FIG.3. Reduction in path due
t o inclination of the cylinder
In figure 3 it is seen that the reduction in path due to an inclination a of the cylinder is *r/4 sin a. Therefore the average reduction of path due to random distribution of the cylinders is
THE RELATION OF THE MEAN FREE PATH TO THE VOLUME OF THE POROUS SOLID AND THE VOLUME OF SOLID SUBSTANCE
Let V = the volume of 1 g. of the porous solid and b = the volume of 1 g. of the solid substance. Then the projected area per gram of the porous solid = V A , = A,. Also, if as a close approximation to the true value of C we assume a cylindrical structure, b = ~ T * s , where s is the length of the cylinder. The fact should be recorded here that the surface area of any flat or convex solid object is exactly four times its average projected area (4). Now we can substitute these values into equations 2 and 4 and obtain
22
S. 6 . KISTLER
Since A , for the cylinders is one-fourth of the surface area of 1 g. of substance, which in this case is 2 ~ T S ,
Equation 6 is general in form and varies only in the value of the coefficient of b. That this coefficient is very insensitive to large changes in shape of the structural elements is shown by the fact that for elements of rectangular cross section with a width five times the thickness, it is 0.73, while for spheres it is 0.59. CALCULATION OF THE MEAN FREE PATH
In the former paper (5) it was shown that the heat conductivity of a porous solid may be resolved into three conductivities: viz., the conductivity of the solid structure ka, the conductivity of the gas in the pores k,, and the film coefficient kz due to imperfect fit between calorimeter plates and the porous solid. These three conductivities are related to each other and to the over-all conductivity k by the equation
The conductivity of the gas, k,, is the only one that is pressure-dependent, and its dependence is given by the equation
k,=K-
L
where K is the normal conductivity of the gas, l o is the normal mean free path a t a pressure of 1 mm., and p is the gas pressure in millimeters. In the former publication the calculated values for K were used rather than the observed, in order to include dichlorodifluoromethane among the gases used. Although this does not greatly influence the calculated value of L , it is deemed advisable now to use the observed values in calculating L. Formerly kl and k , were converted to conductivities for a 1-cm. thickness of silica aerogel, but it is now considered to be better practice to solve equation 7 using the measured conductivities for the actual thickness of the specimen, Le., 0.818 cm. One more reason exists for recalculating L : viz., owing to the laboriousness of an exact solution of equation 7, the former published calculations were simplified by the approximation that the asymptotic value for k a t low pressures can be taken for ka. Since the most unreliable of the conductivity measurements are those taken at low pressures, it seems desirable to calculate L from observations at pressures well above the toe of the curve. The data upon which these calculations are based are reproduced in table 1 and in figure 4. By inspection of the curves, the values a t 43,183, and 740 mm. were chosen for air, and those a t 100,206,and 745 mm. were chosen for carbon dioxide.
23
SURFACE AREA OF MICROPOROUS SOLIDS
The exact solutions of the equation for both air and carbon dioxide, for which I am greatly indebted to Kenneth F. Whitcomb, are given in table 2. The units in which conductivities2are expressed are calories/cm. X sec.
x 'C.
TABLE 1 Observed conductivity of silica aerogel
I
AIR D
2 43 94 183 206 363 387 740 743
k
x 105
2.93 3.10 3.26 3.63 3.61 4.16 4.06 4.85 4.83
I
CARBON DIOXIDE
P
k X 10'
9 24 49 100 206 397 745
2.65 2.80 2.98 3.06 3.42 3.84 4.33
Loa P R E ~ S U R I N EMM..
FIQ.4. Conductivity of silica aerogel
Before an exact solution of the equation was found, an attempt was made to find a, simplification that would yield acceptable results. Table 3 gives the results for both air and carbon dioxide, using two methods of approximation. Method I relies upon the fact that as the pressure is reduced k , becomes I n the actual calculations, conductivities are taken for t h e particular thickness (0.818 cm.) of gel in t h e calorimeter, thus eliminating the thickness dimension, since k, has the dimensions calories/cm.' X sec. X "C.
24
8. 5. KISTLER
smaller and smaller and eventually becomes insignificant compared to ka. The unreliability of the method lies in the inaccuracy of the conductivity data a t small values and the uncertainty of the constancy a t low preseures of the contact conductivity between calorimeter plates and the blocks of aerogel. In addition, the value of L is particularly sensitive to error in the estimated asymptotic value of k. This fact is illustrated by the values of L in table 4, where the data for carbon dioxide were used and three different values for k substituted. The asymptotic value is in considerable uncertainty, depending upon whether the TABLE 2 Ezact solutions of the equations L
IO
TABLE 3 Avvrozimationa L
/ A i r I c o r I A i l c o r l
I I1
1
I
6.0 X 10-4 1.7 X 10-8
1
1
Air
I
cor
3.1 X 10-5 2.8 X 10-' 17.0 X 1W6cm. 6.6 X 10-8 am. 3.0 X 10-' 2.7 X lO-& 4.5 X 1W6om. 6.1 X 10-Icm.
TABLE 4 Variation of L t a t h aevmptotic value of k u m n c V A L U OF ~ k
I
L
ml.lcm. X wc. X 'C.
em.
2.6 X 10-' 2.6 X 10-' 2.7 X lW'
11.6 X 10-' 8.6 X 5.6 X le6
measured value a t 9 or 24 mm. pressure is given the greater weight in drawing the curve. O'wing to the fact that the curve should roughly parallel that for air, it is felt that the curve should paas through the 24 mm. point. Method I1 makes the assumption that k2 is so large aa to make the quantity l/kz insignificant compared to l / k . This method permits the solution for L from data a t relatively large premures where the constancy of the contact conductivity is not in doubt, and where the percentage error in the measurements is low. It is evident from tables 2 and 3 that neither method of approximation is reliable. If a number of similar measurements are made with the same calorim-
26
SURFACE AREA OF MICROPOROUS SOLIDS
eter, it would appear that the best simplification would be to use method 11, introducing an approximate value for k1.8 SURFACE AREA CALCULATED FROM MEAN FREE PATH DATA
Substitution of the mean free path of the aerogel, 7.8 X 10-' cm., obtained by averaging the results of the exact calculations for both air and carbon dioxide, in equation 6 yields 1
7.8 X 10- = - (V A*
- 0.62b)
Since the apparent specific gravity of the aerogel was 0.184,and the specific gravity of the silica structure is 2.05,
1 1 T.' = - = 5.44 and b = - = 0.49 0.184 2.05 Therefore, after rearrangement, the equation becomes
A, =
7.8 X 1W
(5.44
- 0.30) = 6.6 X 10' cm?
Multiplying by four, one obtains Surface area = 2.6 X l@ em.* per gram 8 Mr. Whitcomb's solution is aa follows: From three meaaurementa of conductivity, one can set up the three simultaneous equations (see equations 7 and 8):
1 KL k1 +
+ -1 - - ;1 ki k1
LTB
-+ 1
KL
-1= - 1 .___ 1
kl
k"
k s +E+L
'
KL
kr
+
L+G
where B , E , and Q are the values of 20/p at the three pressures. Let
Set up the new quantities:
M = Q N K ( E - G) P = KQ(E B) KN(B U Q N ( E B ) ( E - G)
-
Then
and
- + -
- G)
1
1
+g'E
26
S . S . KISTLER SURFACE AREA BY ADSORPTION
The extensive work of Emmett and Brunauer (1, 2, 3) gives one some confidence in their method of measuring the surface area of a solid by thc adsorption of a gas or vapor on it. Although the experimental work for the prcscnt paper was done long before they proposed their method, wc fortunatcly have adsorption data from a piece of the identical silica aerogel that was uscd in the heatconductivity measurements. Adsorption of water vapor was measured in a LIcBain-Bakr fused-silicnspring apparatus that had bem flushed out several times with watcr vapor, mm. each time to remove foreign heating to 300'C. and pumping down to adsorbed gases or vapors. Adsorption was determined at 33.9"C.up to a relative humidity of 32 per cent, using the aerogel as it came from the calorimeter. ,4ir was then admitted and TABLE 5 Adsorption of water vapor by silica aerogel BEFORE OXIDATION
AFTER OXIDATION
Per cent ndaorbed
Prwsure mm. Hg
0.4 1.8 7.4 12.6
31.7
0.62 1.07 2.14 2.65
Prpsaure
Relative
humidity
m n . Hg
per Cent
4.4 8.9 11.8 18.9 24.4 27.0 28.8 31.8 33.7
11.1 22.5 29.7 48.2 61.5 67.8 72.6 80.2 85.0
Per cent adsorbed
0.88
1.60 2.22 3.10 3.63 4.53 4.88 6.11 7.00
the silica aerogel nas heated to 360-400°C for a few hours to ouidize all organic matter. Silica aerogel as it comes from the autoclave carries several per cent of organic matter, and this sample turned bronn during the initial evacuation a t 300°C. This oxidation treatment reduced the weight by 0 8 per cent, u hich indicates that most of the organic matter had formerly been removed by the heating, pumping, and flushing procedure. Xeverthcless, this remaining 0.8 per cent was enough to influence strongly the coiirse of the adsoiption curve. The data are presented in table 5 and figure 5, in which the ndsorption referred t o is that in addition to the 3.3 per cent of water left in the gel after evacuation. Although the initial parts of the two curves do not agree, they seem to be converging. It is unfortunate that the measurements on the unoxidized gel were not carried to 50 or GO per cent humidity. However, i t is evident that the concavity downward is diminishing rapidly, and it is probable that Emmett and Brunauer's point B , the point a t which the curve either becomes a straight line or is inflected, is not far from 6.3 per cent total adsorption. For the gel after
SURFACE AREA OF MICROPOROUS SOLIDS
n
oxidation, this point is a t 6.5 per cent adsorption. Whether or not the 3.3 per cent water in the gel after evacuation should be included in the adsorption figure from which surface area is calculated is an open question. From the fact that this residual water has been proved to be on or accessible to the surface (6), it was decided to include it in the calculations. The area covered per molecule of water (calculatedJrom the density of ice, using Emmett and Brunauer’s equation) is close to 11 As,from which it follows that 6.5 per cent water in the gel will cover 2.1 X lo6 cm.l in 1 g. of the silica aerogel. Since the water molecules must in some way fit the underlying silica structure, it is probable that the average area per water molecule is well above 11 b? Therefore 2.4 X lo6 cm.2 per gram of gel is a minimum value.
R c ~ n ~ i HUMIDITY, vr PERCENT
FIG.5 . -4dsorption of water by silica aerogel
The closeness of the check between the area calculated from measurements of heat conductivity and that obtained by adsorption is very gratifying and adds to our confidence in the value of both mcthods. It has been expected that the area calculated by adsorption would be substantially larger than that obtained from heat-conductivity data for substances like silica aerogel, since the former method includcs the areas of cracks and crevices into which water molecules can penetrate, whereas the latter calculation is based on the existence of flat or convex surfaces only. However, gaseous molecules in crevices will contribute to the heat conductivity of the structure, and an examination of this contribut,ion and its dependence on area and pressure will have t o be made before a conclusion can be reached. It is interesting to record for comparison the fact that Emmett and Brunauer
28
S. 8. KISTLER
calculated the surface area of a sample of conventional silica gel to be 4.6 X 1 W cm.2 per gram, from the adsorption of nitrogen, argon, oxygen, carbon monoxide, and carbon dioxide at low temperatures. OTHER HEAFCONDUCTIVITY AND ADSORPTION MEASUREMENTS ON SILICA AEROQEL
The heat conductivity of another sample of silica aerogel was measured over a range of pressures. The results are given in figure 6 and table 6. Although the hydrogel was cast in the same manner aa for the above aerogel, viz., by mixing equal volumes of water glass (1.18 sp.gr., “N” brand) and 4 N acetic acid and allowing it to gel in crystallizing dishes, it ha3 subsequently been found (7) that conditions of washing and autoclaving have a profound influence upon the properties of the aerogel. I t is therefore not surprising to find that the mean free paths of different specimens do not agree.
P8g1)UBm
coIIDum.IIR
mm. H E
