T H E RELATION BETWEEN HEAT CONDUCTIVITY AND STRUCTURE I N SILICA AEROGEL S. S. KISTLER Department of Chemistry, University of Illinois, Urbana, Illinois Received J u n e 14, 1934
The structural units of jellies have dimensions of such an order that they have failed to yield to the ordinary methods of mensuration. Electrical conductivity, diffusion, and adsorption measurements have yielded figures of questionable accuracy on account of the uncertainties involved in the interpretation of the results. Ultramicroscopic studies in optically empty jellies are fruitless, while a measurement of the Rayleigh scattering of light, when it is intense enough to be measured, can at best give only qualitative results because the elements of structure are far too close together to obey Rayleigh’s equations. The situation is somewhat better with gelatinous membranes where ultrafiltration of colloidal particles and Bechold’s bubble test are available, but in the former case one is far from certain that the maximum diameter of particle that passes the membrane is within a factor of 10 of the actual average diameter of channel through which it passes, and a corresponding uncertainty enshrouds the bubble test, owing to the fact that only the largest passages can be thus estimated and also owing to uncertainty of the aurface tension a t small radii of curvature. X-ray methods can indicate only the size of crystal in crystalline gels, but can in no way indicate the relation between size of crystal and size of structural units, e.g., fibrils, platelets, spheres, etc., from which the gel is constructed. The development of a process for the replacement of the liquid in a jelly by a gas with little or no change in its structure (2) has made available a powerful new measuring rod, the mean free path of the gas molecules. The kinetic theory of gases has received such substantiation and acceptance in the past two-thirds of a century that the mean free path may be regarded as one of the well-established physical quantities. Two methods of associating dimensions within the aerogel with the mean free path of a gas have been used in this laboratory. By forcing a gas through an aerogel a t different absolute pressures, the pressure at which the type of flow through the gel changes from viscous to molecular can be found and related to the average size of opening through which the gas passes. The results of this work will be published later. The second 79
80
S. S. XISTLER
method relies upon the linear relation between heat conductivity of a gas and its mean free path, other quantities being held constant. RELATION BETWEEN MEAN FREE PATH AND HEAT CONDUCTIVITY O F A GAS
According to the kinetic theory of gases, thermal conductivity of a gas is associated with the mean free path by the equations k =Bc,~
(1)
0.35 pvl
(2)
and 7 =
where k is the coefficient of heat conductivity, B is a constant, cv is the specific heat of the gas at constant volume, 17 is the viscosity, p the density, 8 the arithmetical average velocity of the molecules, and I the mean free path. The constant 0.35 is due to Boltamann. The coefficient k can then be represented by the expression k = k'l
(3)
where k' is a function of the pressure, temperature, and composition of the gas. Since we know nothing of the nature of the fine structure in an aerogel except that the free spaces are much larger than the elements of structure themselves (aerogels with 98 per cent void space have been produced), we will think of a large number, N , of molecules randomly distributed through the aerogel, starting from rest and moving in straight lines in all directions until they collide with structural elements of the gel. If the dn structure is random, which we must assume, the fraction -of the N - n molecules that have not yet struck surfaces will impinge within the distance dx, where n is the number that have already struck. Therefore
N-n
dn = -dx L
(4)
1 where - is the proportionality factor. Integration gives L
which is the familiar equation connecting the number of impacts between molecules with distance in a gas, and L can now be defined as the mean free path of a highly attenuated gas within the aerogel. In other words, e - 1 or 63 per cent of the molecules will hit surfaces in the distance L. e
HEAT CONDUCTIVITY AND STRUCTURE IN SILICA AEROGEL
81
I n the same manner one finds that when the pressure is sufficiently high many molecules strike others in going between surfaces, dn =
(v
”>
+I - dx
( N - n) (1 -I- L ) dx
L1
and the mean free path of the gas in the aerogel is
Ll 1, = -
L+1
Now substituting equation 5 into equation 3
k,
= k’l, = k’
1L =k
L+1
L L+l
where k, is the conductivity coefficient for the gas while in the aerogel. It was in this approximate form that the equation was used in a former publication (3). More careful consideration of the mechanism of heat transfer through an aerogel indicates that the coefficient B in equation 1should be different for the gas in tl& aerogel than for the free gas under the usual measuring conditions where the mean free path is very small with respect to the distance apart of the calorimeter plates. Where the molecules are moving in straight lines between surfaces with only occasional impacts in between, as is the case within aerogels at low pressures, it is readily shown that
k =
Mc,Il 6.06 x 1023
where M is the molecular weight and 1 is the number of impacts of molecules on unit area in 1 see. Measuring length in centimeters and pressure in millimeters of mercury,
I
=
1.99
x
D
1016
drn
A
which yields
where lo is the normal mean free path of the gas at the given temperature and a pressure of 1 mm. Equation 7 assumes that with every impact a molecule comes to thermal equilibrium with the surface on which it strikes before it departs. If one
82
S. S. KISTLER
assumes that only the fraction a of the molecules comes to thermal equilibrium and that the fraction 1, - a is specularly reflected, the equation becomes
Equations 7 and 8 neglect the correction factor proposed by Chapman (1) as being inoperative under these conditions. EXPERIMENTAL PROCEDURE AND RESULTS
The calorimeter and method of operation were described in a former paper (3). The gel was made by mixing E Brand water glass, kindly furnished by the Philadelphia Quartz Company, diluted to a specific gravity of 1.20, with an equal volume of 4.3 N acetic acid and casting over mercury in paraffined crystallizing dishes. After 48 hours the cakes were removed, washed free of salts, and extracted in a Soxhlet extractor with 95 per cent alcohol until the water content of the cakes was less than 10 per cent. They were then placed in an autoclave with excess alcohol and heated to 26OoC., a t which temperature the alcohol was allowed to escape. The cakes of very nearly the same thickness were next trimmed with a sharp knife to shapes that could be fitted closely together on the lower plate of the calorimeter. The calorimeter was then assembled and evacuated to less than 0.01 mm. for half an hour in order to remove adsorbed moisture. Following the evacuation the gas to be used was admitted and measurements made at pressures differing from each other roughly by a factor of 2. The lowest pressure used was generally in the neighborhood of 10 mm. to avoid making measurements a t pressures so low that the conductivity of gas films between the pieces of silica aerogel and the calorimeter plates would be affected. As was pointed out in the earlier publication in which measurements were reported on powdered aerogel, reproducibility of results is good. Only occasionally does a reading in a series fall far from a smooth curve, and those occasions have been traced to a drift of the zero point on the high sensitivity galvanometer that controls the temperature of the guard ring. The results of the measurements with three gases, air, carbon dioxide, and dichlorodifluoromethane, are presented in table 2 and the figure. These three gases were chosen since they represent three very distinct types and possess mean free paths, as calculated by equation 2, differing by a factor of 3. Measurements were also made with hydrogen, but were not satisfactory on account of the high thermal conductivity between guard ring
83
HEAT CONDUCTIVITY AND STRUCTURE IN SILICA AEROGEL
and disc and the consequent uncertainty of readings due to the unusually bad zero point drift of the galvanometer. When the galvanometer suspension has been replaced, this series will be repeated and the results reported later. Table 1gives the calculated constants required in equation 7 for the three gases. The required data were obtained from the International Critical Tables except for the last gas, for which the data were obtained from the literature (4) and from Mr. A. L. Henne. The average temperature of TABLE 1 Calculated constants f o r a i r , carbon dioxide, and dichlorodijluoromethane 0.4s
Air.. . . ; , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.27 X 10-6 1.67 x 10-6 1.20 x 10-6
7.46 >( 10-3 5.04 x 10-3 2.46 X 10-3
coz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CCl2F2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TABLE 2 Observed conductivity of silica aerogel
co2
AIR
P
Conductivity
x 105
mm.
740 363 183 2 43 94 20 6
387 743
P
CClaFz
Conductivity
x
10'
mm.
4.85* 4.16 3.63* 2.93 3.10 3.26 3.61 4.06 4.83
745 397 206 100 49 24 9
P
Conductivity
x
106
mm.
4.33* 3.84 3.42 3.06* 2.98 2.80 2.65
740 406 259 123 59 29 8
3.71* 3.38 3.18 2.86* 2.63 2.59 2.68
* Values used in calculating the results given in table 3.
cal. em. X sec. X T . To demonstrate the reproducibility of the data, two series of measurements were taken on air and distinguished in the figure by crosses and circles.
measurement was 33.3"C.
Conductivities are reported as
CALCULATIONS
Owing to the difficulty of obtaining good thermal contact between the calorimeter plates and the cakes of aerogel, there is a constant thermal resistance in series with that of the aerogel itself. The series conductivity
84
S. S. KISTLER
will be called b. Also in parallel with the conductivity due to the gas in the aerogel, there is a constant conductivity, k3, due to the continuous structure of silica and to the gas in the small spaces between the cakes. The overall Conductivity, _ . IC,. as measured in the calorimeter, is related to the other conductivities by the equation
1
25 0.2
1.0
1
1
+ G = -K
2.0
3.0
Log Pressure, mm. FIG.1. CONDUCTIVITY OF SILICA AEROGEL
To be perfectly general it is evident that equation 8 should be used for E
in
nrrlnr t n
nllnnr
fnr
n
nnaaihln
n n o n m m n r l a t i n n nnnffininnt n lnaa
than 1
From what is known of a gel surface, however, it seems permissible, a t least as a first approximation, to establish a = 1 and therefore gain the decided advantage of the simplicity of equation 7. From the curves in the figure, one can estimate the value of ka for each gas as the asymptotic value a t low pressures. This quantity combined
HEAT CONDUCTIVITY AND STRUCTURE IN SILICA AEROGEL
85
with t,he calorimetric conductivities at two other pressures enables one to solve equation 7 for L. Table 3 gives the results of such calculations, using the values of K starred in table 2. It is gratifying that by the use of three such different gases one should obtain such concordant results, and it gives one confidence both in the method and in the verity of the conclusions. The average distance apart of the structural elements in this batch of silica aerogel whose apparent density was 0.184 g. per cubic centimeter must lie very close to 5 X em., or 500 A.U. There is one second-order souwe of error in these calculations that must eventually be mathematically evaluated. In the derivation of the coefficient 0.058 in equation 7, it was assumed that all of the impacts experienced by gas molecules within the aerogel are with surfaces. This assumption is a very close approximation, particularly a t the lower pressures, but a t the higher pressures, where there are more direct impacts between gas molecules, the coefficient should increase slightly. This dependence of the TABLE 3 Calculated jree spaces within the aerogel QA8
ka
L
2.9 X 2 . 6 X, 10-6 2.5 X
4.8 X 5 . 1 X 10-6 5 . 1 X 10”
Cm.
Air. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . COZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CClzFz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Average.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,I
5 . 0 X 10-6
coefficient upon pressure will slightly increase the calculated value of L. Nevertheless, as a first approximation and as a very acceptable value until this function can be evaluated, the average of table 3 can stand. SUMMARY
The heat conductivity of silica aerogel of density 0.184 filled with three different gases a t various pressures has been measured. The relation between heat conductivity, mean free path of the gas molecules, and the average distance separating the elements of structure in the aerogel has been derived from kinetic theory, and the last quantity has been evaluated em. a t 5.0 X (1) (2) (3) (4)
REFERENCES CHAPMAN, S. : Phil. Trans. 2llA, 433 (1911). S.S.:Nature 127,741 (1931); J. Phys. Chem. 54,52 (1932). KIBTLER, KISTLER, S.S.,AND CALDWELL, A. G . : Ind. Eng. Chem. 26,658 (1934). T., JR., AND HENNE,A. L.: Ind. Eng. Chem. 22, 542 (1930). MIDQLEY,
