Pore-Size Distribution in Porous Materials Pressure Porosimeter and Determination of CornpIet e Macro p o r e 3 ize is t ri butio ns‘
D
H. L. RITTER A N D L. C. DRAKE Socony-Vacuum Oil Co., Paulsboro, .
N. J.
A
method for determining the maerapore-size distribution in a porous solid, as well as the derived distributions for some typical porous materials, is presented. A glass dilatometer placed in a thermostatcd high-pressure bomb is used to measure the small changer i n volume of a mass of mercury, in which the porous material under investigation is immersed, when the mercury is subjected to varying external pressure.
angle of contact greater than 90’ (the common phenomenon of capillary depression) ; that this opposition may be overcome by the application of external pressure; and that the pressure required to fill a given pore is a measure of the size of the pore. Henderson, Ridgway, and Ross (6) have used this principle in a very limited way, and Loisy (9) has proposed the use of the same principle in a study of poresize distributions.
D
The relation (quoted by Washburn) giving the pressure required to force liquid into a pore of given size is
ETERMINATION of total pore volume is a routine measurement in most laboratories dealing with porous materials. The value usually is calculated as the difference of two specific volumes (reciprocal density). Thus the internal pore volume is the difference between the reciprocals of real density and particle density; the intergranular (void) volume is the difference between the reciprocals of bulk density and particle density; and the sum of pore and void volumas is the difference between the reciprocals of bulk and real densities. [The nomenclature of McBain (10) is followed in identifying the several densities, assuming (with some error) that real and true densities are equal.] The total internal pore volume is then calculated from observa tions of the real and particle densities, determined, for example, by the usual pycnometric method using water and mercury, respectively, as the displacement liquids. I n processes involving diffusion rates and the availability of internal surface to large molecules, a knowledge of total pore volume is less important than a knowledge of the fraction of total pore volume contributed by pores in a given size range-Le., of the distribution of pore sizes. It is convenient to classify the internal pores of porous materials roughly in two ranges. Present usage (2)applies the name “micropores” to those having radii smaller than 100 A.; “macropores” to those larger than 100 A The division of the pore volume of a given porous material into micro- and macropores implies the existence of a distribution in size, yet little xork has been done in the determination of such distribution functions. Rabinowitsh and Fortunatow (19)have determined the respective fractions of micro- and macropores in a number of porous solids by means of the Kelvin equation. Pore-size distributions in the micro region have been determined qualitatively using the Kelvin equation by Kubelka (8) whose work is largely invalidated by neglect of adsorption. The adsorption equation of Brunauer, Deming, Deming, and Teller (3) may be of some use in this connection, but is open to the criticism that it does not satisfactorily combine the simultaneous effects of adsorption and capillary condensation. Jellinek and Fankuchen ( 7 ) have used the scattering of x-rays a t very small angles to evaluate pore size (or particle size) but assumed a constant average size. The unpublished work of Shull (14) on low-angle x-ray scattering takes into consideration a pore-size distribution; but, this method not only cannot conveniently be used for pores larger than perhaps 500 A. in radius, but the results in terms of pore size may also be open to question. This paper present8 a method for determining the macropore-size distribution in a porous solid as well as the derived distributions for some typical porous materials. Washburn (16) has pointed out the fact that surface tension opposes the entrance into a small pore of any liquid having an 1
p r = -2 u cos0
(1)
where p is the pressure, r the pore radius, u the surface tension, and e the contact angle. It may be derived as follows: In a pore of circular cross section, the surface tension acts along the circle of contact over a length equal to the perimeter of the circle. This force is 2 rru. Normal to the plane of the circle of contact, the force tending to squeeze the liquid out of the pore is -2rru cos e. (The negative sign arises from the fact that the angle between the direction of action of the surface tension and the positive normal to the plane of contact is t - e. Since 0 > go”, the term - 2 w u X cos e is intrinsically positive.) Opposing this force is the applied pressure acting over the area of the circle of contact with a force At equilibrium these opposing forces are equal: equal to &p. -2rru cos e = r r * p , whence Equation 1 follows immediately. From this relation it appears that a porous material under zero pressure will “absorb” none of any nonwetting liquid in whirh it is immersed. When the pressure is raised to some finite value, the liquid Rill penetrate and fill all pores having radii greater than that calculated from Equation 1. [The authors have arbitrarily excluded from the category of “pores” all those openings having radii greater than that given by 1 for p = 25 pounds per sq. inch (1.75 kg. per sq. cm.). This is the lowest pressure to which their samples are subjected and corresponds to a radius of about 43,000 i., which is below the limit of resolution of the unaided eye.] As the pressure is increased the amount of liquid “absorbed” increases monotonically a t a rate proportional to the differential pore volume due to pores of size corresponding to the instantaneous pressure. Thus, a given pore-size distribution gives rise to a unique pressuring curve; and, conversely, a given pressuring curve affords a unique determination of the pore-size distribution. SOURCES
OF ERROR
Deviation from the assumed circular cross section is an important source of error. It should, however, be noted that this assumption appears in Equation 1 only through the ratio, 2 / r , of perimeter to area. For noncircular cross sections, this ratio will still be of the order of the reciprocal of some average radius although the constant, 2, may change. The effect here will be only to change the radii calculated from Equation 1 for various pressures by a constant factor. The shape of the distribution curve and order of magnitude of the calculated radii will not be appreciably different. The question of the correct contact angle between liquid and solid is uncertain. The authors have adapted this experimental method to the use of mercury as the working liquid, inasmuch as mercury has a contact angle greater than 90’ for most solids, and have measured contact angles for merbry on a large variety of
First part of paper on “Pore-Sire Diatrlbution in Poroua hIaterials”.
Yeaond part ia found on page 787.
782
ANALYTICAL EDITION
December, 1945
783
materials by the heighhtdf-sessile-drop method ( l l ) ,obtaining values ranging from 135" t o 142" (I). While the surfaces used were in equilibrium with air at normal humidity, and the contact angles thus meastsored are probably high becsuse of adsorbed moisture, it would seem that the internal surfaces of the porous materials must be similarly contaminated; and that the measured contact angle is close t o the true value required in applicacion of Equation 1. The authors have assumed an average conf,act angle of 140" in this study. Differentiation of Equation 1 and elimination of p give
4' =
- A 8 hn 0
,
. (2)
ILJ the fractional error incurred in oaleulsted pore radius by an error of AB in contact angle. For 0 in the neighborhood of 1404 I r l t for a 1' error in contact angle is only about 1.5%. This error is probably smaller than that caused by taking the perimeter to cross-sectional area ratio BS 2/r, and moreover operates in the same manner in its ultimate effecton the distribution curve. If 0 is not in the neighborhood of 140', the only effect is the same magnification or reduction in the scale of pare-radius values, although the size of the error may be considerably more serious. Table I shows the variation in calculated pore radius with a p plied pressure for contad angles of 140' (used in this investigation), 112" (found by U'zshburn for mercury on glass, 18,and 180" (theextreme case). The pore radius calculated by this method is the radius of the opening t o the pore. If there exist in the material pores which are considerably larger than the largest entrances t o them, these pores will be measured as of the s h e of the largest opening. Inasmuch RR mailabilitv of internal surface is determined hv the size
size. In point of-availability, if one is concerned with molecules 20 k.in diameter, it is less important to h o w that a certain pare has a diameter of 40 b. than that the largest entrance t o it is only 15 1. In this connection, see (1s). PRESSURE POROSIMETER
Experimentally, t h e essential problem is t o measufe the small changes in volume of 8. mass of mercury, in which the porous material under investigation is immersed, when the mercury is subjected to varying external pressure. The volume changes are .easured electricslly in B glass dilatometer placed in a therma.ated high-pressure bomb and subjected t o fluid pressure up to ),000poundspersq.inch(700kg.persq.om.). (Using0 = 140' id G = 480 dynes per cm, Equation 1 gives 107 b. as the pore dius in equilibrium a t 10,ooO pounds per sq. inch. Thus, inatigations up to 10,OIN pounds per sq. inch just cover the lacropore range.)
'sm&er capillary tubing may be used for very porous or slighily wous material. The dilatometer is provided with a device for observing the :ight of mercury in the capillary when enclosed in an opaque eta1 bomb. A metal Wire is strune taut along the inside of the
-
bb.- ..
.
Prensure. P.8.i.
25 100
_..
~
.
.
A.
A.
A.
42.680
20.840
55 680 13:920 6,960 4,640
200
10.67D 5.330 3.560 2,670
700
2.135 1,520
300 400 500
......
Contact Angle Pore Radiua for Contact Angles of: I?O. 112180' 5.210 2.600 1.740 1,300 1.040 744
3.480
2.780
1,990
-.
784
Vol. 17, No. 12
INDUSTRIAL AND ENGINEERING CHEMISTRY
If the tubing is of uniform diameter and the wire of uniform resistance, then the change in resistance of the wire-mercury conductor will be a direct measure of the change in volume of the mercury. Figure 1 is a diagram of the dilatometer. The wire is looped over a bridge fused across the junction of capillary to bulb, and passes up through the capillary and out through side holes in the tubing. The ends are separately anchored between cushioned nuts threaded on an insulating screw. Another nut provided with locknut is threaded on the inner end of the screw, so that when this nut is tightened the screw is backed out of the tube and the wire thereby stretched taut. The smooth glass bridge allows the tension to e ualize over both branches and obviates separate tightening. T i e authors have used No. 32 platinum wire. Platinum is used because it is the only common metal with an air-stable surface not attacked by mercury, and this size is a compromise between sturdiness and flexibility. The dilatometer is calibrated directly in cubic centimeters per ohm by taking coordinated readings of resistance and weight of mercury buretted from a stopcock sealed temporarily to the bottom of the dilatometer, while the latter is held a t 0” C. in an ice bath, For convenience, only those dilatometers exhibiting a constant cubic centimeter per ohm conversion factor are retained for use. Dilatometer 5 , for example, had an average conversion factor of 0.608 cc. per ohm with an average deviation over its useful length of 0.002 cc. per ohm.
value. Corrections to the observed gage pressure are made by adding atmospheric pressure plus the average mercury height in the dilatometer. Table I1 gives typical data for five porous materials investigated. The final pressuring curve is obtained by plotting against observed pressure (corrected), p , the volume of mercury “absorbed” per gram of porous material, Va V. Typical pressuring curves are shown in Figures 3 and 6. The compressibility of mercury and the change of resistance with pressure of platinum are both negligible in this pressure range. This supposition was satisfactorily checked by filling the dilatometer with mercury only and pressuring to 10,000 pounds per sq. inch.
-
PORE-SIZE DISTRIBUTION
+
Let the total volume of all pores having radii between r and r dr be
dV = D(r)dr
(3)
where D ( r ) is the distribution function for pore size. From Equation l, assuming constant u and e,
pdr -4- rdp = 0
(4)
Eliminating T and dr from Equations 1,3, and 4 gives 2
COS
e
dV = D ( T ) 7 d p = -D(T) P
‘DILATOMETER
f dp
(5’)
The volume measured by the dilatometer is the volume of a1 pores having radii greater than r-Le., the total pore volume, Vo, decreased by the volume, V , of pores smaller than T. Thus the pressuring curves plot V , - V as a function of p . The slope of )‘ = is then an experimenthe pressuring curve, ’(“ dP dP’ tally determinable quantity and Equation 5’ may now be rewritten in the form
. -
-
MERCURY
-
Figure 2. Filling Pistol for Dilatometer
The dilatometer is filled with sample through its open bottom, sealed off, and placed in the filling pistol as shown in Figure 2. ’The pistol is evacuated at about 10-3 mm. of mercury for 30, minutes, during which time the mercury is poured back and forth several times between the reservoir and the barrel of the pistol. Finally, the pistol is up-ended with the dilatometer head down, isolated from the vacuum line, and the vacuum broken by removing the stopcock plug. Atmospheric pressure forces mercury through the dilatometer head and fills the entire vessel with mercury. In operation, the filled dilatometer is placed in a highpressure bomb with one end of the resistance wire grounded and the other brought through an insulated lead in the bomb head. (The high-pressure insulated lead is a product of the American Instrument Company Silver Spring Md.) Pressures up to 2000 pounds per sq. inch are suppded from a full cylinder of nitrogen and read on a dial gage ( * 10 pounds per sq. inch) calibrated against a dead weight gage. Pressures from 2000 to 10,000 pounds per sq. inch are generated by forcing oil into the bomb with a hand-operated Bosch fuel-injection pump and read directly on the dead weight gage. The initial 2000 pounds per sq. inch of gas pressure provide a cushion a t the top of the bomb which prevents oil from the Bosch pump from spilling over into the top of the dilatometer a t the higher pressures and consequent fouling of the capillary tube.
in which all the terms on the right are known or determinable. Values of the derivative in Equation 5 required to evaluate D ( T ) are readily obtained by graphical differentiation. For a number of values of p , the pressuring curve is differentiated to obtain d(V0 - V ) / d p , is calculated from Equation 1, and D ( r ) is calculated from Equation 5. Plotting D ( T ) against r gives the distribution curve. Curve A of Figure 3 is the pressuring curve for a sample of diatomaceous earth (Celite catalyst carrier Type 296 furnished
0.90
0.15
f0.60
0.10
2. ”
cL*ss
I >O
Pressuring is stopped from time to time and coordinated readings of pressure and resistance are made. With some materials there is a measurable rate of penetration and time is ,allowed for the resistance to rise to its equilibrium value. When the pressure is rapidly applied there is a noticeable rise in temperature and time is allowed for the system to cool and the resistance to fall to its equilibrium
B.uT F l l T i E D
5 0 30
0.00 0
0 05
-
500 1000 ABSOLUTE PRESSURE,
1500 P.6.i
Figure 3. Pressuring Curves for Some Porous Materials
2 000
.
ANALYTICAL EDITION
December, 1945
'185
which may be solved for n. Equations 7 and 8 then serve to find ro and A . For the diatomaceous earth of Figure 4, r, and D, were estimatedto be4100A. and 17.5 X 1 0 - 6 ~ perd.,respectively. ~. Table I1 gives Vo= 1.045 cc. From these data, n = 0.58, ra 5380 A., and A = 4.16 X lo-' cc. per A. The upper curve of Figure 4 is a plot of Equation 6 with these values for the constants. This distribution may perhaps be regarded as the "normal" distribution for pores in a porous solid, and the indicated agreement between the postulated distribution and that found experimentally is satisfactory. The area under this curve is the total pore volume The average pore radius is defined as
-
ve.
F 0.00,
Figure 4.
20M)
4000 6090 P O R E R A D I U S , A.
8000
Distribution Functions for Diatomaceous Earth and Fritted Glass
by Johns-Manville Corp.) heated to 550' C. for 16 hpurs. By differentiating and ap lying Equation 5, the circled points in the plot of Figure 4 were ogtained. The points appear to form a modified Maxwellian distribution, and an attempt was made to fit a curve of the form
= r o f i
(7)
Placing this value of r, in Equation 6 gives
D, = A(n/e)"
+ +)/2
