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Contact angle and the Rootare—Prenzlow equation in mercury intrusion porosimetry
Contact Angle and the Rootare-Prenzlow Equation in Mercury Intrusion Porosimetry H. F. HUISMAN PD Magnetics, 1 Molenstraat 37, 4902 N M Oosterhout, The Netherlands Received M a y 4, 1982; accepted October 7, 1982 F r o m mercury intrusion experiments specific surface areas are calculated with the R o o t a r e - P r e n zlow equation and pore radii with the W a s h b u r n equation. Normally, the contact angle of mercury on the solid material u n d e r investigation is used in these equations. It is shown that this contact angle is not correct for porous systems consisting o f aggregates or clusters o f small particles. In this case an apparent contact angle (0) m u s t be used. This angle is described mathematically by the CassieBaxter equation cos 0 = fl cos 01 - f2, in which f l a n d f2 are the fractional areas o f m e r c u r y in contact with solid material a n d air (vacuum), respectively, a n d 0l is the real contact angle of m e r c u r y on the solid material. This equation, however, cannot be used in the Rootare-Prenzlow equation because the fractional areas are usually u n k n o w n . For this reason a semiempirical equation was developed, which relates the contact angle o f the porous system (0) with the real contact angle a n d the porosity of the aggregates. The Rootare-Prenzlow a n d the W a s h b u r n equations modified with the apparent contact angle now give a plausible explanation for the observations that specific surface areas calculated from Hg intrusion data are often too large a n d pore radii are often too small compared with the real values. The modified Rootare-Prenzlow equation also takes into account in a simple way the variation of the contact angle with the penetration volume.
face area from BET and mercury porosimetry measurements. This densification effect has been investigated.
INTRODUCTION
As part of the evaluation of (magnetic) pigment powders mercury intrusion and BET surface measurements are performed on a routine basis at PD Magnetics. In nearly all cases the specific surfaces measured with the BET method and calculated from the intrusion mercury porosimetry by means of the Rootare-Prenzlow equation (1) can be brought in good agreement by adjusting the contact angle. The adjusted contact angles are then normally between 130 and 160 ° , and the differences can be understood from the differences in the surface chemistry of the particles. To improve the output of magnetic tapes and the dispersibility of the magnetic powders, densification of the powders has proven to be beneficial. With these densified powders, a greater adjustment of the contact angle is needed to achieve agreement in sur-
EXPERIMENTAL
Mercury porosimetry measurements were done with a Carlo Erba (type 200) apparatus. This apparatus was improved to make it more sensitive, and for data handling it is connected with a Philips minicomputer (PM4410). All necessary calculations are also executed on the same minicomputer. A full description of the apparatus and data handling is given in (2). The BET surfaces were measured with a Carlo Erba BET apparatus type 1800, also for data sampling and handling connected with a Philips minicomputer. MATERIALS
The magnetic metal powders were made by pseudomorphic conversion of a-FeOOH
A joint venture o f Philips and DuPont. 25
Journal of Colloid and InterfaceScience, Vol. 94, No. 1, July 1983
0021-9797/83 $3.00 Copyright© 1983 by AcademicPress, Inc. All rightsof reproduction in any form reserved.
26
H. F. HUISMAN
(goethite) in a stream of hydrogen at a temperature between 350 and 400°C. To stabilize the powder against spontaneous oxidation in air, the outer part of the particles is slightly oxidized in a fluid bed process. This process covers the particles with a very thin oxide layer. A detailed description of the preparation can be found in (3, 4). Starting from a single magnetic metal powder a series of powders was made by varying the time of densification. The powders were densified in the presence of toluene in a Cabot-kneader (Plasti-corder, Brabender, Duisburg). The kneader was filled with metal powder and some toluene and after certain time intervals samples were taken. After evaporation of the toluene in vacuo powders with a different degree of densification were obtained. To give the original powder the same solvent treatment the powder was, without exerting any force, saturated with toluene. After equal contact time the toluene was evaporated in the same way as for the densified samples. MEASUREMENTS
Pore volumes. The pore volumes of the series were calculated from the Hg intrusion
porosimetry curves. A typical example of the intrusion curve is given in Fig. 1. For the calculation of the pore volume, corrections have to be made for compressibilities of the mercury, of the glass of the dilatometer, of the sample itself and for the interparticle voids (2)• In Fig. 2 the corrected curve for the pore volume vs pressure is given. Pore sizes. The Washburn equation relates the pressure to the radius of cylindrical pores intruded with mercury: r~q = - 2
O'LV COS 0
p
In this equation aLv is the surface tension of mercury (0.485 N/m), P is the applied pressure, 0 is the contact angle for which in all cases 160 ° is taken, and req is the radius of an equivalent cylindrical pore which should have been intruded at the same pressure as the actual pore is. From the corrected intrusion curve (Fig. 2) the pore size distribution is derived by differentiating. An example of such a distribution is given in Fig. 3. Specific surface areas by mercury porosimetry. The specific surface areas (A) were calculated from the Hg intrusion data with the formula of Rootare and Prenzlow (1).
SO00 -AU/counLs
~
° ol
2000
i
iO00 r • t
U i ntcp
/ /
/." P/at I
0 400
800
1200
FIG. 1. Mercury intrusion curve. Journal of Colloid andlnterface Science, VoL 94, No. 1, July 1983
[11
1600
_ _ J
2000
CONTACT
ANGLE/MERCURY
27
POROSIMETRY
-~U~/~ 10 -4
rn3/~,g
P/IO 7 Pa 2
4
6
I 10
8
FIG. 2. Corrected m e r c u r y intrusion curve.
A _
- 1
~vm P d V
O'LV COS 0
[2]
ao
in which Vm is the maximum penetration volume for 1 kg of sample. A few assumptions (5, 6) have to be fulfilled in order for Eq. [2] to be valid. The first assumption is that there are no pores with a diameter smaller than about 40
~ . As can be seen from Fig. 2 there are no pores intruded above 1000 kg/cm 2 and the slope of the curve between 1000 and 2000 kg/cm 2 is in perfect agreement with the slope calculated from the Hg and glass compressibilities and volumes (2). This indicates there are no pores filled between 1000 and 2000 kg/cm 2 and the powder has no detectable effect on compressibility. This makes it
dU/dri1/~
10 4 m 2 / k g
[3oPePsd i u s / n r n 10
1000
100
10000
FIG. 3. Pore size distribution. Journal of Colloid and Interface Science, Vol. 94, No. 1, July I983
28
H. F. HUISMAN
improbable but not impossible that there are pores smaller than 40/~. Absolute proof can be found by analysis of the nitrogen adsorption isotherm by means of the de Boer's t method (7-9). The second assumption is that the surface of the particles is smooth. Otherwise Youngs' equation, which is the basis for the Rootare equation, must be replaced by the YoungWenzel equation. The roughness of our particles is unknown but in any case this roughness is the same for all powders in the series leading to a systematic error in the specific surfaces, which is compensated for by the contact angle. The third assumption, the one of constant contact angle, will be discussed later.
nonporous reference samples and the materials under investigation. The standard t curve, obtained from the reference samples, obeys the empirical equation
lnpo]p = 2.3026(13.99 t 2 + 0.034)
[3]
in which t is the standard adsorbed layer thickness, calculated from the reference isotherms with
t(A) = 15.47(Va/A)
[4]
where Va = adsorbed volume N2 gas (ml (STP)/g) and A = specific surface of the sample (m2/g). In Figs. 5 and 6 the t plots for the nondensified and for a densified sample are shown. These results prove that the samples BET surface areas and porosity by analysis have no pores with pore dimensions between of nitrogen isotherms. Figure 4 shows an ad- 10 and 50 A. Because the extrapolated t sorption and desorption isotherm which is curves do not go through the origin, a small typical for our magnetic metal powders. amount of ultramicropores (dimensions < 10 From the slopes between P/Po = 0.05 and A) may be present. The amounts do not dif0.30 the BET surface areas are calculated fer for the densified and nondensified sam(10). A convenient method of assessing po- ples and represent the porosity of the primary rosity from nitrogen adsorption data has particles. These results show that the first been reported by De Boer and co-workers assumption for applying the Rootare-Pren(7-9). The method is based on graphical zlow equation is fulfilled. comparison of the isotherms obtained from Contact angles of powders. The value of 400
% 320 v v
240
160
x x
80
/ x+x
.2
.4
o6 ~/1>o
.8
FIG. 4. BET adsorption and desorption isotherm. Journal of Colloid and Interface Science,
Vol. 94, N o , i, J u l y 1983
29
CONTACT ANGLE/MERCURY POROSIMETRY 40
32
x
v x
x
lg
_ _ 3
4
8
I2 tl(R)
16
2O
FIG. 5. t plot of a nondensified powder.
the contact angle of powders plays an important role in m a n y physicochemical processes and therefore its measurement has been subjected to a great deal o f research for many years. However, the direct measurement o f the contact angle o f mercury on powder has not been solved fundamentally (5).
The method mostly used is a direct measurement on compressed powders. The powder is compacted under high pressure (>5000 kg/cm 2) in a die. The forces involved in the compression are so high that plastic deformation of the particles is almost inevitable. So the compression is likely to render the contact surface unlike the surface of the orig-
40 x
x/
~2 I'--Or?
~ 2q
16
J
4
_ _
I
8
I
_
_
12 t/(~)
±
I
16
20
FIG. 6. t plot of a densified powder.
JoumalofColloidandlmerfaceScience,
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H. F. H U I S M A N
inal powder particles. Especially in our case (a metal particle covered with a very thin oxide layer) the surface chemistry of the particles will be changed in a unknown way. Further the density of the compacts is always lower than the specific gravity of the material. This means that the compacts are porous and the surface cannot be perfectly smooth. Also the contact angle of mercury on the die itself is nearly 180 °, which is higher than the literature data for this material. This difference is caused by the microscopic roughness of the die surface. From this one must conclude that the contact angles measured on compressed powders are determined by the (unknown) surface chemistry of the primary particles, the microscopic roughness of the contact surface, and the remaining porosity of the compacts. Recently a new instrument especially designed for mercury contact angle measurements has been described (11). A 800-#m cylindrical hole is made in samples of compacted powders. The pressure of which mercury penetrates the hole is determined. With the Washburn equation, which is derived for this situation, the contact angle is calculated. One may expect that this contact angle has the same uncertainties as the one measured on compacted powders. It depends on the microscopic roughness of the cylindrical hole, on the porosity of the powder particles which is likely to be different from the original noncompacted powder particles, and on the surface chemistry of the primary particles. Another uncertainty is caused by the dimensions of the particles and the pores. Particles and pores are often in a region where deviations from the macroscopic contact angles can be expected. This has led to the introduction of a microscopic contact angle, which is usually smaller than the macroscopic contact angle (12-14). It appears that the true value of the contact angle is a rather elusive quantity and direct measurement gives only qualitative information. An elegant procedure for estimating the Journal of Colloid and Interface Science, Vol. 94, No. 1, July 1983
average contact angle of a porous powder is the one proposed by Rootare (15). It is based on the equalization of BET and Hg porosimetry surface areas. Equation [2] is the basis for this equalization and the only unknown is the average contact angle. It is also the basis of our investigation of the compaction effect. R ES U LTS
The pore volume and specific surface area results for the densified powder calculated from Hg porosimetry are given in Table I. As seen from this table the specific surface area decreases as the densification increases (decreasing pore volume). In BET measurements this decrease was not found. The BET surface area was within the experimental error constant for all powders and equal to (31.5 _+ l) × lO3 (m2/kg). DISCUSSION
The fact that the surface areas calculated from mercury porosimetry experiments decrease with increased densification while the BET surface area is constant, can be because: (a) The pore diameters are becoming progressively smaller and the smallest diameters are no longer intrudable by mercury (at the TA B LE I Specific Surfaces Calculated with Eq. [2] a nd Pore Volumes as a F unc t i on of Densification T i m e
Densification time (h) 0 1 1.25 1.65 1.85 2.0 2.18 2.30 2.50
Pore volume (m~/kg) 1.18 7.10 5.75 4.50 3.56 2.78 2.61 2.35 2.14
× × × X × × × X ×
10 -3 10 -4 10 -4 10 -4 10 -4 10 -4 10 -4 10 -4 10 -4
Spee. surface a (m~/g)
Contact angleb (°)
(V~ .p)213
32.3 25.9 24.5 22.5 21.2 20.0 19.3 18.8 18.4
160 138.9 135.5 130.9 128.1 125.6 124.2 123.2 122.4
3.84 2.74 2.38 2.02 1.73 1.47 1.41 1.31 1.23
0 = 160 °. b Specific surface = 32.3 m2/g,
31
CONTACT ANGLE/MERCURY POROSIMETRY
applied pressures). This situation is not very likely as already discussed. (b) Not all solvents have been removed under the vacuum conditions o f the mercury porosimetry sample preparation. For this reason the powders were analyzed for their carbon content. No evidence was found indicating presence of solvents. (c) The contact angle is not constant but decreases as a function of densification. This can then either be caused by chemical changes in surface characteristics o f the particles or that the contact angle changes as function of pore volume. The first of these mechanisms is not likely because adsorption and dispersion characteristics of the powders are not a function of densification, and in ESCA measurements all powders are alike. Thus, the discrepancy of surface area between BET and mercury porosimetry can best be explained as use o f an inappropriate contact angle in the calculation.
The Cassie-Baxter Equation If the magnetic powders are examined in a scanning electron microscope, it can be seen that the powders consist of more or less spherical clusters in which the acicular primary particles are packed at random. F r o m this it is obvious that cylindrical or other easy describable pores are not present and that the intruding mercury faces along its intruding meniscus a combination of solid material (the primary particles) and voids. In this situation the contact angle can best be described by the Cassie-Baxter equation (16, 17). COS 0 = f l COS 01 -- f 2
[5]
where f t and f2 are the fractional areas o f the solid material and the voids in contact with mercury. 01 and 0 are the contact angle of mercury on the solid and the apparent contact angle for the intruding Hg, respectively. With Eq. [2] the contact angles to achieve a constant specific surface area were calcu-
lated. Because the highest value in the series (the original powder) is close to the BET surface area (31.5 _+_1) we used this as a starting value. In Table I, Column 4 the calculated contact angles are given. The qualitative trend o f decreased contact angle with decreased pore volume is in agreement with the Cassie Eq. [5]. A direct quantitative test is impossible, because the values of f l and f2 are unknown.
Model of Kossen and Heertjes For a simple model of equal spheres and uniform distribution in space Kossen and Heertjes (18) have transformed Eq. [5] into an equation with the volume porosity. cos 0t = - 1 +
(1
~v) 1
cos
[6]
In this equation ev is the volume porosity per unit o f volume. In Table I1 the Eq. [6] calculated contact angles 01 are given together with ~v and 0. The calculated values are reasonably constant, though there is probably an indication that the values of 01 pass through a m i n i m u m . In Fig. 7 the calculated curve of cos 0 vs volume porosity (ev) is drawn according to Eq. [6]. In this calculation 0t was taken as 107 ° . In the same figure the experimental results of Table II are plotted. F r o m this Fig. 7 one can conclude TABLE II Porosity and Calculated Contact Angles According to Kossen and Heertjes. (Equation [6]) Contact angle Densification time (h)
Porosity, ~v
Apparent
Calculated
0 1 1.25 1.65 1.85 2.0 2.18 2.30 2.50
0.833 0.820 0.786 0.742 0.695 0.640 0.626 0.601 0.578
160 138.9 135.5 130.9 128.1 125.6 124.2 123.2 122.4
125.4 106.2 106.1 105.4 106.3 107.5 107.0 107.5 107.8
Journal of Colloid and lnlerface Science, Vol. 94, No. 1, July 1983
32
H. F. HUISMAN
.8 i;. ~
L3 i
.9
/
1
.2
0
I
I
I
.2
.4
.G
m
I .8
DOLUrlE ? 0 R O S I T Y
FIG. 7. Cosines of apparent contact angles versus volume porosity according to Kossen and Heertjes. Solid line, experimental curve; dashed line, Kossen and Heertjes curve.
(a) The value of the original, nondensifled, powder deviates significantly from the other results. T o be in line with the other data the contact should have been 147 ° instead of 160 ° . This is beyond the experimental error. (b) In the low porosity region (~v < 0.4) the contact angle calculated with Eq. [6] is m u c h too low, it is even lower than 0~, which is impossible. The model used by Kossen and Heertjes is an oversimplification of the actual situation and not very likely to occur in practice as they already stated themselves. Clearly it does not apply in this case. Perhaps it can be used as an approximation in a limited porosity region.
Empirical Equation For the reason mentioned above we have replaced Eq. [5] by an empirical equation that is easier to handle cos 0 = a cos 01 - b x f
[7]
in which 0 and 01 have their usual meaning, Journa[ofCol[oidandlnterface Science. Vol.
94, No. 1, July 1983
f is the ratio of the void and solid surface areas in contact with mercury, and a and b are proportionally constants. This equation is based on the idea that the apparent contact angle is a function of the original contact angle and that for the voids (expressed as the ratio f ) a correction is needed and as a first approximation this correction is first order in f. Expressing all volumes and surface areas on the basis of 1 kg of solid material, the solid surface area in contact with mercury is constant and only the void/surface area ratio varies. For this situation a is equal to 1 in Eq. [7]. Bearing in m i n d the tendency o f mercury to form spherical surfaces the assumption is made that the ratio void/solid surface area is proportional with the 2/3 power o f the ratio of the volumes involved. Equation [7] now becomes COS 0 = COS 01 -- b2 X
(vm] kl/pl
[81
where p is the specific gravity o f the material and bz is a constant which depends on the geometry of the particles and on the arrange-
CONTACT ANGLE/MERCURY POROSIMETRY
33
.8
I
.4
.2
0
P
L ~ 3
_ _ 3 _
1
2
I
4
I
5
(Pore volume x Spec. Srau. ) , , 2 / 3
FIG. 8. Cosines of apparent contact angles versus correction factor f for LMP 172.056. m e n t of the particles in the clusters. Only in the case o f uniform particles in a regular array can b2 be calculated. For densest and loose packings of uniform spheres and of cylinders b2 varies between 0.12 and 0.33. In Table I the values for (V~. 0) 2/3 are given in column 5. In Fig. 8 the cosine o f the calculated contact angles (0) are plotted vs f = (Vm" p ) 2 / 3 . The regression equation for the best fit2 is cos 0 = - 0 . 3 5 9 - 0.146 X ( V ~ . o ) 2/3
[9]
with a coefficient of regression o f 0.997 and an s of 0.006. This means that a linear relation describes well the relation between the contact angles and the porosity of the samples. The extrapolated value for the contact angle of mercury on the solid is 111 °. Because the real contact angles plotted as function o f (Vm'O) 2/3 reaches the ultimate value (cos (180 °) = - 1 ) asymptotically the linear relation cannot hold for high porosity powders. This is illustrated in Fig. 9 where 2 In the calculation of the regressionequation the original powder has not been taken into account. In Figure 8 and 10 these points are marked with a X.
the Cassie-Baxter equation, our linear approximation, and the real values are given. F r o m this diagram it is clear that our empirical linear relation will only be valid whenever the porosity of the powder particles is not too high. This deviation from the linear relation can be seen in Fig. 10 where we have plotted another set of data. Also here the best fit is a linear relation between cos 0 and (Vm" p)2/3. Only the original powder itself with a porosity (ev) of 0.91 does not fit on the linear regression graph that can be drawn through the other data points. The regression equation fitting the data points (except for the original powder) is cos 0 = - 0 . 2 6 6 - 0.233(Vm" 0) 2/3
[10]
with a coefficient of regression of 0.98. The extrapolated value for the contact angle of mercury on the solid is 105 ° . The values for the extrapolated contact angles 111 and 105 ° are not unrealistic for this kind o f material (a metal covered with an oxide layer). However due to the assumptions discussed earlier and the uncertainty about the curve at low porosity (for Ev < 0.5 there are no data) these values must Journal of Colloid and Interface Science, Vol. 94, No. 1, July 1983
34
H. F. HUISMAN
i
.8 //
(.3
z!
I
.4
.2
D
I
_ _ J -
2
L
4
I
6
8
(Pore
Uolbme
I
I0 x Spec. Grau.),,2/3
FIG. 9. Diagram showing the relationship of cos Oversus f(= Vm. 0)2/3) for the Cassie-Baxter equation, our linear relation, and the real data.
not be taken too absolutely, but they are of the right order of magnitude. This makes it very likely Eq. [8] can be used in the porosity regime from 0 up to 0.8. This corresponds with a void surface area fraction between 0 and 0.6. In other words Eq. [8] can be used for cor-
recting the contact angle and the mechanism which is the basis of the Cassie equation is very likely to occur in mercury porosimetry.
The Modified Rootare-Prenzlow Equation Equation [8] also gives information about the dependence of the contact angle with the
J .8
3
/
v I
.4
,2
0
1
i
I
I
2
I
3 (Pore
4 Uolume
I
5 x Spec, Grav.),,2/3
FIG. 10. Cosines of apparent contact angles versus correction factor f for LMP 172.054. Journal of Colloid and Interface Science, Vol. 94~ No. 1, July 1983
35
CONTACT ANGLE/MERCURY POROSIMETRY
penetration of the mercury and it can be used to modify (5) Eq. [2] into A = -1 fo rm PdV aLV COS 0(V)
TABLE II1 Pore Radii r ~ (ore)
[11] Densification time (h)
Pore volume, V~ (m3/kg)
0 = 160°
0 = variable°
rhb (nm)
72.6 59.4 46.3 40.3 32.9 25.8 25.7 23.8 23.0
72.6 47.6 35.1 28.1 21.6 16.0 15.4 13.9 13.1
74.9 45.1 36.5 28.6 22.6 17.6 16.6 14.9 13.6
and replacing cos 0(V) by Eq. [8]
A -
-1 O'LV COS 01
{f0vmPdV + constant COS 01
X foVm ve/3pdV} . [12] The second term of Eq. [12] is always negative and this may be one of the reasons specific surface areas calculated from Hg porosimetry in which only the first term is taken into account are often too large (19, 20). If in general the Cassie mechanism is important in explaining the intrusion of mercury in powder/clusters or porous media, one may expect that for the retraction of mercury from these systems it also can be of importance. For the retraction experiment the equation becomes more complex because the composite surface can now consist of solid material (contact angle: OR(receding)), air (vacuum) (contact angle: 180°), and areas where Hg remains (contact angle: 0°). This situation always leads to a smaller contact angle for the retraction experiment than for the intrusion experiment. This mechanism can give an additional reason (21) for hysteresis and mercury trapping after retraction.
The Modified Washburn Equation In the same way that the Rootare-Prenzlow equation is modified for porous systems, the Washburn equation [1] can be adapted to these systems. This leads to re" = - 2
O'LV { --P-- tCOS 01 -
- [ Vm~2/3~ U2t I ~ P /
J" [13]
In Table III we have tabulated the rm.~ (the value of the pore radius whereby the maximum in the pore size distribution curve is
0 1 1.25 1.65 1.85 2.0 2.18 2.30 2.50
1.18 7.10 5.75 4.50 3.56 2.78 2.61 2.35 2.14
X X X X × X X X X
10 -3 10-4 10 4 10 -4 10 -4 10-4 10-4 10 -4 10 -4
" T h e values from Table I are used. b rh = 2 V m / A and A = 31,500 rn2/kg.
situated) calculated with a fixed contact angle of 160 ° and with a variable contact angle (see Table I), which is the same as calculated with Eq. [ 13]. Because the calculations are based on the assumption of a cylindrical pore model, a model independent radius (the "hydraulic radius") (6) is tabulated in column 5 of the same table. This "hydraulic radius" (r0 is calculated from the total pore volume obtained from mercury porosimetry, Vm, and the total surface area obtained from BET sorption isotherms, A(--31500 m2/kg) according to
rh = 2V/A.
[14]
The agreement between the corrected rmax and the "hydraulic radius" is remarkable. This indicates that the procedure followed for adapting the Rootare-Prenzlow and the Washburn equations to these porous systems is consistent. CONCLUSIONS
(1) The experiments make it very likely that the mechanism underlying the CassieBaxter equation is operative in the intrusion process of mercury in powders of the kind studied here (acicular magnetic powders). Journal of Colloid and Interface Science, Vol. 94, No. 1, July 1983
36
rL F. HUISMAN
However, there is a reason to believe that it is more generally applicable for mercury porosimetry as already discussed by Good and Mikhail (5). Especially for compacted powders or granulates (14) and for porous materials (22), it may be a very important mechanism. (2) By comparing Hg porosity and BET specific surface areas it is possible, in special cases, to find a semiempirical relation between contact angle and intrusion volume. This relation can be used to modify the Rootare-Prenzlow equation for the change in contact angle with penetration. (3) From this analysis it becomes clear why specific surface areas calculated from Hg porosimetry experiments are often too high compared with BET specific surface areas. With the same reasoning the pore diameters calculated from the intrusion curve with the Washburn equation are too small compared with pore diameters measured with other techniques (e.g., from permeability, or nitrogen capillary condensation) (23). (4) For practical application, the Rootare-Prenzlow equation can be used to calculate the specific surface from Hg intrusion porosimetry with reasonable accuracy if (a) The contact angle is an average contact angle in which the influence of porosity and/or roughness is taken into account. (b) All surface areas are counted in the Hg porosimetry experiment. (In practice pore diameters must be larger than about 10 nm.) (5) The Washburn equation with the correct contact angle gives for these systems the same average pore radius as the one calculated from the total pore volume and the total specific surface area from BET experiments.
Journalof Colloidand lnterJaceScience.Vol.94, No. 1, July 1983
ACKNOWLEDGMENTS I wish to thank PD magnetics management for permission to publish this paper and Mr. T de Laat from Philips Central Research for the BET measurements. REFERENCES 1. Rootare, H. M., and Prenzlow, C. F., J. Phys. Chem. 71, 2733 (1967). 2. Huisman, H. F., Rasenberg, C. J. F. M., and v. Winsum, J. A., to be published in Powder Technol.
3. v.d. Giessen, A. A., and Klomp, G. J., IEEE Trans. Mag. MAG-5, 317 (1969). 4. v.d. Giessen, A. A., IEEE Trans. Mag. MAG-9, 191 (1973). 5. Good, R. J., and Mikhail, R. S. H., Powder Technol 29, 53 (1981). 6. v. Brakel, J., Madry, S., and Svat~, M., Powder Technol. 29, 1 (1981). 7. de Boer, J. H., J. Catal. 11, 46 (1968). 8. Broekhoff, J. C. P., Thesis, Delft, 1969. 9. Linsen, B. G., "Physical and Chemical Aspects of Adsorbents and Catalysts." Academic Press, New York, 1970. 10. Brunauer, S., Emmett, P. H., and Teller, E., d. Amer. Chem. Soc. 60, 309 (1938). 11. Shields, J. E., and Lowell, S., Powder Technol. 31, 227 (1982). 12. Jameson, G. J., and Del Cerro, M. G., J. Chem. Soc. Faraday Trans. 1 72, 883 (1976). 13. Pethica, B., J. Colloid Interface Sci. 62, 567 (1977). 14. Amaral Fortes, M., J. Chem. Soc. Faraday Trans. 1 78, 101 (1982). 15. Rootare, H. M., Aminco Laboratory News, Aminco Reprint, No. 439 (1968). 16. Cassie, A. B. D., Discuss. FaradaySoc. 3, 11 (1948). 17. Matijevie, E. (Ed.), "Surface and Colloid Science," Vol. 2, Chapter 2. Wiley Interscience, New York, 1969. 18. Kossen, N. W. F., and Heertjes, P. M., Chem. Eng. Sci. 20, 593 (1965). Kossen, N. W. F., Dissertation, Delft, 1965. Witvoet, W. C., Dissertation, Delft, 1971. 19. Dees, P. J., and Polderman, J., Powder Technol. 29, 187 (1981). 20. Spitzer, Z., Powder Technol. 29, 177 (1981). 21. Lowell, S., and Shields, J. E., J. Colloid Interface Sci. 80, 192 (1981); 83, 273 (1981). 22. Moscou, L., and Lub, S., Powder TechnoL 29, 45 (1981). 23. de Wit, L. A., and Scholten, J. J. F., J. Catal. 30, 30 (1975).
face area from BET and mercury porosimetry measurements. This densification effect has been investigated.
INTRODUCTION
As part of the evaluation of (magnetic) pigment powders mercury intrusion and BET surface measurements are performed on a routine basis at PD Magnetics. In nearly all cases the specific surfaces measured with the BET method and calculated from the intrusion mercury porosimetry by means of the Rootare-Prenzlow equation (1) can be brought in good agreement by adjusting the contact angle. The adjusted contact angles are then normally between 130 and 160 ° , and the differences can be understood from the differences in the surface chemistry of the particles. To improve the output of magnetic tapes and the dispersibility of the magnetic powders, densification of the powders has proven to be beneficial. With these densified powders, a greater adjustment of the contact angle is needed to achieve agreement in sur-
EXPERIMENTAL
Mercury porosimetry measurements were done with a Carlo Erba (type 200) apparatus. This apparatus was improved to make it more sensitive, and for data handling it is connected with a Philips minicomputer (PM4410). All necessary calculations are also executed on the same minicomputer. A full description of the apparatus and data handling is given in (2). The BET surfaces were measured with a Carlo Erba BET apparatus type 1800, also for data sampling and handling connected with a Philips minicomputer. MATERIALS
The magnetic metal powders were made by pseudomorphic conversion of a-FeOOH
A joint venture o f Philips and DuPont. 25
Journal of Colloid and InterfaceScience, Vol. 94, No. 1, July 1983
0021-9797/83 $3.00 Copyright© 1983 by AcademicPress, Inc. All rightsof reproduction in any form reserved.
26
H. F. HUISMAN
(goethite) in a stream of hydrogen at a temperature between 350 and 400°C. To stabilize the powder against spontaneous oxidation in air, the outer part of the particles is slightly oxidized in a fluid bed process. This process covers the particles with a very thin oxide layer. A detailed description of the preparation can be found in (3, 4). Starting from a single magnetic metal powder a series of powders was made by varying the time of densification. The powders were densified in the presence of toluene in a Cabot-kneader (Plasti-corder, Brabender, Duisburg). The kneader was filled with metal powder and some toluene and after certain time intervals samples were taken. After evaporation of the toluene in vacuo powders with a different degree of densification were obtained. To give the original powder the same solvent treatment the powder was, without exerting any force, saturated with toluene. After equal contact time the toluene was evaporated in the same way as for the densified samples. MEASUREMENTS
Pore volumes. The pore volumes of the series were calculated from the Hg intrusion
porosimetry curves. A typical example of the intrusion curve is given in Fig. 1. For the calculation of the pore volume, corrections have to be made for compressibilities of the mercury, of the glass of the dilatometer, of the sample itself and for the interparticle voids (2)• In Fig. 2 the corrected curve for the pore volume vs pressure is given. Pore sizes. The Washburn equation relates the pressure to the radius of cylindrical pores intruded with mercury: r~q = - 2
O'LV COS 0
p
In this equation aLv is the surface tension of mercury (0.485 N/m), P is the applied pressure, 0 is the contact angle for which in all cases 160 ° is taken, and req is the radius of an equivalent cylindrical pore which should have been intruded at the same pressure as the actual pore is. From the corrected intrusion curve (Fig. 2) the pore size distribution is derived by differentiating. An example of such a distribution is given in Fig. 3. Specific surface areas by mercury porosimetry. The specific surface areas (A) were calculated from the Hg intrusion data with the formula of Rootare and Prenzlow (1).
SO00 -AU/counLs
~
° ol
2000
i
iO00 r • t
U i ntcp
/ /
/." P/at I
0 400
800
1200
FIG. 1. Mercury intrusion curve. Journal of Colloid andlnterface Science, VoL 94, No. 1, July 1983
[11
1600
_ _ J
2000
CONTACT
ANGLE/MERCURY
27
POROSIMETRY
-~U~/~ 10 -4
rn3/~,g
P/IO 7 Pa 2
4
6
I 10
8
FIG. 2. Corrected m e r c u r y intrusion curve.
A _
- 1
~vm P d V
O'LV COS 0
[2]
ao
in which Vm is the maximum penetration volume for 1 kg of sample. A few assumptions (5, 6) have to be fulfilled in order for Eq. [2] to be valid. The first assumption is that there are no pores with a diameter smaller than about 40
~ . As can be seen from Fig. 2 there are no pores intruded above 1000 kg/cm 2 and the slope of the curve between 1000 and 2000 kg/cm 2 is in perfect agreement with the slope calculated from the Hg and glass compressibilities and volumes (2). This indicates there are no pores filled between 1000 and 2000 kg/cm 2 and the powder has no detectable effect on compressibility. This makes it
dU/dri1/~
10 4 m 2 / k g
[3oPePsd i u s / n r n 10
1000
100
10000
FIG. 3. Pore size distribution. Journal of Colloid and Interface Science, Vol. 94, No. 1, July I983
28
H. F. HUISMAN
improbable but not impossible that there are pores smaller than 40/~. Absolute proof can be found by analysis of the nitrogen adsorption isotherm by means of the de Boer's t method (7-9). The second assumption is that the surface of the particles is smooth. Otherwise Youngs' equation, which is the basis for the Rootare equation, must be replaced by the YoungWenzel equation. The roughness of our particles is unknown but in any case this roughness is the same for all powders in the series leading to a systematic error in the specific surfaces, which is compensated for by the contact angle. The third assumption, the one of constant contact angle, will be discussed later.
nonporous reference samples and the materials under investigation. The standard t curve, obtained from the reference samples, obeys the empirical equation
lnpo]p = 2.3026(13.99 t 2 + 0.034)
[3]
in which t is the standard adsorbed layer thickness, calculated from the reference isotherms with
t(A) = 15.47(Va/A)
[4]
where Va = adsorbed volume N2 gas (ml (STP)/g) and A = specific surface of the sample (m2/g). In Figs. 5 and 6 the t plots for the nondensified and for a densified sample are shown. These results prove that the samples BET surface areas and porosity by analysis have no pores with pore dimensions between of nitrogen isotherms. Figure 4 shows an ad- 10 and 50 A. Because the extrapolated t sorption and desorption isotherm which is curves do not go through the origin, a small typical for our magnetic metal powders. amount of ultramicropores (dimensions < 10 From the slopes between P/Po = 0.05 and A) may be present. The amounts do not dif0.30 the BET surface areas are calculated fer for the densified and nondensified sam(10). A convenient method of assessing po- ples and represent the porosity of the primary rosity from nitrogen adsorption data has particles. These results show that the first been reported by De Boer and co-workers assumption for applying the Rootare-Pren(7-9). The method is based on graphical zlow equation is fulfilled. comparison of the isotherms obtained from Contact angles of powders. The value of 400
% 320 v v
240
160
x x
80
/ x+x
.2
.4
o6 ~/1>o
.8
FIG. 4. BET adsorption and desorption isotherm. Journal of Colloid and Interface Science,
Vol. 94, N o , i, J u l y 1983
29
CONTACT ANGLE/MERCURY POROSIMETRY 40
32
x
v x
x
lg
_ _ 3
4
8
I2 tl(R)
16
2O
FIG. 5. t plot of a nondensified powder.
the contact angle of powders plays an important role in m a n y physicochemical processes and therefore its measurement has been subjected to a great deal o f research for many years. However, the direct measurement o f the contact angle o f mercury on powder has not been solved fundamentally (5).
The method mostly used is a direct measurement on compressed powders. The powder is compacted under high pressure (>5000 kg/cm 2) in a die. The forces involved in the compression are so high that plastic deformation of the particles is almost inevitable. So the compression is likely to render the contact surface unlike the surface of the orig-
40 x
x/
~2 I'--Or?
~ 2q
16
J
4
_ _
I
8
I
_
_
12 t/(~)
±
I
16
20
FIG. 6. t plot of a densified powder.
JoumalofColloidandlmerfaceScience,
Vol. 94, No.l,
July1983
30
H. F. H U I S M A N
inal powder particles. Especially in our case (a metal particle covered with a very thin oxide layer) the surface chemistry of the particles will be changed in a unknown way. Further the density of the compacts is always lower than the specific gravity of the material. This means that the compacts are porous and the surface cannot be perfectly smooth. Also the contact angle of mercury on the die itself is nearly 180 °, which is higher than the literature data for this material. This difference is caused by the microscopic roughness of the die surface. From this one must conclude that the contact angles measured on compressed powders are determined by the (unknown) surface chemistry of the primary particles, the microscopic roughness of the contact surface, and the remaining porosity of the compacts. Recently a new instrument especially designed for mercury contact angle measurements has been described (11). A 800-#m cylindrical hole is made in samples of compacted powders. The pressure of which mercury penetrates the hole is determined. With the Washburn equation, which is derived for this situation, the contact angle is calculated. One may expect that this contact angle has the same uncertainties as the one measured on compacted powders. It depends on the microscopic roughness of the cylindrical hole, on the porosity of the powder particles which is likely to be different from the original noncompacted powder particles, and on the surface chemistry of the primary particles. Another uncertainty is caused by the dimensions of the particles and the pores. Particles and pores are often in a region where deviations from the macroscopic contact angles can be expected. This has led to the introduction of a microscopic contact angle, which is usually smaller than the macroscopic contact angle (12-14). It appears that the true value of the contact angle is a rather elusive quantity and direct measurement gives only qualitative information. An elegant procedure for estimating the Journal of Colloid and Interface Science, Vol. 94, No. 1, July 1983
average contact angle of a porous powder is the one proposed by Rootare (15). It is based on the equalization of BET and Hg porosimetry surface areas. Equation [2] is the basis for this equalization and the only unknown is the average contact angle. It is also the basis of our investigation of the compaction effect. R ES U LTS
The pore volume and specific surface area results for the densified powder calculated from Hg porosimetry are given in Table I. As seen from this table the specific surface area decreases as the densification increases (decreasing pore volume). In BET measurements this decrease was not found. The BET surface area was within the experimental error constant for all powders and equal to (31.5 _+ l) × lO3 (m2/kg). DISCUSSION
The fact that the surface areas calculated from mercury porosimetry experiments decrease with increased densification while the BET surface area is constant, can be because: (a) The pore diameters are becoming progressively smaller and the smallest diameters are no longer intrudable by mercury (at the TA B LE I Specific Surfaces Calculated with Eq. [2] a nd Pore Volumes as a F unc t i on of Densification T i m e
Densification time (h) 0 1 1.25 1.65 1.85 2.0 2.18 2.30 2.50
Pore volume (m~/kg) 1.18 7.10 5.75 4.50 3.56 2.78 2.61 2.35 2.14
× × × X × × × X ×
10 -3 10 -4 10 -4 10 -4 10 -4 10 -4 10 -4 10 -4 10 -4
Spee. surface a (m~/g)
Contact angleb (°)
(V~ .p)213
32.3 25.9 24.5 22.5 21.2 20.0 19.3 18.8 18.4
160 138.9 135.5 130.9 128.1 125.6 124.2 123.2 122.4
3.84 2.74 2.38 2.02 1.73 1.47 1.41 1.31 1.23
0 = 160 °. b Specific surface = 32.3 m2/g,
31
CONTACT ANGLE/MERCURY POROSIMETRY
applied pressures). This situation is not very likely as already discussed. (b) Not all solvents have been removed under the vacuum conditions o f the mercury porosimetry sample preparation. For this reason the powders were analyzed for their carbon content. No evidence was found indicating presence of solvents. (c) The contact angle is not constant but decreases as a function of densification. This can then either be caused by chemical changes in surface characteristics o f the particles or that the contact angle changes as function of pore volume. The first of these mechanisms is not likely because adsorption and dispersion characteristics of the powders are not a function of densification, and in ESCA measurements all powders are alike. Thus, the discrepancy of surface area between BET and mercury porosimetry can best be explained as use o f an inappropriate contact angle in the calculation.
The Cassie-Baxter Equation If the magnetic powders are examined in a scanning electron microscope, it can be seen that the powders consist of more or less spherical clusters in which the acicular primary particles are packed at random. F r o m this it is obvious that cylindrical or other easy describable pores are not present and that the intruding mercury faces along its intruding meniscus a combination of solid material (the primary particles) and voids. In this situation the contact angle can best be described by the Cassie-Baxter equation (16, 17). COS 0 = f l COS 01 -- f 2
[5]
where f t and f2 are the fractional areas o f the solid material and the voids in contact with mercury. 01 and 0 are the contact angle of mercury on the solid and the apparent contact angle for the intruding Hg, respectively. With Eq. [2] the contact angles to achieve a constant specific surface area were calcu-
lated. Because the highest value in the series (the original powder) is close to the BET surface area (31.5 _+_1) we used this as a starting value. In Table I, Column 4 the calculated contact angles are given. The qualitative trend o f decreased contact angle with decreased pore volume is in agreement with the Cassie Eq. [5]. A direct quantitative test is impossible, because the values of f l and f2 are unknown.
Model of Kossen and Heertjes For a simple model of equal spheres and uniform distribution in space Kossen and Heertjes (18) have transformed Eq. [5] into an equation with the volume porosity. cos 0t = - 1 +
(1
~v) 1
cos
[6]
In this equation ev is the volume porosity per unit o f volume. In Table I1 the Eq. [6] calculated contact angles 01 are given together with ~v and 0. The calculated values are reasonably constant, though there is probably an indication that the values of 01 pass through a m i n i m u m . In Fig. 7 the calculated curve of cos 0 vs volume porosity (ev) is drawn according to Eq. [6]. In this calculation 0t was taken as 107 ° . In the same figure the experimental results of Table II are plotted. F r o m this Fig. 7 one can conclude TABLE II Porosity and Calculated Contact Angles According to Kossen and Heertjes. (Equation [6]) Contact angle Densification time (h)
Porosity, ~v
Apparent
Calculated
0 1 1.25 1.65 1.85 2.0 2.18 2.30 2.50
0.833 0.820 0.786 0.742 0.695 0.640 0.626 0.601 0.578
160 138.9 135.5 130.9 128.1 125.6 124.2 123.2 122.4
125.4 106.2 106.1 105.4 106.3 107.5 107.0 107.5 107.8
Journal of Colloid and lnlerface Science, Vol. 94, No. 1, July 1983
32
H. F. HUISMAN
.8 i;. ~
L3 i
.9
/
1
.2
0
I
I
I
.2
.4
.G
m
I .8
DOLUrlE ? 0 R O S I T Y
FIG. 7. Cosines of apparent contact angles versus volume porosity according to Kossen and Heertjes. Solid line, experimental curve; dashed line, Kossen and Heertjes curve.
(a) The value of the original, nondensifled, powder deviates significantly from the other results. T o be in line with the other data the contact should have been 147 ° instead of 160 ° . This is beyond the experimental error. (b) In the low porosity region (~v < 0.4) the contact angle calculated with Eq. [6] is m u c h too low, it is even lower than 0~, which is impossible. The model used by Kossen and Heertjes is an oversimplification of the actual situation and not very likely to occur in practice as they already stated themselves. Clearly it does not apply in this case. Perhaps it can be used as an approximation in a limited porosity region.
Empirical Equation For the reason mentioned above we have replaced Eq. [5] by an empirical equation that is easier to handle cos 0 = a cos 01 - b x f
[7]
in which 0 and 01 have their usual meaning, Journa[ofCol[oidandlnterface Science. Vol.
94, No. 1, July 1983
f is the ratio of the void and solid surface areas in contact with mercury, and a and b are proportionally constants. This equation is based on the idea that the apparent contact angle is a function of the original contact angle and that for the voids (expressed as the ratio f ) a correction is needed and as a first approximation this correction is first order in f. Expressing all volumes and surface areas on the basis of 1 kg of solid material, the solid surface area in contact with mercury is constant and only the void/surface area ratio varies. For this situation a is equal to 1 in Eq. [7]. Bearing in m i n d the tendency o f mercury to form spherical surfaces the assumption is made that the ratio void/solid surface area is proportional with the 2/3 power o f the ratio of the volumes involved. Equation [7] now becomes COS 0 = COS 01 -- b2 X
(vm] kl/pl
[81
where p is the specific gravity o f the material and bz is a constant which depends on the geometry of the particles and on the arrange-
CONTACT ANGLE/MERCURY POROSIMETRY
33
.8
I
.4
.2
0
P
L ~ 3
_ _ 3 _
1
2
I
4
I
5
(Pore volume x Spec. Srau. ) , , 2 / 3
FIG. 8. Cosines of apparent contact angles versus correction factor f for LMP 172.056. m e n t of the particles in the clusters. Only in the case o f uniform particles in a regular array can b2 be calculated. For densest and loose packings of uniform spheres and of cylinders b2 varies between 0.12 and 0.33. In Table I the values for (V~. 0) 2/3 are given in column 5. In Fig. 8 the cosine o f the calculated contact angles (0) are plotted vs f = (Vm" p ) 2 / 3 . The regression equation for the best fit2 is cos 0 = - 0 . 3 5 9 - 0.146 X ( V ~ . o ) 2/3
[9]
with a coefficient of regression o f 0.997 and an s of 0.006. This means that a linear relation describes well the relation between the contact angles and the porosity of the samples. The extrapolated value for the contact angle of mercury on the solid is 111 °. Because the real contact angles plotted as function o f (Vm'O) 2/3 reaches the ultimate value (cos (180 °) = - 1 ) asymptotically the linear relation cannot hold for high porosity powders. This is illustrated in Fig. 9 where 2 In the calculation of the regressionequation the original powder has not been taken into account. In Figure 8 and 10 these points are marked with a X.
the Cassie-Baxter equation, our linear approximation, and the real values are given. F r o m this diagram it is clear that our empirical linear relation will only be valid whenever the porosity of the powder particles is not too high. This deviation from the linear relation can be seen in Fig. 10 where we have plotted another set of data. Also here the best fit is a linear relation between cos 0 and (Vm" p)2/3. Only the original powder itself with a porosity (ev) of 0.91 does not fit on the linear regression graph that can be drawn through the other data points. The regression equation fitting the data points (except for the original powder) is cos 0 = - 0 . 2 6 6 - 0.233(Vm" 0) 2/3
[10]
with a coefficient of regression of 0.98. The extrapolated value for the contact angle of mercury on the solid is 105 ° . The values for the extrapolated contact angles 111 and 105 ° are not unrealistic for this kind o f material (a metal covered with an oxide layer). However due to the assumptions discussed earlier and the uncertainty about the curve at low porosity (for Ev < 0.5 there are no data) these values must Journal of Colloid and Interface Science, Vol. 94, No. 1, July 1983
34
H. F. HUISMAN
i
.8 //
(.3
z!
I
.4
.2
D
I
_ _ J -
2
L
4
I
6
8
(Pore
Uolbme
I
I0 x Spec. Grau.),,2/3
FIG. 9. Diagram showing the relationship of cos Oversus f(= Vm. 0)2/3) for the Cassie-Baxter equation, our linear relation, and the real data.
not be taken too absolutely, but they are of the right order of magnitude. This makes it very likely Eq. [8] can be used in the porosity regime from 0 up to 0.8. This corresponds with a void surface area fraction between 0 and 0.6. In other words Eq. [8] can be used for cor-
recting the contact angle and the mechanism which is the basis of the Cassie equation is very likely to occur in mercury porosimetry.
The Modified Rootare-Prenzlow Equation Equation [8] also gives information about the dependence of the contact angle with the
J .8
3
/
v I
.4
,2
0
1
i
I
I
2
I
3 (Pore
4 Uolume
I
5 x Spec, Grav.),,2/3
FIG. 10. Cosines of apparent contact angles versus correction factor f for LMP 172.054. Journal of Colloid and Interface Science, Vol. 94~ No. 1, July 1983
35
CONTACT ANGLE/MERCURY POROSIMETRY
penetration of the mercury and it can be used to modify (5) Eq. [2] into A = -1 fo rm PdV aLV COS 0(V)
TABLE II1 Pore Radii r ~ (ore)
[11] Densification time (h)
Pore volume, V~ (m3/kg)
0 = 160°
0 = variable°
rhb (nm)
72.6 59.4 46.3 40.3 32.9 25.8 25.7 23.8 23.0
72.6 47.6 35.1 28.1 21.6 16.0 15.4 13.9 13.1
74.9 45.1 36.5 28.6 22.6 17.6 16.6 14.9 13.6
and replacing cos 0(V) by Eq. [8]
A -
-1 O'LV COS 01
{f0vmPdV + constant COS 01
X foVm ve/3pdV} . [12] The second term of Eq. [12] is always negative and this may be one of the reasons specific surface areas calculated from Hg porosimetry in which only the first term is taken into account are often too large (19, 20). If in general the Cassie mechanism is important in explaining the intrusion of mercury in powder/clusters or porous media, one may expect that for the retraction of mercury from these systems it also can be of importance. For the retraction experiment the equation becomes more complex because the composite surface can now consist of solid material (contact angle: OR(receding)), air (vacuum) (contact angle: 180°), and areas where Hg remains (contact angle: 0°). This situation always leads to a smaller contact angle for the retraction experiment than for the intrusion experiment. This mechanism can give an additional reason (21) for hysteresis and mercury trapping after retraction.
The Modified Washburn Equation In the same way that the Rootare-Prenzlow equation is modified for porous systems, the Washburn equation [1] can be adapted to these systems. This leads to re" = - 2
O'LV { --P-- tCOS 01 -
- [ Vm~2/3~ U2t I ~ P /
J" [13]
In Table III we have tabulated the rm.~ (the value of the pore radius whereby the maximum in the pore size distribution curve is
0 1 1.25 1.65 1.85 2.0 2.18 2.30 2.50
1.18 7.10 5.75 4.50 3.56 2.78 2.61 2.35 2.14
X X X X × X X X X
10 -3 10-4 10 4 10 -4 10 -4 10-4 10-4 10 -4 10 -4
" T h e values from Table I are used. b rh = 2 V m / A and A = 31,500 rn2/kg.
situated) calculated with a fixed contact angle of 160 ° and with a variable contact angle (see Table I), which is the same as calculated with Eq. [ 13]. Because the calculations are based on the assumption of a cylindrical pore model, a model independent radius (the "hydraulic radius") (6) is tabulated in column 5 of the same table. This "hydraulic radius" (r0 is calculated from the total pore volume obtained from mercury porosimetry, Vm, and the total surface area obtained from BET sorption isotherms, A(--31500 m2/kg) according to
rh = 2V/A.
[14]
The agreement between the corrected rmax and the "hydraulic radius" is remarkable. This indicates that the procedure followed for adapting the Rootare-Prenzlow and the Washburn equations to these porous systems is consistent. CONCLUSIONS
(1) The experiments make it very likely that the mechanism underlying the CassieBaxter equation is operative in the intrusion process of mercury in powders of the kind studied here (acicular magnetic powders). Journal of Colloid and Interface Science, Vol. 94, No. 1, July 1983
36
rL F. HUISMAN
However, there is a reason to believe that it is more generally applicable for mercury porosimetry as already discussed by Good and Mikhail (5). Especially for compacted powders or granulates (14) and for porous materials (22), it may be a very important mechanism. (2) By comparing Hg porosity and BET specific surface areas it is possible, in special cases, to find a semiempirical relation between contact angle and intrusion volume. This relation can be used to modify the Rootare-Prenzlow equation for the change in contact angle with penetration. (3) From this analysis it becomes clear why specific surface areas calculated from Hg porosimetry experiments are often too high compared with BET specific surface areas. With the same reasoning the pore diameters calculated from the intrusion curve with the Washburn equation are too small compared with pore diameters measured with other techniques (e.g., from permeability, or nitrogen capillary condensation) (23). (4) For practical application, the Rootare-Prenzlow equation can be used to calculate the specific surface from Hg intrusion porosimetry with reasonable accuracy if (a) The contact angle is an average contact angle in which the influence of porosity and/or roughness is taken into account. (b) All surface areas are counted in the Hg porosimetry experiment. (In practice pore diameters must be larger than about 10 nm.) (5) The Washburn equation with the correct contact angle gives for these systems the same average pore radius as the one calculated from the total pore volume and the total specific surface area from BET experiments.
Journalof Colloidand lnterJaceScience.Vol.94, No. 1, July 1983
ACKNOWLEDGMENTS I wish to thank PD magnetics management for permission to publish this paper and Mr. T de Laat from Philips Central Research for the BET measurements. REFERENCES 1. Rootare, H. M., and Prenzlow, C. F., J. Phys. Chem. 71, 2733 (1967). 2. Huisman, H. F., Rasenberg, C. J. F. M., and v. Winsum, J. A., to be published in Powder Technol.
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