The contact angle in mercury intrusion porosimetry

Powder Technology. 29 (1961) 53 - 62 0 Elsevier Sequoia S.A., Iausanne - Printed in The Netherlands

The Contact

Angle

in Mercury

Intrusion

53

Porosimetry

ROBERT J. GOOD Dcpartmcnt

ofChemical

Engineering,

State University of New

I-ark at Buffalo.

Amherst.

NY 11260

(U.S.A.)

and RAOUF SH. MIKHAIL Department

of Chemistry.

Faculty

of Sciences,

Ain Shams iJnir,crsity, Cairo (Egypt)

SUMMARY The general theory of the contact angle of mercury on complex. internal surfaces of solids, i.e. within pores that cannot be approximated as circular cylinders, is examined_ A model for such pore surfaces, in which the pores are represented as cylinders with surfaces that are very rough and that contain void spaces corresponding to the entrances of bmnch pores, is developed_ This model enables us to conclude that in the Washburn equation, for the pressure required to force mercury into a pore of radius r. the contact angle that should be used is 180”, particularly for macropores and mesopores in most practical solids_

examination of the mercury intrusion technique that the complex topography is, for the purpose of the experiment, equivalent to roughness and the presence of void areas in internal surfaces whose more gross configuration is simple. This will enable us to use eqn. (1) with a corrected contact angle that is appropriate to the rough surface, to deduce an equivalent radius, r. The distribution of these equivalent radii will then serve to characterize the solid or the compacted powder. The general equation for the pressure drop across a curved liquid-vapor interface, such as exists when mercury is driven into a porous body, is the Laplace equation: 9P=yLv

l_ INTRODUCTION A question that has plagued mercury intrusion porosimetry, ever since the technique was first proposed [ 11, is what value to use for the mercury solid angle. The familiar equation (sometimes attributed to Washburn) for the pressure drop AP across the surface of a liquid in a cyiindrical capillary tube whose radius is r, is [ 1 - 31 AP=

2yLV cos r

e

(1)

where yL,, is the surface tension of the liquid and 8 is the contact angle. The radius of curvature of the meniscus (assuming gravitational effects to be negligible) is r/cos 0. It is, of course, very rare that pores in practical solids or compacted powders are even approximately cylindrical, or even that a circular cross-section exists at any location in the material. It will be the thesis of this

1 -RI

(

+-

1

R2 > where (l/R, + l/R,) is the mean curvature at any point on the curved surface. The radii, R, and Rg, are functions of the geometry of the space in which the curved surface exists, and also of the contact angle around the contact line formed by the liquid at the solid surface. It is commonly assumed that the contact angle is necessarily a constant at all points on the perimeter, provided only that the solid surface is chemically homogeneous and isotropic. But this principle would, in some cases, conflict with the overriding principle of constancy of hydrostatic pressure, and hence of the mean curvature over all the liquid surface. When such a conflict exists, then eqn. (1) must be modified by the inclusion of a two-dimensional pressure +rrm, ~SLV = YSLVIRSLV

(3)

where rsLV is the line tension and RsLV is the local radius of the three-phase line [4, 5]_ At the three-phase line, the sum of the force due

to the Laplace pressure and the two-dimensional pressure, zSLx-, must be equal to the value of AP that esists far from the solid, because otherwise a pressure gradient would esist locally within the liquid. The line tension is generally small [6], and so the two-dimensional pressure makes a negligible contribution to capillary behavior in macropores. In micropores, line tension might be espected to have a large influence on capillary behavior, if the local void space does not have asial symmetry and constant perimeter. If a homogeneous solid surface is not a plane, or if the void space is not bounded by a surface of revolution such that the contact line of a drop of liquid would be circular, the deviation of the line from circularity leads to the local conflict between the Laplace equation and the condition of constancy of the contact angle. (This particular aspect of the mathematics of liquid surfaces has not received a complete analysis as yet. Reference [5] gives a qualitative treatment of a related problem for heterogeneous, smooth surfaces_) For mesopores, it would be espected that line tension effects would be minimal for voids in the neighborhood of 300 .X diameter, and that for voids at the small end of a mesopore range (depending, of course, on geometry) line tension effects could be appreciable_ Parenthetically, an open question that needs to be dealt with is the value that should be used for the surface tension of mercury. Uncertainty exists because of (a) impurities in the mercury; (b) adsorption of components fiorn the solid via the gas phase, e-g_ water desorbed from the solid; (c) for micropores, the depression of surface tension that Tolman has predicted [7] as occurring when the radius of curvature is very small. Now: (a) Impurities in the best instrument-grade mercury that is currently available should be sufficiently dilute that they would not diffuse to the surface (which is continuously espanding) at an appreciable rate, and so this effect can probably be ignored. (b) Impurities released by desorption from the solid would manifest themselves in a dependence of apparent pore distribution on out-gassing time. Two effects may occur: a decrease in the surface tension of mercury and a change in the contact angle. -An example of ‘he latter will be discussed in the next section. (c) The Tolman effect is

hard to estimate with any generality, beyond the fact that it can be expected to manifest itself only at the very smallest pore sizes, e-g_ pore diameters corresponding to about five atom diameters_ For this range of pore sizes, an uncertainty also arises (for the same reason) in gas adsorption studies of pore distribution_ See the last section of this paper. In view of the multiple uncertainties cited above, we can only recommend that Kernball’s value of 7 [S] , i.e. 485 dyn/cm, should be employed, in routine studies. In non-routine studies there is generally a need to examine the possibility that gases desorbed from the solid may cause a decrease in yLv _

2. CONTACT ANGLE DATA Good and Pascheck [9] have recently reported that the contact angle of mercury on fused silica varies with the water content of the gas phase. For relative humidity increasing from 20 to lOO%, the advancing contact angle, k, increased from 135O to 138”, and the retreating angle, 0,. increased from 119.5” to 121”. (The advancing angle is the angle observed when the liquid front is advancing very slowly, or has just been caused to advance; the retreating angle is the angle that is observed when the liquid front is retreating slowly, or has just been caused to retreat. The hysteresis, N, is equal to 0, - O,.) A much more serious variation occurred when silica was outgassed at 750° under vacuum: 0, was observed to be 105” -and 0, was 90 - 95”. Similar changes were obtained with Pyrex glass. The reason why the contact angle is lower after out-gassing is a change in ysv , the solid/ vapor surface free energy, and/or ysL, the solid/liquid interfacial free energy. This may be seen from Young’s equation: cos f3 = (Ysv

_YSL)/YLV

(4)

Outgassing glass or silica, to remove adsorbed water, will increase both ysv and ysL. though at present it is impossible to predict a priori which will increase the more. The results in ref. [9] show that ysv and 7sL must be nearly equal for mercury on fully dehydrated silica and Pyrex glass. The variation of yLv with water content of the vapor was also discussed in ref _ [ 9]_

In very recent work at Buffalo found that intensive outgassing

[lo] it was of porous

Vycor, even in the temperature range where sintering occurs, had little effect on the apparent pore distribution, when the outgassed solid was compared with solid that had been equilibrated with water vapor before pumping on it in the porosimeter cell. This could mean

that the surface became rehydrated very rapidly during transfer from the vacuum oven to the porosimeter. An alternative esplana-

tion, which we will develop below, is that because of roughness, the effective contact angle is 180” for the large majority of systems. Good and Koo [ll] have recently observed apparent contact angles of mercury on the surface of cement paste (which is always microscopically rough and porous even when prepared by molding against a smooth solid) of 17C - 175O. These angles were scarcely distinguishable from 180° ; and since cos 170” is -0.9848, cos 0 was within 1.5% or less of

cos 180”.

Oliver et al. [12j

have reported

SEM photographs of mercury grooved nitrocellulose surfaces,

greatly elevated

apparent

drops on which showed

contact

angles

relative to the macroscopic envelope of the ridge-tops_ Roughness may cause a retreating contact angle to be lower than the equilibrium angle on a flat surface [ 131, particularly if the latter is less than 90” ; cf the behavior of water on sandblasted glass, on which the retreating angle is zero. In ref. 123 it is stated that contact angles ranging ikom 112” to 142” have been observed for mercury on various solids. The commonest reported angles are between 130” and 140”. Dumore and Schols [14] have recommended 180” for porous materials such as sandstone, but they do not explain the basis for their conclusion. Morrow et al. [15, IS] have measured the capillary rise in roughened, cylindrical capillaries. For a solid on which the ‘intrinsic’ contact angle was believed to be 130”, they found an effective advancing angle of about 175” and a retreating angle close to 125”. In a current bulletin on porosimetry [ 171, it is recommended that “the k-ue value of the contact angle should be employed when it is known to be other than 130”“. It was not :;tated how this measurement is to be made.. and it has been pointed out recently [18] that the methods that have

been developed for measuring contact angles on Bat, estemal surfaces are in general inappropriate for internal surfaces_ It has long been known that roughness must have a major effect on the contact angle [13, 18 - 251. The various theoretical approaches that have been made yield the same qualitative result, which is in agreement with experiment: the contact angle of mercury on a rough or porous surface will be greater than that on a microscopically smooth surface having the same composition. Three ‘equilibrium’ theories of the roughness effect are those of Wenzel [ 201, Shuttleworth and Bailey [ 221 and Cassie [ 24, 25]_ The Young-Wenzel eqllation is G(Ysv

_YSL)

= YLV

cos

o\v

where the roughness factor, 6, is the ratio of the actual area of the ‘apparent’ area_ If n? = 1, eqn. (5) is Young’s equation, and 0, = 0: cos O,/cos

0 = 6:

(6)

To illustrate eqns. (5) and (6), if 0 is, say, 135O, then a roughness factor of 1.30 leads to a predicted effective angle Olc- = 157” ; and if bc is above 1.42 (which is not a very large value), 01%-would be 180” _ Shuttleworth and Bailey [ 221 employed a model surface of sawtooth ridges parallel to the liquid front, with slope angle Q relative to the macroscopic ‘envelope’ of the surface. They pointed out that for this model structure, the advancing angle, Oa , is equal to 0 f o, and the retreating angle, 0 =, is 0 r = 0 - o _ The hysteresis, H, is equal to 20. The Wenzel equation, on the other hand, ignored hysteresis_ Hence, the Wenzel treatment gives the minimum effect of roughness on contact angle. However, because of the incompatible assumptions of the two treatments, one cannot simply add the predictions of the two treatments together_ More recent theoretical discussions of hysteresis (13, 19, 21, 231 show that, for more general configurations than parallel, sawtooth ridges, the advancing angle will always be greater than that predicted by eqn. (5). (Only for roughness in this form of perfect, parallel grooves or ridges, lying parallel to the direction of motion of the liquid front, will roughness not contribute to hysteresis_) For general, rough surfaces, or for general, heterogeneous surfaces, energy

56

barriers exist between successive locations of the three-phase line_ These are caused by the local contortion of the liquid-vapor surface_ In passing from one stationary location to another, the liquid-vapor surface must pass through configurations in which the liquidvapor area is increased; this requires activation energy, E * = yLVIIAfL,, _ (If line tension is important, there will be an additional term, where L is length of the three-phase 7sL\~lL*, iine.) This energy of activation can be furnished from thermal energy in the system, or by vibrations that have non-thermal sources in the laboratory_ If we turn to the esperimental measurement of the ‘true’ angle that should be used in the pressure equation, we find that very serious difficulties arise. Most fundamentally, measurements on a macroscopic surface of a solid will be relevant only if the external or macroscopic surface and the internal surfaces have identical roughness_ fractional void and chemical composition. For space, example, measurement in which a powder is compacted and the estemal surface of the aggregate is employed will be misleading [see ISI Rootare [ 26,273 has suggested a method that has some promise, and which in addition casts some light on the theoretical problem of measurement_ Rootare and Prenzlow [26] pointed out that the energy, PdV, that must be put into the system when an increment of mercury volume, dV, is forced into the pore space is converted into surface work equal to Here, dA’ is the area of solid (%v - ysL)b4’_ that is covered when the increment dV of volume is introduced. Then, PdV = -(ysrr

-

ysL)dA’

Young’s equation shows cient of 64’ is equal to yLv

that the coefficos 8. Hence,

PdV

d4’=YLV

Integrating, obtained A’=--

(7)

CO.sB

Root-are

and

Prenzlow

PdV

where V, is the maximum penetration volume_ If V, isnot appreciably greater than V,., the iimit accessible with the inslxurnent

employed, and if we can assume A' to be equal to the area, ABET, determined by gas adsorption, we can write:

cos

0 =

-1 YLVABET

VI-II PdV

_I-

(10)

o

The assumptions that have just been made will be valid with respect to measurements using an instrument whose upper limit is 50 000 psi or 7250 kpa, provided the volume and area of pores smaller than about 40 A diameter are negligible_ Rootare 1271 has estimated contact angles on a number of solids; his results ranged from 115” to 160”. His value for porous Vycor was 153”, which is different from the results of Good and Pascheck for silica and Pyres glass, mentioned above. (The surface of porous Vycor is thought to resemble that of Pyres (283, because of residual boron that remains after the etching step that makes the material porous_) There are three criticisms that can be made in regard to eqn. (10). The first is one that we have alluded to above, and which Rootare mentions explicitly. Practical mercury porosimetry is limited to the study of pores larger than about 40 A diameter [17] _ The volume of pores smaller than this size is usually small relative to the volume in the larger ranges of pore size, so V,- = V, . It is not so generally true that A' = A=T, particularly for high-area solids. If there is a large area in the internal surfaces of the pores that are smaller than the 40 A limit, then the value of cos 19 computed by eqn. (10) will be too small. The second is that, for a rough solid, the relation between (ysv - ysL) and 0 is not the Young equation, but the Young-Wenzel equation; see above_ A third defect of eqns. (9) and (10) is that they employ the assumption that the contact angle is a constant over the whole range of pore sizes. This assumption will not in general be strictly valid, even when the surface composition is uniform_ As will be shown below, in a porous solid or powder containing pores that have anything but the narrowest distribution of sizes, the contact angle appropriate to the larger pores wiu, in general, be considerably greater than the angle appropriate to the smallest pores.

If the contact angle varies with penetration volume, V, and correspondingly, with area A’ already covered with mercury, i.e. if 8 = O(V), then Rootare’s equation may be modified:

YLVA’ = ---

/ 0

vm PdV -

(11)

cos e

(12) The differential

form,

eqn. (12),

is not direct-

ly useful, because there is at present no esperiment by which dV/dA’ can be measured. We may note in passing that the Rootare and F’renzlow approach should not be applied to the retraction curve obtaiued in mercury porosimetry. Structural hysteresis, in which mercury is left behind in the porous body by detachment of mercury from the retreating mass, as the pressure is reduced, invalidates the assumptions of the theory. Indeed, this effect renders any employment of retraction data in porosimetry dubious 129, 301.

3_ GEOMETRY

OF

REAL

CONTACT

ANGLES

COMPLEX

GEOMETRY

PORES:

ON SURFACES

MERCURY WITH

It has already been remarked that pores in real solids only rarely are circular cylinders, so eqn_ (1) is rarely exact For elliptical cylinders,

(13) where 2r, and 2rz are the minor and major axes of the ellipse. For slit pores, r2 is infinity, and 9P = yLv cos e/r,

= 2yLv

cos

e/d

(14)

the observed ilP will lead to too large an estimate of r, _ ‘Ink-bottle’ pores have been frequently discussed in the past, and the existence of small pores extending off the sides of large pores is commonly recognized_ The curve of mercury instrusion volume uerslcs pressure may be used to deduce an apparent pore size distribution, but any apparent distribution may be achieved in a number of different ways. For example, if the pores are wedge-shaped slits (for which eqn. (14) is appropriate), the P

uersus V curve may be indistinguishable that for a pore system in which smaller

from pores

branch off larger, cylindrical ones. A similar ambiguity esists in regard to the interpretation of gas adsorption data for porous solids. It is clear that the use of a single parameter, r, that arises from an unrealistic model (pores as circular cylinders) is responsible for many of the shortcomings of mercury porosimetry. Two approaches can be employed to remedy this defect. One is to employ more parameters, e.g. two radii, as in eqn. (13). This remedy, if carried to completion, would require the introduction of at least two or three more geometric parameters in addition to one of the principal radii and the ratio, rl/rl_ Such an approach would almost certainly lead to a ‘working’ theory that would be too complex to be very useful. The additional parameters could be determined only by means of independent measurements, and not on the basis of the mercury intrusion data per se. A better remedy, which we will now develop, is to improve the model in a simple and rather realistic way. Electron micrographs of pores in solids generally show the interior surfaces to be exceedingly rough and irregular_ For example, Fig. 1 shows two recently published SEM photographs by J. W. Neasham 131, 321, and drawings that give his structural interpretations_ Quite obviously, the interior surface of these pores, particularly those with ‘porelining’ chlorite, is covered with a structure that could be called a fuzz, and the interior

where d is the width of the slit_ For a more complex case, consider a ‘capillary’ pore formed by a triangular cluster of three parallel cylinders or spheres that are not necessarily in contact, and let r, be the radius of the largest cylinder than can be inscribed in the space within the cluster. The mean radius of curvature at the apex of the meniscus will be larger than r,/cos 8, and AP will be smaller than ZyLv cos e/r,. If eqn. (1)

Scholten et al. have pu’Aished electron microscope investigations of chrysotile 133 353 _ They report that, in the interior of the primary fibers of this mineral, cylindrical channels 80 A in diameter are present. The

is used in connection

rarity

with such a structure_

surface

is not chemically

of obserriations

homogeneous.

such as this serves to

SAND

(b)

(a)

Fig. 1. Two types OF pore morphology of clnb- in sandstone_ Electron (a) ‘Discrete particle’ kaolinite, (b) ‘pore-lining’ chlorite.

‘prove the rule’ of the generalization that the interior surfaces of pores are rough, and topologically complex. It is clear that the structure of the interior surfaces in porous solids will, in generaI, be such that roughness and the void spaces in these surfaces will dominate the contact angle behavior of mercury inside the pores. We may implement this conclusion by knodelling’ porous solids as containing cylindrical pores that have rough internal surfaces, and that there are void seas in those internal surfaces. The Cassie equation [ 241, for a surface in which void areas are presenti, is* COSO =uz

GRGfN

case,

-cl2

(15)

*Ref_ [IS] in which the Cassie equation is discussed contains two misprints. In eqqnr (29) and (30). p_ 275, the last terms on the right should be negative_

micrographs

of J_ W. Neasham

1313 _

where u1 and u2 are the fractional areas of the surface of type 1 solid and of the voids. (This equation can easily be generalized to chemically heterogenous solids.) Since u1 + tJ2 = 1, cos 0’ = Ol(COS 8 + 1) -

1

(16)

If, say, crl = 0.5 and 0 = 135”, 8’ will be 140°. Two further modifications of eqn. (16) are needed. The first arises because, while the type 1 solid may itself be smooth, it wiIl more commonly have a Wenzel roughness ratio that is greater than unity. Then cos 0 in eqn. (16) must be multiplied by A. Alternatively, if we take into account the slope angle 9, then 0 must be replaced by (0 + 0) in eqn. (16). As already mentioned, the Wenzel theory and the Shuttleworth theory cannot be used simultaneously in a rigorous theory_

The

second

modification

arises

because

to a flat surface with voids in it, if the liquid surface across all pore mouths is flat. This can be true if the mouth of every pore or branch pore (void) has a sharp edge in the plane of the surface that is external to it, and if the eqn.

(16)

will

hydrostatic

only

be valid,

pressure

on the

with

liquid

respect

is zero.

The

increment of energy due to liquid-vapor area, associated with covering the mouth of a branch pore whose mouth area is A, (with equivalent radius, r,) will be yLVAm, if the mouth-covering liquid surface is flat. If the hydrostatic pressure is greater than zero, the liquid surface across the pore mouth will be conves, and the increment of liquidvapor area will be larger than A, by a factor, a, lying between 1 and 2; it will be 2 if the surface of the intruding liquid is hemispherical. If the mercury front is advancing into a cylindrical pore whose radius is ri (see Fig. 2) in 2 region where a side-pore with radius r,,, exists, then the radius of curvature in the sidepore will be the same as that in the main pore. If, say, rm < O.lri, Q will be close to 1. If, then, the liquid is forced into main-line pores with radius, say, ri = 0.2ri, the liquid surface radius within the side-pore will be one-fifth as large, and Q will be appreciably greater than l_ Clearly, Q will be a function of the ratio of the size of the side-pore to that of the main pore; the larger r,,,/r, the greater Q will be. Putting this argument quantitatively, we write cos 0’ = 67 cos 0, -

u~(Y(P. r)

The condition 0’ to be 180”

must

for

0 = -1

cos

if ul(ti

that is

If LY= 1, this condition l5?= -l/cos independent tion

on ol,

a,

(17)

be met

cos t?, + a) -[I

in order

< -1

(18)

reduces to (1%

of crl _ The corresponding condiratio is not unity,

if the roughness

is I31 =

a-1 Q COST,

(a)

(20) +-Q

If, say, 61 = 1.30, u1 = 0.50, 8 = 135” and Q = 1, then employing eqn. (17), 0’ will be 164”. If Q = l-1,0’ wiu be 180”. Clearly, the contact angle 8’ will vary with the roughness factor, and it will also be strongly dependent

(b) Fig. 2. Effect of ratio main pore. (a) Radius

mouth

of side-pore

of size of side-pore to that of of curvature, R,, of liquid in

(whose

radius is r,)

is same as

that in large pore, where radius is ri; r,.,, =ri/lO. Surface of liquid across main pore is nearly flat, and Q = 1. (b) Radius of curvature of liquid in side-pore mouth is same as in smaller region of main pore; r,,., =rj/3_ Surface of liquid in side-pore is appreciably nonplanar, and Q > 1.

60

on the fractional area of pore mouths, and the ratio of side-pore diameter to main-pore diameter_ Equations such as these can be developed to estimate the equilibrium angle for a variety of model systems_ It turns out that only a modest degree of roughness and fractional void area are needed to produce an effective 1 SO’ contact angle. But non-equilibrium ccnsiderations lead to even stronger predictions of a 160’ contact angle, in real systems. The the05 of Eick eC al. [ 131 indicates that, even with a 1’er-y slow rate of penetration, the advancing angle for mercury on a rough surface will always be appreciably greater than given by eclns. (6) or (1’7). This conclusion is in agreement with the results of Morrow ec al. mentioned above_ These arguments lead to the conciusion that the interior of a pore does not need to be veq rough, arci there need not be a great deal of area that -iould correspond to branch pores, for the effective contact angle to be close to lSO’_ The aduarzcing mercury contact lS0” under angle, 0,. will be effectively conditions of mean slope roughness much less than 40’ _ (A family of sawtooth ridges with mean slope 409 has a roughness factor l-3_) ?Vith surfaces more comples than sawtooth ridges, the prediction of a 130” contact angle is obtained at lesser rougbnesses than 1.3, Le. with internal surfaces that could be regarded as rather smooth. We can now make a recommendation as to the angle that should be used in eqn. (1): 1SO’. This recommendation as to 0 applies to the surfaces of most macropores and mesopores. In solids with very well-developed. fiat faces, such as mica, kaolinite and montmorillonite, mesopores and even macropores could have faces such that the contact angle would not be increased appreciably on account of roughness and side-pore mouths. But if a solid is known in advance to contain slit-shaped pores, the cylindrical approrimation, eqn. (I), is clearly not applicable anyway. The pressure-pore size relation should be eqn_ (14)_ In general, when the pore size range is reached where the surfaces are those of primary crystallites, the contact angle should be treated as that on a flat, void-free surface, as just discussed with respect to micaceous

solids. If micropores are present in a solid, and if the porosimeter is capable of forcing mercury into such pores, a contact angle of I35 - 140” will be appropriate, rather than 180” _ Just how far this applies above the 40 _A limit for micropores cannot be estimated in any general way, since it will depend strongly on the form and degree of perfection of the primary crystallites. (However, see below, Section 4, in regard to some recent studies of chrysotile-)

4. APPLICATION

Scholten et al. 133 - 353 have investigated the pore distribution for four materials, using three different methods: mercury intrusion, nitrogen adsorption, and electron microscopy. Reference 1343 can be used as a test for the theories developed above. In ref. [ 341, these authors pointed out that when gas adsorption is used to determine pore distribution, a modified form of the Kelvin equation must be employed, which takes into account the ‘tan der Waals forces emanating from the pore walls”_ The correction increases with decreasing pore size; for diameters around 100 A, it is about +23X. When this correction was employed 134, 353, the distribution ma-sima were found to lie significantly above the masima obtained by mercury intrusion. This was true of compressed Aerosil powder, of an iron oxidechromium oxide catalyst, and of synthetic chrysotile powder, Mg,(OH)a - SizOs. In the case of chrysotile, the distribution was bimodal, and both peaks were shifted. The peak at the smaller diameter was observed at 80 A both by electron microscopy and by corrected nitrogen adsorption, and at about 50 A by mercury intrusion. De Wit and Scholten 1353 state that “Intioduction of 0 = 180” into eqn_ (1) would bring our results into somewhat better, though not full, agreement with the EM and nitrogen results, but such a contact angle has never been found for mercury on silicates, and hence must be regarded as very unlikely”. Our present conclusion, that 180” is the proper angle to use cn account of roughness and branching, thus has empirical justification.

61

Scholten’s EM work also points to certain limitations on the generalization of using 0 = 180” (cf. above, Section 3). These authors observed, by electron microscopy, that for chrysotile the primary particles were hollow, needle-shaped crystallites with uniform internal channels_ On such internal surfaces, the contact angle should not be raised on account of roughness and void area, so a 135 - 140” angle must be used, and another explanation must be sought If pores are truly circular cylinders, line tension should not influence the contact angle, because no work is done by or against the line tension, when the liquid front advances_ The length of the circular threephase line will remain constant. The suggestion of de Wit and Scholten, involving the Tolman radius-of-curvature effect, is implausible because of the pore dimensions. It would be expected that the 30 - 40% decrease in yLv that would be needed to explain the shift in the peak of the distribution would be observed only when the radius of curvature was far less than 40 X. Another suggestion appears more reasonable_ White [ 361 has shown that the liquidvapor interface can be espected to approach the solid surface at a different local angle from that formed by the tangent to the macroscopically observable liquid surface. The departzze from the macroscopic trend will occur at a distance only a few (one, two, three) molecular diameters from the solid surface (see Fig. 3). For a low macroscopic angle, i.e. less than SO”, the microscopic angle in micropores would be smaller than the angle, 0, that appears in Young’s equation, and for a macroscopic angle greater than 90” (as with mercury) the microscopic angle in the smallest pores should be greater than the macroscopic angle. The reason for this variation is the same van der Waals forces due to the solid, that were involved in correcting the Kelvin equation; see above_ The corrections reported in ref. 1351 for nitrogen adsorption indicate that the 30 - 40% anomaly with mercury might be accounted for in this way. A final alternative is, to consider the apparent 180” contact angle in the internal channels of the crysotile fibers as an empirical fact, regardless of mechanism. This conclusion would indicate that the restriction indicated

(b) theory of microscopic Fig. 3. White’s (a) Macroscopic angle, f? < 90" ; (b) angle. 0 = 135'.

contact angles. macroscopic

in Section 3, that for micropores a contact angle of 130 - 140” should be employed, may be correct. Hence we must leave the question of the contact angle in micropores open, for the present. Fortunately for mercury porosimetry, this question will arise only very rarely, because of the general inaccessibility of the micropore range, which was noted above. Finally, we must treat the results that de Wit and Scholten obtained with ZrOz powder [ 351: they found the mercury intrusion determination of pore distribution to be in satisfactory agreement with the corrected nitrogen adsorption results. They remark, “This might be due to the fact that we are dealing here with a very loosely packed system in which the dimensions of entrances of cavities are practically equal to the dimensions of the cavities themselves”_ Compare above, Section 3. Very probably, the ZrOe powder bed was in a condition of packing that was at the outer border of the region where description as a “porous body” is appropriate_ The electron micrograph shown in ref. [ 351 is consistent with this view. Now, for isolated powder particles that lack internal pores, characterization by “pore radii” is certainly inappropriate_ If we consider the continuum variation of packing, from a tightly compacted aggregate,

62 to a loosely

packed system, to a very loosely packed powder in which the particles are effectively isolated, we can see that the Zel powder could, by coincidence, have been in the region where mercury porosimetry with 0 assumed to be 130 - 140” would yield approximately the same distribution as obtained by corrected Nz adsorption. It has been evident for some time that there is a need for a tabulation of the values of contact angles of mercury on various solids that is more estensive than tables that are currently available_ One of us (R. J. G.) has undertaken this; it was intended to include it as an Appendis to the present paper_ It has not been possible to complete this survey in time for the deadlines in this special issue_ We have concluded that a separate paper k this journal would be appropriate_ It is planned that the paper would contain some evaluative comments, which are needed because some reported values seem (on the basis of methods used) to be more appropriate than others, with respect to mercury porosimetry.

XCKNOWLEDGENENTS This work vas supported by the Xational Science Foundation under Grants INT 76189’77 and IXT 77-24340_ The espansion of this paper, as suggested by Dr. van Brakel, was conducted while the senior author was Visiting Professor at the Department of Chemical Engineering and Chemical Technology, Imperial College, London.

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