Volume and contact angle hystereses in mercury porosimetry of ASC-whetlerite

Volume and Contact Angle Hystereses in Mercury Porosimetry of ASC-whetlerite J. K L O U B E K The d. Heyrovsky Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, 121 38 Prague 2, Czechoslovakia Received July 28, 1981; accepted J a n u a r y 26, 1983 O n a sample of ASC-whetlerite, local hysteresis in mercury porosimetry both for reintrusion and extrusion courses was investigated. In this way, some important characteristics o f the porous body were evaluated: contact angles a n d their hysteresis, adhesion tension a n d its intermediate values, differences o f interfacial energies, a n d line tension.

3~R COS OR = 7~ -- %L -- "r/r,

INTRODUCTION

[2]

where OR is the equilibrium receding contact angle, "r~ and ~'~L are surface and interfacial energies, respectively, r is the line tension (10), and r is the radius of the respective pore. The expression 3's - 7SL for intrusion deviates from 3'~ - "r~t for extrusion. The difference may arise from changes of the density and ordering of molecules in the transient region of the three-phase interface which is influenced by the preceding position of the meniscus (8). Similarly, the value of r may be changed for the same reasons. Some authors (9, 11) stated that T depends on r. However, such dependence has not been found in porosimetry and, therefore, is neglected in Eq. [2]. A special type of volume hysteresis has been observed in active carbon and is called local hysteresis (12). It sets in when the effective receding angle 0{~ = 90 °. Then, the three-phase interface in the pores does not ~/R COS 0 A = "Ys -- "YSL, [l ] move during reintrusion from zero pressure where '~R is the surface tension of mercury and 0 = 90 ° up to the pressurepz correspondcorrected for the radius of curvature R, 0A ing to 0A (PI being the m a x i m u m pressure is the equilibrium advancing contact angle, reached in the first intrusion run from which "Ys is the surface energy of the solid, and 3'SL the extrusion started). However, the volume is the interfacial energy between the solid and of mercury in the sample increases owing to mercury. On the other hand, the following the influence of pressure p on the radius of equation conforms to extrusion, curvature of the meniscus R. The depenVolume hysteresis in mercury porosimetry has long been known (1). Freundlich (2) first suggested that the difference between mercury volumes inside a porous body at a given pressure for intrusion and extrusion runs may be caused by contact angle hysteresis. Nevertheless, "ink bottle" pores were considered as the only cause of the volume hysteresis (3-5) until Lowell raised his objections (6). Experiments of Liabastre and Orr (7) with porous glass of controlled structure made it possible to determine quantitatively the contact angle hysteresis of mercury in pores along with the line tension (8). The latter quantity may be considered as a correction in Young's equation for the transient region at the three-phase interface (9). Its value was found to be negligible for intrusion (8) so that the following equation is valid.

135

Journal of Colloid and InterfaceScience, Vol. 95, No. 1, September 1983

0021-9797/83 $3.00 Copyright © 1983 by AcademicPress, Inc. All rightsof reproduction in any form reserved.

136

J. KLOUBEK

dence of the a m o u n t of m e r c u r y reintruded in this way, V~, on p from zero to p~ makes it possible to evaluate 0{, which approximates the average contact angle at p~ for all accessible entrances of e m p t y pores with various radii r. The m a x i m u m value of O{ f r o m a series of m e a s u r e m e n t s referring to various values ofp~ was taken for 0g. Such measurements of reintrusion were performed with a sample of active carbon (12) and ASC-whetlerite (13). In the present paper, attention is paid to the intermediate adhesion tension of reintrusion and the process of extrusion is examined with a sample of ASC-whetlerite. PROCEDURE

The experimental details were given earlier (13). Two examples of extrusion and reintrusion dependence of Vm (mm3/g) on p (MPa) with a sample o f ASC-whetlerite are shown in Fig. I. The hysteresis loops are closed at a pressure higher than zero because the pressure p could be lowered only to the atmospheric pressure enhanced by the colu m n o f mercury in the dilatometer burette. Volumes of local hysteresis are given as V~x = V I - V0 and V~ = V - Vo, where Vt is the volume of mercury intruded in 1 g of the sample at a pressure p~ in the first intrusion run, V is the same during extrusion or reintrusion at the respective pressure p < p~, V0 is the volume obtained by extrapolation o f the reintrusion curve to p = 0. The lowest applied pressure was about 0.16 MPa. After the first run o f intrusion, which is not d e a r with in this paper, extrusion and reintrusion followed. The experimental points of reintrusion, i.e., the dependence of Vm on p, were fitted by the curve calculated for -f = 480 m N / m on the basis of a cylindrical pore model with the use o f the following iteration procedure (12). The meniscus radius R~ at a pressure p~ is given by R~ = (23,/p~) - 0.533. [3] Here, a correction of the mercury surface tenJournal of Colloid and Interface Science,

VoL 95, No. 1, September1983

0,5

P

2O

/z,,"°"

',"f,;

9 /

7 "

S/l~ o

1

p

2

FIG. 1. Hysteresis volume, Vm (mm3/g), in dependence on pressure, p (MPa), for reintrusion (open circles) and extrusion (full circles). Initial points of extrusion are given by half full circles which belong to reintrusion, too. Two typical examples are shown only which are denoted by numbers of measurements in accordance with the tables.

sion -y for its dependence on R is involved (8). The value of ~ (in m N / m ) refers to the plane surface, p is given in M P a a n d R in nm. Then, f r o m an arbitrarily chosen value of 0i~ we calculate r + = - R ~ cos 0i~

[4]

and Vd = (2 - 3 sin

O{ + sin 3 0{)7rR3/3,

[51

where r + is the average radius of accessible pores and Vc~ is the respective volume of a single meniscus cap at the assumed average contact angle O{ relating to the range of pores in question. The effective n u m b e r o f pores n + is given by n + = Vmi/Vei.

[6]

The end points of the curve are Vmi = n + V~I at p = PI and Vm = n + V~ = 0 at p = 0. Other points for Pl > P > 0 are obtained by using the above values of r + and n + in the following equations, analogous to [3]-[6]: R = (23,/p) - 0.533 - c o s 0+ = Vc = (2 - 3 sin 0+ +

r+/R sin 3 0+)7rR3]3

[7] [8] [9]

137

HYSTERESES IN M E R C U R Y P O R O S I M E T R Y

Vm : n+Vc.

[101

The calculated dependence of Vm on p was compared with the measured one and the procedure was repeated for various O{ values to obtain the best fit. Similarly, the dependence of Vc/Vc~ on P/PI m a y be used to determine the correct value o f 0{ by comparing the calculated and measured values of

vm/ v.~. The results with the present sample were previously published (13). In the example shown in Fig. l, the values o f 0~ are 142 and 132 ° for curves 2 and 9, respectively, while 0A = 154° (cf. Table I). Our new data are concerned with extrusion and they are fitted in Fig. 1 by a smoothed curve. R E S U L T S A N D DISCUSSION

Reintrusion The familiar equation for evaluation o f intrusion into cylindrical pores is

1~r/2 = --3"R COS OA,

[11]

where 3"R is the surface tension o f mercury with a radius o f curvature R. It may be assumed that the same relation is valid for the intermediate points o f reintrusion, i.e., TABLE 1 Adhesion Tension of Reintrusion of Mercury into Pores o f ASC-whetlerite a Measurement number

Pl (MPa)

0+

1 2 3 4 5 6 7 8 9 10 11

0.462 0.657 0.753 0.948 1.141 1.337 1,630 1,826 2.021 4,943 13.885

140 142 154 150 153 148 136 136 132 113 145

-~/R cos 0A A ( r a N / m ) (raN/m)

S^ (naN/m)

431.31 431.27 431.24 431.19 431.15 431.10 431.03 430.98 430.94 430.24 428.10

6.36 16.35 6.57 13.87 5.03 13.28 14.66 8.88 8.57 7.70 18.25

431.70 432.43 427.70 419.37 432.62 434.66 438.27 436.24 436.47 426.28 419.13

a A is the average value o f (--TR COS 0+)(COS 0A/COS 0{) X (P/PO, SA is the standard deviation o f A, 0{ is the respective wetting angle at PI.

pr+l 2

= --3"R

COS 0+.

[ 12]

The above iteration procedure is based on this assumption. Because r + is constant for all points of a given reintrusion curve (the three-phase interface cannot move in the range o f contact angles from 90 ° to 0i~), the intermediate adhesion tension A + = -3"R X cos 0÷ should be a linear function o f p. The values of 0{ differ for various p~ because o f different ratios o f the n u m b e r o f pores with the largest radius r (corresponding t o / h according to Eq. [11]) to the rest o f smaller pores with entrances accessible to mercury. Therefore, all A + values are multiplied by the ratio of cos 0A/COS 0{ to bring the measurements for different p~ in accord with the actual adhesion tension A. Values of 0÷ corresponding to the measured Vm data were calculated by an iteration procedure using Eq. [9] with the respective R (Eq. [7]) and Vc = Vm/n + (n + according to Eq. [6]). In accord with Eq. [7] and p = 2 3"R/R, 3"R = 3' -- 0.533p/2.

[13]

where 3, = 480 m N / m . The dependence of--3'R COS 0 = (--3"R X COS 0+) (COS 0A/COS 0{) on p is shown in Figs. 2 and 3. The values of--3"R COS 0A are given in Table I along with the respective PIThey decrease somewhat with PI owing to changing "YR. If we assume that the correct value of--3"R COS 0g is independent o f the pressure, then at p~ = 13.9 MPa (the highest value measured) 0A should rise from the original 154 ° to about 155 °, which makes a negligible difference. The average value o f A = (--3"RCOS0+) (COS0A/COS0~) (PdP) and their standard deviation SA are given in Table I along with 0{. The average of all A values is 430.44 + 1.99 m N / m . Thus, the linear dependence shown in Figs. 2 and 3 along with a good accord between A and --3"R COS 0 A confirm that the intermediate adhesion tension o f reintrusion is given in the whole range of the local hysteresis by Journal of Colloid and InterfaceScience, Vol. 95, No. 1, September 1983

138

j. KLOUBEK

/?///f

,oo[/ i f 0h

0.8 12 P FIG. 2. Dependence of--~/R COS0 (mN/m) on p (MPa) for reintrusion. The straight lines are denoted by numbers of measurements according to the tables.

r/r + = cos 0A/COS 0i~

"~R COS 0 = (~R COS 0+)(COS 0A/COS 0 { ) = (TR COS OA)(p/pI).

[ 14]

Hence, the reintrusion curve may be extrapolated to zero pressure when experimental points at pressures near to zero are not available. As follows from the preceding item, the experimental points deviating from Eq. [ 14] could have been omitted from the evaluation. Such points, e.g., the lowest one o f the curve 9 in Fig. 1 (which is not included in Fig. 3) give higher A + owing to crossing the hysteresis loop from the lowest pressure of extrusion which could not be lowered to p=0. It follows from Eq. [ 14] that cos 0+ = cos O~(p/pi)

hysteresis and a chosen 0~ instead of px and 0{, respectively, the same iteration procedure as described above can be done: the values obtained are r~ (Eq. [4]), VcM (Eq. [5]), and the best fitting value of 0{~. In analogy to Eq. [15], cos O{ = cos 0{~ (PI/PM). Then, after calculating Vci (Eq. [5]), the corresponding theoretical Vm~ is determined as Vm~ = VM V j VcM, where VM is the experimental value at a pressure PM and V ~ is the volume of the single meniscus at the same pressure. The difference between the theoretical and the actually measured Vm~ values represents the volume of the largest pores with a radius r, which can be calculated from the equation [ 16]

(cf. Eqs. [1 1] and [12]), where r + is obtained from Eq. [4]. Thus, the reintrusion curve may be utilized to obtain more information about the porous body under investigation. It should be noted that the shift of the meniscus into the pore can be avoided in some cases of reintrusion if the extrusion is stopped at a higher relative pressure. Such a reintrusion curve can be extrapolated to zero pressure and evaluated as described above. The method of calculating other characteristics of the porous structure from the local hysteresis curves, i.e., n u m b e r of pores with a given radius and their average length, was described earlier (12, 13).

[ 15]

provided that 7R may be considered constant. Indeed, variation of "fR is negligible in the measured range o f p. Hence, alternatively to Eq. [8], 0+ can also be determined in this way. It can happen that the reintrusion does not follow the local hysteresis curve up to p~ and the meniscus of mercury starts to shift into the largest pores. Even in such a case, the reintrusion curve for the local hysteresis may be extrapolated from the lower Vm values up to the corresponding VmX. With some mean value o f the pressure PM in the region o f local Journal of Colloid and InterfaceScience, Vol.95, No. 1, September1983

1

400

2

P

3

4

5

,0/ ./

'%°II/ 1 0 0 ~

8

12

P

FIG. 3. See the legend to Fig. 2.

139

HYSTERESES IN MERCURY POROSIMETRY

Extrusion

As already mentioned above, the basic assumptions are that r is constant in the whole range of local hysteresis loop and the mercury meniscus always makes a sphere cap. In such a case, the purely geometrical relations [4] and [8] hold. On the other hand, the fact that Vm values at the same p are higher for extrusion than for reintrusion contradicts Eq. [7] because only one R and, consequently, single 0 and Vm values correspond to the respective p. However, the experimental results indicate that R should be smaller for extrusion than the value according to Eq. [7]. Two reasons may be considered: (a) surface tension 3' is decreased substantially by some other more efficacious factor than the meniscus curvature, and (b) the curvature is influenced by still another factor than 3, and p. The problem cannot be solved at present, but, intuitively, the latter possibility will be considered which means that 3'R is given by Eq. [13] both for reintrusion and extrusion. Accordingly, 0+ values are obtained from the measured Vm values by the iteration procedure using Eq. [9] in which R is substituted by r+/cos 0+, i.e., the curvature is not determined by pressure but by contact angle. As shown in the previous section, the intermediate adhesion tension of reintrusion,

There is no analogy to contact angles of intrusion; O{ deviates from 0A due to the pore size distribution while the effective receding angle 0{~results from the pore geometry. The lowest 0 is reached at p = 0 when the mercury surface is plain so that its effective value should be always 0~ = 90 ° irrespective of the OR value. In Fig. 4, two examples of the relation between --3"R COS 0 and p for the measured extrusion data are shown. For comparison, the course of the corresponding reintrusion is also included. In the range of lower relative pressures, the course of extrusion is approximated by a straight line, There are not enough experimental points to determine it precisely, but this circumstance is not important here. As it could be expected, the dependence of--3"R COS0 on p deviates substantially from Eq. [18] at pressures from p~ down to PE because of the transition stage of meniscus between 0Aand OR. Therefore, the experimental points for higher relative pressures are fitted by a curve calculated from the empirical relation Ae = Ar + k[(p/pi)(1 - p/pi)] ply.

[ 19]

The constant k is given by the slope of the dependence of Ae - Ar on [(P/P3 (1

Mr -- (--3"R COS 0)r, (el. E q . [14]), is d i r e c t l y

proportional to p, and the intermediate contact angles (in the range from 90 ° up to 0A) are simply related to 0A by cos 0/COS 0A = P/PI (cf. Eqs. [14] and [15]). Similar relations may be proposed for extrusion in the range of intermediate contact angles between 90 ° and 0R, i.e., Ae =

( - - 3 " R COS 0 ) e

= (--3"R COS 0+)(COS 0A/COS0F)

=

(3"Rcos O)e(pE/p),

c, 300

P

[18]

where PE is the pressure at which 0 = OR. Unfortunately, PE and OR a r e not known, and no relation between OR and 0~ exists.

1.0

1.5

""

o~

,,'

/'i

100

[17]

and 3"~ cos 0~

0.5

~

/

/

// •

i

0

5

i 10 P

FIG. 4. Dependence of--tR cos 0 (mN/m) on p (MPa) for extrusion (full circles and dashed lines) and reintrusion (open circles and straight lines). Two typical examples are shown only and they are denoted by numbers of measurements in accordance with the tables. Journal of Colloid and Interface Science, Vol. 95, No. 1, September 1983

140

J. K L O U B E K

'

200

'

T A B L E II

6'

Constants k of Eq. [14], their Standard Deviation Sk, and the Respective Radii of Pores r

3 2 100

I

0.2

0.4

0.6

f[p)

FIG. 5. Dependence of (Ae - Ar), i.e. (--3~R COS 0)e -- (--TR COS 0)r, on a function o f p, tip) = [(p/Pt)(1 - p / l ~ ) y / ~ . Measurements are numbered in accordance with the tables, the points belong to numbers as follows: O, 1; O, 2; ~, 3; O, 4; O, 5; O, 6.

- p / p i ) ] p/~ (Figs. 5 and 6). The k values and their standard deviations Sk along with the respective radii r (Eq. [16]) are given in Table II. The value ofA~ = (--TR COS0)~ in the range between PE a n d / ~ should be influenced both by --'YR COS 0 A a n d - - Y R COS OR . Therefore, it may be assumed that k in Eq. [19] equals to --'YR COS OR. In this case, k should include the line tension r according to Eq. [2]. The dependence of k on 1/r was linearized as

[201

y = a + b/r,

where y = k. Values of k belonging to smaller

Measurement number

r (nm)

k (mN/m)

Sk (mN/m)

1 2 3 4 5 6 7 8 9 10 11

1867 1313 1145 910 756 645 529 472 426 174 62

137.19 248.20 299.27 305.11 327.42 340.36 330.31 328.89 342.89 314.35 295.49

3.58 7.58 3.65 4.27 15.00 9.11 14.56 13.70 9.38 8.14 4.14

radii ( m e a s u r e m e n t s N o . 5 - 1 1 ) give a = 337.79 m N / m and b = - 2 . 3 7 × 1 0 -6 m N with a standard deviation Sa = +6.15 m N / m and coefficient of d e t e r m i n a t i o n Ca = 0.809. By comparing Eqs. [2] and [20], we obtain a = "r~L -- % and b = r. Evaluation of pores with larger radii does not give reasonable results. A more complicated equation could be found to fit the k values in the whole range of radii but its physical interpretation would be difficult. As shown in Fig. 4, the extrusion curve consists of two branches. The point of their intersection at p = PE gives (--TR COS0)i which

T A B L E III Points of Intersection in the Course of Extrusion and the Respective Receding Angles 0x

200

I00 I

o12

f {p)

0.'4

o16

FIG. 6. Dependence of(Ae - At) on f ( p ) (see the legend to Fig. 5). The points belong to measurements as follows: O, 7; O, 8; O, 9; O, 10; O, 11. Line 8 merges in line 7. Journal of Colloid and Interface Science, Vol. 95, No. 1, September 1983

Measurement number

pr~ (MPa)

(-TR cos O)i

OR

2 3 4 5 6 7 8 9 10 11

0.31 0.36 0.33 0.40 0.47 0.47 0.55 0.55 1.14 2.80

332.34 360.28 332.17 346.04 353.94 333.50 335.50 337.97 310.20 290.78

133.83 138.65 133.80 136.14 137.53 134.03 134.36 134.78 130.29 127.35

HYSTERESES IN MERCURY POROSIMETRY

is equal to --'YR COS OR. The values ofpE were estimated graphically and the values of (-2rR cos 0)i were calculated as Ae in Eq. [19] using the estimated values of PE, and --3'R COS 0A from Table I and k from Table II. The results along with the corresponding values of OR are given in Table IlL The differences between (-3'R cos 0)i and k are negligible (up to 2% only) for measurements No. 7-11 suggesting that the above assumption o f k -- --3'R COS OR is correct. Higher deviations for larger pores may result from inaccuracy of PE a n d / o r k values. Nevertheless, the dependence of (--3'R COS 0)i (from Table III) on l / r can be reasonably evaluated in the whole range o f measurements according to Eq. [20] with y = (--3'R COS 0)i leading to a = 345.71 m N / m , b = - 3 . 6 8 × 10 -6 mN, Sa = +9.53 raN/m, and Ca = 0.729. These values are in a good agreement with those obtained above from the dependence o f k on 1/r. Thus, in correspondence with Eq. [2].

k

141

courses of reintrusion was suggested. Unlike reintrusion, the basic relation between p and R (Eq. [7]) does not hold true, the adhesion tension changes and the line tension exerts an influence on extrusion. In the region o f small relative pressures, a direct proportionality o f the intermediate adhesion tension, Ae, to p was found with a slope different from that for Ar. For higher relative pressures, an empirical relation between A~ and f ( p ) was found. Evaluation o f both branches o f extrusion curves gave the line tension, ~- ~ 3 × 10 -6 mN, and "YSL - - ~fS '~" 340 m N / m which deviates substantially from the value of'YSL -- "Yscorresponding to reintrusion (431 mN/m). Contact angles found are 0g = 154 ° and 0~ = 135 ° (extrapolated to the plane surface). In contradiction to the effective contact angle 0~, the actual o n e (OR) is m u c h higher. Hence, the volume hysteresis in mercury porosimetry is caused only partly by the contact angle hysteresis. The other reason consists in the pore geometry.

= ( - - ' Y R COS 0)i = - - ~ R COS OR

REFERENCES

= ~ L -- ~ + ¢/r. SUMMARY

On the basis o f the previous results of mercury porosimetry on a sample of ASCwhetlerite (13), the reintrusion curves in the region of local hysteresis were evaluated from the point o f view of intermediate adhesion tension, At. The direct proportionality between Ar and p was verified which justifies extrapolations for parts o f the reintrusion curve not available. This is an important condition for the contact angle determination from the shape of the mentioned curve and for evaluation o f the pore size distribution. The extrusion curves were examined and an explanation o f their deviation from the

1. Drake, L. C., and Ritter, H. L., lnd. Eng. Chem., Anal, Ed. 17, 787 (t945). 2. Freundlich, J., Electrochim. Acta 6, 35 (1962). 3. Rootare, H. M., in "Perspect. Powder Metall." (J. S. Hirshorn and K. H. Roll, Eds.), Vol. 5, p. 237. Plenum Press, New York, 1970. 4. Orr, C., Jr., Powder Technol. 3, 117 (1970). 5. Adamson, A. W., "Physical Chemistry of Surfaces," 3rd. Ed., p. 531. Wiley, New York, 1976. 6. Lowell, S., Powder Technol. 25, 37 (1980). 7. Liabastre, A. A., and Orr, C., J. Colloid Interface Sci. 64, 1 (1978). 8. Kloubek, J., Powder Technol. 29, 63 (1981). 9. Starov, V. M., and Churaev, N. V., Kolloid. Z. 42, 703 (1980). 10. Pethica, B. A., J. Colloid Interface Sci. 62, 567 (1977). 11. Rusanov, A. I., Kolloid Z. 39, 704 (1977). 12. Kloubek, J., Powder Techno[. 29, 89 (1981). 13. Kloubek, J., Carbon 19, 303 (1981).

Journal of Colloid and Interface Science, Vol. 95, No. 1, Selatember 1983