S.LEVEE AXD G, AI. BELL
1408
Vol. 67
THE DISCRETE-IOK EFFECT ASD COLLOID STABILITY THEORY TWO PARALLEL PLATES IX A SYMMETRICAL ELECTROLYTE’ BY S.LEVINEAND G. pvl. BELL Department of JIathemafics, Uniz’ersity of Manchester, and Department of Mathematics, Chelsea College of Science and Technology, London, S.W.S, England Received September i7, 1962
.4 general expression based on Grahame’s double layer model is derived for the interaction energy of the electric double layers of two parallel colloidal plates immersed in an aqueous binary symmetrical electrolyte. The results constitute a further development of the stability theory of Derjaguin and Landau and Verwey and Overbeek. The specific character of the adsorbed counterions in the Stern layer is taken into account through the specific (chemical) adsorption energy, the distance of the adsorption (inner Helmholtz) plane from the colloidal w-all, and the number of absorption sites occupied by each counterion. Any variation of the dielectric constant of the “inner region” with surface potential is, however, ignored and the Gouy-Chapman theory is applied to the diffuse layer. Two types of discrete-ion effect relating to the adsorbed counterions are considered. One is the entropy correction due to the ion size and the other (which is the more important) is the difference between the actual electric potential a t an adsorbed counterion and the average potential on the adsorption plane. This difference, termed the “discreteness-of-charge” effect, adds a new negative term, very nearly proportional to the density of adsorbed counterions, t o the adsorption energy. In consequence, as the potential-determining ion concentration and hence the interfacial potential vary, the potential a t the limit of the diffuse layer (tho outer Helmholtz plane) at the critical coagulation condition passes through a maximum. A corresponding maximum is found in the concentration of coagulating electrolyte. A convenient method of calculating the interaction energy for moderately large outer Relmholtz plane potentials is developed and the linear superposition approximation is also used. By comparison and by considering correction terms the two methods are shown to be reasonably accurate. The conclusion is that none of the simple formulas previously proposed for the variation of coagulating electrolyte Concentration with valency can be generally valid and that the Schulze-Hardy rule must describe a range of ratios of coagulating concentration dependent on the specific adsorption characteristics of the counterion. The theoretical results are compared with experiment particularly for the silver halide sols. 1. Introduction
In two recent papers2 (called I and 11) the authors incorporate the so-called discreteness-of-charge effect (or discrete-ion effect) into the stability theory of lyophobic colloids of Derjaguin and Landau4 and Verwey and Overbeekj (D.L.V.O. theory). In viev of, the complexity of the problem the model of two parallel plate-like particles of infinite thickness, immersed in a large voIume of aqueous, biliary, symmetrical flocculating electrolyte. is chosen. The concentration in the dispersion medium of the potential-determining anions, responsible for the negative plate charge, is assumed negligibly small. The Stern-Grahame picture of the electric double layer (Fig. I), originally applied to the mercury-electrolyte interface, is modified to correct for the discrete character of the adsorbed counterions and is assumed to apply to the surfaces of the colloidal plates. In a recent communication6 (111), the authors show that this discrete-ion effect can be interpreted in terms of a “self-atmosphere” potential a t an adsorbed counterion. An important consequence of this innovation is that the electrostatic potential at the outer Helmholtz plane (O.H.P.), the boundary betweeii the Stern “inner region” and the diffuse layer, exhibits a maximum as the potential across the interface varies at constant electrolyte concentration. This is in accord ~ i t experiment h for the mercury system? and a maximum at other interfaces is suggested by the (1) Part of this paper w&s presented at the Unilever Research Symposium, held at Noordwijk, Netherlands, from September 19th to 22nd, 1961. (2) S. Levine, G. M. Bell, and D. Calvert, Nature, 191, 699 (1961). (3) S.LeTine and G. M. Bell, J . Collozd Scz., 17, 838 (1962). (4) B. V. Derjaguin and L. Landau, Acta Physzcochzm. U R S S , 14, 633 (1941). (5) E. J. W. Verwey and J. Th. G. Overbeek, “Theory of the Stability of Lyophobic Colloids,” Elsevier, Amsterdam, 1948. (6) S.Levine, G. M. Bell, and D. Calvert, Can. J . Chem., 40, 518 (1962); see also G. M. Bell, S. Levine, and B. Pethioa, Trans. Faraday Soc., 68, 904 (1962).
observation of a maximum in the { poteiitial of silver bromide glass-electrolyte interface,lo and O/M7 emulsions stabilized by surface active ions.ll The inner region is treated as a homogeneous uniform medium with both dielectric constant and byidth fixed, independent of surface charge densities. Colloid stability theory should include dielectric saturation and electrostriction in the inner regioii but for brevity these will not be discussed here. However an entropy term in the adsorption isotherm due to the size of the partially hydrated ions will be introduced. The GouyChapman theory is applied to the diffuse layer. According to the conventional D.L.V.O. theory the electrolyte concentration required for flocculation increases with the surface potential of the colloidal particles. In contrast, our theory predicts that the flocculating concentration increases with this potential when it is small but then attains a maximum and decreases. With uiiivaIent coaguIatiiig ions this maximum is found a t high potentials, which are seldom attainable in coagulating hydrophobic sols but the maximum is reached with polyvalent coagulating ions. This is a consequence of the maximum in the potential a t the O.H.P. and therefore will not occur if the “selfatmosphere” effect for an adsorbed ion is ignored. Qualitative agreement is obtained between our theory and the experimental work on the dependence of flocculating electrolyte concentration on surface potential, most of which has been done with silver halide sols in “statu nascendi” by Tezak, Matijevi6, Mirnik, (7) D . C. Grahame, Chem. Rev.,41, 441 (1947); Z. Elektrochem., 62, 264 (1958); D. C. Grahame and B . A. Soderberg, J . Chem. Phys.. 22, 449 (1954). (8) K. N. Davies and A. K. Holliday, Trans. Faraday SOC.,48, 1066 (1952). (9) R. H. Ottewill and R. F. Woodbridge, to be published. (10) S.A. Greenberg, T. N . Chang, and R . Jarnutowski, J . Polymer Sei., 68, 147 (1962). (11) P. J. Anderson, Trans. Faraday SOC.,66, 1421 (1959). (12) D. A. Haydon, Proo. Roy. Soc. (London), A268, 319 (1960).
DISCRETE-IOK EFFECT
July, 1963
aiid their s ~ h o o l s ~but ~ - there ~ ~ ~ is one difference for h g I sols. In I and I1 a simple approximation to the electric double layer interaction was obtained by linear superposition (L.S.) of the potential distributions of the interpenetrating diffuse layers. A noticeable drop in t,he coagulating concentration on each side of the maximum was found whereas, according to IIirnik, et u Z . , ’ ~ the coagulating concentration in Agl systems is almost independent of the plate potential. I n the present paper, a general expression will be obtained for the electrochemical free energy of interaction per unit area of the plates, based on the electric double layer model described above. A practicable method of calculating this energy for large potentials a t the O.H.P. is developed by employing an expansion due to Derjaguin and Landau4 aiid Hoskin and Levi11e.l~ It is found that over a large range of surface potential the variation in the coagulating electrolyte concentration is reduced by choosing large potentials a t the O.H.P. Comparison is made with linear superposition and a method of improving upon the latter is also given. The different formulations of the Schulze-Hardy rule, as suggested by Tezakl3--I5l 8 and by the D.L.V.O. theory are also discussed. 2.
1409
\
e
Notation and Preliminary Relations
2h separation of the two plates d thickness of the inner region h’ = h - d distance between the median plane and O.H.P. p distance between the wall and the plane of centers of adsorbed counterions (inner Helmholtz plane or I.H.P.) y = d - p distance between the I.H.P. and O.H.P. e, €1 dielectric constant of the (aqueous) dispersion medium:aiid inner region K = e1/4ad capacity per unit area of the inner region e, k, T electronic charge, Boltzmann’s constant, absolute temperature $o potential a t the colloidal wall, independent of h $d = $d(h), $p = $p(h), li/, = +,,(/I) potentials at the O.H.P., I.H.P., and median plane 7 0 = e h / k T , l/d = m(h) = e+d(h)/kT, virn = lirn(h) = e$,(h)/ k T dimensionless quantities q d = $d( m ) , H d = q d ( a) values of $d and ?/d a t infinite plate separation a $ d = $d(h) - q d , Al/d = q d ( h ) - H d k = sech ‘/zzVrn(h), v = sech ZHd z p , zs valencies of coagulating cation and potential-determining anion z - z valency type of coagulating electrolyte. For negative plates 2 = z p n number per unit volume. of z-z electrolyte (dissociated) mole. cules in dispersion medium no number per unit volume of solvent (water) molecules K Debye-Huckrl parameter. defined by K~ = 8anz2e2/ekT 0 = ,th’, X = 1te/4aK dimensionless quantities ’ 8 = Y p ( h ) number of adsorbed cations per unit area of I.H.P. up = zpevp = up(h) charge per unit area of adsorbed cations on the I.H.P. u,j = ud( h) net charge contained in a column of diffuse layer, of unit cross-section and contained b e t w e n the 0 . H .P. and median plane. From the Goug-Chapman theory of the diffuse layer the value of ud at infinite plate separation is given by eq. 2.1 (13) B. Teeak, E. MatijeviC, and IC. F. Schulz, J . P h y s . Chem., 55, 1367 (1951); 69, 769 (1955). (14) J. P. Kratohvil,
M. Orhanovib, a n d E. Rlatijevib, ibid., 64, 1216 (1960). (15) 11. Alirnik, F. Flajsman, IC. F. Scliulz, and B. Tezak, i b i d . , 6 0 , 1473 (1933); M. Mirnik, F. Flajsman, and B. Tezak, Kolloid Z . , 185, 138 (1962).
( l e ) M . BIirnik, J . Phys. Chem., 65, 1638 (1961). (17) K. E. Hoskin and 8. Levine, Phil. Trans. Roy. Soe. (London), 248, 449 (19RG). (18) E. Rlatijevii-, D. Rroadhurst, a n d 31. Kerker, J . Phys. Chem., 63, 15.52 (1969).
L
Fig. 1.-Grahame
model of electric double lager, applied to two parallel plates at separation 2h.
= u,(h), U, = z,evs = uo(h) number of potential-determining ions and charge per unit area on plate wall Y p = ’p‘ , a), Zp = up( a): Y, = v,( m j values of v p , upi and u s at infinite plate separation Aup = up(h) - zp,Aud = ud(h) - Xd qo = +o - x’ where x’ IS the potential across inner region when u0 = up = 0, due t o the Lange x potential \-a number of adsorption sites per unit area of plate Q = kTd2K/epr quantity having dimencions of cJarge per unit area up0 = Q/zpg surface charge density which is approximately equal to the value of Zp a t which $‘dl is a maximum. In this paper, the maximum refers to the magnitude of the potential, irrespective of sign. I = zp/upO dimensionless parameter measuring the density of adsorbed cations @p = ->% pygup = kTgug self-atmosphere potential a t an ad-
-e&
sorbed cation, caused by its disturbing influence on neighboring adsorbed ions. This is defined as the potential a t the center cf a circular area in the I.H.P. of radius r0 where m 0 % p = zpe, due t o a charge density -up uniformly distributed over this area. (From above definition ZpePp’kT = --T ) g factor occurring in the “self-atmosphere” potential of an adsorbed ion. It depends on the dielectric constants of particle, inner region, and diffuselayer, on the electrolyte concentration, and decreases somewhat with surface density of adsorbed ions.6 For the silver halide systems it probably has a value in the vicinity of d / p . We assume g independent of T~ and also e
>> €1.
Pp =
+
xpe+p
+ + @p’
kT
zpe5bp
+
111 [ ~ p / ( N a
- p ~ p ) ” ] (2.2)
chemical potential of adsorbed cation (counterion) with center on the I.H.P. [ p is the chemical (short-range) energy of adsorption, and a,$ is due to the electrostatic image potential of the ion. The last term is due to the entropy of distribution
S. LEVINEAND G. M. BELL
1410
of the ions on the adsorption sites of the I.H.P. This includes a co-area" term, which is obtained by adapting the volume fraction statistics of Florylg and Hugginsto t o the hydrated cations each occupiing p adsorption sites and thus allows for the size of the adsorbed ions I,
- lii r
R(T)= pa =
ES
+ z,e$n
K ( k ) ,E ( k )complete elliptic integrals of the first and second Binds, respectively
G(k)
fJ
{ 2 E ( k )- 2k
=
H(k)
+ p In (1 - p b r ) , b = upO/.Vazpe (2.3)
Vol. 67
=
ii (E)
= --
3(1
- (1 - Ic2)K(k)j / k
- k 2 ) K ( k )+ 4k
-
4E(k)
dimensionless quantity
chemical potential of potential-determining ion
on plate wall
+
kl' In (%/no)chemical potential c.f counterion in solution, where activity coefficient is equated to unity pLsb chemical potential of potential determining ion in the solution, having a form similar to P L ~ O .The volume of electrolyte is so large that ~ p oand kLaO are independent of any variation in vp the surface potential $0 and and u s . On equating pCsto hence 70, are constant in the equilibrium state, being independent of h and n. @p = [ao - s p $p' specific absorption potential energy of cation in the I.H.P. pp* = - cpe$d, ps* = pq -z,e$ 100 mv.) and is still reasonable for IC < 0.80 (1% \kdl > 60 mv.). It will be designated as the H.P. (highpotential) approximation and most of the calculations in section 6 hare been performed with this approximation. A value of k is chosen and the remaining relevant quantities are expressed in terms of k , namely, v by (4.9), ~ h by ' (4.6), X by (4.10), r by (4.12), by (4.11), and n by (4.13). In the original D.L.V.O. theory the double layer interaction per unit area of plates a t separation 2h' is given by I * ( h ) provided $d is replaced by $0. Thus v becomes vo = sech '/2 q0and the terms involving X are absent on the right-hand sides of (4.6) and (4.9). Making use of the expansion by Hoskin and Levine" of 1*(h)up to terms in u05, the latter two relations on the D.L.V.O. theory read (dn/d+o)r = 7,
~ h '= liK(k) -
ug
1 61c2
- - (1
and 00
=
kH(k) uo3 1 - - - (7 + 2k2 - k4)uO5+ . . . 1 - k2 312' 40k4 (4.18)
and these determine $0 and ~ h 'as functions of the parameter 12 at the flocculating conditions. The dependence of the flocculating electrolyte concentration on IC and therefore on $o is obtained from (4.10) and (4.13). It is observed that In n does not involve the capacity K , since K does not occur in the D.L.Y.0. theory. 5. Large Separation Approximation I n papers I and I1 the three relations (4.10)-(4.12) were expressed rather differently in terms of the L.S.
+ ...]
(3.1) Also we need to substitute - 1 = 16 t02e-2einto the expression (4.2) for z A q d . If now (5.1) is expanded in powers of A q d and (4.2) is substituted, then up to terms of order e-*e, pe(h) proves to have the same form as (5,l) provided ?ld is replaced by H d and by
c1
r
2x
1
=
+ lc2)vO3-
(C, - 1)
+ S ( r ) + x cosh
l12xHd
I ( h ) is now easily obtained by integrating pe(h) with respect to h a t coilstant H d and the first stability condition in (4.8) can be expressed as ta4(28- I)
+ t3*(128 - S P - 3) - 28 + 1 +
(8 - l)e2' =
8P)to' -
1
+XS ( r )
(e - l)e2'
[(z) + + (3 - 128
+ (1 - 20)(to4- 6ta2
l)]
(5.3) Also (4.10) becomes
x=--96ec2Kc2A T
to2L,
(">2
ze
and (4.12) is
2 111 (QL)
+
+ D(x)
(5.5)
mhereD(z) = B(x) 4 - 2ln[(96/n)(kT/~e)~(r~/KA)]. The linear approximation is obtained b17 putting 0 = 1 and L = 1 and then the constant D ( z ) becomes identical with that introduced in l I e 3 I n this approximation (5.5) is solved for r as a function of H d and then rF.0 and n a t coagulation can be found from (4.11), (5.4), and (4.13). I n the D.L.V.O. theory we replace to by tanh l/4zq0 and equate the left-hand side of (5.3) to zero. This yields 6' as a function of $oand the flocculating concentration a t specified #o is gil-en by (5.4) and (4.13). In the theory presented here, the simultaneous equations (5.3)-(5.5) need only be solved (by an iteration process) for moderate values of \ k d , since the H.P. approximation is more accurate a t high potentials. 6. Numerical Results and Comparison with
Experiment From the results of the previous sections we can plot the coagulating concentration of added electrolyte against the surface potential $o for given values of the (27) S. Levine and A. Suddaby, Proc. Phys. Soc. (London), A64,287, 431 (1951).
DISCRETE-IOS EFFECT
July, 1963
1415
relevant parameters. This coagulating concentration has been expressed by the dimensionless parameter X and a t T = 25' ( E = 78.54) this may be converted to concentration in mole/l. by using the relation K = 0.3291zdc X lo* to give
--Po
300
(niT.,). 200
0
100
log c = 2 log K X / Z - 16.6264
(6.1) Experimentally it is not $o but the concentration or activity of the potential-determining ion which is measured; by the Nernst equation the natural logarithm of this activity contains the term z,e$o/k7'. Accordingly, for a typical colloidal system, negative AgI sol, we transform from $0 to PI (minus the common logarithm of the activity of the potential-determining I- ion). From (5.4)) (4.11), and (4.12), when P d = 0, X = 0, r = 0 and hence Z P = 0, $o - x' = 0, and by electrical neutrality uO( m ) = 0. Since experimentally PI = 10.6 a t the isoelectric pointz8for AgI we may write the Kernst equation as
PI
=
10.6
+ 0.4343\ko/25.71
i
-
I( 3
-3
-4
4
(6.2)
where \ko is measured in millivolts. Since the isoelectric points for AgBr and AgCl are a t pBr = 6.9 and pC1 = 5.7,28we may change to pBr or pC1, respectively, instead of PI simply by shifting the log c-PI curves by 3.7 or 5.8 units to the left. Now Mirnik, et a l . , I 5 find PI = 9.8, pBr = 6.0, and pC1 = 3.3 at their so-called negative stability limit, which we identify with the condition \kd = 0. One reason for these differences is the dependence of the isoelectric point on the age and mode of preparation of the silver halide crystals. Several a ~ t h o r s ~find ~ - that ~ ~ the pBr increases with crystal size at the isoelectric point for freshly prepared AgBr sols. This may be related to the existence of a Schottky or Frenkel defect diffuse layer inside the crystal. 3 1 In Fig. 2 and 3 we give log c-PI curves calculated for the D.L.V.O. theory a t A = 2 X erg The accuracy of the L.S. and H.P. approximations is illustrated by comparing the results they give with exact results derived by using various suitable expansion^^',^^ referred to in previous sections. From Fig. 2 it can be seen that the L.S. approximation overestimates the coagulation concentration which however tends to a limiting value cm, proportional to 2-6, on both the exact and L.S. curves at small PI (large \ko). In Fig. 3 the exact curves for z = 1 and those given by the H.P. approximation, the L.S. approximation, and its extension described in section 5 are compared. The L.S. and H.P. curves cross at IC = 0.79 (Iz\kol = 64 mv.) and, as would be expected, the H.P. approximation is better at higher, and the L.S. a t lower, potentials. Typical log c-PI coagulation curves for the extension to the D.L.V.O. theory with the H.P. approximation (solid curves) and L.S. approximation (broken curves) are shown in Fig. 4, 5 , and 6, the relevant parameters being given in Tables I and 11. To reduce computation N , has been chosen to gire the same values of b as in some of the results of IL3 The constants B(z) (28) H. R. Kruyt, "Colloid Science," Vol. 1, Elsevier, Amsterdam, 1952. (29) P. F. J. A. Julien, Thesis, Utrecht, 1933; G. H. Jonker, Thesis, Utreoht, 1943. (30) J. Barr and €1. 0. Diokinson, J. Phot. Sci., 9, 222 (1961); see also reference 9. (31) T. B. Grimley and N. F. M o t t , Discusstons Faraday Soc., 1, 3 (1947): T. B. Grimley, Proc. Roy. Soc. (London), A'201, 40 (1950). (32) S. Levine and G. M. Bell, Can. J . Chem., 38, 1346 (1960).
t
-2
5
6
7
8
9
1
0
1
1
PI.
Fig. 2.-Logarithm of coagulating concentration against PI or plate potential for 1-1,2-2, and 3-3 electrolytes in the D.L.V.O. theory. For each valency, the lower curve represents the exact calculation and the upper one the L.S. approximation. A = 2 x 10-12ergcm.2and T = 25". --Po
-0.6
-r
250
200
(mv.). 150
200
so
1
L
-l..i
-3.0
-
I
I
I
I
b
7
8
9
10
PI
Fig. 3.-Logarithm of coagulating concentration against PI or plate potential for a 1-1 electrolyte in the D.L.V.O. theory. The lower curve is the exact result and the upper one the L.S. approximation. The points inside the triangles represent the H.P. approximation and those inside the circles are the extension to the L.S. approximation. Figure 3a (inset).-Experimental curves forAgI based on reference 16, Fig. 5 .
for curves A, 13, and C in Fig. 4 were determined by choosing convenient values for the parameter k at the maximum in I \ k d . The Tezak-RIirnik (henceforward T.M.) rule is that the flocculating value of log c is a linear function of z, making the log c-PI curves for z = 1, 2, and 3 eyuidistant. In the present theory the T.M. rule can be satisfied at one particular value of PI by proper choice of the parameters; for example, the constants B(z). rl'his value of PI is marked with arrows, further details
s. LEVISE,4ND G. 11.BELL
1416
v01. 67
TABLE I CHARACTERISTICS OF PLOTS BETWEEN ELECTROLYTE CONCENTRATION ASD SURFACE POTEYTIAL AT CRITICAL COAGULATION erg cm.?; Na = (3/8) X 1OI6 cm.-*; T = 25’; K , pfarad./cm.z; qo,\Ed, mv.; upo, pcoul./cni.2; c, mole/l.; Zp = TU&”’ A = 2 X 78.54; r/d = 1/4; g = 4/3 High potential approximation
e =
Curve
z
p
B(r)
K
A B
1 2 3 3
1 3 5 5
11.39 12.96 10.97 8.75
30 20 20 20
Fig.4(solidcurves)
C
D
b
3.084 1.028 0.685 0.685
-
0.0514 ,00856 ,00380 ,00380
1,601 0.600 ,267 ,267
,
.
6.33 7.16 7.16
Maximum
k
r
0.67 .60 .58 ,595
1,057 1.086 1,108 1,108
r1
W
7
h’x
-pd
-*O
115 82 62 56
669 419 241 185
1.004 0,967 ,954 ,964
Tezak-Mirnik form of Schulze-Hardy rule Curve
k
r
A B C
0.705 ,603 ,713
0.031 0.83 13.6
Fig. 4(solid curves)
D ( z ) = B(a) Broken curve
Z
D(z)
E F G F G
1
8.92 12.44 12.07 8.89 13.03
Fig. 4
Fig. 6
2 3 2 3
-*d
-*o
-1ogc
Curve
k
r
99 78 31
394 394 394
1.30 2.76 4.21
-4 B C
0.724 ,610 ,710
0,0090 0,092 10.6
Lincar superposition approximation = 2 log (0.910 x l O - l * / A 9 )
+ 2 In Kz2 - 9.29, log c
--
AIaximum-
r
111 79 60 70 63
-‘PO
-log
91 77 32
319 319 319
1.37 2.78 4.18
C
+ 4 log to
Curve with same paramerers
- ‘I,,, 666 417 235 283 263
-*Po
$0 76 32
241 241 241
-log c Curve
1.49 A 2.83 B 4.18 J
r
k
0,724 0,0090 ,610 0.092 .710 10.8
-xI,d
--‘Yo
91 76 32
319 319 310
-loa
c
1.38 2.78 4.18
this difference is not’ quite constant, taking the values 1.14 and 0.96. The experimental result’s by the “statu nascendi” method quoted by Mirnik33for silver halide sols and given in Table IV also obey the T.M. law only approximately. TABLE I11 TEZAK-MIRNIK LAWNEAR MAXIMUM 2
B(z)
K
b
k
-qo
-*d
-1OgC
rm
T
4.50 30 0.0514 0.78 69 266 1.64 1.180 1.057 9.05 30 .00963 .61 76 266 2.78 0.834 1.137 3 11.93 20 ,00380 ,576 65 266 3.74 1.108 1.108
1 2
TABLE IV VALUES O F A
.4gI
K-Ba Ba-La
log C
-Ghosh---
------Mirnik---
AgBr
AgCl
1 . 9 1 1 . 4 4 0.88 K-Ba 1 . 7 7 1.62 1.18 Ba-AI
(33) hl. Mirnik, private eommunication.
2-6
law
AgI
1.62 6log 2 = 1 . 8 1 1.11 6 l o g 1 . 5 = 1 . 0 6
I~ISCRETE-IOS EFFECT
July, 1963
1417
TABLE V LIKEARSUPERPOSITIOS
CORRECTIONS TO :HIGH POTENTIAL AND
APPROXIMATIONS (Corrections denoted by letter 6, q
d
in millivolts)
H.P. approximation Curve A (Fig. 4 a n d 6) B (Fig.4)
1
SB(s) -0.08
2
-0.05
C (Fig.4)
3
-0.02
2
k
-Pd
0.67 .72 .6 .7 .7 .58 .82 .82
115 92 82 50 50 62 18
18
pI -log c -0.70 1.17 4.99 1.36 3.51 2.75 3.09 8.76 3.09 0.67 6.54 3.75 10.24 4.73 7.73 2.63
6(pI)
0.10 17 .04 .02 .I2 ,004 - .06
.22
6(-logc) 0,006 .02 ,002 .01 -- .01 ,002 .01 -- . 0 4
L.S. approximation CUrYe E (Fig. 4 and 6) F (Fig.4)
G (Fig.4)
2
8o(Z)
Hd
1
-0.08
2
-0.05
1.5 4 1.0 2 5 2.5 0.8 0.8
3
-0.02
PI 9.63 2.39
-Fd
39
103
10.0
26 64 64 21 21
7.06 1.81 10.18 2.88
-1OK
C
2.47
1.16 3.83 2.78 2.78 4.63
6(pI)
0.03 1.08 0.04 .38 - .23
-
4.683
-
.OB
.18
6(-lOgC) 0.08
.l5 .11
.15 .l5 .13 .15
dL
+ ro
P
Cl
,,,I =
;[l
- 2+-)1'*+TT(Tp0
..
-
2
3
4
>
0
7
S
9
1
0
1
1
Fig. 4.-Logarithm of coagulating concentration against pJ or plate potential for 1-1, 2-2, and 3-3 electrolytes in the present theory. Solid curves are the H.P. approximation and the broken curves are the L.S. approximation. On the solid curves (A, B, C and A, B, D) the T'ezak-Mirnik formulation of the SchulzeHardy rule is valid at the arrows. The L.S. approximation for curve I) ie not shown. Parameter values in Tables I and 11.
2
4
10
8
6
PI.
Fig. 5.-Logarithm of coagulating concentration against p l for a 3-3 electrolyte (H P. approximation) in the present theory. The arrows mark the value of PI where the T.M. law is valid between each of the three curves and the curves A ( z = l ) and B ( 8 = 2) in Fig. 4. Parameter values in Tables I and 11.
.]
--Pa
600
500
400
(mv.). 300
200
0
100
+
[ l - 0.089 ( T C T ~ ~ I X ) ~ " . ] (6.3) P if y = 1 8. and up0 is measured in pcoul. 'em. Such a decrease in g with increasing r will also iiicrease the value 73, defined by (3.19) a t which, d$d/dh = 0, and indeed this point, may not be observed in practice. To illustrate the effect of varying the capacity K , curve J in Fig. 5 which differs from curve D in Fig. 4 by an increase in K froin 20 to 30 pfarad./cm.2 has been drawn. The constant B(3) in cur7.e J has been so chosen that J intersects D a t k = 0.71 (marked by an arrow) where the T.M. law is obeyed. Apart from a shift iii the position of the maximum, it is observed that there is little change in the log c-pI plot. The well known rqle that the flocculating concentration is proportional to x ( r 6 law) is also only obeyed a t isolated values o€ pT. From (4.6), (4.9), and (4.10), in the H.P. approximation X K has the fuiictional form z - y ( k ) and hence from (4.13) n has the form ~ - ~ g ( k ) . If the zF6 law applies theii k must hare the same value for z = 1 , 2 , and 3 at a given or pl. This is illustrated in Fig. 6 where the parameters for curves A, B, and C are the same as for A, B, and C, respectively, of Fig. 4 except for the constants B ( 2 ) and B(3) which are fixed by as=
1
r)I.
Mirnik16 claims that, as well as obeying the T.M. rule, the flocculating concentration is constant over a wide range of pI. (Experimental rlesults of Mirnik, et a1.,15 are shomii in Fig. 3a.) I n our t'heory (see XI3) the slope of the log c-pI curve on the low p I (high potential) side decreases &h ?Id and thus a reduction in y l d gives bettier agreement with Mirnik's condition. This is illustrat,eld for x = 3 in Fig. 5 where for curves H aiid I the rat'io y / d has been reduced from '/4 to i/o As g = d/B, this means that g has been reduced from 4//3 to 6/4. The capacity K is adjusted to keep b .onaltered and the constants B(x) are so chosen that t'he T.M. rule (marked by arrows) applies at assigned values of PI. The other parameters and conditioiis relating to the above rule are given in Table TI. The result of correct'ing for t.he decrease in g wit11 iiicrease j.n r is similar. In the particular case where E >> el, a:nd the dielectric constants of the inner rep;ion aiid particle medium are equal, it may be shown that g = - 1-22
0
/'
-51
I
0
1
2
1
3
4
5
0
7
s
,
9
1
0
1
1
PI.
Fig. 6.--Similar plots t o those in Fig. 4, with different parameters for 2-2 and 3-3 electrolytes. The curves B and E for the 1-1 electrolyte are identical with those in Fig. 4. The L.S. approximation for curve D is not shown. On the solid curves (A, B, C ) the z m 6 law is valid at the arrows. Parameter values in Tables 1and 11.
suming that the r6 law holds a t k = 0.70; further details are given in Ta,ble 11. Curve D differs from C in
S. LEVIXEASD G. AI. BELL
1418 200,
160
3 e
v
5 I
loo
50
C
I
100
1
I
1
200
300
400
-qo
600
(mi-).
Fig. 7.-Variation of potential a t the 0.II.P. with plate potential a t coagulation. Solid curves are a re-plot of -4,B, C (H.P. approximation) and broken curves a re-plot of E, F, G (L.S.approximation) in Fig. 6 . The common point of intersection marks the x -6 law.
Fig. 6 only in the change of the c a p a d y K from 20 to 15 pfarad./cm.2. There is no value of PI a t which the r6 law is obeyed exactly between the curves A, B, and C in Fig. 6. Finally, in Fig. 7, the three curves A, B, and C and the corresponding L.S. plots have been redrawn to show the variation of &d with Qko. The x-6 law is obeyed a t the point of common intersection. By decreasing the van der Waals constant A all the curves can be displaced parallel to the log c axis with no change in shape and in the directioii of increasing c, provided the capacity K and the density of adsorption sites ATa are simultaneously increased so that K A and K/Na remain constant. From (4.6), (4.9), and (4.10) the product K X depends only on A and k and it follows from (4.13) that if k (or q d ) is unaltered, a change in In n and therefore log c depends oiily on a corresponding change in A. To ensure that no and hence PI are unaffected in this operation, it is seen from (4.11) and (4.12) that, as mil as \ k d , the quantities b, X,and y l d must not vary and this is achieved by keeping K A and KIL’V,fixed. Figures 2 and 3 suggest that the niaximum in the log c-PI plot becomes flatter as the potential q d increases. Indeed, for the smaller values of q d considered in 11,3 steeper maxima than those shown in Fig. 4 , 5 , and 6 were obtained, particularly on the low potential (large PI) side. Although the agreement with the experimental results for A4gI(Fig. 3a) is improyed by increasing q d our theory still predicts a larger slope in the (log c-PI) curve at small PI. Since the correctioiis to the H.P. aiid L.S.approximations derired in Fig. -2 and 5 will be small it is sufficient to indicate their magnitudes with the aid of a few typical examples in Table IT, ‘The corrections are found to be consistent with those illustrated in Fig. 3 aiid with differences between the H . P . and L. S. curves in Fig. 4 and 6. We have concentrated on the experiments of the “statu nascendi” schools and on their interpretations of the Schulze-Hardy rule for silver halide sols. However, the 2-6 law is consistent with Ostmald’s empirical “activity coefficient” rule3?which seems to predict approximately the coagulating concentration in a number of (34) R-.Ostnald, J. Phys. Chem., 42, 981 (1938).
Vol. 67
sols. Our theory indicates a range of flocculating concentrations, having a complicated dependence on a whole number of factors. One aspect of coagulation worthy of comment is the relation between the potential \kd and the valency z. If the z - ~law is obeyed, then for both the H.P. and L. S. approximations, q d is proportional to x-l. With the T.M. law q d is not simply related to x , although it does decrease with increasing valency. Xow according to Ghosh and his schoo13j coagulation occurs a t a definite [-potential irrespective of the valency of the counterions. If the {-potential may be equated to q d this means that \ k d is independent of x a t flocculation. This is approximately true in the calculations of Table 111, but not in the other examples given here and in 1 1 . 3 Furthermore, the corresponding values for A log c obtained by Ghosh and shown in Table IT’ indicate that l q d l should diminish with increase in x. The situation may be summed up by saying that from the theory presented in this paper none of the simple rules which have been proposed for the variation of coagulating concentration with valency can be generally valid. This applies to the T.M. law that A log c is independent of x and PI, to the law, and to the rule of Ghosh that the [-potential is independent of x.
7. Discussion Our theoretical results agree better with the “statu nascendi” experiments on AgBr and AgCl (see IT3)than 011 AgI in respect to the values of A log c, the maximum iii log c, and the limited range of pBr and pC1 over which t’heT.M. law holds. Mirnikl6maintains that the interpretation of the “statu nascendi” flocculating experiments 011 AgBr and AgCl is uncertain because of other processes, such as recry~tallizat~ion~~ competing with coagulation, but the difference between AgI aiid AgBr may be partly explained in another way.. A maximum in the [-potential of AgBr sol under variation of pBr a t constant ionic strength8mgimplies a maximum in the (log e-pBr) coagulation curves. However, a maximum in the [-pot’ential of AgI is not definitely est.ablished experimentallysalthough a very shallow maximum from the low PI side is indicated on the experimental coagulation curves in Fig. 3a,36a n-hich is consistent with the results of Kruyt and K l 0 m p 6 . ~ ~Lijklema38has suggested that the Stern (univalent) ions may be located a t some distance from the AgI surface so that r/d would be small. Indeed our present’ form for vanishes m-hen r/d = 0 and for p = 1 the theory reduces to the classical Stern one, where the I.H.P. and O.H.P. coincide and the potential at the O.H.P. has no maximum. However, the expression for +p is only a first approximation (1116)3 9 and a more accurate treatment will yield & small but, different from zero at y / d = 0. It should be noted that our theory predicts (35) B. N. Ghosh, S. C. Rakshit, and D. IG. Chattoraj, Trans. Faradau Soe., 6 0 , 729 (1964); Kolloid Z., 164, 48 (1967). (36) R. H. Ottewill and A. Watanabe, ibid., 171, 33 (1960). (36a) NOTEADDED I N PRoos.-Recently Lyk1e:na (Trans. Faraday Soc., 69, 418 (1963)) has calculated IL minimum in the surface excess of co-ions for the negative AgI-simple 1-1 electrolyte system when the pI is changed a t constant ionic strength, particularly a t 0.1 M . This indicates a maximum in the pctential 1 q d a t the O.H.P. (37) H. R . Kruyt and hl. A. M. KlompB, Kolloid Beih., 64, 484 (1943). (38) J. Lijklema, Kolloid Z., 176, 129 (1961). (39) Compare V. G. Levioh, V. A. Kiryanov, and V. S. Krylov. Dolcl. Alcad. A‘auk S.S.S.R., 136, 1425 (1960); V. G. Levich and V. S. Krylov, ibid., 140, 518 (1962).
1
COSTACT ASGLESIx
July, 1963
THE
~IERCVRY-WATER-BESZEXE SYSTEM
that the maximum becomes more pronounced as the valency of the counterion increases. A number of other factors ignored or inadequately treated in this paper need to be included in a more detailed comparison with experiment. For example, the function R(r) would be affected by the non-uniform distribution of so-called active spots on which the potential-determining ions are adsorbed, and various corrections to the Gouy-Chapman theory of the diffuse
1419
layer should be considered. -4part from the iiiadequacies of our double-layer model, the theory has been developed for two parallel plates. whereas the “statu nascendi” experiments apply to particles with mean radius comparable with the double-layer thickness, particularly with polyvalent couiiterions. This should not affect the general shape of the (log c-PI) coagulatioii curves but there will be some relative displacement between the different valencies.I1
CQKTACT ANGLES AND ISTERFACIAL TESSIQNS IS THE APERCUWY-WATERBEKZENE SYSTEM BY EDWARD B. BUTLER Pan American Petroleum Gorporation, Tulsa, Oklahoma Received September 86,1962 The interfacial tensions between mercury and benzene and between mercury and vater IT ere measured by the pendent drop method. The values did not change either with time or with temperatuie over the range LO to 50”. The interfacial tension between mercury and water saturated with benzene changes with temperature as does the interfacial tension between mercury and benzene saturated with water and as does the interfacial tension between water and benzene. None of the interfacial tensions changed with time. The values for the contact angles (measured through the water phase) were calculated from the interfacial tension data. The contact angles were also measured. Although the experimental values tended to drift toward the calculated equilibrium values, the two never coincided. The structures a t the mercury-benzene and at the mercurywater interfaces are inferred from the interfacial tension data. The extremely long times apparently required for the measured contact angles to reach the equilibrium values are then explained in terms of the structures a t the interfaces.
Introduction Contact angles are a measure of the degree to which a solid is wet by one liquid in preference to another. Such measurements find application in fields such as ore flotation, detergency, and lubrication. At Pan American contact angles are used to determine whether or not a reservoir rock is oil-wet. This information is applied in reservoir engineering to obtain more complete oil recovery from the porous rock in which the oil is st0red.l unfortunately the iiiterpretation of wettability in terms of contact angles is not entirely certain, for the numerical. value of the contact angle depends on the way it is formed. Further progress in the application of contact angles as a measure of wettability will be aided by a better understanding of the way contact angles approach equilibrium. The purpose of the present work is to examine the reasons for the existence of an advancing contact angle which differs from the receding angle. The relationship between the contact angle e, the solid-liquid interfacial tensions yos and yws, and the liquid-liquid interfacial tension yow is stated by Young’s equationO2
Since all of the quantities on the right-hand side of eq. 1 are thermodynamic free energies, their values are determined by the stake of the system, consequently e is uniquely defined. But a unique value of e is not found when contact angles are actually measured. Almost every investigator has found two contact angles
-an advancing angle and a receding one. Moreover, the value of the angles in any particular system has varied with the iiivestigator. Harkins4 is one of the few workers to claim a single value for the experimentally measured contact angle. .A number of explanations I3ai.e been advanced to account for this discrepancy between Young’s equation and experiment. Among these are surface roughness, lack of surface cleanliness, and surface heterogeneity, but none has been adequate. An answer to the discrepancy between theory and experiment could be obtained if measured -\Talues for the interfacial tensions could be substituted in Young’s equation so that could be calculated. Unfortunately, except for a few special cases, solid-liquid interfacial tensions cannot be measured. Experimental verifization of Young’s equation does not exist and Kewmaii’s triangle, which may be viewed as a generalized case of Young’s equation, has been verified only 0nce.j Another attempt at a verification of Kewman’s triangle was not entirely satisf actory.6 Since interfacial tensions cannot be measured between solids and liquids, a tlii ee-phase liquid system in which all the interfacial tensions could be measured was chosen for this investigation. This system was mercury, water, and benzene. The equilibrium contact angle in this system can be calculated from the measured interfacial tensions and compared with the experimental angle. Such a comparison will help differentiate among the many possible causes for the existence of an advancing angle different from the
(1) R. 0. Leach, 0. R . Wagner, H. W. Wood, a n d C. F. Harpke, J . Petrol. Tech., 14, 114 (1962). (2) T. Young, Phzl. T~ans.,64, (1805); Works (Ed. Peacock), 1’01. 1, p. 432. (3) A Dupree, “Theorie Mwhanique de la Chaleur,” 1896, p. 393.
(4) W. D. Harkins, “Physical Chemistry of Surfaces,’’ Reinhold Publ. Corp , New York, K Y , 1952, pp. 278-251. (5) C. 8 . Smolders, Doctoral Thesis, Unirersity of Utreoht, Netherlands, 1961. (6) D J Donahue and F. E Rartell, J Phus. Chem., SG, 480 ‘1952).
cos 61
=
yo0
- YNB Yow
(1)
