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Particle interactions and rheological effects in kaolinite suspensions
Aduances in Colloid and fnferface Science, I7 (1982) 219-232 Else&r Scientific Publishing Company, Amsterdam -Printed
219 in The Netherlands
PARTICLE INTERACTIONS AND RHEOLOGICAL EFFECTS IN KAOLINITE SUSPENSIONS A.E. JAMES'and D.J.A. WILLIAMS" University College, Singleton Park, Swansea SA2 8PP, U-K_
CONTENTS ABSTRACT _._.__________________________________________________________~lg I. INTRODUCTION -_____.__._...-.-___________-___-_______________..__________2lg II. PARTICLE INTERACTIONS __~__________________.__________--___~_...._______.221 III_ A. Electrostatic interaction _____._......._.__________.._..__-_______~21 B. van der Waals attraction -______.__._..._______________.___________223 C. Total energy of interaction ____._....______.__________.___________223 IV. RHEOLOGY AND PARTICLE INTERACTIONS ...................................225 CONCLUSIONS ....................._................__..___......__.___._.2~8 V. APPENDIX I .__________.__.___.._______________.__________~____________.228 VI_ VII. NO~?&~~CLATURE *_*~~___***._.~~*-~****~~*~..*..**~--~_*__~_~___***~___*~ 230 VIII. ACKNOWiEDGE~E~~T..........._._._.___......~..........._.....*___.______231 IX_ REFERENCES ..____..............__________________---___._____.________231 I.
ABSTRACT
?article interactions in kaolinite suspensions are modelled by representing the edge and face of a kaolinite platelet as a cylinder and flat plate, respectively_ Computations of total energy of interaction show that at pH 6, 7 and 8 and in the presence of 10v4 -10-I M NaCl both face-edge and edge-edge modes of interaction are likely. Rheological parameters for flocculated suspensions (extrapolated shear stress and plastic viscosity) for dilute sodium kaolinite suspensions (volometric ccncentration 0.02) are interpreted in terms of the proposed interaction models. INTRODUCTION II. Although considerable theoretical and practical interest has been shown in the rheology of heterogeneous suspensions containing kaolinite (ref. l-5), an unequivocal interpretation of the effect of interparticle forces on their rheological properties has proved elusive. This phenomenon is not entirely unexpected in 'Department of Oceanography, University College, Swansea. " Department of Chemical Engineering, University College, Swansea OOO~-8686/82/000~0000~$03_50 0 1982Elsetier ScientificPublkhingCompa&
220 view
of the complexity of such materials not only in regard to their size and shape, but also because of the difficulty of obtaining reliable electrokinetic data for clay minerals (ref. 3,6)- Difficulties also remain in formulating a satisfactory model for the surface electrical properties of an anisometric heteropolar crystal such as kaolinite (ref_ 7). The pseudoplastic behavior of kaolinite suspensions (ref. 3) suggests that the development of structure associated with clay flocculation is an important factor influencing the rheological behavior of these suspensions and that it is the progressive breakdown of the floe structure with increasing shear rate (v) which gives rise to the shear-thinning behavior noted by a number of investigators (see, e.g., ref_ 1,3,8,g). An analysis of the flocculation process and the precise manner in which flocculation affects flow behavior, particularly in heterogeneous clay suspensions, is a complex problem; heterogeneity may arise due to the presence of impurities such as montmorillonite, mica, muscovite (ref. 10) or by deliberate addition of other clay minerals such as illite (ref. 5) and quartz (ref. 11) in predetermined amounts. Despite the practical importance of such suspensions, relatively few systematic rheological measurements have been reported using well characterized mixtures (ref. 5, II). In general, an analysis of the stability of colloid particles against flocculation in the presence of a velocity gradient requires simultaneous consideration of interparticle and hydrodynamic forces. Such an analysis has at present been done only for the relative motion of two equi-sized, uniformly charged spheres in a laminar velocity gradient (ref. 12). however, particles of kaolinite are anisometric and dissimilarly charged and present seemingly retractable problems in terms of such an analysis. ClearL!y,at the present time simplification is necessary if any progress is to be made- Herein in a first approximation we examine the flocculation process and its influence on rheological parameters such as residual stress (~5) and plastic viscosity (Q_ ) (ref. 13) in terms of the mode of interaction of particles and inter-particleenergies at equilibrium as a function of chemical environment (pH, salt concentration)_ Accordingly, stability is assessed with the aid of a potential energy diagram (ref. 14), in which the free energy of interaction VT (the sum of the van der Waals attraction energy VA and the coulombic energy V,) . is plotted as a function of the distance of separation of tbJ0 particles. Calculation of VT for kaolinite particles requires specification of particle shape, mode of interaction of particles and zeta potential (<)_ Interaction models wherein the kaolinite face is represented by a plate and the platelet edge as a sphere have been developed by Flegmann et al. (ref. 3); these models and associated equations for VR and VA for various interactions have subsequently been used by Vong et al(ref. 5). Herein an attempt has been made to formulate realistic models for kaolinite interactions in which the platelet edge is modelled as a cylinder rather than a
sphere. Despite a number of approximations, these models provide a satisfactory basis for the interpretation of certain features of the rheology of dilute (volume fraction ~0.02) sodium-kaolinite suspensions. Attention is focussed on suspensions at pH 6, 7 and 8 since these pH values appear to embrace the zero point of charge of the edge (ref. 4) and also because it is possible to make reasonable estimates of zeta potential for the face and edge (ref. 6). III.
PARTICLE INTERACTIONS
Depending on the physico-chemical environment, various geometric modes of interaction have been suggested for kaolinite particles, namely face-face (ff), faceedge (fe) and edge-edge fee) modes (ref. 10,15)_ The flat hexagonal plate-like shape of kaolinite is well established from electron microscopy (ref. 15); electron microscopic evidence in support of treating the platelet edge as a cylinder is given typically by Weaver (ref- 16) who observed kaolinite particles with 'rounded' edges. I?atheMatiCalrepresentation of these various interactions are now made using the geometrical simplifications mentioned above. Electrostatic interaction If we regard the face-face interaction of a kaolinite particle as a flat plateflat plate interaction at constant surface potential _I, then VRf is given by (ref. 14):
A"
Vff = R
4nk T Es:exp (- sHof .-r(l+exp(-cH157)
'
where n = number of ions/m 3 , k = Boltzmann's constant, T = absolute temperature, = Tieo:>l/kT, c = Debye-Huckel parameter, Ho = distance between surfaces, w = t0 valency of the counter-ion and e. = electronic charge. In extremum, there are two possible modes of edge-edge interaction, namely an interaction with the axes of the cylinders parallel and an interaction with the axes at right angles. The crossed cylider mode is neglected here in favor of the parallel Mode since the latter results in a deeper net interaction energy at close distance of separation (%l nm) between surfaces_ These remarks are readily substantiated through the use of Eqs_ 2.40, Z-41, 3_1f and 3.23 of Sparnaay (ref. ZO), together with the data of Tables 1 and 2. Thus the edge-edge interaction is approximated as the interaction between two parallel cylinders of equal radii at constant surface potential whence the free energy of interaction is given by (ref. 7):
where R = 2b + Ho, b = radius of cylinder and:
where
Ko( ) = modified
Bessel
function
of
the
second
kind .and zero
order
and argu-
ment ( ). For large
(Kb),
Eq. 3 can be written
using
the
asymptotic
expression: (4)
whence
substitution
of
Eq. 3 into
Eq.
2 gives:
-icH Vee = (2~)~ R For large
c%2kb$
4 I (*-R) z
KR, Eq. 5 can be further
give:
ee _ ZcE(zkb)%$E ‘R
-
e
The interaction interaction
surface potentials .i*1 and 32 is given
‘R =
the
between
a cylinder
by Hogg,
Healy
1.
diagram
Definition
I for
for
H f
and H = the method
a
and face
little
a cylinder
kaolinite
loss
and Fuerstenau (~21 f 0:)(1
19) with
platelet
having
two flat
dimensionless
(ref.
of
and a plate
Now VR for
and a2-
2@1+2 cosech
Definition
Fig.
edge
(y]($-)2 {
shown in Fig.
with
of
accuracy
to
(5)
between
We use Derjaguin’s is
simplified
-
*l
where a1 = -Jeoel/kT
(5)
-?cH ’
(1 f Ho/2b)3i
as the
_
dissimilar
plates (ref.
- coth
18)
H)
for
approximated
(but
surface
constant)
potentials
as: ,
(7)
distance between two plates (kh). fe as fol 1 ows _ The interaction VR
to obtain
a dimensionless
cylinder - pIate interaction.
diagram
with
is
cylinder-plate
interaction.
radius
a (=kb) _
The energy
223
of interaction between the double layers is then given by: m
vfe, a .9
s
V&s
,
(8)
0 where VR is defined by Eq. 7.
The formulation and integration of Eq. 8 is discussed in detail in Appendix I and the resultant expression for Vfe is given as: R 2 (2skb)'@,~2e
'
2 .2 -?Ho - - cl+" -2 e
(9)
i
a_ van der Waals attraction The potential energy of attraction between ~PJO flat plates of thickness 6 is given by Eq. 48, Vervey and Overbeek (ref. 141: 1 (HofZ&)2
(10)
where A = Hamaker constant. For two parallel cylinders, Sparnaay's (ref. 20) result may be approximated (see also f!ahantyand Ninham. ref. ‘21): 3/2 '
(11)
where 2 = length of cylinder_ For the interaction between the face and edge of a platelet, we use the foflokfing approximate cylinder-plate expression: fe -A 'A =m
1 + s2- I)
S2_21/1,2 tan
(12)
where S = (Ho+b)/b_ Eq. 12 is readily derived using Derjaguin's method (Fig. 1) wherein Eq. 10 has been approximated by Vr = - A/127iH; which implies that 6 >> Ho. An equation similar to Eq. 12 has also been derived by Callaghan (ref_ 22) but his formula is in error by a factor of 2_
c.
Total energy of interaction The total energy of interaction as a function of distance of separation for the three modes of interaction is given by:
where US represents f-f, e-e, and f-e interactions.
1. Zeto potential. We assume that the surface potentials (a) required for computation of VT are given by zeta potentials. There are few published data on tbe electrokinetic properties of kaolonite (ref. 2,6,7,23,24)_ These data differ from worker to worker ref’lecting the source of clay, experimental technique and the method of converting electrophoretic mobility to zeta potential, Conversion of 'bulk' electrophoretic data to zeta potentiafs for face and edge of a kaolinite platelet has been attempted by Flegmann et al. (ref. 3) using a polymer-adsorption technique and Williams and Williams (ref. 6), who estimated edge potentials by a linear combination of quartz and x-alumina data. For present purposes the electrokinetic
data
and method of
Williams
and Williams
(ref. 6) are used to estimate zeta potentials at pH 6, 7 and 8 for a range of NaCl concentrations IO-*M- 10-4M. The present authors have obtained additional electrokinetic data on the same kaofinite sample in the presence of IO-TM NaCl (ref, 11); a single particle microeTectrophoresis apparatus (ref. 6) employing palladium electrodes to avoid gassing problems at high salt concentrations was used (ref. 25)_ Values of zeta potential used in Eq. 13 are shown in Table 1. TABLE 1 Zeta
NaCl 1o-4 (M) lO-3 lo-* 10-l
potentials
for
sodium
pH 6 13e 9 7 4
kaolinite
Zeta potential (5) mV PH 8 PH 7 e -14 4 e 3 2 1
-9 -7 -4
6,7L f PH -54 -43 -40 -26
These data suggest that the point of zero electrokinetic potential of the edge ia a letter greater than pH 7 which value is in good agreement with several published values (ref. 4). Other data needed for the computations are listed in Table 2_ In the calculation of "FE* VA is computed with respect to the particle surfaces white V is computed allowing O-5 nm for the distance between the particle surface R and slipping plane and 0.5 nm for the distance of minimum approach of slipping planes. Curves exemplifying the total interaction energy against distance of separation are shown in Fig. 2. Note that Fig. 2b represents conditions close to the point of zero charge for the kaolinite edge. -The results of the computations (for 10-l- 10m4M and pH 6,7,8) may be summarized as follows: 1) there i'sa high potential energy barrier to face-face flocculation; 2) there is no barrier to either edge-edge or face-edge flocculation; 3) at any one pH the depth of the primary minimum VT (calculated for Ho = 1.5 nm) for the f-e interaction decreases
225
with
an increase
in
NaCl concentration
(see
also
Fig. 4); and 4) at a particular pH the depth of the primary minimum for the e-e interaction increases with an increase
in NaCl concentration-
Table 2 Data: Total Energy of Interaction Calculations (VT) Temperature, T = 298'K Boltzmann constant, k = 1.38 x 10mz3 JK-I Elementary charge, e. = 1.601 x 10-lgC Permittivity, E = 7-07 x 10-10 kg-l m-3 A2 sec4 Avogadro's number, NA = 6.0225 x IOz6 k mole-1 Debye-Huckel parameter, .L: = I(2NAI ez)/skTf" = 3.29 x 10' I" m-I Particle dimensions: thickness of kaotinite plate = O-07 rim radius of kaolinite plate = 0.4 urn radius of kaolinite edge (b) = 0.035 ilm length of kaolinite edge (e) = 0.4 urn Hamaker constant, A = 4.4 x 10-20 J_(3)_
Fio,
in
2_
Total
the presence
interaction
of
energy
(VT)
as
a function
of
distance
of
separation
(Ho)
10m4M NaCl; a) pH 6; b) pH 7; c) pH 8.
IV. RHEOLOGY AND PARTICLE INTERACTIONS The present computations differ in detail from those of Flegmann et al. (ref. 31, who suggested on the basis of sphere-sphere models for f-e and e-e interactions (their Fig. 7) that e-e interactions at pH 7 (i.e. a z-p-c. of the edge) were the most likely- Their suggestion stems from the appearance of a primary maximum (1-20kf) for f-e interactions at pH 7 in the presence of 10e4M N&J. Calculations
226 using Eqs_ 3 and 6 of Flegmann et al. (ref- 3) with the present data (Tables 1
and 21 resulted in a
primary maximum of w-25kT, only for the e-e interaction in
the presence of 16-k
NaCl. Although it may be argued that estimation of VT is sensitive, not only to the geometrical shapes chosen for the interacting surfaces but also to the estimates of surface properties, representation of the edge as a sphere is not in accord with the observed shape of the kaolinite particle.
It is
"herefore more appropriate to accept estimates of VT based on the cylindrical
geometry which indicate that interactions are likely in both f-e and e-e modes. Additional evidence is needed to discriminate between these two modes in any given chemical environment, Rand and Melton (ref..3) suggest on the basis of rheological data that the z.p_c_ of the edge is pH 7-O_ From their experimental flow curves, they ctaim rR to be invariant with ionic strength at the z.p.c. of the edge, supporting the suggestion of Flegmann et al. (ref. 3) that the preferred interactions are e-e. For ~b to be invariant with I requires an e-e interaction since only at the z,p.c. wi17 the interaction be mainly due to van der Waals forces. This invariance with I at the z_p.c. of the edge is not observed in the rheological data of Williams and Williams (ref. 26). Their data (shown with additional data of James and W-TTTiams.ref. 11, fn Fig_ 3) clearly demonstrate the decline in 'B with salt concentration at pH values near or at the z.p.c. of the edge (ref. 4)- This variation of 'B with I can be attributed to f-e interactions since Eq. 9 (compare Eq. 6) atlOkJS a variation of 'H with electro'lyteconcentration at the z.p.c_ of the edge. Thus the present rheological evidence in conjunction with the results of Fig. 2 strongly suggest that f-e interactions are preferred at pH 6, 7 and 8,
-.- I
1.0
od r26
I i I
10-1
lo-'
Concentration
lo-’ of
lo-’
100
Sadium Chloride.
Fig, 3_ Residual stress (~~1 as a function of salt concentratSon.
22’7
ihe variation of experimentally measured values of 'H with a theoretically determined VT (Fig. 4) calculated for H = l-5 nm for a f-e interaction (Eqs_ 9 0 and 12) for Na-kaolininte suspensions at a volumetric concentration of 0.02 is in accord with the decline of yB with salt concentration (Fig- 3). According to Fig. 4 a unique relationship exists between r13and VT for 6zpHc8 and 10-4K~Iz IO-% NaCl. At pH ==z_p_c. the effect of increasing I is to reduce the electrostatic attraction between the dissimilarly charged faces and edges (Table I), thus reducing the absolute value of VT (Fig. 4). At pH> z.p.c_ the influence of increasing I is best seen through Eq. 9, wherein it is apparent that VH G $(, ij )_ ef Consequently, VH increases with I despite the reduction associated with the product (Qe@f) (see Table 1 for variation in zeta potential with I). Although Eqs. 9 and 12 provide an interpretation of the relationship between interparticle forces anti 'H which is consistent with our experimental data (Fig, 31, a theoretical representation for TH in terms of VT is not forthcoming at present- It is evident that progress in this direction is hampered by the lack of an adequate theoretical description of the hydrodynamic effects arising during the collisions between floes in a flocculated suspension (ref. 13).
1.6 1X 1.2 1.a
-Ye
pH6
0
pH7
P
pH8
IJ
0.6
(Nm-* 1 0.6 0-L 0.2 0 0
Fig.
-s
-10
-15
VT
IaJ)
-20
-25
-30
4_~~ versus VT for face-edge interaction.
The f-e interaction could conceivably lead to a card-house variety of floe structure (ref. 15); however. an alternative structure seems possible. Consider the relative motion of two kaolinite particles in a velocity gradient at an instant when both piates are aligned parallel with the flow, Then immediately an edge of an overtaking plate passes over and beyond the edge of another plate, it is impossible for flocculation to occur in a stepped or overlapping f-e mode_ Presumably such a structure will develop as a chain of overlapping platelets; once formed such a floe wi?l also be stronger than that due to e-e interactions (compare depths of primary minima in Fig- 2)- The foregoing assumption implies that for NaCl concentrations and pH values in which f-e interactions are preferred, then for a given solids concentration the basic flow units at high shear rates will be floes whose dimensions and structure will not vary greatly with chemical environmentExperimental data support this‘ suggestion with plastic viscosities showing little variation about an average value of 1.55 x 10e3 Nm-'3 for Na-kaolinite suspensions at pff6, 7 and 8 in the presence of 1O-1- IO-'M NaCl (ref. II, 26). Flegmann et al. (ref. 3) also found (their Fig. 3) little variation in u for 5~pH(10 in P2 the presence of 10q4 NaCl_ At low shear rate (-2--O)VJe suppose these flow units to entangle in a complex three-dimensional structure which still involves f-e interactions between units. However, under these conditions of shear the possibility of a card-house structure formed from stepped units is likely_ The observed shearthinning behavior of Na-kaofinite suspensions (I, 3, 4, 9) is thus due to the gradual breakdown of this three-dimensional network with the basic flow unit being produced at sufficiently high values of shear rate. V.
CONCLUSIONS Realistic estimates for the total interaction energy between kaolinite particles
in either f-e or e-e modes are obtained by representing the edge as a cylinder_ For well characterized Na-kao?inite suspensions (ref. II, 26). the extrapolated shear stress in not invariant with NaCl concentration at or near the z.p.c. of the kaolinite edge. This assumption suggests caution in the use of rheological measurements as a means of estimating edge isoelectric points (ref. 4). The various assumptions necessary to obtain estimates of the total interaction energy in kaolinite suspensions must be borne in mind and serve to provide stimulus for further effort both in refining electrical models for the kaolinite platelet and in the analysis of hydrodynamic effects_ VI.
APPENDIX I Eq. 8 has been formulated with the aid of the small angle approximations: case = l- e2/2, sine = e (see Fig. 1) and can be written following the substitution of Eq. 7 for VR as:
229 H)]d% Integration is TaciJitated by some re-arrangement of Eq. -fi eH+e ,Hle-H and
coth H =
cosech H =
2 eH_emH
Al.
(Al) Since:
'
then: 20102
H +
($3: f
Iij)(I-coth
N)
= Ae
IGJhere Wtence
VR
=
)
-H_&$-2H cosech
.-2H
(0: ?-0:) = W2 and 20102 = A/2_ the expression for VR may be written: E”C
I If22f3102e-H-(0; + @;)e-*HI g-
e-*yl _
(l-
0
(A*?
Usins the binomial theorem, we expand (l- e-2H)-1 as follows: e-2:!)
(I_
-1
=
l+e-*H + .--a~ ~ .-fjH + f.. =fi
+e-2mH
m=o
,
whence Eq_ h2 becomes: VR =
2
( I[ T
tT4
20;02e-H - (0; + @,2)e-ZE f e-2mH, 1m=o
(A31
and Eq. Al becomes:
(A4! where h
0
= CH
o-
Let:
(A51 and
Now
I1
can be written
as:
230 II = i
.-hO
-[&a32/Z
c
e-3h0
_ .-3a62f2
+
_ _ _ _]
&
)
0
whence the general term of II can be integrated as: 2 exp - (Zm+I) FdB _ /0
'Ieta2 = (Zm+l)a 2
we
If
P
J
and note that:
!5
=-"2,2 _ ds =&
,
(A7)
I
0
2
then
exp- (2m+ll
f-
%j-- de =
2(2z+l)a
0
I
4
_
Thus II may be written as: 4
Za
co e T
L1 ii
K1 =
eaho
-2mhO/(2m
f
1)’
(ASI
Similarly it can be shown that the general term in Eq. A6 to be integrated is: ,-2(m+ l)h, 0
The use of Eq. 7 allows 12 to be written: 12=s
i
5i -2h, x e m=o
I
e-2mho/(mt I)%._
(PC)
Insertion of Eq. A8 and Eq_ A9 into Eq. A4 gives (after some re-arrangement):
fe “R
2
5(2sub)2$1$2 e
4
The term involving m converges rapidly with ("Ho)>O_5. VII. a A b 3
NOXENCLATWE dimensionless radius of cylinder Hamaker constant radius of cylinder parameter defined by Eq. 2
(Alo)
231
e
electronic charge
ho
hO H
dimensionless distance between two plates shortest distance between surfaces ionic strength
HO I k KV(zI P
m n NA R S T vA vR vT s
distance between two plates dimensionless distance (=xHo)
Boltzmann's constant modified Bessel function of the second kind and order length of cylinder number 3 number of ions/m Avogadro's number (2b + Ho) in Eq. 2 (Ho f b)/b in Eq. 12 absolute temperature van der Waals attractive energy coulombic energy total energy of interaction (=VA + VR)
v and argument z
thickness of plate permittivity Debye-litickel parameter valency of counter-ion plastic viscosity dimensionless potential surface potential Bingham stress zeta potential
V-111. ACKNOWLEDGEF!ENT The authors gratefully acknowledge the financial support of the National Environment Research Council (Contract F60/B4/02). REFERENCES IX. 1 A.S. ?lichaelsand J. C. Bolger, Ind. Eng. Chem., Fund., 1(1962)153. 2 S.K. Nicol and R-J. Hunter, Aust. J. Chem., 23(1970)2177. 3 A.W. Flegmann, J-W. Goodwin and R-H. Ottewill, Proc. Brit. Ceramic Sot., No. 13(1969)31. 4 B-Rand and I.E. Felton, J. Colloid Interface Sci., 60(1977)308. 5 R.N. Yqng, A-J. Sethi, H.P. Ludwig and 11-H.Jorgensen, J. Geotech. Engng. Div., A_S_C_E_,
6 7 8 9
105(1979)1193.
D-J-A_ Williams and K-P. Williams, J_ Colloid Interface Sci., 65(1978)79. A.P. Ferris and W-B. Jepson, J. Colloid Interface Sci-, 51(1975)245. R.J. Hunter and S.K. Nicol, J. Colloid Interface SC-i.,28(1968)250. J-W_ Goodwin, Trans_ Brit_ Ceramic Sot., 70(1971)66_
232 10 II I2 13 I4
R-r<_Schofield and H-R. Sampson. Disc. Faraday Sot., 18(1954)135. A.E. James and D.J.A. k!illiams.in preparation_ G.R, Zeichner and W-R. Schowalter, A.I.Chem.E.J., 23(1977)243. W-8. Russel, J. Rheol. 24(1980)287. E-J-W_ Verwey and J.Th.G. Qverbeek, Theory of the Stability of Lyophobic Cofloids, Elsevier. Amsterdam. X948_ I5 H-67" Olphen, Introduction t& Clay Colloid Chemistry, Interscience, New York, 16 17 18 19 20 21 22 23 24 25 26
C-E.-?feaver,Clays, Clay ?'.inerals, 24(1976)215. A.E. James and D-3-A. !?illaims,J. Colloid Interface Sci., 79(198X)33. R. Hogg, T-G!.Healy and D.W. Fuerstenau, Trans. Faraday Sot., 62(1966)1638. B-V_ Derjaguin. Kolloid 2.. 69(193431X M-J. Sparnaay, Rec. Trav. Chim., 78(1959)680. J. Mahanty and B-G!.Ninham, Dispersion Forces, Academic Press, London, 20, 1376. I. Callaghan, Ph.D. Thesis, University of Bristol, 1975. N. Street and A. 5. Buchanan, Aust. J. Chem., 9(1956)450. R.F_ Packham, J. Co7loid Interface Sci., 20(1956)81. R-A. Neihof, J_ Colloid and Interface Sci., 30(1969)128_ K-P. Williams and D-J-A_ Williams, Proc. Brit. Ceram. Sot.. 1981, in press.
219 in The Netherlands
PARTICLE INTERACTIONS AND RHEOLOGICAL EFFECTS IN KAOLINITE SUSPENSIONS A.E. JAMES'and D.J.A. WILLIAMS" University College, Singleton Park, Swansea SA2 8PP, U-K_
CONTENTS ABSTRACT _._.__________________________________________________________~lg I. INTRODUCTION -_____.__._...-.-___________-___-_______________..__________2lg II. PARTICLE INTERACTIONS __~__________________.__________--___~_...._______.221 III_ A. Electrostatic interaction _____._......._.__________.._..__-_______~21 B. van der Waals attraction -______.__._..._______________.___________223 C. Total energy of interaction ____._....______.__________.___________223 IV. RHEOLOGY AND PARTICLE INTERACTIONS ...................................225 CONCLUSIONS ....................._................__..___......__.___._.2~8 V. APPENDIX I .__________.__.___.._______________.__________~____________.228 VI_ VII. NO~?&~~CLATURE *_*~~___***._.~~*-~****~~*~..*..**~--~_*__~_~___***~___*~ 230 VIII. ACKNOWiEDGE~E~~T..........._._._.___......~..........._.....*___.______231 IX_ REFERENCES ..____..............__________________---___._____.________231 I.
ABSTRACT
?article interactions in kaolinite suspensions are modelled by representing the edge and face of a kaolinite platelet as a cylinder and flat plate, respectively_ Computations of total energy of interaction show that at pH 6, 7 and 8 and in the presence of 10v4 -10-I M NaCl both face-edge and edge-edge modes of interaction are likely. Rheological parameters for flocculated suspensions (extrapolated shear stress and plastic viscosity) for dilute sodium kaolinite suspensions (volometric ccncentration 0.02) are interpreted in terms of the proposed interaction models. INTRODUCTION II. Although considerable theoretical and practical interest has been shown in the rheology of heterogeneous suspensions containing kaolinite (ref. l-5), an unequivocal interpretation of the effect of interparticle forces on their rheological properties has proved elusive. This phenomenon is not entirely unexpected in 'Department of Oceanography, University College, Swansea. " Department of Chemical Engineering, University College, Swansea OOO~-8686/82/000~0000~$03_50 0 1982Elsetier ScientificPublkhingCompa&
220 view
of the complexity of such materials not only in regard to their size and shape, but also because of the difficulty of obtaining reliable electrokinetic data for clay minerals (ref. 3,6)- Difficulties also remain in formulating a satisfactory model for the surface electrical properties of an anisometric heteropolar crystal such as kaolinite (ref_ 7). The pseudoplastic behavior of kaolinite suspensions (ref. 3) suggests that the development of structure associated with clay flocculation is an important factor influencing the rheological behavior of these suspensions and that it is the progressive breakdown of the floe structure with increasing shear rate (v) which gives rise to the shear-thinning behavior noted by a number of investigators (see, e.g., ref_ 1,3,8,g). An analysis of the flocculation process and the precise manner in which flocculation affects flow behavior, particularly in heterogeneous clay suspensions, is a complex problem; heterogeneity may arise due to the presence of impurities such as montmorillonite, mica, muscovite (ref. 10) or by deliberate addition of other clay minerals such as illite (ref. 5) and quartz (ref. 11) in predetermined amounts. Despite the practical importance of such suspensions, relatively few systematic rheological measurements have been reported using well characterized mixtures (ref. 5, II). In general, an analysis of the stability of colloid particles against flocculation in the presence of a velocity gradient requires simultaneous consideration of interparticle and hydrodynamic forces. Such an analysis has at present been done only for the relative motion of two equi-sized, uniformly charged spheres in a laminar velocity gradient (ref. 12). however, particles of kaolinite are anisometric and dissimilarly charged and present seemingly retractable problems in terms of such an analysis. ClearL!y,at the present time simplification is necessary if any progress is to be made- Herein in a first approximation we examine the flocculation process and its influence on rheological parameters such as residual stress (~5) and plastic viscosity (Q_ ) (ref. 13) in terms of the mode of interaction of particles and inter-particleenergies at equilibrium as a function of chemical environment (pH, salt concentration)_ Accordingly, stability is assessed with the aid of a potential energy diagram (ref. 14), in which the free energy of interaction VT (the sum of the van der Waals attraction energy VA and the coulombic energy V,) . is plotted as a function of the distance of separation of tbJ0 particles. Calculation of VT for kaolinite particles requires specification of particle shape, mode of interaction of particles and zeta potential (<)_ Interaction models wherein the kaolinite face is represented by a plate and the platelet edge as a sphere have been developed by Flegmann et al. (ref. 3); these models and associated equations for VR and VA for various interactions have subsequently been used by Vong et al(ref. 5). Herein an attempt has been made to formulate realistic models for kaolinite interactions in which the platelet edge is modelled as a cylinder rather than a
sphere. Despite a number of approximations, these models provide a satisfactory basis for the interpretation of certain features of the rheology of dilute (volume fraction ~0.02) sodium-kaolinite suspensions. Attention is focussed on suspensions at pH 6, 7 and 8 since these pH values appear to embrace the zero point of charge of the edge (ref. 4) and also because it is possible to make reasonable estimates of zeta potential for the face and edge (ref. 6). III.
PARTICLE INTERACTIONS
Depending on the physico-chemical environment, various geometric modes of interaction have been suggested for kaolinite particles, namely face-face (ff), faceedge (fe) and edge-edge fee) modes (ref. 10,15)_ The flat hexagonal plate-like shape of kaolinite is well established from electron microscopy (ref. 15); electron microscopic evidence in support of treating the platelet edge as a cylinder is given typically by Weaver (ref- 16) who observed kaolinite particles with 'rounded' edges. I?atheMatiCalrepresentation of these various interactions are now made using the geometrical simplifications mentioned above. Electrostatic interaction If we regard the face-face interaction of a kaolinite particle as a flat plateflat plate interaction at constant surface potential _I, then VRf is given by (ref. 14):
A"
Vff = R
4nk T Es:exp (- sHof .-r(l+exp(-cH157)
'
where n = number of ions/m 3 , k = Boltzmann's constant, T = absolute temperature, = Tieo:>l/kT, c = Debye-Huckel parameter, Ho = distance between surfaces, w = t0 valency of the counter-ion and e. = electronic charge. In extremum, there are two possible modes of edge-edge interaction, namely an interaction with the axes of the cylinders parallel and an interaction with the axes at right angles. The crossed cylider mode is neglected here in favor of the parallel Mode since the latter results in a deeper net interaction energy at close distance of separation (%l nm) between surfaces_ These remarks are readily substantiated through the use of Eqs_ 2.40, Z-41, 3_1f and 3.23 of Sparnaay (ref. ZO), together with the data of Tables 1 and 2. Thus the edge-edge interaction is approximated as the interaction between two parallel cylinders of equal radii at constant surface potential whence the free energy of interaction is given by (ref. 7):
where R = 2b + Ho, b = radius of cylinder and:
where
Ko( ) = modified
Bessel
function
of
the
second
kind .and zero
order
and argu-
ment ( ). For large
(Kb),
Eq. 3 can be written
using
the
asymptotic
expression: (4)
whence
substitution
of
Eq. 3 into
Eq.
2 gives:
-icH Vee = (2~)~ R For large
c%2kb$
4 I (*-R) z
KR, Eq. 5 can be further
give:
ee _ ZcE(zkb)%$E ‘R
-
e
The interaction interaction
surface potentials .i*1 and 32 is given
‘R =
the
between
a cylinder
by Hogg,
Healy
1.
diagram
Definition
I for
for
H f
and H = the method
a
and face
little
a cylinder
kaolinite
loss
and Fuerstenau (~21 f 0:)(1
19) with
platelet
having
two flat
dimensionless
(ref.
of
and a plate
Now VR for
and a2-
2@1+2 cosech
Definition
Fig.
edge
(y]($-)2 {
shown in Fig.
with
of
accuracy
to
(5)
between
We use Derjaguin’s is
simplified
-
*l
where a1 = -Jeoel/kT
(5)
-?cH ’
(1 f Ho/2b)3i
as the
_
dissimilar
plates (ref.
- coth
18)
H)
for
approximated
(but
surface
constant)
potentials
as: ,
(7)
distance between two plates (kh). fe as fol 1 ows _ The interaction VR
to obtain
a dimensionless
cylinder - pIate interaction.
diagram
with
is
cylinder-plate
interaction.
radius
a (=kb) _
The energy
223
of interaction between the double layers is then given by: m
vfe, a .9
s
V&s
,
(8)
0 where VR is defined by Eq. 7.
The formulation and integration of Eq. 8 is discussed in detail in Appendix I and the resultant expression for Vfe is given as: R 2 (2skb)'@,~2e
'
2 .2 -?Ho - - cl+" -2 e
(9)
i
a_ van der Waals attraction The potential energy of attraction between ~PJO flat plates of thickness 6 is given by Eq. 48, Vervey and Overbeek (ref. 141: 1 (HofZ&)2
(10)
where A = Hamaker constant. For two parallel cylinders, Sparnaay's (ref. 20) result may be approximated (see also f!ahantyand Ninham. ref. ‘21): 3/2 '
(11)
where 2 = length of cylinder_ For the interaction between the face and edge of a platelet, we use the foflokfing approximate cylinder-plate expression: fe -A 'A =m
1 + s2- I)
S2_21/1,2 tan
(12)
where S = (Ho+b)/b_ Eq. 12 is readily derived using Derjaguin's method (Fig. 1) wherein Eq. 10 has been approximated by Vr = - A/127iH; which implies that 6 >> Ho. An equation similar to Eq. 12 has also been derived by Callaghan (ref_ 22) but his formula is in error by a factor of 2_
c.
Total energy of interaction The total energy of interaction as a function of distance of separation for the three modes of interaction is given by:
where US represents f-f, e-e, and f-e interactions.
1. Zeto potential. We assume that the surface potentials (a) required for computation of VT are given by zeta potentials. There are few published data on tbe electrokinetic properties of kaolonite (ref. 2,6,7,23,24)_ These data differ from worker to worker ref’lecting the source of clay, experimental technique and the method of converting electrophoretic mobility to zeta potential, Conversion of 'bulk' electrophoretic data to zeta potentiafs for face and edge of a kaolinite platelet has been attempted by Flegmann et al. (ref. 3) using a polymer-adsorption technique and Williams and Williams (ref. 6), who estimated edge potentials by a linear combination of quartz and x-alumina data. For present purposes the electrokinetic
data
and method of
Williams
and Williams
(ref. 6) are used to estimate zeta potentials at pH 6, 7 and 8 for a range of NaCl concentrations IO-*M- 10-4M. The present authors have obtained additional electrokinetic data on the same kaofinite sample in the presence of IO-TM NaCl (ref, 11); a single particle microeTectrophoresis apparatus (ref. 6) employing palladium electrodes to avoid gassing problems at high salt concentrations was used (ref. 25)_ Values of zeta potential used in Eq. 13 are shown in Table 1. TABLE 1 Zeta
NaCl 1o-4 (M) lO-3 lo-* 10-l
potentials
for
sodium
pH 6 13e 9 7 4
kaolinite
Zeta potential (5) mV PH 8 PH 7 e -14 4 e 3 2 1
-9 -7 -4
6,7L f PH -54 -43 -40 -26
These data suggest that the point of zero electrokinetic potential of the edge ia a letter greater than pH 7 which value is in good agreement with several published values (ref. 4). Other data needed for the computations are listed in Table 2_ In the calculation of "FE* VA is computed with respect to the particle surfaces white V is computed allowing O-5 nm for the distance between the particle surface R and slipping plane and 0.5 nm for the distance of minimum approach of slipping planes. Curves exemplifying the total interaction energy against distance of separation are shown in Fig. 2. Note that Fig. 2b represents conditions close to the point of zero charge for the kaolinite edge. -The results of the computations (for 10-l- 10m4M and pH 6,7,8) may be summarized as follows: 1) there i'sa high potential energy barrier to face-face flocculation; 2) there is no barrier to either edge-edge or face-edge flocculation; 3) at any one pH the depth of the primary minimum VT (calculated for Ho = 1.5 nm) for the f-e interaction decreases
225
with
an increase
in
NaCl concentration
(see
also
Fig. 4); and 4) at a particular pH the depth of the primary minimum for the e-e interaction increases with an increase
in NaCl concentration-
Table 2 Data: Total Energy of Interaction Calculations (VT) Temperature, T = 298'K Boltzmann constant, k = 1.38 x 10mz3 JK-I Elementary charge, e. = 1.601 x 10-lgC Permittivity, E = 7-07 x 10-10 kg-l m-3 A2 sec4 Avogadro's number, NA = 6.0225 x IOz6 k mole-1 Debye-Huckel parameter, .L: = I(2NAI ez)/skTf" = 3.29 x 10' I" m-I Particle dimensions: thickness of kaotinite plate = O-07 rim radius of kaolinite plate = 0.4 urn radius of kaolinite edge (b) = 0.035 ilm length of kaolinite edge (e) = 0.4 urn Hamaker constant, A = 4.4 x 10-20 J_(3)_
Fio,
in
2_
Total
the presence
interaction
of
energy
(VT)
as
a function
of
distance
of
separation
(Ho)
10m4M NaCl; a) pH 6; b) pH 7; c) pH 8.
IV. RHEOLOGY AND PARTICLE INTERACTIONS The present computations differ in detail from those of Flegmann et al. (ref. 31, who suggested on the basis of sphere-sphere models for f-e and e-e interactions (their Fig. 7) that e-e interactions at pH 7 (i.e. a z-p-c. of the edge) were the most likely- Their suggestion stems from the appearance of a primary maximum (1-20kf) for f-e interactions at pH 7 in the presence of 10e4M N&J. Calculations
226 using Eqs_ 3 and 6 of Flegmann et al. (ref- 3) with the present data (Tables 1
and 21 resulted in a
primary maximum of w-25kT, only for the e-e interaction in
the presence of 16-k
NaCl. Although it may be argued that estimation of VT is sensitive, not only to the geometrical shapes chosen for the interacting surfaces but also to the estimates of surface properties, representation of the edge as a sphere is not in accord with the observed shape of the kaolinite particle.
It is
"herefore more appropriate to accept estimates of VT based on the cylindrical
geometry which indicate that interactions are likely in both f-e and e-e modes. Additional evidence is needed to discriminate between these two modes in any given chemical environment, Rand and Melton (ref..3) suggest on the basis of rheological data that the z.p_c_ of the edge is pH 7-O_ From their experimental flow curves, they ctaim rR to be invariant with ionic strength at the z.p.c. of the edge, supporting the suggestion of Flegmann et al. (ref. 3) that the preferred interactions are e-e. For ~b to be invariant with I requires an e-e interaction since only at the z,p.c. wi17 the interaction be mainly due to van der Waals forces. This invariance with I at the z_p.c. of the edge is not observed in the rheological data of Williams and Williams (ref. 26). Their data (shown with additional data of James and W-TTTiams.ref. 11, fn Fig_ 3) clearly demonstrate the decline in 'B with salt concentration at pH values near or at the z.p.c. of the edge (ref. 4)- This variation of 'B with I can be attributed to f-e interactions since Eq. 9 (compare Eq. 6) atlOkJS a variation of 'H with electro'lyteconcentration at the z.p.c_ of the edge. Thus the present rheological evidence in conjunction with the results of Fig. 2 strongly suggest that f-e interactions are preferred at pH 6, 7 and 8,
-.- I
1.0
od r26
I i I
10-1
lo-'
Concentration
lo-’ of
lo-’
100
Sadium Chloride.
Fig, 3_ Residual stress (~~1 as a function of salt concentratSon.
22’7
ihe variation of experimentally measured values of 'H with a theoretically determined VT (Fig. 4) calculated for H = l-5 nm for a f-e interaction (Eqs_ 9 0 and 12) for Na-kaolininte suspensions at a volumetric concentration of 0.02 is in accord with the decline of yB with salt concentration (Fig- 3). According to Fig. 4 a unique relationship exists between r13and VT for 6zpHc8 and 10-4K~Iz IO-% NaCl. At pH ==z_p_c. the effect of increasing I is to reduce the electrostatic attraction between the dissimilarly charged faces and edges (Table I), thus reducing the absolute value of VT (Fig. 4). At pH> z.p.c_ the influence of increasing I is best seen through Eq. 9, wherein it is apparent that VH G $(, ij )_ ef Consequently, VH increases with I despite the reduction associated with the product (Qe@f) (see Table 1 for variation in zeta potential with I). Although Eqs. 9 and 12 provide an interpretation of the relationship between interparticle forces anti 'H which is consistent with our experimental data (Fig, 31, a theoretical representation for TH in terms of VT is not forthcoming at present- It is evident that progress in this direction is hampered by the lack of an adequate theoretical description of the hydrodynamic effects arising during the collisions between floes in a flocculated suspension (ref. 13).
1.6 1X 1.2 1.a
-Ye
pH6
0
pH7
P
pH8
IJ
0.6
(Nm-* 1 0.6 0-L 0.2 0 0
Fig.
-s
-10
-15
VT
IaJ)
-20
-25
-30
4_~~ versus VT for face-edge interaction.
The f-e interaction could conceivably lead to a card-house variety of floe structure (ref. 15); however. an alternative structure seems possible. Consider the relative motion of two kaolinite particles in a velocity gradient at an instant when both piates are aligned parallel with the flow, Then immediately an edge of an overtaking plate passes over and beyond the edge of another plate, it is impossible for flocculation to occur in a stepped or overlapping f-e mode_ Presumably such a structure will develop as a chain of overlapping platelets; once formed such a floe wi?l also be stronger than that due to e-e interactions (compare depths of primary minima in Fig- 2)- The foregoing assumption implies that for NaCl concentrations and pH values in which f-e interactions are preferred, then for a given solids concentration the basic flow units at high shear rates will be floes whose dimensions and structure will not vary greatly with chemical environmentExperimental data support this‘ suggestion with plastic viscosities showing little variation about an average value of 1.55 x 10e3 Nm-'3 for Na-kaolinite suspensions at pff6, 7 and 8 in the presence of 1O-1- IO-'M NaCl (ref. II, 26). Flegmann et al. (ref. 3) also found (their Fig. 3) little variation in u for 5~pH(10 in P2 the presence of 10q4 NaCl_ At low shear rate (-2--O)VJe suppose these flow units to entangle in a complex three-dimensional structure which still involves f-e interactions between units. However, under these conditions of shear the possibility of a card-house structure formed from stepped units is likely_ The observed shearthinning behavior of Na-kaofinite suspensions (I, 3, 4, 9) is thus due to the gradual breakdown of this three-dimensional network with the basic flow unit being produced at sufficiently high values of shear rate. V.
CONCLUSIONS Realistic estimates for the total interaction energy between kaolinite particles
in either f-e or e-e modes are obtained by representing the edge as a cylinder_ For well characterized Na-kao?inite suspensions (ref. II, 26). the extrapolated shear stress in not invariant with NaCl concentration at or near the z.p.c. of the kaolinite edge. This assumption suggests caution in the use of rheological measurements as a means of estimating edge isoelectric points (ref. 4). The various assumptions necessary to obtain estimates of the total interaction energy in kaolinite suspensions must be borne in mind and serve to provide stimulus for further effort both in refining electrical models for the kaolinite platelet and in the analysis of hydrodynamic effects_ VI.
APPENDIX I Eq. 8 has been formulated with the aid of the small angle approximations: case = l- e2/2, sine = e (see Fig. 1) and can be written following the substitution of Eq. 7 for VR as:
229 H)]d% Integration is TaciJitated by some re-arrangement of Eq. -fi eH+e ,Hle-H and
coth H =
cosech H =
2 eH_emH
Al.
(Al) Since:
'
then: 20102
H +
($3: f
Iij)(I-coth
N)
= Ae
IGJhere Wtence
VR
=
)
-H_&$-2H cosech
.-2H
(0: ?-0:) = W2 and 20102 = A/2_ the expression for VR may be written: E”C
I If22f3102e-H-(0; + @;)e-*HI g-
e-*yl _
(l-
0
(A*?
Usins the binomial theorem, we expand (l- e-2H)-1 as follows: e-2:!)
(I_
-1
=
l+e-*H + .--a~ ~ .-fjH + f.. =fi
+e-2mH
m=o
,
whence Eq_ h2 becomes: VR =
2
( I[ T
tT4
20;02e-H - (0; + @,2)e-ZE f e-2mH, 1m=o
(A31
and Eq. Al becomes:
(A4! where h
0
= CH
o-
Let:
(A51 and
Now
I1
can be written
as:
230 II = i
.-hO
-[&a32/Z
c
e-3h0
_ .-3a62f2
+
_ _ _ _]
&
)
0
whence the general term of II can be integrated as: 2 exp - (Zm+I) FdB _ /0
'Ieta2 = (Zm+l)a 2
we
If
P
J
and note that:
!5
=-"2,2 _ ds =&
,
(A7)
I
0
2
then
exp- (2m+ll
f-
%j-- de =
2(2z+l)a
0
I
4
_
Thus II may be written as: 4
Za
co e T
L1 ii
K1 =
eaho
-2mhO/(2m
f
1)’
(ASI
Similarly it can be shown that the general term in Eq. A6 to be integrated is: ,-2(m+ l)h, 0
The use of Eq. 7 allows 12 to be written: 12=s
i
5i -2h, x e m=o
I
e-2mho/(mt I)%._
(PC)
Insertion of Eq. A8 and Eq_ A9 into Eq. A4 gives (after some re-arrangement):
fe “R
2
5(2sub)2$1$2 e
4
The term involving m converges rapidly with ("Ho)>O_5. VII. a A b 3
NOXENCLATWE dimensionless radius of cylinder Hamaker constant radius of cylinder parameter defined by Eq. 2
(Alo)
231
e
electronic charge
ho
hO H
dimensionless distance between two plates shortest distance between surfaces ionic strength
HO I k KV(zI P
m n NA R S T vA vR vT s
distance between two plates dimensionless distance (=xHo)
Boltzmann's constant modified Bessel function of the second kind and order length of cylinder number 3 number of ions/m Avogadro's number (2b + Ho) in Eq. 2 (Ho f b)/b in Eq. 12 absolute temperature van der Waals attractive energy coulombic energy total energy of interaction (=VA + VR)
v and argument z
thickness of plate permittivity Debye-litickel parameter valency of counter-ion plastic viscosity dimensionless potential surface potential Bingham stress zeta potential
V-111. ACKNOWLEDGEF!ENT The authors gratefully acknowledge the financial support of the National Environment Research Council (Contract F60/B4/02). REFERENCES IX. 1 A.S. ?lichaelsand J. C. Bolger, Ind. Eng. Chem., Fund., 1(1962)153. 2 S.K. Nicol and R-J. Hunter, Aust. J. Chem., 23(1970)2177. 3 A.W. Flegmann, J-W. Goodwin and R-H. Ottewill, Proc. Brit. Ceramic Sot., No. 13(1969)31. 4 B-Rand and I.E. Felton, J. Colloid Interface Sci., 60(1977)308. 5 R.N. Yqng, A-J. Sethi, H.P. Ludwig and 11-H.Jorgensen, J. Geotech. Engng. Div., A_S_C_E_,
6 7 8 9
105(1979)1193.
D-J-A_ Williams and K-P. Williams, J_ Colloid Interface Sci., 65(1978)79. A.P. Ferris and W-B. Jepson, J. Colloid Interface Sci-, 51(1975)245. R.J. Hunter and S.K. Nicol, J. Colloid Interface SC-i.,28(1968)250. J-W_ Goodwin, Trans_ Brit_ Ceramic Sot., 70(1971)66_
232 10 II I2 13 I4
R-r<_Schofield and H-R. Sampson. Disc. Faraday Sot., 18(1954)135. A.E. James and D.J.A. k!illiams.in preparation_ G.R, Zeichner and W-R. Schowalter, A.I.Chem.E.J., 23(1977)243. W-8. Russel, J. Rheol. 24(1980)287. E-J-W_ Verwey and J.Th.G. Qverbeek, Theory of the Stability of Lyophobic Cofloids, Elsevier. Amsterdam. X948_ I5 H-67" Olphen, Introduction t& Clay Colloid Chemistry, Interscience, New York, 16 17 18 19 20 21 22 23 24 25 26
C-E.-?feaver,Clays, Clay ?'.inerals, 24(1976)215. A.E. James and D-3-A. !?illaims,J. Colloid Interface Sci., 79(198X)33. R. Hogg, T-G!.Healy and D.W. Fuerstenau, Trans. Faraday Sot., 62(1966)1638. B-V_ Derjaguin. Kolloid 2.. 69(193431X M-J. Sparnaay, Rec. Trav. Chim., 78(1959)680. J. Mahanty and B-G!.Ninham, Dispersion Forces, Academic Press, London, 20, 1376. I. Callaghan, Ph.D. Thesis, University of Bristol, 1975. N. Street and A. 5. Buchanan, Aust. J. Chem., 9(1956)450. R.F_ Packham, J. Co7loid Interface Sci., 20(1956)81. R-A. Neihof, J_ Colloid and Interface Sci., 30(1969)128_ K-P. Williams and D-J-A_ Williams, Proc. Brit. Ceram. Sot.. 1981, in press.