1920
Langmuir 1991, 7, 1920-1922
Criterion for Electrostatic Stability of Dispersions at Low Ionic Strength Ian D. Morrison Webster Research Center, Xerox Corp., Webster, New York 14580 Received January 2, 1991.I n Final Form: March 29, 1991 Dispersions in nonaqueous media are often stabilized by electrostatic charge. DLVO theory, which explains charge stabilization in aqueous media quite well, cannot be applied to dispersions at low ionic strength because the Debye length, a necessary parameter for DLVO theory, is usually not known. A theory suitable for dispersionsat low ionic strength can be obtained when the repulsion between particles is taken as simple Coulombic repulsion rather than the repulsion due to electrical double layer overlap. With a few approximations an analytical expression for the stability ratio is obtained. The criterion for dispersion stability for many nonaqueous, electrostaticallystabilized dispersions can be approximated by the expression Oo2> 103/Da,where the surface potential is in millivolts, the particle radius is in micrometers, and D is the dielectric constant of the medium.
Introduction The stability of a dispersion against flocculationdepends on the rate of particle collisions and the probability of a collision leading to sticking. Smoluchowski derived an expression for the rapid rate of flocculation where interparticle repulsive forces are ignored and particles are assumed to stick on every collision. Fuchs extended the analysis by including the effect of a long range interparticle potential.' For spheres of radius a with a distance dependent interparticle potential, Ubt, the rate of rapid flocculation is reduced by a factor
where R is the distance between particle centers, k is the Boltzmann constant, and T i s the absolute temperature. W is called the stability ratio; the higher the value of W, the more stable the dispersion. The half-lifeof a dispersion is related to its stability ratio by the expression1 31 w t1/2 = 4kT no
where 1 is the viscosity of the medium and no is the initial number of particles per unit volume. The half-life of a dispersion is an experimentally accessible number so that the stability ratio can be determined. What is needed is an understanding of the nature of the interparticle potential. The classic theory for the stability of colloids is due to Derjaguin and Landau2 and Verwey and Overbeek (DLVO).3 The results of DLVO theory are equations that relate the interparticle potential, Ubt, to four important factors: the magnitude of the attractive forces between particles, the sizes of the particles, the magnitude of the surface potentials, and the Debye length, 1 / ~where ,
[
-1 = -xni+:] e2
-112
(3) DtokT i and e is the electronic charge, D is the dielectric constant K
( 1 ) Ross,S.; Morrison, I. D. Colloidal Systems and Interjaces; Wiley:
New York, 1988; p 255f. (2) Derjaguin, B.; Landau, L. Theory of t,hestabilityof strongly charged lyophobic sols and of the adhesion of strongly charged particles in solutions of electrolytes. Acta Physicochim. URSS 1941, 14 (6), 663-667. (3) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: New York, 1948.
of the medium, t o is the permittivity of free space, nio is the number of ions of type i per unit volume far from any surfaces, and zi is the valence of ions of type i. The Debye length is often interpreted as a measure of the thickness of the cloud of counterions surrounding a charged particle. Equation 3 shows the Debye length to be inversely proportional to the square root of the ionic strength of the medium, I
I=' / 2 ~ M i z ~
(4)
i
where Mi is the molar concentration of ions of type i. Therefore the Debye length is essentially related to the ionic strength of the medium. The ionic strength is sometimes determined from the dispersion formulation or sometimes gotten from conductivity measurements if the ionic concentrations, valences, and mobilities are known. For many nonaqueous dispersions, the concentrations, charges, and mobilities of the charge carrying species are not known; therefore neither the ionic strength nor the Debye length can be calculated. The conductivities of many nonaqueous dispersions are quite low, so that the ionic strengths are obviously low and the Debye lengths long. When the ionic strength is small, the Debye length can be larger than interparticle distances for any reasonable particle concentrations. In this case counterions do not form a diffuse layer around individual particles but are more uniformly distributed throughout the solution. In the limit where the counterion distribution is uniform throughout the solution, the interaction of the charged particles can be described by simple Coulomb forces. This is the Hiickel approximation of electrokinetics.4 Ranges of the applicability of the Hiickel approximation have been given.5 van der Minne and Hermanie showed the important role electrostatic charges can have on the stability of nonaqueous dispersions.6~7Koelmans and Overbeekaproposed (4) Hunter, R. J. Zeta Potential in Colloid Science; Academic: New York, 1981; p 69. Hockel, E. Phys. 2. 1924,25, 204. (5) Reference 1, Table C.4, p 349. (6) van der Minne, J. L.; Hermanie, P. H. J. Electrophoresis measurements in benzene-correlation with stability, I. Development of method. J. Colloid Sci. 1952, 7, 600-615. (7) van der Minne, J. L.; Hermanie, P. H. J. Electrophoreaia measurements in benzene-correlation with stability, 11. Results of electrophoresis, stability, and adsorption. J. Colloid Sci. 1953, 8, 38-52.
0 1991 American Chemical Society
Langmuir, Vol. 7, No. 9,1991 1921
Electrostatic Stability of Dispersions a theory for this stability by combining the equations for Coulombic repulsion and Hamaker attraction and calculating the magnitude of the potential barrier as particles approach each other. They showed that 1000-nmparticles and coarser with a Hamaker constant of lO-lg J can be stabilized by surface potentials of the magnitude observed by van der Minne and Hermanie and that smaller particles would be stabilized only with much larger surface potentials. McGown and Parfittg calculated values of W for spheres of equal radius over the range 20-500 (in steps of 20) and 500-1000 nm (in steps of 50) and surface potential over the range 5-80 mV (in steps of 5)) with Hamaker constants of (0.5, 1.0, 2.5, 5.0, and 10.0) X lO-19 J. Both the particle size and the surface potential have a major influence on the stability of the dispersion,but the stability ratio is relatively insensitive to the value of the Hamaker constant.1° McGown et al." have also shown that the 50mV potentials they measure for rutile dispersions in xylene solutions of Aerosol OT are sufficient to stabilize 100-500-nm particles. They used a Hamaker constant of 5x J.
(7)
Debye's derivationcontinued by using eq 6. The derivation here continuesby using eq 7, but the derivationsare similar. When the surface potential is high enough, eq 7 has a maximium that corresponds to a potential energy barrier over which particles must diffuse in order to collide. The total potential can be expanded as a Taylor series around the position of the maximum, SO, and truncated after the quadratic term to give
u,, = uo+ 1/2u/(s- soy
(8) where UOand UO"are the total potential and its second derivative evaluated at SO.The first derivative is zero at SO. Following Debye, the stability ratio, eq 1, can be approximated by
Substituting eq 8 into eq 9 gives
Derivation The purpose of this paper is to derive an analytic expression for the stability ratio for dispersions at negligible ionic strength (thick double layers.) This derivation uses ideas analogous to those of Debye, who analyzed the stability at high ionic strength (thin double layers).I2 The total energy of interaction between two particles is the sum of the attractive force, here estimated by the Hamaker equation,13and the electrostatic repulsion between two charged, insulating sphered4
where A is the Hamaker constant, S = R - 2a, is the shortest distance between particle surfaces, and @po is the surface potential. Debye simplified the electrostaticterm by assuming that significant interactions at high ionic strength (thin double layers) occur only when particles are close, so that the preexponential factor, which is not as dependent on distance as the exponential factor, can be evaluated at R = 2a, and eq 5 simplified to
+ 2~Dc,aq;e-~ 12s This approximation is not appropriate for the interaction of particles at low ionic strength because electrostatic interactions are long range. A more appropriate approximation is to keep the preexponential term and evaluate the exponential factor with an infinite Debye length, that is, K equal to zero. In this case eq 5 simplifies to U,, = --
(8).Koelmam,H.; Overbeek, J. Th. G. Stability and electrokinetic depositionof suspensionsin non-aqueous media. Diecues. Faraday SOC. 1954,18,52-63. (9)McGown, D. N. L.; Parfitt, G. D. Stability of non-aqueous disperaions,Part 5. Theoreticalpredictionsfor dispersionsin hydrocarbon media. Kolloid 2.2.Polym. 1967, 219, 48-51. (10) Parfitt,G.D.;Peacock, J. Stability of colloidal dispersionsin nonaqueous media. Surf. Colloid Sci. 1978, 10, 163-226. (11) McGown, D. N.L.; Parfitt, G. D.; Willis, E. Stability of nonaqueous dispersions I. The relationshipbetween surface potential and stability in hydrocarbon media. J. Colloid Sei. 1965, 20, 650-664. (12) Debye, P. J. W. Stability conditionsfor colloids. In Chu, B. Molecular Forces: Based on theBakerLectures of P. w.Debye; Interacience: New York, 1967. (13) Fbference 1, eq A.21, p 213. (14) Reference 1, eq B.51, p 246.
or
W I!
's" e x p - Uo+ z1U / x 2 ) dx 2a klT(
(11)
0
Integrating gives
V,
1 -TkT Wexp 2a 2u," (kT)
Equation 12 is Debye's. The stability ratio can be and its second calculated once the total potential, UO, derivative, UO", at SOare known. The first derivative of eq 7 evaluated at SOis
where SO= Ro- 2a. The second derivativeof eq 7 evaluated at SOis
The first and second derivatives of the electrostatic repulsion can be evaluated at R = 2a instead of Ro = 2a SOsince SOis small. Therefore eqs 13 and 14 become
+
and
U0" I! - Aa +-TDfo@: 6s:
a
The first derivative is zero at SO,therefore Aa S,2 = 127rDe0O,2 The total potential, eq 7, can be evaluated at SOwith the approximation that R = 2a to give
and the second derivative, eq 16, can be rewritten in the
Morrison
1922 Langmuir, Vol. 7,No. 9, 1991 form
Table I. Surface Potential To Stabilize Particles as a Function of Radius
v,l/=-Aa+-A 6s:
12s:
The position of the maximum, So, is close to the solid surface, so that the ratio of the radius to the position of the maximum, a/&, is large. Therefore in eq 18,the second term dominates, and in eq 19, the first term dominates. Or
U, N +Aa2/6S:
(20)
and
U," N -AaI6St Substituting eqs 20 and 21 into eq 12 gives
(21)
w- -( 1 3 r Aa S t k T) l l 2 e x p ( H ) 2a
surface potential, mV
radius, wm
T
6Sk:
0.01 0.05 0.1 0.3
224
100 71 41
radius, Mm
surface potential, mV
0.5 0.75 1.0 5.0
32 26 22 10
derivation can be proposed based on the approximation that only the electrostatic repulsion is significant for the total energy of interaction. The idea is that the electrostatic repulsion hinders particles from colliding, but if they collide, they stick. One advantage of this approximation is that the stability ratio (eq 1) can be calculated without any further approximations. Assume that
(22) Substituting eq 26 into eq 1 gives
Substituting eq 17 into eq 22 gives
w = 2a$2,
meXp(
4r~t~a~9: RkT
) dR
R2 This is the expression sought, the stability ratio in terms of material-dependent quantities.
Example If the temperature is 298 K, the Hamaker constant is in joules, the radius is in micrometers, and the surface potential is in millivolts, then
W
(1.67 X 10')
(D3at3:)1' exp((1.35 4
X
Equation 27 can be integrated directly to give
W=
Daq: > lo3 (a in pm and 9, in mV) (25) as the criterion for stability. This expression can be used to specify the surface potential necessary to stabilize a dispersion as a function of particle radius. For hydrocarbons, D = 2, and typical values of the necessary surface potential as a function of radius are shown in Table I. The values in this table compare well with experimental evidence.6-11J5
Alternate Derivation Equation 23 shows that the stability ratio depends weakly on the Hamaker forces of attraction. An alternate (15) Kornbrekke, R. E.; Morrison, I. D.; Oja, T. Electrophoretic mobilities of particles in low conductivity media. Submitted for publication in Longmoir.
(2rDeoa(po2 kT )[ exp( 2r%")
- 11
(28)
Evaluating eq 28 at 298 K and dropping the insignificant second term give
(4) Dave exp((1.35
W=
10-2)Da(p,2)
(24) Equation 24 shows the stability to be a strong function of particle size and surface potential but a weak function of the attractive force. As the Hamaker constants for many nonaqueous dispersions are of the order of J and the stability ratio depends on the 4th root, not much error is introduced by assuming this value as a general approximation. Furthermore, if a stable dispersion is defined as one whose stability ratio, W, is greater than some large value, a commonly chosen one is lo5, then the necessary value of Dacpo2 for stability can be calculated from eq 24 to give
(27)
X
(29)
lO-')Da(p,2)
where the radius is in micrometersand the surface potential is in millivolts. Again with a stability ratio of 105 chosen as the criterion for stability, eq 29 can be used to calculate the necessary value of Dam2 for stability Dav: > lo3 (a in pm and 3, in mV) Exactly as found before!
(30)
Conclusion The stability ratio for electrically charged, insulating spheres dispersed in a medium of negligible ionic strength can be approximated by
W-
(
3072D3e:a33,6
)
2rDc0a(p,2 )lI4exp(
kT
(23)
or by
If lo5 is chosen as the minimum value of W above which dispersions are stable, then for many common nonaqueous dispersions both these approximations give
> 103/Da (26) as a good approximation relating the surface potential in millivolts needed to stabilize particles as a function of their radius in micrometers.
