Langmuir 2001, 17, 2167-2171
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Strong Interaction of Colloidal Particles: An Extension of Langmuir’s Method Luo Genxiang,† Wang Hao Ping,*,† and Jin Jun‡ Department of Materials Science, Fushun Petroleum Institute, Fushun, Liaoning, 113001, People’s Republic of China, and Beijing East Heavy Oil Technical Development Ltd., Beijing, 100081, People’s Republic of China Received September 29, 2000. In Final Form: January 4, 2001
According to Langmuir’s suggestion, that is, sinh y ≈ ey/2 in the nonlinear Poisson-Boltzmann equation for high surface potentials of the particles, we derive simple approximate expressions for the repulsive energy and force between two dissimilar plates with constant high surface potentials. By use of Derjaguin’s method and the improved Derjaguin method, the expressions of the repulsive energy between two dissimilar spheres with constant high surface potentials have been derived. These formulas are applicable in the regime of a repulsive system and can only be used at κh < 2π, and the accurate location is at ∼κh < 4. These formulas are considerably in agreement with the exact numerical values of the interaction of dissimilar plates given by Devereux and de Bruyn for high surface potentials.
Introduction Derjaguin1 first solved the Poisson-Boltzmann equation (PB) for a system of two dissimilar charged plates under the boundary condition that the surface potential of the plates remains constant, independent of plate separation. The interaction force was obtained in terms of elliptic integrals. Afterward, Hogg, Healy, and Fuerstenau2 (HHF) derived a simplified formula for the interaction of two dissimilar plates at a constant surface potential using the Debye-Hu¨ckel linear approximation to the PB equation. The formula of HHF gives a good approximation to the exact values for low potentials. Ohshima, Healy, and White3 made corrections to the fourth and sixth powers of the surface potential in the HHF formula for the doublelayer interaction at a constant potential between two dissimilar plates. These formulas, which are applicable for low and moderate potentials, are tedious, however. Using the concept of “the effective surface potential”,4 we derived the approximate expressions of the interaction free energy and force, valid up to the low to the moderate potential regime.5 Recently, Ohshima has proposed a novel linearization method for simplifying the nonlinear PB equation to derive accurate analytic expressions for the interaction energy between two identical parallel plates.6-8 However, the equations work well only for small separations unless the potential is low, and the equations are limited to the case of the low to the moderate potential regime. More complete and detailed treatments were done * To whom correspondence should be addressed. E-mail:
[email protected]. † Fushun Petroleum Institute. ‡ Beijing East Heavy Oil Technical Development Ltd. (1) Derjaguin, B. V. Discuss. Faraday Soc. 1954, 18, 85. (2) Hogg, R.; Healy, T. W.; Fuerstenau, D. W. Trans. Faraday Soc. 1966, 62, 1638. (3) Oshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 89, 484. (4) Sader, J. E.; Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1995, 171, 46. (5) Wang, H. P.; Jin, J. J. Colloid Interface Sci. 1996, 177, 380. (6) Ohshima, H. Colloid Surf., A 1999, 146, 213. (7) Ohshima, H. J. Colloid Interface Sci. 1999, 212, 130. (8) Ohshima, H. J. Colloid Interface Sci. 2000, 225, 204.
by Devereux and de Bruyn9 and McCormack, Carnie, and Chan.10 On the basis of the nonlinear PB equation, their work provided extensive numerical tabulations of the force and interaction free energy per unit area between dissimilar plates under a constant surface potential9 and presented a general algorithm for calculating the doublelayer force and interaction energy between two dissimilar plates with constant potentials, constant charge, or in mixed cases.10 The problem of determining the interaction energy between two dissimilar charged plates under high surface potential conditions has received comparatively little attention. There is no simple expression applicable to the interaction force and interaction energy except numerical analysis. This lack of attention is partly associated with mathematical difficulties. Langmuir11 suggested that hyperbolic sine was simplified to an exponential function for the interaction of identical flat plates at high surface potentials. Further, he obtained the relation among midway potential, surface potential, and plate separation; however, an approximate expression had not been derived. It is the purpose of this paper to consider the same problem as Langmuir did and extend it to the interaction of two dissimilar parallel flat plates and of two spheres at constant high surface potentials. In this paper, we confine ourselves to interaction under the assumption of constant potential at the particle surfaces. Interaction of Two Dissimilar Plates at High Surface Potential Figure 1 gives a schematic presentation of the system under consideration. Two parallel plates are h apart in a symmetrical electrolyte of valence ν. An x-axis is perpendicular to the given plates, and one plate located at its origin. (9) Devereux, O. F.; de Bruyn, P. L. Interaction of Plane Parallel Double Layers; MIT Press: Cambridge, MA, 1963. (10) McCormack, D.; Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1995, 169, 177. (11) Langmuir, I. J. Chem. Phys. 1938, 6, 893.
10.1021/la0013796 CCC: $20.00 © 2001 American Chemical Society Published on Web 03/08/2001
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Case 1: y is monotonic. Because y1 < y2, y is a monotonic increasing function, that is, dy/dξ > 0 for 0 e x e h. In other words, we choose the positive square for 0 e x e h to yield explicitly
∫yy
dy
2
xe
1
y
) κh
(7)
+C
This system is attractive. Case 2: y is not monotonic. In this case, the derivative of y must vanish at an interior point. Let this point be x ) s, namely,
(dξdy)
)0
ξ)κs
Figure 1. Schematic representation of the dimensionless potential profile between two dissimilar plates with a potential minimum u located at s.
In this case, the integral constant C ) -eu; u is the dimensionless potential in the minimum, and u ) eνψmin/kT at ζ ) κs. Note that when y1 ) y2, s ) h/2 for any h. Thus, for 0 e x e s the solution y is decreasing monotonically, whereas for s e x e h it is increasing. This system is repulsive. A simple manipulation gives
∫uy
Then, the PB equation for potential relative to the bulk solution is
νeψ d2ψ 2νen ) sinh 2 kT dx 0 r
(1)
where e is the elementary electric charge, n is the electrolyte concentration, 0 is the permittivity of vacuum, r is the relative permittivity of the solution, k is the Boltzmann constant, and T is the absolute temperature. We can arrange that the surface potentials ψ1 and ψ2 of the plates 1 and 2, respectively, are constant and ψ2 is always greater than ψ1, which has the same sign as ψ2. Equation 1 may be simplified by introduction of the dimensionless parameters y and ξ:
ξ ) κx
νe ψ yi ) kT i
1 d2y ) sinh y ) (ey - e-y) 2 dξ2
dy
1
xe
y
+
-e
u
(3)
∫uy
dy
2
xe
y
) κh
(9)
- eu
The general solution of eq 9 then becomes
1 tan-1(xey2-u - 1) + tan-1(xey1-u - 1) ) κheu/2 2
(10)
In eq 10,
eyi-u . 1 then eyi-u - 1 ≈ eyi-u
(10-1)
Let us expand the arctangent in a power series:
tan-1(xeyi-u) ≈
(2)
where κ is the Debye-Huckel parameter and i is 1 or 2. Substitution of those parameters in eq 1 shows that
(8)
1 π 2 x yi-u e
i ) 1 or 2
(11)
Equation 11 may then be substituted into eq 10 without any further assumptions or approximations and then manipulated and simplified to give a quadratic equation:
[
]
κh2 - (e-y1/2 + e-y2/2)2 eu - πκheu/2 + π2 ) 0 4
(12)
Abandoning the illogical root, we obtain
The boundary conditions are
x ) 0 ψ ) ψ 1 x ) h ψ ) ψ2
(4)
For present purposes, yi . 1; Langmuir11 neglected e-y in eq 3 and obtained
d2y 1 ) sinh y ) ey 2 dξ2
(5)
eu/2 )
2π κh + 2(e-y1/2 + e-y2/2)
(13)
or the dimensionless potential in the minimum
u ) 2 ln
(
2π -y1/2
κh + 2(e
)
+ e-y2/2)
(13-1)
The repulsive force per unit area (or disjoining pressure), P, is given by Langmuir11 as
Equation 5 is integrated:
(dξdy) ) ( xe + C y
(6)
where C is an integral constant. Hence, one must consider two cases:
P ) 2nkT(cosh u - 1)
(14)
Combining eq 12 with eq 14, we find that the repulsive force per unit area of two dissimilar plates with high
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Langmuir, Vol. 17, No. 7, 2001 2169
Figure 2. The nondimensional force (P′, solid line) and the minimum potential (u, dashed line) between two dissimilar plates with reduced potentials, y1 ) 10 and y2 ) 20, as a function of the reduced separation κh.
surface potentials is
[
P ) nkT
2π
κh + 2(e
-y1/2
+ e-y2/2) 1 (κh + 2(e-y1/2 + e-y2/2)) 2π
]
2
Figure 3. The dimensionless repulsive energy (dashed line) and force (solid line) between two dissimilar plates with constant surface potentials y1 ) 5 and y2 ) 10 as a function of the reduced separation κh. Symbols (0, 2) are the exact values given by ref 9.
(15)
For case 1, the solution of eq 7 is
tanh
(
)
C1/2(xey2 + C - xey1 + C) κhC1/2 )0 2 y2 y1 (e + C)(e + C) C x
(16)
The repulsive force per area in constant surface potentials is P ) -(C + 2);6 however, eq 16 is a transcendental equation for C and we cannot derive a simple analytic solution for C. To solve the equation is not as easy as numerical integration. The Results and Discussion The free energy of interaction per unit area ought to be
VR )
∫h∞P dh
(17)
The approximation sinh(y) ≈ ey/2, which is applied to the one-dimensional PB equation, is exact only in the region near the plate surface where yi . 1. However, in the region distant from the surface of the plate eq 5 is not exact. So, u and P are divergent at κh f ∞, so eq 17 would be divergent if eq 15 should be substituted (see Figure 2). From eqs 11 and 15, we find u ) P ) 0 at κh ≈ 2π, which is independent of the values of yi (i ) 1 or 2) in the main. This predicates the upper limit of the integration as equal to h ) 2π; then, the integration of eq 17 can converge. Thus, we obtain
VR )
{ (
1 2nkT 1 1 2π2 [(κh + b)3 κ κh + b 2π + b 24π2
)
(2π + b)3] + (κh - 2π)
}
{ (
(2π + b)3] + (κh - 2π)
)
zi - z × 100 z
(19)
The dimensionless force and energy have been defined by
P1 )
1 2nkT 1 1 2π2 [(κh + b)3 κ κh + b 6.9 + b 24π2
)
where b ) 2(e-y1/2 + e-y2/2). By replacement of one 2π with 6.9, eq 18 has better precision. Because u and P are divergent at κh f ∞, eqs 15 and 18 cannot be applied for large κh. However, large κh goes beyond the range of the strong interaction. Figures 3 and 4 make the comparisons of eqs 15 and 18 with the exact numerical values of the interaction of dissimilar plates given by Devereux and de Bruyn for high surface potentials.9 The percentage relative error has been defined by
(17-1)
or
VR )
Figure 4. The dimensionless repulsive energy (dashed line) and force (solid line) between two dissimilar plates with constant surface potentials y1 ) 10 and y2 ) 20 as a function of the reduced separation κh. Symbols (0, 2) are the exact values given by ref 9.
}
(18)
P nkT
V1 )
κVR 2nkT
(19-1)
Whereas zi is the value calculated with eq 19-1, z is the exact numerical value from ref 9. The agreement of the approximation, eq 15, with the exact numerical results is good. For y1 g 5 and κh e 3.0,
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Table 1. Deviations of Equation 18 for Two Identical Plates at Kh f 0 relative errors (%) κh
y ) 10
y ) 20
0 0.001 0.002 0.01 0.02
23.76 21.5 19.5 9.6 4.5
23.37 0.07 ∼0 ∼0 ∼0
the relative errors of eq 15 are less than 5%. For κh e 3.5, the relative errors are less than 10%. For the interaction energy, eq 18, the applicable range extends to κh ) 4.0. Beyond this range, the error increases rapidly. The higher the surface potential, the better the precision of eqs 15 and 18. When the distance between the plates is made smaller, a point is reached where the minimum in the potential falls on the low potential surface, y1. Then, its surface charge is zero; maximum repulsive force is reached. In this case, κh is
κhmin )
2π - 2(1 + e e
(y1-y2)/2
y1/2
)
≈ 2(π - 1)e-y1/2
(20)
At y1 ) 5, κhmin ) 0.352; at y1 ) 10, κhmin ) 0.0288. This is the applicable upper limit of eqs 15 and 18. For very high surface potential y1, y2 . 1 and κhmin f 0, so eqs 15 and 18 may be used safely for practically all values of κh. It must be pointed out that for two identical plates the system is always repulsive, and maximum repulsive energy is reached at κh ) 0; for this case, the error of eq 10-1 is large, so the equations of interaction energy and interaction force cannot give good results at κh ∼ 0. Table 1 shows this feature clearly. We can apply Derjaguin’s method12 or the improved Derjaguin method13 to derive expressions for the interaction energy of two dissimilar spheres at κh e 4.0. Consider two spheres, having radii a1 and a2 and surface potentials ψ1 and ψ2, respectively. Let H0 be the shortest distance between the two spheres and assume that ψ1 and ψ2 are constant, independent of H. By use of Derjaguin’s method, viz.,
VS )
2πa1a2 a1 + a2
∫H∞ VR dh 0
(21)
The upper limit of the integration of eq 21 is put equal to H ) ∞, whereas the highest value having any physical sense should be H ) a1 + a2 + H0; the value ∞ is chosen as it gives the most simple expression.14 This approximation tends to make the value of VR too high. However, when κh f ∞, eq 18 is still divergent. We note that the zero point value of eq 18 is still at κh ≈ 2π; thus, to avoid divergence of eq 21 it is necessary to integrate VR from κH0 to 2π, namely,
VS )
2πa1a2 a1 + a2
∫κH2π VR dh 0
(22)
(12) Derjaguin, B. V. Kolloid-Z. 1934, 69, 155. (13) Wang, H. P.; Jin, J.; Blum, L. Colloid Polym. Sci. 1995, 273, 359. (14) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the stability of lyophobic colloids; Elsevier: Amsterdam, 1948.
Figure 5. Comparison of numerical results of Honig and Mul (O) with present results of eqs 23 (solid line) and 26 (dashed line) for spherical particles with y ) 10 and κa ) 10.
Substituting eq 18, we obtain
VS )
4πkTa1a2 2
{ [(
)
]
2π - κH0 2π + b κH0 + b 6.9 + b
2π2 ln
κ (a1 + a2) 1 [(2π + b)4 - (κH0 + b)4 - 4(2π + b)3(2π 96π2 1 κH0)] - (κH - 2π)2 (23) 2
}
where b ) +e ). In particular, the force, f, between two large spheres at the distance of closest approach, H0, is simpler than the interaction energy. It is 2(e-y1/2
-y2/2
f)
2πa1a2 V a1 + a2 R
(24)
where VR is from eq 18. Up to now, numerical computer calculations of the interaction energy between two dissimilar spheres with high potentials on the basis of the PB equation have not been authoritatively reported. Using Derjaguin’s method, Honig and Mul15 obtained the repulsive energy of two spheres with equal radii a for which surface potential is y ) 10 by numerical integration. In Figure 5, we show the comparison of the present results with those of Honig and Mul. The greatest deviation is equal to 1.8% in κH0 e 1, and the errors quite rapidly increase when κH0 > 2. For example, at κH0 ) 2 the error is 4.3%; at κH0 ) 2.4, 7.5%; at κH0 ) 2.8, 12.5%; at κH0 ) 3.2, 19.8%. The applicable range of eq 23 is narrower than that of eqs 15 and 18. The reason is that 2π < κH ) κ(a1 + a2 + H0). The improved Derjaguin method11 cannot extend the useful range of eq 23, and the precision of eq 23 may be improved in larger κH (see Figure 5).13 Equation 23 is applicable only to repulsive systems, and we note that in the case of y1 ) y2, s ) h/2 for any h, and a minimum ymin ) u always exists. Thereinafter, we derive a formula of the interaction energy between two identical spheres with high surface potentials by the improved Derjaguin method. According to the improved Derjaguin method,
VS ) πa
1 κh + dh ∫H2π VR(1 - 2κa 2κa)
(25)
Substituting eq 18, we obtain a dimensionless repulsive energy S: (15) Honig, E. P.; Mul, P. M. J. Colloid Interface Sci. 1971, 36, 258.
Strong Interaction of Colloidal Particles
S)
Langmuir, Vol. 17, No. 7, 2001 2171
κ2VS ) S1 + S2 + S3 2πankT
[
(26)
π2 B (2κa + b + 1) ln S1 ) κa κH + b 2π(2κa + π + b + 1) ... + 6.9 + b κH(2κa + 2π + b - 0.5κH + 1) 6.9 + b
(
[(
)[
)
]
(26-1)
4 -1 1 B4 (κH + b) 1+ - B3(2π 2 2κa 4 4 24π 5 B3 B2 bB 1 (κH + b) - 2π2 κH) ... + 2κa 5 4 2κa 5 4 (κH + b) b B3κH2 (26-2) 4 2
S2 )
]
S3 )
(
)
[
]]
2 2 2π3 (κH - 2π) (1 + 2κa) κH (κH - 3π) + 3κa 4κa 6κa (26-3)
where B ) 2π + b which is a constant in the case of constant surface potential. Figure 5 shows a comparison of the
results of Honig and Mul with those from eqs 23 and 26. Conclusions We have presented a simple method of calculating the interaction force and energy per unit area between two dissimilar plates with high potentials at a constant surface potential. By use of Derjaguin’s method and the improved Derjaguin method, the expressions for the interaction free energy between two dissimilar spheres with high surface potentials have been derived. These formulas should be applicable to the case of a repulsive system; namely, in this case the derivative of y must vanish at an interior point, and a minimum ymin ) u always exists. The repulsion or attraction between dissimilar planar surfaces exists a turning point at ∼κh ≈ 2(π - 1)e-y1/2. These formulas are divergent when κhf∞, and the zero point is at κh ≈ 2π. This means that they can be used only at κh < 2π, and the accurate location is at ∼κh e 4. The condition of the strong interaction has been met. When the distances between two dissimilar plates with high potentials is large, the Debye-Hu¨ckel approximation may be exact for eq 3. This belongs to the area of weak interactions of plates with high potentials; we shall undertake detailed consideration of such a system elsewhere. LA0013796
