КОЛЛОИДНЫЙ ЖУРНАЛ, 2007, том 69, № 4, с. 496-500
УДК 541.18
THE INTERACTION ENERGY BETWEEN IDENTICAL PARTICLES © 2007 Genxiang Luo*, Qingdao Wang**, Haoping Wang*, Jun Jin***
*Department of Chemistry, Liaoning University of Petroleum & Chemical Technology Fushun, Liaoning 113001, PR China **Vocational &Technology College, Liaoning University of Petroleum & Chemical Technology
Fushun, Liaoning 113001, PR China ***Beijing East Heavy Oil Technical Development Ltd. Beijing 100081, PR China Поступила в редакцию 25.09.2006 г.
The improved Derjaguin's method is used to derive the approximate expressions for the electrostatic interaction energy between spherical colloidal particles. The approximate results are in good agreement with the exact numerical solutions. The results from the product of original Derjaguin's approximation with the curvature correction, 2a/R, can be satisfactorily used at different Ka and Kh, no matter high or low the potential of spherical particle is.
INTRODUCTION
The interaction energy between colloidal particles provides the basis for the theoretical understanding of a wide rang of colloidal phenomena, including particle deposition, heterocoagulation, and transport of colloidal particles and macromolecules through porous media. Frequently, calculations of the interaction energy are performed using the classical DLVO theory [1, 2] for simple geometrical shapes, like plane parallel surfaces, spheres and cylinders. The interaction energy between two plane parallel plates has been considered in detail in [1-5]. Nevertheless, colloidal particles are not infinitely large flat plates in reality, and it seems worthy to consider them as spheres. The interaction of two spherical particles will give a very good approximation for reality, irrespective of the shape of the real particles.
Although the sphere is also a simple geometrical shape, for the spherical particles, the fundamental differential Poisson-Boltzmann (PB) equation could not be solved explicitly yet, except for the Debye-Huckel approximation. Generally, the interaction energy between two spheres has been determined in an approximate form, or by a full numerical solution of a nonlinear partial differential equation. At present, there exist only two approximate approaches, i.e. Derjaguin's method [6] and the linear superposition approach introduced by Bell et al. [7].
The problem of obtaining the numerical solutions of the nonlinear PB equation has a long time [8-12]. The energy of interaction of spheres was calculated using the bi-spherical coordinates and finite difference method. However, the numerical solution of PB equation is a difficult computational problem, one requires specialized skills. Therefore, attempts have been made to improve the present approximate approaches, i.e. Derjaguin's method and the linear superposition.
In Derjaguin's method the interaction between spheres is built up from the interaction of pairs of rings with radius h, thickness dh at a distance H. The rings interaction is calculated as if they are parallel flat surface with the same area and distance. Thus, the interaction energy between two spherical colloidal particles, VS, calculated by Derjaguin's method is represented by
Vs =
2na, a2 a, + a
J VR (h) dh.
(1)
Here VR(h) is interaction energy per unit area of two parallel plates, a1 and a2 are radii of the spheres,
a, a2
is the geometrical Derjaguin's factor equal to
0.5a for identical spheres. Due to recent interest in interaction of anisotropic particles, Derjaguin's method was generalized by White [13] and Adamczyk et al. [14-16] to convex bodies. Generalized Derjaguin's factor includes four principal radii of curvature, it is very inconvenient to use it in the three dimensional situations, except for the crossed-cylinder problem, when
it equals to —R, R2, where P is the angle formed
by cylinder axes and Rj, R2 are radii of the cylinders.
After 60 years, Papadopoulos et al. [17] proposed a method to improve the old approximate assumption suggested by Derjaguin's in 1934. The principal feature of Papadopoulos's improvement is that two interacting rings behave as two parallel ones, divided by a distance equal to not a straight-line segment, but to the arc of a circle, which is perpendicular to both rings. Their method has inherited the approximations induced by Derjaguin's method and is expected to give good results for large
H
2
Ka and small kH (k is Debye-Huckel parameter.). Wang et al. [18] gave a correction to the classical Derjaguin's method and obtained simple analytic expressions for the case of low surface potentials of two spheres (improved Derjaguin's method). The expressions can be satisfactorily applied to the case of small Ka and large kH.
The purpose of this paper is to derive approximate expressions for interaction energy between two spherical particles using improved Derjaguin's method. Since Derjaguin's approximation tends to overestimate the energy at large separations and the linear superposition tends to overestimate the energy at small separations, we attempt to combine Derjaguin's approximation with the linear superposition approximation. In this paper we confine ourselves to the interaction under the assumption of constant potential at the particle surfaces in a symmetrical electrolyte solution.
elementary electric charge, the k is the Boltzmann constant, T is the absolute temperature, e0 is the electric constant, £r is the dielectric constant of the solution.
Generally speaking, Eqs. (2) are applicable to the case of the plates with high surface potentials. However, the equations have limitations. The equations can be only used at Kh < 2n and accurate location is at Kh < 4. Equations (2) are divergent when h —- Therefore we can't directly to calculate the interaction energy between two spherical colloidal particles by Derjaguin's method. We note that zero point value of the Eqs. (2) is at Kh ~ 2n, thus in order to avoid divergence of the Eq. (1) it is necessary to integrate VR from Kh to 2n, namely
2n
2naa r T. , , S--— VRdKh.
S a1 + a2 J
K =
(1a)
SPHERICAL PARTICLES WITH HIGH SURFACE POTENTIAL
The advantage of the Derjaguin's method is that it can be applied to both linear and nonlinear regimes of plate interactions. For plate particles with high surface potentials we simplified the Poisson-Boltzmann equation on the basis of Langmuir's suggestion [19], and derived interaction energy and force equations between two dissimilar plane parallel plates with high surface potential or high surface charge in the case of constant surface potential or constant surface charge. Agreement with the exact numerical values of the interaction of dissimilar or similar plates is good [20-22]. For the energy of interaction of dissimilar plates in the case of constant potentials we have
2nkT I Vr = —— l2n
K
1
1
Kh + b 2 n + b
(2a)
1 ;[(Kh + b)3- (2n + b)3] + (Kh -2n)
24 n2
Substituting Eqs. (2a) and (2b) into Eq. (1a), we obtain respectively
T, 2nnkTA L ; V? = -;— 1 2 n
k I 1
ln
2n + b\ 2n- kH
kH + b) 2 n + b
96 n2
[( 2 n + b )4-(kH + b )4-
(3a)
- 4(2n + b)3(2n - kh)] - j-(kh-2n)2},
Vs =
2nnkTA L 2 -2— 1 2n
k I 1
ln
2n + b \ 2n - kH"
kH + b) 6.9 + b
96n2
[( 2 n + b )4-(kH + b )4-
(3b)
- 4(2n + b)3(2n - kH) ] - 2(kH- 2n)2}.
or
Vr =
2nkT\ 2 ( 1 1
'{ n Uh + b 6. 9 + b
K
24 n2
(2b)
1 ; [(Kh + b )3 - (2n + b )3 ] + (Kh - 2n) L
Here b = 2(e yi'2 + e yi'2), andyt = (i = 1 or 2; when Ji = y2, we define the surface potential as y0) is the scaled
,,, 2 2x1/2
12nu e \
surface potential of the plates, k = -— is the De-
ler £q kT J
bye-Huckel parameter, Kh is the dimensionless distance of two plates, ^ is the surface electric potentials, e is the
2 a1 a2
Here A =- is the reduced particles radius. Equa-
a1 + a2
tion (3b) has better precision except that kh approaches zero.
The interaction energy between identical spheres with constant surface potentials y0 = 4, 5, 6 and 8 is sketched in Fig. 1. Verwey and Overbeek [1] and Honig et al. [4] have given some results of repulsive energies between two spherical particles at various potentials and distances using exact equation of two parallel plats (it contains four elliptical integrals.) by Derjaguin's method. We recalculate the values of repulsive energies between two spherical particles, and found that the results calculated by Honig et al. [4] are basically exact and the results calculated by Ver-wey and Overbeek [1], limited by condition of the time, have many errors. In the following we use the values of the interaction energy which are recalculated between two identical spheres at the case of constant surface potential.
2
5 KOnnOHAHblH XyPHAn tom 69 < 4 2007
LUO et al.
I 0.7
Fig. 1. The dimensionless repulsive energy vs distance for two identical spheres with different surface potential: 1 -y0 = 4, 2 - 5, 3 - 6, and 4 - 8. Solid lines are calculated by Eq. (3b), symbols are the exact values calculated by Eq. (10).
03 ■ 4
Fig. 3. The relative dimensionless repulsive energy vs distance calculated using Eq. (9) (1, 2) and (5) (3) at y0 = 2 (1), 4 (2) and Ka = 5. The exact values (4) are from Ref. [9].
We carried out the calculations at kH < 3, and the relative errors are maximal at kH = 0 and equal to 7.3, 4.9, 3.3, and 1.8%, respectively. Equations (3) can be only used at kH < 3, that is limited by Eqs. (2).
Verwey and Overbeek [1] indicated that two approximations are introduced in the deduction of original Derjaguin's method, both of which tend to make the value of VS higher. Firstly, the upper limit of integration is put equal to H = whereas the highest value having any physical sense should be H = 2a + H0. Secondly, the surfaces of the spheres are not parallel to each other, as was assumed in the deduction. Wang et al. [18] gave
I
0.7
0.5 1.0 1.5 2.0 2.5
Fig. 2. The relative dimensionless repulsive energy vs distance calculated using Eq. (9) at Ka = 5 (1), 1 (2) and y0
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