Langmuir 1998, 14, 5383-5388
5383
Electrical Interactions between Two Charged Spheroids in a Symmetric Electrolyte Solution Jyh-Ping Hsu* and Bo-Tau Liu Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China Received February 25, 1998. In Final Form: June 11, 1998 The interactions between two charged spheroids immersed in a symmetric electrolyte solution are evaluated. We show that the more closely two particles resemble spheres, the higher the electrical interaction energy between them. The variation in the total interaction energy between two particles, which includes the electrical and the van der Waals energies, reveals that the greater the radius of curvature of a particle, the higher the primary maximum or the harder it is for two particles to undergo an irreversible coagulation. If the double layer around a particle is thin compared with its radius of curvature, the critical coagulation concentration (CCC) of counterions is independent of the latter. This implies that the relative orientation between two particles is insignificant as far as the CCC is concerned. In this case, the result obtained in the present study reduces to that predicted by the Derjaguin-Landau-Verwey-Overbeek theory.
1. Introduction One of the most significant characteristics of a colloid suspension is its stability, which depends largely on the interactions between two dispersed entities. The classic Derjaguin-Landau-Verwey-Overbeek (DLVO) model,1 for example, provides an analytical description of the variation in the total interaction energy between two charged particles, which includes the electrical and the van der Waals interaction energies, as a function of the separation distance between particles. The calculated total interaction energy is readily applicable to the estimation of the dependence of the critical coagulation concentration of counterions on its valence, the SchulzeHardy rule. The derivation of the DLVO model was based on spherical particles and symmetric electrolytes. The geometric nature of this type of particle leads to an essentially one-dimensional problem, and the relative orientation between two particles need not be considered, a dramatic simplification for the mathematical treatment. Several attempts have been made to modify the DLVO model so that a more general case can be simulated. Hsu and Kuo, for example, considered the problem of planar2 and spherical3 particles in an asymmetric electrolyte solution for an arbitrary level of electrical potential. The analysis was also extended to the case of particles covered by an ion-penetrable membrane.4,5 In general, describing the problem about the interactions between two arbitrary particles is nontrivial. In particular, for nonspherical particles, the relative orientation between two particles becomes significant. Also, information about the electrical potential distribution, which is governed by the PoissonBoltzmann equation, is necessary for estimation of the electrical interaction energy between two particles. Solving this equation analytically, however, is almost impossible, except for some extremely limited cases. If the electrical potential is low, the Poisson-Boltzmann equation can be approximated by a linear expression, which * To whom correspondence should be addressed. Fax: 886-23623040. E-mail:
[email protected]. (1) Hiemenz, P. C. Principles of Colloid and Surface Chemistry, 2nd ed.; Dekker: New York, 1986. (2) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1995, 171, 254. (3) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1997, 185, 530. (4) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1995, 174, 250. (5) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1996, 183, 184.
Figure 1. Schematic representation of the problem under consideration. {X, Y, Z} and {Xi, Yi, Zi}, i ) 1, 2, are the global and the local Cartesian coordinates, respectively, and θi and φi are the polar and azimuthal angles, respectively. The origin of the global Cartesian coordinates is located at the line segment joining {X, Y, Z} ) {h2, 0, 0} and {X, Y, Z} ) {h1, 0, 0}, with h ) h2 - h1 being the closest surface-to-surface distance between two particles. The line segment lies on the X axis.
is more readily soluble. For example, Ohshima and Kondo6 examined the interactions between two ionpenetrable spheres at low electrical potential. Hsu and Liu7 considered the case of two ion-penetrable, charged spheroids. Hsu and Tseng8 discussed the interactions between two thin disks of arbitrary orientation. In the present study, the analysis of Hsu and Liu9 is extended to the case of two rigid, charged spheroids in a symmetric electrolyte solution. The electrical potential can assume an arbitrary level, and a nonlinear PoissonBoltzmann equation is considered. Choosing spheroidal geometry has several advantages, such as the ability to simulate a wide class of shapes and to estimate the effects of both the relative orientation between two particles and the curvature of the particle surface. 2. Modeling Referring to Figure 1, we consider two spheroids immersed in an electrolyte solution. (6) Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1993, 155, 499. (7) Hsu, J. P.; Liu, B. T. J. Colloid Interface Sci. 1997, 190, 371. (8) Hsu, J. P.; Tseng, M. T. Langmuir 1997, 13, 1810. (9) Hsu, J. P.; Liu, B. T. J. Colloid Interface Sci. 1997, 192, 481.
S0743-7463(98)00227-3 CCC: $15.00 © 1998 American Chemical Society Published on Web 08/19/1998
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Hsu and Liu
Figure 2. Local spheroidal coordinates adopted for particle i: (a) prolate spheroid; (b) oblate spheroid.
2.1. Orientation of Two Spheroids. Let {X, Y, Z} be the global Cartesian coordinates and {Xi, Yi, Zi}, i ) 1, 2, be the local Cartesian coordinates for particle i. For convenience, the origin of the global Cartesian coordinates is located at the line segment joining {X, Y, Z} ) {h2, 0, 0} and {X, Y, Z} ) {h1, 0, 0}, with h ) h2 - h1 being the closest surface-to-surface distance between the two particles. This line segment lies on the X axis. The global and the local coordinates are related by10
[][
Zi ) ai[(1 - ηi2)(ξi2 - j)]1/2 sin Φi and
Xi2
][ ]
Xi cos θi cos φi sin θi cos φi sin φi X - Pi1 Yi ) -sin θi cos θi Y - Pi2 , 0 Zi -cos θi sin φi -sin θi sin φi cos φi Z - Pi3 i ) 1, 2 (1)
where {Pi1, Pi2, Pi3} denotes the coordinates of the center of particle i, and θi and φi are the polar and azimuthal angles, respectively. The local Cartesian coordinates {Xi, Yi, Zi}, i ) 1, 2, can be transformed into the local spheroidal coordinates {ξi, ηi, Φi}, i ) 1, 2, shown in Figure 2, through the following relations:11
Xi ) aiηiξi
(2a)
Yi ) ai[(1 - ηi2)(ξi2 - j)]1/2 cos Φi
(2b)
+ 2
Yi2
ai2ξi
ai2(ξi2 - j)
Xi2
Yi2
ai2ηi
2
ai2(1 - ηi2)
Zi2
+
ai2(ξi2 - j)
-
)1
(3a)
)j
(3b)
Zi2 ai2(1 - ηi2)
tan Φi ) Zi/Yi
(3c)
In these relations, j is a shape index (j ) 1 for prolates, j ) -1 for oblates), ai is the distance between the focus and the center of particle i, i ) 1, 2, ξi, ηi, and Φi denote respectively the radial, the angular, and the rotational coordinates for particle i. If the unperturbed electrical potential due to the presence of particle i is independent of Φi, eqs 3a-c can be simplified as
{ [ ( )] } { [ ( )] }
ξi ) Ω + Ω2 - 4j
Xi ai
ηi ) ( Ω - Ω2 - 4j (10) Meirovitch, L. Methods of Analytical Dynamics; McGraw-Hill: New York, 1994. (11) Moon, P.; Spencer, D. E. Field Theory Handbook; SpringerVerlag: Berlin, 1961.
(2c)
where
2 1/2
Xi ai
1/2
/2
2 1/2
/2j
, i ) 1, 2, j ) -1, 1 (4a)
1/2
, i ) 1, 2, j ) -1, 1 (4b)
Interactions between Two Charged Spheroids
Ω)
() () Xi ai
2
+
Yi ai
2
Langmuir, Vol. 14, No. 19, 1998 5385
+ j, i ) 1, 2, j ) -1, 1 (4c)
Z} ) {h1, 0, 0}, the derivatives of eq 6 with respect to both Y and Z must vanish. We have
Pi1 ) hi - Lbi{Ai/{(Ai + Ui2Di + Ui3Fi)2 - [Ui22(Di2 -
The surface of a spheroid can be represented by
AiBi) + Ui32{Fi2 - AiCi) + 2Ui2Ui3(DiFi - AiEi)]}}1/2 (8a)
Ai(X - Pi1)2 + Bi(Y - Pi2)2 + Ci(Z - Pi3)2 + 2Di(X - Pi1)(Y - Pi2) + 2Ei(Y - Pi2)(Z - Pi3) + 2
2Fi(X - Pi1)(Z - Pi3) - Lbi ) 0 (5)
Pi2 ) Ui2(Pi1 - hi)
(8b)
Pi3 ) Ui3(Pi1 - hi)
(8c)
where the sign before the square root is negative for particle 1 and positive for particle 2 and
with
Ai ) cos2 θi(Ri2 cos2 φi + sin2 φi) + sin2 θi Bi ) sin2 θi(Ri2 cos2 φi + sin2 φi) + cos2 θi
(5a) (5b)
Ui3 ) Ci ) Ri2 sin2 φi + cos2 φi
-CiDi + FiEi
Ui2 )
-BiFi + EiDi
(5d)
2.2. Electrical Potential Distribution of a Spheroid. A spheroidal surface can be characterized by the following two parameters:
λ) 2
Ei ) (Ri - 1) sin θi cos φi sin φi
(5e)
1 c[ξs2 + (1 - j)/2]1/2
Rs ) 2
Fi ) (Ri - 1) cos θi cos φi sin φi
(5f)
Ri ) Lai/Lbi
(5g)
where Lai and Lbi denote respectively the lengths of the semimajor and the semiminor axes. If Lai > Lbi, then eq 5 represents a prolate surface. If Lai < Lbi, then eq 5 denotes an oblate surface, while if Lai ) Lbi, then eq 5 describes a spherical surface. Equation 5 can be rewritten as
(8e)
BiCi - Ei2
(5c)
Di ) cos θi sin θi(Ri2 cos2 φi + sin2 φi - 1)
(8d)
BiCi - Ei2
[
]
ξs2 - (1 + j)/2
ξs2 + (1 - j)/2
(9a)
1/2
(9b)
where c ) κa, and κ2 ) 2F2I/RT. Rs is the ratio length of minor axis/length of major axis, λ denotes the ratio Debye length/length of semimajor axis, ξ ) ξs represents a spheroidal surface, denotes the dielectric constant, κ and R are the reciprocal Debye length and the gas constant, respectively, T is the absolute temperature, and F and I are respectively the Faraday constant and the ionic strength. It can be shown that the Poisson-Boltzmann equation governing the electrical potential distribution for a charged spheroidal surface in a symmetric electrolyte solution at an arbitrary level of electrical potential is9
X(i) ) Pi1 + {-[Di(Y - Pi2) + Fi(Z - Pi3)] (
∞
y(i) )
{(Di2 - AiBi)Y2 + (Fi2 - AiCi)Z2 + 2(DiFi AiEi)YZ - 2Ti1Y - 2Ti2Z + Wi } /Ai (6) 2 1/2
where the superscript i denotes the surface of particle i, the sign before the square root is positive for particle 1 and negative for particle 2, and
Ti1 ) Pi2(Di2 - AiBi) + Pi3(DiFi - AiEi)
(7a)
Ti2 ) Pi3(Fi2 - AiCi) + Pi2(DiFi - AiEi)
(7b)
Wi2 ) Pi22(Di2 - AiBi) + Pi32(Fi2 - AiCi) + 2Pi2Pi3(DiFi - AiEi) + AiLbi2 (7c) Since the end points of the line segment (corresponding to the closest surface-to-surface distance between two particles) are located at {X, Y, Z} ) {h2, 0, 0} and {X, Y,
λny(i) ∑ n n)0
(10)
where y(i) is the unperturbed scaled electrical potential due to the presence of particle i, y(i) n denotes the nth-order perturbation term of particle i, and λ is a perturbation parameter. It can be shown that9
[
y(i) 0 ) 2 ln
[ () ()
]
1 + R exp(-x) 1 - R exp(-x)
(11a)
( ) ( ( )) ( ) ( ( ))] () [ () ( ) ( ( ))] ()
y(i) y(i) y(i) 3 ln(R) 0 0 0 sinh + ln tanh 2 2 2 4 y(i) y(i) y(i) ln(R) 0 0 0 coth - sinh ln2 tanh 2 2 2 4 y(i) y(i) g1 0 0 + coth + 2f2 csch 2 2 2 y(i) y(i) y(i) 0 0 0 ln tanh + g2 sinh (11b) sinh 2 4 2
y(i) 1 ) f1 -csch
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Hsu and Liu
where
R ) tanh(ys/4)
(11c)
x ) chs(ξ - ξs)
(11d)
f1 )
(
1 f -1 cλhs h 2 s
)
(11e)
f2 ) f/cλhs
(11f)
g1 ) f1(2 + ln(R)) + 4f2
(11g)
(
( )) ( )
ys ys csch2 g2 ) (f1 + 2f2) 1 - cosh 2 2 ln(R)(f1 ln(R) + f1 + 2f2) (11h) f ) (ξs2 - j)/ξs2 2
2
(11i)
2
hs ) [(ξs - jη )/(ξs - j)]
1/2
2.3. Special Cases. Under some conditions, eqs 15 and 16 can be simplified and become more readily estimable. Let us consider three special cases. Case 1. If two particles are symmetric about the Y-Z plane, eq 15 leads to
VDL ) π
∂y 2κ [cosh(y) - 1] + ( ) } dr dh′ (RT ZF ) ∫ ∫ { ∂r (17) h
[
∫Ξ (∆π + E2 )bn - (EB‚nb)EB 2
]
(12)
where Ξ represents the Y-Z plane, b n is the unit outer normal, and E B is the electrical field vector with strength E. The osmotic pressure, ∆π, can be evaluated by
∆π )
κ [cosh(y) - 1] (RT ZF ) 2
2
(13)
F)
( ) ∫ {2κ [cosh(y) - 1] + (∂Y∂y ) + (∂Z∂y ) - (∂X∂y ) } dΞ (14)
RT 2 ZF
2
0
2
1
2
2
1
[
(β - γ ) + d (β2 + γ2)
2 2
2
]
{[(β + γ)2 + d2][(β - γ)2 + d2]}1/2
dL1 dL2 (18)
where
h ) h 2 - h1
(18a)
d ) h + La1 + La2 + L1 + L2
(18b)
β ) Lb1(1 - L12/La12)1/2
(18c)
γ ) Lb2(1 - L22/La22)1/2
(18d)
Case 3. According to the DLVO model, the total interaction energy between two particles, V, comprises the electrical interaction energy and the van der Waals interaction energy; the former can be estimated by applying the Deryaguin approximation.13 Suppose that κh is small and that the double layer is thin. Then
V) VVDW + VDL )-
Substituting this expression into eq 12 gives
2
2
∫-aa ∫-aa 4d1 4 (β2 + γ2) -
VVDW ) -A132
(11j)
dΞ
∞
Case 2. If two particles are at a head-to-head orientation, then the van der Waals interaction energy becomes
The electrical potential is scaled by y ) ZFψ/RT, while ψ and Z are the electrical potential and the valence of the electrolyte, respectively. 2.3. Electrical Interaction between Two Spheroids. Suppose that the electrical potential at the center plane between two particles is low and that the superposition principle is applicable.12 Then the electrical repulsive force between two particles can be calculated by
F)
∞
2
A132 12π
∫ΞL12 dΞ + Br∫Ξexp(-κL) dΞ
(19)
ZFψ 4κRT
(19a)
2
Ξ
2
2
The electrical interaction energy between two particles can be evaluated by
∫h F(h′) dh′ ∞
VDL )
(15)
The unretarded van der Waals interaction energy between two particles, VVDW, can be calculated by13
VVDW ) -
A132 2
π
∫V ∫V r16 dV2 dV1 1
2
where
2
(16)
where A132 is the Hamaker constant, and Vi is the volume of particle i. In general, a numerical scheme is necessary for the evaluation of this integral. (12) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, The Netherlands, 1948. (13) Hunter, R. J. Foundations of Colloid Science; Oxford University: London, 1989; Vol. I.
Br ) 64C∞RT tan2
L ) h + X(2) - X(1)
(19b)
Here, C∞ is the bulk electrolyte concentration. 3. Results and Discussion The variations of the electrical interaction energy between two identical prolates, VDL, as a function of the closest surface-to-surface distance, h, for two special orientations and two types of prolate surfaces are illustrated in Figure 3. Those for a higher surface potential are illustrated in Figure 4. These figures reveal that, for a constant h, the side-to-side orientation has a higher VDL than the head-to-head orientation. This is because the average distance between two surfaces for the former is closer than that for the latter. Figures 3 and 4 also show that, for a fixed closest surface-to-surface distance h, the
Interactions between Two Charged Spheroids
Figure 3. Variation on electrostatic interaction as a function of h: (a) ξs ) 1.25 (Rs ) 0.6); (b) ξs ) 1.02 (Rs ) 0.197). Solid line: head to head. Dashed line: side to side. Key: ys ) 1, the length of major axis ) 10-6 m, T ) 298 K, Z ) 1, and κ ) 108 m-1.
Langmuir, Vol. 14, No. 19, 1998 5387
Figure 5. Variation in the ratio of the electrical interaction energy between two particles on the basis of eq 20 and that based on eq 20 without the last term on its right-hand side, ω, as a function of κh. Key: the minimum curvature radius ) 10, θ1 ) 0, and θ2 ) 0. (a) φ1 ) π/2 and φ2 ) -π/2; (b) φ1 ) 4π/10 and φ2 ) -4π/10; (c) φ1 ) 3π/10 and φ2 ) -3π/10; (d) φ1 ) 2π/10 and φ2 ) -2π/10; (e) φ1 ) π/10 and φ2 ) -π/10.
is on the order of 10.15,16 Suppose that the double layer is thin compared with the curvature radius of the particle surface; in other words, let us focus on the local behavior of a surface. In this case, eq 6 can be simplified as
X(i) = hi - (BY2 + CZ2 + 2EYZ)/2W + (ABDY3 + ACFZ3 + (2AED + ABF)Y2Z + (2AEF + ACD)YZ2)/2AW2 (20)
Figure 4. Same as Figure 3 except that ys ) 4.
more the particles are closer to spheres (larger ξs), the higher the VDL if the surface potentials are constant. This is because, for a fixed h, the larger the ξs, the shorter the mean surface-to-surface distance between two prolate spheroids. Note that, for the case of oblate spheroids, the reverse is true. Deryaguin Approximation. The analysis can be simplified significantly if the Deryaguin approximation13 is applied. As pointed out by Stankovich and Carnie,14 the electrical interaction energy between two particles becomes important if κh is in the range 0.3-2. In this case, the deviation in the electrical interaction energy arising from use of the Deryaguin approximation is less than 10% at constant surface potential, provided that κa (14) Stankovich, J.; Carnie, S. L. Langmuir 1996, 12, 2453.
The second term on the right-hand side of this expression represents the contribution from a symmetric surface, and the third term arises from the asymmetry of the relative orientation of two surfaces; the latter is almost always neglected in the literature.18,19 The significance of the asymmetric term is illustrated in Figures 5 and 6. Here the electrical interaction energy between two particles on the basis of eq 20 and that based on eq 20 without the last term on its right-hand side are calculated; the variation in the ratio of these interaction energies, ω, as a function of κh at various relative orientations between two spheroids is plotted. As can be seen from Figures 5 and 6, the asymmetric term has a significant influence on the electrical interaction energy, in general. If the double layer is thin, however, its effect can be neglected. Figures 5 and 6 also show that ω is independent of κh. This can be elaborated as follows: The scaled surface-to-surface distance between two particles L can be decomposed as
L ) h + Lres
(21)
(15) Sader, J. E.; Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1995, 171, 46. (16) Carnie, S. L.; Chan, D. Y. C.; Stankovich, J. J. Colloid Interface Sci. 1994, 165, 116. (17) O’Neil, P. V. Advanced Engineering Mathematics; Wadsworth: Belmont, CA, 1983. (18) Belouschek, P.; Lorenz, D.; Adamczyk, Z. Colloid Polym. Sci. 1991, 269, 528. (19) White, L. R. J. Colloid Interface Sci. 1983, 95, 286.
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Figure 6. As in the case Figure 5, except that the minimum curvature radius is 20.
According to eq 19, the electrical interaction energy between two particles is
curvature and orientation relationship between two spheroids were not derived explicitly in their studies. The total interaction energy between two particles can be written in the following analytic form:
∫Ξexp(-κLres) dΞ
VDL) Br exp(-κh) ) Θ exp(-κh)
Z h2 Y h2 + 2ay 2az
(23)
where Y h and Z h are the principal axes resulting from the rotation of the Y and Z axes in the Y-Z plane, and
ay ) az )
[
(22)
where Θ ) Br∫Ξexp(-κLres) dΞ. This expression reveals that the dependence of VDL on κh can be factored out, and, therefore, the value of ω illustrated in Figures 5 and 6 is independent of κh. Note that eq 22 also implies that if the Derjaguin approximation is applicable, the dependence of VDL on particle shape is reflected by the factor Θ. If κa is large, the surface-to-surface distance between two particles L can be simplified by employing the principal axis theory17 to give
L)h+
Figure 7. Variation in the scaled total interaction energy as a function of the scaled surface-to-surface distance between two particles κh at two curvature radii of the particle surface for the case of A132/Br ) 10: (a) κay ) κaz ) 20; (b) κay ) κaz ) 10.
1 (B + C) + [(B - C)2 + 4E2]1/2
(23a)
1 (B + C) - [(B - C)2 + 4E2]1/2
(23b)
V ) 2πxayaz -
]
A132 1 Br + exp(-κh) 12π h κ
Figure 7 shows the variation in the scaled total interaction energy as a function of the scaled surface-tosurface distance between two particles κh at two curvature radii of the particle surface. This figure reveals that the greater the curvature radius, the lower the second minimum, and, therefore, the easier it is for two particles to undergo a reversible coagulation. On the other hand, a greater radius of curvature also leads to a higher primary maximum, and, therefore, the harder it is for two particles to undergo an irreversible coagulation. When the concentration of electrolyte reaches the critical coagulation concentration, both the total interaction energy between two particles and its derivative with respect to the separation distance between them vanish. By applying these conditions to eq 24 and solving the resultant expressions, we obtain
( )
737283(RT)5 tanh4
In these expressions
B)
B2 B1 + 2W1 2W2
(23c)
C)
C1 C2 + 2W1 2W2
(23d)
E)
E1 E2 + 2W1 2W2
(23e)
It should be pointed out that although similar results were obtained by Belouschek et al.18 and White,19 the local
(24)
C∞ )
exp2(F6A2Z6)
ZFψs 4RT
(25)
Here, the units of C∞ is mol/m3. If the surface potential on a particle is high, since tanh(s) = 1, eq 25 leads to C∞ ∝ Z-6. On the other hand, if the surface potential is low, since tanh(s) = s, eq 25 gives C∞ ∝ Z-2. These results are identical with those predicted by the DLVO model for the case of spherical particles. In this case, the critical coagulation concentration is not a function of the curvature radius. In other words, if the double layer around a spheroid is thin, the critical coagulation concentration is independent of the relative orientation between two spheroids. Acknowledgment. This work was supported by the National Science Council of the Republic of China. LA980227W
