6244
Langmuir 1999, 15, 6244-6255
Electrical Properties of Charged Cylindrical and Spherical Surfaces in a General Electrolyte Solution Yung-Chih Kuo and Jyh-Ping Hsu* Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China Received February 9, 1999. In Final Form: May 20, 1999 The electrical properties of a system containing cylindrical or spherical charged surfaces in a general a-b electrolyte solution are analyzed theoretically. Two perturbation methods, one for low to medium potentials and the other for medium to high potentials, are proposed for the resolution of the PoissonBoltzmann equation governing the spatial variation of the electrical potential. The results obtained are used to derive the basic thermodynamic properties of an electrical double layer, which include Helmholtz free energy, entropy, and surface excess of co-ions. The rate of convergence of the perturbation solution is satisfactory: if the second-order solution is adopted, the deviations in the electrical potential and the thermodynamic properties examined are less than 5%. The results obtained facilitate subsequent prediction of the behavior of a dispersed system, for example, its critical coagulation concentration and stability.
1. Introduction The determination of the spatial variation of the electrical potential for a system comprising a charged surface and an electrolyte solution is the key step to the estimation of its basic physicochemical properties. At equilibrium, this variation can be described approximately by the Poisson-Boltzmann equation.1 The difficulty of solving this equation depends largely on the geometry and the conditions of the charged surface and on the type of electrolyte solution. For an arbitrary level of electrical potential, the only exactly solvable problem is an infinite planar surface remaining at a constant potential in a symmetric electrolyte solution.1 Various approaches have been proposed for the resolution of the Poisson-Boltzmann equation under a more general condition. These approaches lead to either a complicated expression2-13 or an approximate form.14-16 In an analysis of the thermodynamic properties of a planar charged surface in an arbitrary a-b electrolyte solution, Hsu and Kuo,17 suggested using an approximate perturbation method which * To whom correspondence should be addressed. Fax: 886-223623040. E-mail:
[email protected]. (1) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: London, 1989; Vol. 1. (2) La Mer, V. K.; Gronwall, T. H.; Greiff, L. J. J. Phys. Chem. 1931, 35, 2245. (3) Dukhin, S. S.; Semenikhin, N. M.; Shapinskaya, L. M. Dokl. Phys. Chem. 1970, 193, 540. (4) Dukhin, S. S.; Semenikhin, N. M.; Shapinskaya, L. M. Dokl. Akad. Nauk SSSR 1970, 193, 385. (5) Sigal, V. L.; Shamansky, V. E. Dopov. Akad. Nauk Ukr. RSR 1970, 4, 346. (6) Lekkerkerker, H. N. W. Physica A 1989, 159, 319. (7) Stokes, A. N. J. Chem. Phys. 1976, 65, 261. (8) Natarajan, R.; Schechter, R. S. J. Colloid Interface Sci. 1984, 99, 50. (9) Abraham-Schrauner, B. J. Colloid Interface Sci. 1973, 44, 79. (10) Brenner, S. L.; Roberts, R. E. J. Phys. Chem. 1973, 77, 2367. (11) White, L. R. J. Chem. Soc., Faraday Trans. 1977, 273, 577. (12) Overbeek, J. Th. G.; Verhoeckx, G. J.; de Bruyn, P. L.; Lekkerkerker, H. N. W. J. Colloid Interface Sci. 1987, 119, 422. (13) van Aken, G. A.; Lekkerkerker, H. N. W.; Overbeek, J. Th. G.; de Bruyn, P. L. J. Phys. Chem. 1990, 94, 8468. (14) Parlange, J. Y. J. Chem. Phys. 1972, 57, 376. (15) Ohshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 90, 17. (16) Bentz, J. J. Colloid Interface Sci. 1981, 80, 179. (17) Hsu, J. P.; Kuo, Y. C. J. Chem. Soc., Faraday Trans. 1993, 89, 1229.
is based on a semiempirical polynomial function characterized by the degree of asymmetry of the electrolyte solution. The result obtained was also used in the discussion of cylindrical and spherical surfaces.18,19 In a recent study, a perturbation method was proposed by Kuo and Hsu20 to solve the Poisson-Boltzmann equation for a planar charged surface in an arbitrary a-b electrolyte solution. Analytical expressions for thermodynamic properties such as Helmholtz free energy, entropy, and surface excess of co-ions are derived. It should be pointed out that, although a purely numerical approach seems to be straightforward, obtaining analytical expressions for these properties is more desirable for subsequent analysis to predict the behavior of a colloidal dispersion. In practice, since the curvature of a charged surface may not be ignored, a systematic approach, which is capable of estimating the electrical properties of a nonplanar, charged surface in an electrolyte solution, is highly desirable. In the present study, we consider the possibility of extending the analysis of Kuo and Hsu20 to the case of cylindrical and spherical surfaces. This is also an extension of our previous work17 where a less mathematically rigorous approach was adopted. 2. Analysis The distribution of the scaled electrical potential ψ in the electrical double layer near a cylindrical or spherical charged surface in an a-b electrolyte solution at equilibrium is governed by1
g d2ψ m dψ ) + 2 X dX a + b dX
(1)
where ψ ) Fφ/RT, X ) κr, κ2 ) F2a(a + b)Ca°/r0RT, and g ) exp(bψ) - exp(-aψ). In these expressions, φ is the electrical potential, F and R denote respectively the Faraday constant and the gas constant, r and 0 are the relative permittivity of the liquid phase and the permittivity of a vacuum, respectively, Ca° is the bulk concentration of cations, T is the absolute temperature, κ is the (18) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1994, 167, 35. (19) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1995, 170, 220. (20) Kuo, Y. C.; Hsu, J. P. Chem. Phys. 1997, 236, 1.
10.1021/la990138z CCC: $18.00 © 1999 American Chemical Society Published on Web 07/30/1999
Electrical Properties of Charged Surfaces
Langmuir, Vol. 15, No. 19, 1999 6245
reciprocal Debye length, r is the distance from the surface, and m denotes the geometry of a surface (m ) 1 for a cylindrical surface and 2 for a spherical surface). Note that if m ) 0, eq 1 reduces to the case of a planar charged surface.20 We assume that the boundary conditions associated with eq 1 are
ψ ) ψ0 at X ) 0
(2a)
ψ f 0 and dψ/dX f 0 as X f ∞
(2b)
where ψ0 denotes the scaled surface potential. Equation 1 can be rewritten as17
dY 2 af ) - 2mY2 dX k
( )
∫1
Y
1 dY dY XY2 dX
( )
)
Y ) Y0 at X ) X0
(4)
where Y0 ) exp(aψ0/2) and X0 ) κr0, r0 being the radius of the cylinder or the sphere. For convenience, we define the scaled surface charge density p as17
p ) -aFσ/20rRTκ
(5)
where σ denotes the surface charge density. The value of p can be evaluated by
a dψ 2 dX
( )
)
X)X0
dY (Y1 dX )
X)X0
1/2
( )
+
(
)( )
aψ0 2
p ) exp -
1/2
( )
m + (F0 + F1 + 2F2) X
k-4 b-a ) k b+a
1/2
+
m (F + F4 + 2F5) + X0 3 2m2 (H3 + H4 + 2H5) X02
aψ0 m + B1 + 2B2 + (F + F4 + 2 X0 3 2F5) +
ψ0 )
[
(8)
(8a)
Note that, according to its definition, || < 1. Substituting eq 8 into the second term on the right-hand side of eq 3
]
L1 L2 2 ln(q - p) + C1 + 2C2 + + 2 a X0 X 0
(11)
where q ) (p2 + 1)1/2 and C1, C2, L1, and L2 are defined in the Appendix. Substituting eq A4 into eq 11 provides the relation between ψ0 and p. The free energy of an electrical double layer Fel can be calculated by18,19
=
(7)
2m2 (H3 + H4 + 2H5) (10) 2 X0
where B1 and B2 are defined in the appendix and f0 ) f(Y0), Fn+3 ) Fn(Y ) Y0) exp(-aψ0/2), and Hn+3 ) Hn(Y ) Y0) exp(-aψ0/2), for n ) 0, 1, and 2. Inverting eq 10 yields
(6)
where F0, F1, and F2 are defined in the Appendix and
)
af0 k
( )
∫0σ φ0 dσ
(
)
U3 U4 4(a + b) Ca°RT U0 + U1 + 2U2 + + aκ X0 X 2 0
(12)
where U0, U1, U2, U3, and U4 are defined in the Appendix. The entropy of an electrical double layer Sel can be calculated by18,19
( )
Sel ) =-
∂Fel ∂T
Ci
[
3Fel 2(a + b)Ca°R U3 + V0 + V1 + 2V2 2T aκ X0 2U4
Substituting this expression into the second term on the right-hand side of eq 3 and performing the integration, we obtain the first-order approximation to eq 3 as
af dY ) dX k
m (F + F1 + 2F2) + X 0 2m2 (H0 + H1 + 2H2) (9) X2
where H0, H1, and H2 are defined in the Appendix. On the basis of eqs A3, 6, and 9, we obtain
Fel )
We consider two perturbation methods. The first method is for low to medium-high electrical potentials, and the second method is for medium-high to high electrical potentials. Method 1. Low to Medium Potentials. The zerothorder approximation to eq 3 is obtained by letting m ) 0 on its right-hand side. This is equivalent to assuming X0 f ∞, that is, the surface is planar. We have
af dY ) dX k
1/2
(3a)
The boundary condition associated with eq 3 becomes
p)
( )
) - sinh
k 2 1 Y2 + Yk 1a k-2 k-2
(
dY af ) dX k
(3)
where k ) 2 + 2b/a, Y ) exp(aψ/2), and
f)
and conducting the integration leads to the second-order approximation
2
X0
+p
(
)]
V4 V3 + 2 X0 X 0
(13)
where V0, V1, V2, V3, and V4 are defined in the Appendix. The surface excess of co-ions Γ can be calculated by18,19
Γ)
(
)
aκFel -2Ca0 dFel κp p+ κ 4(a + b)Ca°RT 2RTkCa° dp
=-
[
(
)]
2Ca° U4 V4 V3 W0 + W1 + 2W2 - 2 - p + 2 κ X X0 X0 0 (14)
where W0, W1, and W2 are defined in the Appendix.
6246 Langmuir, Vol. 15, No. 19, 1999
Kuo and Hsu
Method 2. Medium to High Potentials. As in the case of the previous method, the zeroth-order approximation to eq 3 is obtained by letting m ) 0 on its right-hand side. To derive the first-order approximation, we note that, for the case of a negatively charged surface, 0 < Y < 1. Therefore, (af)1/2 is expanded into its Taylor series in terms of Y around Y ) 0. This gives
k 2 4 k 1 Y Y2 8 k-2 2(k - 2) k k 3 6 1 1 Yk + Y + Yk+2 2 16 k - 2 k-2 2(k - 2) 1 Y2k + ... (15) 2(k - 2)2
(
(af)1/2 ) 1 -
(
)
)
Substituting this expression into eq 7 and the resultant expression into the right-hand side of eq 3, we obtain
af dY ) dX k
1/2
( )
+
D1 X
(16)
On the basis of eqs A3, 6, and 18, we have
(
)( ) ( )
p = exp ) -sinh
aψ0 af0 2 k
1/2
+
D 3 D4 + X0 X 2 0
D3 D 4 aψ0 + B1 + 2B2 + + 2 X0 X 2
(19)
0
where D3 ) D1(Y)Y0) exp(-aψ0/2) and D4 ) D2(Y)Y0) exp(-aψ0/2). Inverting eq 19 yields (Appendix)
ψ0 =
[
]
E1 E2 2 ln(q - p) + C1 + 2C2 + + 2 a X0 X 0
where E1 and E2 are defined in the Appendix. Following the same procedure as that employed in the derivation of eq 12, we obtain
Fel =
(
D1 = (af)
[
G1
(16a)
k 1 k 2 4 Y + Y2 + 24 k - 2 2(k - 2) 1 k 3 6 1 Y Yk 80 k - 2 (k - 1)(k - 2) 1 k Yk+2 + Y2k (16b) 2(k + 1)(k - 2)2 2(2k - 1)(k - 2)2
(
G1 ) Y 1 +
(
)
)
]
The second-order approximation is obtained by first expanding (af)1/2 as
k 3 k 2 4 Y + Y2 + 8 k 2 2(k - 2) 1 k 3 6 3k 5 Yk Y Yk+2 16 k - 2 k-2 2(k - 2)2
(
(af)-1/2 ) 1 +
(
)
0
(21)
3 15k2 Yk+4 + Y2k + ... (17) 8(k - 2)2 2(k - 2)2 Substituting eqs 15 and 17 into eq 16, (dY/dX) can be expressed as a polynomial of Y. Substituting this polynomial into the right-hand side of eq 3 and integrating the resultant expression, we obtain
af dY = dX k
1/2
( )
+
D1 D2 + 2 X X
(18)
where
D 2 = m2
[
() ( k af
1/2
G2 -
)
G12 2af
(18a)
k 2 4 k 1 Y2 + Y k-2 6 k-2 1 Yk (k - 1)(k - 2) k(2k2 + k - 2) Yk+2 (18b) (k - 1)(k + 1)(k + 2)(k - 2)2
G2 ) -Y2 ln Y +
(
)
]
where U5 and U6 are defined in the Appendix. The entropy of an electrical double layer becomes
Sel ) =-
( ) ∂Fel ∂T
Ci
[
3Fel 2(a + b)Ca° U5 + V0 + V1 + 2V2 2T aκ X0 2U6 2
+
X0
)]
(
V6 p V5 + 2 q X0 X 0
where V5 and V6 are defined in the Appendix. The surface excess of co-ions is
Γ=-
)
)
4(a + b)Ca°RT U5 U 6 U0 + U1 + 2U2 + + aκ X0 X 2
where -1/2
(20)
[
(
(22)
)]
U 6 p V5 V6 2Ca° W0 + W1 + 2W2 - 2 + 2 κ q X0 X X0 0 (23)
The amount of co-ions adsorbed near the particle-liquid interface can be evaluated by
Γ)
aCa°
∫∞[exp(bψ) - 1]Xm dX
bκX0
m 0
(24)
3. Discussion Note that if m ) 0, eq 1 reduces to the governing equation for a planar surface. In this case, it can be shown that both methods 1 and 2 lead to the results of our previous study for a planar surface.20 The results obtained in the previous section can be modified to yield the corresponding results for a positively charged surface. This is done by interchanging a and b and replacing φ with -φ, and σ with -σ. Tables 1 and 2 show the performance of the present perturbation method for a lower ionic strength; that for a higher ionic strength is illustrated in Tables 3 and 4. Here, the electrical potential and thermodynamic properties of an electrical double layer calculated by the present method and those based on the exact numerical calculation of eq 1 are compared and the percentage deviations of the former from the latter are presented. The reference state for the evaluation of the thermodynamic properties is an uncharged surface. For simplicity, the second-order
Electrical Properties of Charged Surfaces
Langmuir, Vol. 15, No. 19, 1999 6247
Table 1. Averaged Percentage Deviations (%) in the Thermodynamic Properties of an Electrical Double Layer Estimated by the Present Approaches from Those Based on the Exact Numerical Calculationa p ) 0.5
X0
method
EFel
ESel
EΓ
2
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
2.98 19.82 2.46 5.17 2.16 2.96 4.07 4.04 3.22 3.17 2.55 2.63 13.85 2.93 4.11 2.42 3.04 2.01
3.17 21.77 2.73 5.63 2.46 3.10 4.36 4.41 3.50 3.67 2.74 2.91 14.20 2.99 4.87 2.69 3.12 2.38
2.51 18.32 1.87 4.97 1.68 2.62 3.51 3.75 2.89 2.96 2.31 2.43 12.63 2.41 4.29 1.77 2.78 1.62
5 10 p ) 10
2 5 10
p ) 20
2 5 10
Table 3. Averaged Percentage Deviations (%) in the Thermodynamic Properties of an Electrical Double Layer Estimated by the Present Approaches from Those Based on the Exact Numerical Calculationa X0
method
EFel
ESel
EΓ
2
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
3.26 16.94 2.71 4.33 2.24 2.76 4.11 4.08 3.43 3.32 2.70 2.75 11.08 3.29 4.05 2.83 2.96 2.21
3.10 19.02 2.67 5.03 2.42 2.91 4.22 4.30 3.39 3.53 2.58 2.79 12.67 3.51 4.26 3.14 3.02 2.60
2.72 15.63 2.06 4.12 1.75 2.27 3.84 4.01 3.03 3.46 2.42 2.55 10.15 2.94 4.10 2.03 2.97 1.86
p ) 0.5
5 10 p ) 10
2 5 10
p ) 20
2 5 10
For the Case of a cylindrical surface. Parameters used: ionic strength ) 1 mol/m3, T ) 298.15 K, and r ) 78.
a For the case of a cylindrical surface. Parameters used: ionic strength ) 100 mol/m3, T ) 298.15 K, and r ) 78.
Table 2. Averaged Percentage Deviations (%) in the Thermodynamic Properties of an Electrical Double Layer Estimated by the Present Approaches from Those Based on the Exact Numerical Calculationa
Table 4. Averaged Percentage Deviations (%) in the Thermodynamic Properties of an Electrical Double Layer Estimated by the Present Approaches from Those Based on the Exact Numerical Calculationa
a
p ) 0.5
X0
method
EFel
ESel
EΓ
2
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
2.69 17.24 2.31 4.76 2.13 2.70 3.87 3.82 3.01 2.98 2.44 2.43 15.01 3.15 4.48 2.68 3.15 2.12
3.02 18.69 2.63 5.47 2.32 3.07 4.03 4.20 3.27 3.32 2.47 2.74 15.30 3.12 5.12 2.87 3.42 2.68
2.47 16.15 1.79 4.84 1.62 2.56 3.38 3.56 2.67 2.78 2.14 2.19 12.93 2.67 4.60 1.90 2.97 1.79
5 10 p ) 10
2 5 10
p ) 20
2 5 10
a For the case of a spherical surface. Parameters used: ionic strength ) 1 mol/m3, T ) 298.15 K, and r ) 78.
perturbation solution is used. The percentage deviation in the thermodynamic property P, EP, is estimated by
∑ ∑| 3
EP )
3
a)1 b)1
* Pa-b
|
- Pa-b
* Pa-b
EFel
ESel
EΓ
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
2.98 15.42 2.67 4.88 2.31 3.14 3.78 3.36 3.41 2.77 2.89 2.35 13.07 2.80 4.43 2.26 3.60 1.89
2.83 16.04 2.45 5.20 2.26 2.83 4.04 4.12 3.29 3.20 2.63 2.56 12.73 3.08 4.92 2.81 3.37 2.24
2.54 14.75 2.06 4.29 1.79 2.22 3.26 3.71 2.57 3.05 2.14 2.30 12.58 2.71 4.10 2.19 3.05 1.62
10 p ) 10
2 5 10
p ) 20
2 5 10
a For the case of a spherical surface. Parameters used: ionic strength ) 100 mol/m3, T ) 298.15 K, and r ) 78.
The expressions below are useful:
[
∞
1
+2
4
(25)
* denotes the value of P for the case of an a-b where Pa-b electrolyte based on the exact numerical solution and P is Fel, Sel, or Γ. Tables 1-4 reveal that if the double layer is thin (large X0), method 1 is satisfactory for low to medium surface potentials, and method 2 becomes appropriate for high surface potentials. For medium to thin double layers, the performance of both methods is satisfactory. Note that, to apply method 1, the distribution of electrical potential for a planar surface needs to be known. In contrast, method 2 does not require this information. Also, since Y < 1, this method is applicable to essentially all levels of electrical potential. However, the higher the electrical potential, the faster the rate of convergence of eqs 15 and 17, as is suggested for the case of a medium to high electrical potential. For a highly charged surface, p is large and Ω approaches zero. The expressions for β0, β1, and β2 converge rapidly.
method
2 5
C12 )
× 100%
X0 p ) 0.5
(-1)i(2i - 1)Ω2i ∑ i)1 ∞
{1 +
]
-
[-4iΩ-2 + (4i + 1)]Ω4i} ln Ω + ∑ i)1 ∞
iΩ4i+4) ln2 Ω ∑ i)1
4( C12 2
{
+ C2 ) ∞
4
∑
1
-
4
19 4
∞
Ω2 +
∑ (-1) (2i - 1)Ω i
}{
2i
-
i)1
[2i - (2i + 1)Ω2]Ω4i + 1 -
i)1
(26a)
∞
∑ [-4iΩ
-2
+ (4i +
i)1
1)]Ω4i - 2(1 + 3Ω2 + 4Ω4 - 4Ω6 - 13Ω8 ∞ ∞ (-1)i (i + 1)(i + 2)Ω2i+4j ln Ω + 7Ω10) 2 j)0 i)0
∑∑
∞
2{
∑ [i
2
i)1
}
+ 4i + 2 + (i + 1)(i + 2)Ω4]Ω4i+4} ln2 Ω (26b)
6248 Langmuir, Vol. 15, No. 19, 1999
Kuo and Hsu
In practice, only the first few terms on the right-hand side of these expressions are necessary. Although the mathematical manipulations of the present perturbation approach seem tedious, the analytical results obtained, that is, the free energy, the entropy, and the surface excess of co-ions of an electrical double layer, expressed respectively in eqs 12-14 (or eqs 21-23), are simple. It should be pointed out that the derivation of these fundamental relations is for subsequent analyses. The evaluation of the critical coagulation concentration (or stability) of a colloidal dispersion, for instance, involves the differentiation (or integration) of the total interaction energy between two charged entities, and the analytical expressions derived in the present study are readily applicable. In contrast, solving the Poisson-Boltzmann equation numerically, although straightforward, is disadvantageous. This is because both the maximum of the total interaction energy and its derivative with respect to the separation distance between two particles must vanish at the critical condition. If the Poisson-Boltzmann equation is solved numerically, these conditions need to be examined through a trial-and-error procedure, which is extremely time-consuming, even if a high-speed computer is available. Acknowledgment. This work is supported by the National Science Council of the Republic of China.
where
A0 ) (1 - K02)/2
(A2a)
A1 ) -K0K1 - (1 - 4K02 + 3K04 - 4K04 ln K0)/8A0 (A2b) A2 ) -K0K2 - [(1 - 3K02)K12 - 4K0 (1 - K02 + 2K02 ln K0) K1 + 2K02 (1 - K02 + 2K02 ln K0 2K02 ln2 K0)]/4A0 - A12/2A0 (A2c) Equations 4, A2, and 6 lead to
p = - sinh(aψ0/2) + B1 + 2B2
(A3)
where
exp(-aψ0) - 4 + 3 exp(aψ0) - 2aψ0 exp(aψ0)
B1 )
8 sinh(aψ0/2)
(A3a)
Appendix For a planar surface (m ) 0), it can be shown that20
B12 + 1 - exp(aψ0) + aψ0 exp(aψ0) - a2ψ02 exp(aψ0)/2
∞
Y ) K0 +
nKn ∑ n)1
(A3b) On the basis of eq A3, we obtain
K0 )
1+Z 1-Z
(A1a)
K1 ) K1,Z - K1,Z(X)X0)
(A1b)
K2 ) K2,Z - K2,Z(X)X0)
(A1c)
Z ) [tanh(aψ0/4)] exp[-(X - X0)]
(A1d)
2
(
)
8 5 217 4 187 5 Z + Z + ... Z2 + Z3 + 2 45 30 3(1 - Z)2 11 3 13 4 211 5 Z + Z + Z + ... K1,Z(X)X0) Z2 + 12 8 120 3 3 3 2 + Z ln Z Z K1,Z2 (X)X0) (A1f) 16 8 16
[(
)
)
(
)
]
Expanding (af/k)1/2 in terms of Y and substituting the resultant expression into eq 7, we have
dY dX
ψ0 =
2 [ln(q - p) + C1 + 2C2] a
(A4)
where
4Z 2 1 22 4 Z + ... 2 + Z + Z 2 + Z3 + 2 5 5 175 3(1 - Z) (A1e)
K2,Z )
(
2 sinh(aψ0/2)
(A1)
where
K1,Z )
B2 )
C1 )
1 q q p 1 p + + 1ln(p + q) 2 2p 2q 2pq 2p 2q (A4a)
C2 )
(
(
(
)
p p2 q3 q2 q p3 q -1+ - 2+ 3- 3- 2+ 3 2p 2q 4q 8p 4p 8q 4p
)
1 1 1 1 1 p + + + + + 4q3 4p2 2pq 4q2 8p3q 8pq3
p p2 q2 q p3 q q3 -1+ - 2+ 3- 3- 2+ 3 2p 2q 2p 2p 2q 2q 2p
)
(
)
2 q3 q p p3 ln (p + q) p 1 ln(p + q) + + + 2 8 2q3 pq p3 p q q3
(A4b) Equations A1 and A2 lead to
∞
) A0 +
∑ nAn n)1
(A2)
dY (Y1 dX )
2
= M0 + M1 + 2M2 ≡ M
(A5)
Electrical Properties of Charged Surfaces
Langmuir, Vol. 15, No. 19, 1999 6249
Equations A1 and 8 lead to
where
() A0 K0
M0 )
(
2
)
4Z2 (1 - Z2)2
dY (Y1 dX )
2
)
A 1 K1 64 3 Z + ... + f A 0 K0 3 4Z(-1 + 2Z + ...)K1,Z(X)X0) (A5b)
M1 ) 2M0
(
=M+
where
)
2
N0 )
2
2A2 4K1A1 2K2 3K1 + + f 2 A0 K0A0 K0 A0 K02 A1
M 2 ) M0
(
)
N 1 ) N0
64 2 64 3 Z + ... + Z + ... K1,Z(X)X0) + (1 - 4Z + 3 3 ...)K1,Z2(X)X0) + (-4Z + ...) K2,Z(X)X0) (A5c)
N 2 ) N0
If X0 is large, eq 3 can be approximated by
dY
)
dX
2m 2m (N0 + N1 + 2N2) ≡ M + N X X (A8)
() af
1/2
k
mY2
+
∞
X[A0 +
nAn] ∑ n)1
∫0Z
( ) 1 dY
Y dX
2
dZ Z
0
∫0Z NZ dZ = Q0 + Q1 + 2Q2
Q0 ) (A7a)
P1 )
∫0
P2 )
∫0
M2 64 3 32 2 dZ f Z + ... + Z + ... × Z 9 3
(
)
K1,Z(X)X0) + (ln Z - 4Z + ...)K1,Z2(X)X0) + (-4Z + ...) K2,Z(X)X0) (A7c) Substituting eq A7 into eq A6 and collecting terms of the same order in yield eq 8 with
(
P0K02 A0
Q1 )
Q2 )
∫0Z
∫0Z
)
2K1 A1 P1 + K0 A 0 P0
0
(A9)
)
(A9a)
N2 64 3 16 2 dZ f Z + ... + Z + ... × Z 27 3 1 K1,Z(X)X0) + ln Z - 4Z + ... K1,Z2(X)X0) + 2 (-4Z + ...) K2,Z(X)X0) (A9c)
∫0Z
(
(
)
)
Substituting eq A9 into eq A6 and collecting terms of the same order in , eq 9 can be recovered with
H0 ) H 1 ) H0
(A7e)
P1A1 P2 + (A7f) P0A0 P0
N0 dZ ) -ln(1 - Z2) Z
N1 64 3 dZ f Z + ... + Z 27 2Z(-2 + Z + ...) K1,Z(X)X0) (A9b)
(A7d)
2K2 K12 2K1A1 A2 A12 2K1P1 + 2+ + K0 K0A0 A0 A 2 K0P0 K 0
)
2K2 3K12 + ) P2 (A8c) K0 K2
where
M1 64 3 dZ f Z + ... + Z 9 4Z(-1 + Z + ...) K1,Z(X)X0) (A7b)
F2 ) F0
(A8b)
(A7)
∫
(
)
Following the same procedure as that employed in the derivation of eq A7, we obtain
Z M0 2Z2 dZ ) P0 ) 0 Z 1 - Z2
F 1 ) F0
(A8a)
A1 2K1 F1 + ) P1 A0 K0 F0
(
where
F0 )
) P0
(
(A6)
∫0Z MZ dZ = P0 + P1 + 2P2
Z
K02
A2 A1F1 F2 2K1A1 2K1F1 + + A0 A0F0 F0 K0A0 K0F0
Substituting eq A5 into the second term on the righthand side of this expression gives
Z
A0F0
H 2 ) H0
(
(
Q0K02 A0
(A9d)
)
2K1 A1 Q1 + K0 A0 Q 0
2K2 K12 2K1A1 A2 A12 + 2+ + K0 K0A0 A0 A 2 K 0
0
(A9e)
)
2K1Q1 Q1A1 Q2 + (A9f) K0Q0 Q0A0 Q0
6250 Langmuir, Vol. 15, No. 19, 1999
Kuo and Hsu
If we define
Z0 ) tanh S1 ) -p + B1 + 2B2 +
m(F3 + F4 + 2F5) + X0
X0
)
-1 + K0(X)X0) 1 + K0(X)X0)
4
)
∞
(-1)n+1K0n(X)X0) ∑ n)1
-1 + 2
2m2(H3 + H4 + 2H5) 2
( ) aψ0
(A10)
) β0 + β1 + 2β2
(A13)
with then the inversion of eq 10 is
∞
(-1)n+1(q - p)n ) ∑ n)1
β0 ) -1 + 2 2 ψ0 ) ln[S1 + (S12 + 1)1/2] a
(A11) ∞
for method 1. The symbols L1 and L2 in eq 11 are defined as 2
L1 ≡ mL1,0 + mL1,1 + mL1,2 2
2
2
(A12a)
2
L2 ≡ m L2,0 + m L2,1 + m L2,2
(-1)n+1n(q - p)n ) ∑ n)1
β1 ) 2C1
[(
β 2 ) 2 C2 +
F3 L1,0 ) q
(A12c)
B1F3 F4 + L1,1 ) q (q - p)q3
(A12d)
[
[
2(q - p)q4
2
]
Ω2C12
(A13c)
(1 + Ω)3
(A14a)
1 3 1+q ln + q - p ln(p + q) + 2q - p + 2 p 2
(
]
)
[
]
1+q 1 q3 2 ln + 2-qln2(p + q) + 16 p q p
[
(
)
]
4 1 1 1+q 8 ln - 6q - + 8p ln(p + q) + 8 p q p
(
(
)
)
4 1 -18q - + 22p - 8R1 - R4 + 8 q
2
+
6p(2B1F3F4 + 2B12H3 + B2F32)]/4(q -p)q3 + (q - p)2q6
(A12h)
Let Z0 ) Z(X)X0). Expanding Z0 in the perturbation series of , we have
(
)
7π3B3/2 1 2 45 - 4π (A14c) 16 3
[2F42 + 4F3F5 + 8B1H4 + 8B2H3 - 3B12F32 +
B12F32
]
(q - p)n
R1 π2 -2+ (A14b) 2 8
U2 )
2H5 F42 - F3F5 L2,2 ) + + q q2 2B1F3F4 - 2B2F3 - 3pB1 F3
(1 + Ω)2
-
n(n - 1)
U0 ) p ln(p + q) - q + 1
(A12f)
(A12g)
∑ (-1)n+1 n)1
Ω(C2 + C12/2)
U1 ) -
2
2
(-1)n+1n(q - p)n +
where Ω ) q - p, which is less than unity for a charged surface. Note that if p f 0, then Z0(β0,β1,β2) f 0. The symbols U0, U1, U2, U3, and U4 in eq 12 are defined by
2H4 2F3F4 + 4B1H3 + 3pB1F32 F3F4 + - 2 L2,1 ) q 2(q - p)q3 q
2
n)1
(A13b)
(1 + Ω)2
∞
F5 2B1F4 + 2B2F3 + 3pB12F3 + (A12e) L1,2 ) q 2(q - p)q3
(q - p)q4
2
(A13a)
1+Ω
2ΩC1
C12 )2
B1F3
∞
(A12b)
In these expressions
F32 2H3 F32 L2,0 ) + q 2(q - p)q3 2q2
)∑
C12
-1 + Ω
U3 ) m
∫0
p
(β0 + β1 + 2β2) dp ) m(γ0 + γ1 + 2γ2) q (A14d) U4 ) mU3
(A14e)
To derive the thermodynamic properties of a double layer, eqs A4a and A4b are first rewritten as
Electrical Properties of Charged Surfaces
C1 ) )
p 1 p q - + 1ln(p + q) 2 q 2p 2q
(
(
)
µ2 ≡
)
3 2 1 -2 1ln Ω 2 1 + Ω2 1 - Ω4
[
C2 ) -
)]
(
[
2
ln(p + q) +
(
] [
µ3 ≡
)
6
]
8
]
Substituting eqs A15a and A15b into eqs A13a-A13c, the integral in eq 14d can be conducted. We have
Ω ln Ω + ln + ln 2 (A17b) 1+Ω 1+Ω
∫1Ω (1ln+ ΩΩ)3 dΩ ) - 2(1ln+ΩΩ) + 2(1 1+ Ω) +
Ω 1 1 1 ln - + ln 2 (A17c) 2 1+Ω 4 2
µ4 ≡
1 + 3Ω + 4Ω - 4Ω - 13Ω - 7Ω10 ln Ω + (1 - Ω2)2(1 + Ω2)5 4Ω4(1 + Ω4) 2 ln Ω (A15b) (1 - Ω4)3
2 1-
4
-
p q ln2(p + q) 8p3 8q3
21 9 + 19Ω2 - 21Ω4 - 15Ω6 + + ) 8 4(1 - Ω2)(1 + Ω2)3 2
∫1Ω (1ln+ ΩΩ)2 dΩ )
(A15a)
(p2 - 1)2 1 1 3 9p q + + 2+ + + + 3 2 8q 2q 2p 2pq 8pq
1 q p p3 + 2q3 4 p q
[
Langmuir, Vol. 15, No. 19, 1999 6251
∫1Ω (1ln+ ΩΩ)4 dΩ ln Ω Ω2 2Ω + + 3 2 3(1 + Ω) 3(1 + Ω) 6(1 + Ω) Ω 7 1 1 ln + + ln 2 (A17d) 3 1 + Ω 24 3
)-
µ5 ≡
∫1Ω
ln Ω
dΩ ) ln Ω tan-1 Ω +
2
1+Ω
∞
p β0 dp ) ln Ω - 2 ln(1 + Ω) + 2 ln 2 γ0 ≡ 0 q
∫
γ1 ≡
β
1
[
5
- ln(1 + Ω2) + 2 -
)
+
2(1 + Ω) 2(1 + Ω) 1 1 1 5 - ln(1 + Ω) + ln(1 - Ω) + ln(1 + 4 2 2(1 + Ω)2 4 n ∞ Ω Ω2) ln Ω + [5(-1)n+1 + 1 + (-1)nΩn] 2 n)1 4n π2 1 + ln 2 (A16b) 4 8
]
γ2 ≡
∑
∫0
p
{
β2 dp ) q
}
∫1
Ω
1-Ω C12 dΩ (1 + Ω)3 2
∫1
Ω
∫1Ω (1 ln+ ΩΩ2)2 dΩ
[
Ω
+
∞
∑
1 Ω -9 + - tan-1 Ω + 4(1 + Ω) 4(1 + Ω)2 1 + Ω2 1 9 + tan-1 1 + 2 ln(1 + Ω) + ln(1 + Ω2) + 4 16 7 15 7 ln 2 + 2µ1 µ + 5µ3 - µ4 + µ5 - 2µ6 4 2 2 2 17 75 45 11 1 5 µ + µ µ + µ - µ - µ + 32 8 16 9 8 10 4 11 2 12 4 13 1 1 µ µ + µ16 (A16c) 2 14 32 15
µ7 ≡ )
2
1
[
Ω
+
4 (1 + Ω )
2 2
-
1 2
8(1 + Ω2)
µ8 ≡
3Ω
ln Ω
∞
(-1)nΩn
n)1
n2
∑
+
12
(A17a)
2
2
2(1 + Ω ) tan-1 Ω -
3
∞
∑ 8n)0
(A17e)
tan-1 Ω +
1+
π2E1/2 6
(A17f)
]
tan-1 Ω ln Ω -
(-1)nΩ2n+1
1
+ 16 (2n + 1)2 1 3 2 tan-1 1 + π E1/2 (A17g) 2 64 +
2
∫1Ω 1ln+ ΩΩ dΩ ∞
∑
(-1)n+1Ωn
n2 n+1 n (-1) Ω
ln Ω +
n)1 ∞
2
∑1
µ10 ≡ π2
3
+ 2
) ln(1 + Ω) ln2 Ω - 2
µ9 ≡
∫1Ω 1 + Ω dΩ ) ln Ω ln(1 + Ω) +
tan
-1
1
∫1Ω (1 ln+ ΩΩ2)3 dΩ
Ω
µ1 ≡
+
2n)0 (2n + 1)2
)
where
]
1
π2E1/2 8
tan-1 Ω ln Ω -
(-1)n+1Ω2n+1
Ω
1 C2 dΩ (1 + Ω)2
2
2(1 + Ω2) 1
1
+
(2n + 1)2
n)0
µ6 ≡
∫0p q1 dp
)
∑
(A16a)
(-1)n+1Ω2n+1
ln2 Ω (1 + Ω)2
∑ n)1
dΩ ) -
3
n
ln2 Ω 1+Ω
2
- π3B3/2 (A17h)
+ 2δ1
(A17i)
2
ln Ω ln Ω Ω dΩ ) + ∫1Ω (1ln+ Ω) 3 2(1 + Ω)2 1 + Ω ln
(1 +Ω Ω) + δ
1
- ln 2 (A17j)
6252 Langmuir, Vol. 15, No. 19, 1999
Kuo and Hsu
In these expressions E1/2 and B3/2 are, respectively, the Euler number of 1/2 order and the Bernoulli number of 3/2 order, and
2
Ω dΩ ∫1Ω (1ln+ Ω) 4
µ11 ≡
[
]
-ln2 Ω 2 2Ω Ω2 + ln Ω + 3 2 3 1 +Ω 3(1 + Ω) 2(1 + Ω) 1 2 1 + δ + - ln 2 (A17k) ln(1 + Ω) 3(1 + Ω) 3 1 6
)
[
-ln2 Ω 1 Ω2 1 + + 4 3 2 3(1 + Ω) 4(1 + Ω) 2(1 + Ω)2
)
]
∫
R2 ≡
∫ tanq
dp ) -2
q -1
2
Ω dΩ ∫1Ω (1ln+ Ω) 5
µ12 ≡
∞
ln[(1 + q)/p]
R1 ≡
∑
+ 1)2
n)0(2n
p
(A18a)
dp
) -[tan-1 p + 2 tan-1(q - p)] ln(p + q) +
Ω2 Ω 1 2Ω ln Ω + + 1+Ω 12(1 + Ω)2 3(1 + Ω) 4(1 + Ω)
( )
∫
R3 ≡
(A171)
R1 q
∞
(-1)nΩ2n+1
n)0
(2n + 1)2
∑
2
3 ln(1 + Ω) 1 11 1 Ω 1 - ln ln 2 + δ1 4 6 1+Ω 2 48 12
∞
dp ) 2
Ω2n+1
∑
+ 1)3
n)0(2n
(A18b)
(A18c)
2
µ13 ≡
∫1Ω 1ln+ ΩΩ2 dΩ ) (tan-1 Ω) ln2 Ω - 2δ2
µ14 ≡
∫1Ω (1ln+ ΩΩ2)2 dΩ ) 2(1 +Ω Ω2) +
[
2
(A17m)
]
1 tan-1 Ω ln2 Ω - µ5 - δ2 (A17n) 2
R4 ≡
R5 ≡
∫1Ω 1ln- ΩΩ dΩ
)∞
) -ln(1 - Ω) ln2 Ω - 2
∑
Ωn
n)1 n
2
ln Ω + 2
∞
Ωn
n)1
n3
∑
4π3B3/2 3
∫0
p
µ16 ≡
[
( ) 1+Ω
( )]
ln Ω -
∞
(-1)nΩ2n+1
n)0
(2n + 1)2
∑
1 2
∞
ln2 Ω +
ln Ω -
∑ n)1
(-1)
∞
(-1)nΩ2n+1
n)0
(2n + 1)3
∑
)
n+1
Ω
+
n
-
n)0
(2n + 1)2
∑
ln Ω -
∫
R7 ≡
ln(p/q) q
∫
R6 q
dp )
dp ) -
∞
(-1)nΩ2n+1
n)0
(2n + 1)3
∑
1
∞
∑
Ω2(2n+1)
4n)0(2n + 1)2
1
∞
∑
Ω2(2n+1)
8n)0(2n + 1)3
(A18e)
(A18f)
(A18g)
R1(p)0) ) -π2/4
(A18h)
R2(p)0) ) π2E1/2/4
(A18i)
R3(p)0) ) 7π3B3/2/6
(A18j)
R4(p)0) ) -7π3B3/2/6
(A18k)
R5(p)0) ) -π3/16
(A18l)
R6(p)0) ) π2/32
(A18m)
R7(p)0) ) R3(p)0)/16
(A18n)
π2
12 (A17q)
n2
(-1)nΩ2n+1
(A17o)
q 1 1 9 1-q 1+q + ln + + 4 2 2 8q 64 p 32p 64p 16q 1 p ln ln2(p + q) (A17p) 8 q
(
∞
2
]
()
Ω
ln Ω +
R6 ≡
] [
δ1 ) ln
R2
-
q-1 C2 dp qp2
[
ln[(1 + q)/p] ln(p + q) dp ) -R1 ln Ω - R3 q (A18d)
R2 dp q
2
4p - 9 5 1 1 + q - 2/q 1 - q + 2+ + ln q + + 2p 16q 8 32p2 4q 9 1 1 1 R - R + R + (178 + 6π2 32 3 2 6 4 7 384 1 1 - q 1 - q 69 - 58q + 2+ + + 119π3B3/2) + 3 2 96p 16p 2p 4q 9 p 1 5p - 24 1 + ln R - R ln(p + q) + 48q 2 q 32 1 4 6 )
∫
∫
2
µ15 ≡
δ2 )
Ω2n+1
π3 32 (A17r)
Electrical Properties of Charged Surfaces
Langmuir, Vol. 15, No. 19, 1999 6253
The symbols V0, V1, V2, V3, and V4 in eq 13 are defined by
as a function of p. Equation A4 leads to
V0 ) p ln(p + q)
Y0n ) (q - p)n 1 + (nC1) + 2 nC2 +
1 p2 p - p ln(p + q) + (A19b) 2q q 2
(
)
V1 ) q V2 )
(A19a)
(
)
[
)
2 2p 4 1 q2 + 1 - 6q + - 2 - 3 + 7p ln(p + q) + 8 p q q q
(
)
1 13 4p 4 - 10q - 2 - 3 + 16p (A19c) 8 q q q V3 )
m (β + β1 + 2β2) q 0
W0 ) p - q + 1
)]
(
[
E1,1 )
(A20a)
1 p2 1 1+q - ln W1 ) ln(p + q) + 2q - - p + 2q 2 p q R1 π2 -2+ (A20b) 2 8
[
E1,0 )
(A19e)
The symbols W0, W1, and W2 in eq 14 are defined by
E1,2 )
]
)
[
1 1+q q2 + 2 6 2p - + 2+ 8 ln ln (p + q) + 8 p p q q 2
]
(
( ) (
)
S2 ) -p + B1 + 2B2 +
E2,1 )
E2,2 )
If we define
D 3 D4 + X0 X 2
E2 ) E2,0 + E2,1 + 2E2,2
(A25b)
[ ( )
[
(
)
]
m 3k - 10 2 -1 + Ω + q k-2 k(k + 2) Ωk + ... (A25d) 2(k - 1)(k - 2)
(
)
]
[
3m 2 14k - 31 2 Ω 13 k-2 16qΩ2 k Ωk + ... (A25e) (k - 1)(k - 2)
]
E2,0 )
)
1 17 4p 4 4 + 2 + 3+ + p ln(p + q) + -8q 3 8 q q q q 3 7π B 1 3/2 6p - 8R1 - R4 + 45 - 4π2 (A20c) 16 3
(A25a)
m k k 2 4 2 1+ Ω2 + Ω q k-2 3 k-2 k Ωk + ... (A25c) (k - 1)(k - 2)
1 3q3 2 2p2 1+q ln + 2 - 3q - - 3 × W2 ) 16 p q p q
(
E1 ) E1,0 + E1,1 + 2E1,2
where
(A19d)
V4 ) mV3
)]
n2 2 C ,n∈R 2 1 (A24)
Substituting this expression into eqs 20a and 20b, we obtain
1 2q3 2p2 2q - 2 + 3 ln2(p + q) + 16 p q
(
(
[
[
xk 2 m2 Ω + ... Ωq 2
]
k-5 m2 3 + Ω+ q 4Ω k-2 3k Ωk-1 + ... (A25g) 2(k - 1)(k - 2)
(
)
]
[
3xk 49k - 161 m2 3 Ωq 8Ω 32 6(k - 2)
The symbols U5 and U6 in eq 21 are defined by
then the inversion of eq 19 is
2 ψ0 ) ln[S2 + (S22 + 1)1/2] a
(A22)
for method 2. The symbols E1 and E2 in eq 20 are defined as 2
B1D3 D3 2B2D3 + 3pB1 D3 + + 2 3 q (q - p)q 2(q - p)q3
(A23a)
D32 D4 2B1D4 + 3pB1D32 + + + E2 ) q 2(q - p)q3 2(q - p)q3
4B2D4 - 3B12D32 + 6pB12D4 + 6pB2D32 2 4(q - p)q3
]
3k Ωk-1 + ... (A25h) 4(k - 1)(k - 2)
(A21)
0
E1 )
(A25f)
E12 2 (A23b)
On the basis of eq A4, B1, B2, D3, and D4 can all be expressed
[
U5 ) U5,0 + U5,1 + 2U5,2
(A26a)
U6 ) U6,0 + U6,1 + 2U6,2
(A26b)
k 1 1 Ω2 Ωk + 2 k-2 (k - 1)(k - 2) k 2 4 1 Ω (A26c) 6 k-2 1 3k - 10 2 U5,1 ) m -ln Ω + Ω + 2 k-2 k+2 Ωk (A26d) 2(k - 1)(k - 2)
U5,0 ) m ln Ω +
)
(
[
U5,2 )
(
(
[
)
) ]
]
3m 1 2 14k - 31 ln Ω 2 16 3 k-2 2Ω k Ωk-2 (A26e) (k - 1)(k - 2)2
(
)
]
6254 Langmuir, Vol. 15, No. 19, 1999
[
U6,0 ) m2 Ω U6,1 ) m2
[
U6,2 ) m2 -
[
Kuo and Hsu
]
xk 2 Ω 4
3
∫ pq dp ) - 3pq
(A26f)
4
]
∫ pq dp ) - 4pq
k-5 3k 3 + Ω+ Ωk-1 4Ω k - 2 2(k- 1)2(k - 2) (A26g)
5
4
∫ pq
4
2
∫ pq
4
dp ) -
(A27a)
V6 ) V6,0 + V6,1 + 2V6,2
(A27b)
[
) ]
]
)
]
3m 1 2 14k - 31 k Ωk-2 16 Ω2 3 k - 2 (k - 1)(k - 2) (A27e)
)
[
V6,0 ) m2 Ω -
[
V6,1 ) m2 -
[
V6,2 ) m2 -
∫ q1 dp ) 2qp
2
1 tan-1 p 2
+
∫ q1 dp ) pq 3
k k Ω2 Ωk + k-2 (k - 1)(k - 2) k 2 4 2 Ω (A27c) 3 k-2
(
2
4
k(k + 2) 3k - 10 2 V5,1 ) m -1 + Ω + Ωk k-2 2(k - 1)(k - 2) (A27d) V5,2 )
p 1 + tan-1 p 2q2 2
4
V5 ) V5,0 + V5,1 + 2V5,2
(
p 3 - tan-1 p 2 2 2q
∫ qp dp ) - 2q1
The symbols V5 and V6 in eq 22 are defined by
[
)
3
dp ) p +
4
]
(
(
4
3k Ωk-1 (A26h) 4(k - 1)2(k - 2)
[ ( )
q 1 1+q + ln 2 8 p 8p
-
∫ p1q dp ) - 3p1 q + 3pq + 3qp
3xk 49k - 161 3 ln Ω Ω8Ω 32 6(k - 2)
V5,0 ) m 1 +
3
]
xk 2 Ω 2
∫
p4 ln(p + q) 4
q
(
dp ) p +
)
p 3 - tan-1 p × 2 2 2q ln(p + q) - q +
∫
p2 ln(p + q) q4
(
dp ) -
∫
q2 ln(p + q) p2
)
)
(
]
)
∫ ln (p + q) dp ) p ln (p + q) - 2q ln(p + q) + 2p ∫
2
p2 ln2(p + q) q2
dp ) (p - tan-1 p) ln2(p + q) 2q ln(p + q) + 2p + 2R2 ln(p + q) - 2R5
∫
3k Ωk-1 (A27h) 4(k - 1)(k - 2)
1 1 - R 2q 2 2
1 ln2(p + q) p 1+q 2 ln(p + q) + 2p - 2 ln ln(p + q) + 2R1 p
(
dp ) p -
2
]
1 3 + R 2q 2 2
p 1 + tan-1 p × 2q2 2 ln(p + q) -
(A27f)
k-5 3k 3 + Ω+ Ωk-1 4Ω k - 2 2(k - 1)2(k - 2) (A27g)
3xk 49k - 161 3 Ω8Ω 32 6(k - 2)
∫ q1 dp ) ln(p + q)
4
(
2
p ln (p + q)
)
p 3 - tan-1 p × 2 2 q 2q ln(p + q) 2 - tan-1 p + ln (p + q) - 2q ln(p + q) + 2p + q 3R2 ln(p + q) - 3R5 4
dp ) p +
∞
Also, we have
G 1 ) Y2 G 2 ) Y2
∫1
Y
K02 ) 1 + 4
(af)1/2Y-2 dY
∫1Y (af)-1/2G1Y-2 dY
∑nZ
n
n)0
(A28a)
K04 ) 1 + 8Z ) 32Z2 + 88Z3 + ... ∞
(A28b)
A0 ) -2
∑nZ
n
n)0
The following relations are useful:
2
A0 ) 4Z (1 + 4Z + 10Z2 + ...)
2 q ) -1 + p 1 - Ω2 2 p ) -1 + q 1 + Ω2 dΩ dp )q Ω
A1 A0 A2 A0
)
)
2
1 - Z2 8 Z + ... K1,Z(X)X0) 3 2Z
(
)
(38 Z + ...) + (- 34 + ...) K (X)X ) + (2Z1 + ...)K (X)X ) - 1 -2ZZ K 1,Z
0
2
2
1,Z
0
2,Z(X)X0)
Electrical Properties of Charged Surfaces
Langmuir, Vol. 15, No. 19, 1999 6255
If Z0 is low, then F3 ) -Z0, F4 f 28Z02/9, F5 f 28Z02/9, H3 f -Z0/2, H4 f 130Z02/27, H5 f 130Z02/27, B1 f -8Z02/3, and B2 f -8Z02/3.
lim pf1
Ω(1 -Ω2) Ωp ∂Ω ) ) ∂T 2T(1 + Ω2) 2Tq
lnn Ω ) (-1)n, n ∈ R lim Ωf1 (1 - Ω)n
2 ∂Ω Ω(1 - Ω ) Ωp ) ) ∂κ κq κ(1 + Ω2)
lim [ln Ω ln(1 - Ω)] ) 0
Ωf1
lim pf0
1-q n 1 ln (p + q) ) - , n ∈ R 2 pn+2
1 + q - 2/q2 3 ) 2 p2
LA990138Z
