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The interaction energy between two parallel plates with constant surface charge density
Journal of Colloid and Interface Science 263 (2003) 684–687 www.elsevier.com/locate/jcis
Note
The interaction energy between two parallel plates with constant surface charge density Haoping Wang,a,∗ Chuangye Hou,b and Jun Jin c a Department of Chemistry, Liaoning University of Petroleum and Chemical Technology, Fushun, Liaoning, 113001 PR China b Shallow-Sea Field Branch of Liaohe Field Branch of CNPC, Panjin, Liaoning, 124010 PR China c Beijing East Heavy oil Technical Development Ltd., Beijing, 100081 PR China
Received 1 May 2002; accepted 19 March 2003
Abstract On the basis of Langmuir’s suggestion we simplify the Poisson–Boltzmann equation and derive the relation of surface potential, potential midway, and the plate distance. Thus we obtain the interaction force and energy equations between two dissimilar plates in the case of constant surface charge density. Agreement with the exact numerical values of the interaction of dissimilar plates is good. This method may not only apply to the cases of high constant potential but to the case of high constant charge density. 2003 Elsevier Inc. All rights reserved. Keywords: Electric double layer interaction; Poisson–Boltzmann equation; Langmuir’s method
The subject of this note is the interaction energy and force between two dissimilar plates of highly charged colloidal particles dispersed in aqueous salt solution. The particles have a Gouy double layer, that is, a uniform surface charge neutralized by a diffuse atmosphere of small ions obeying the Poisson–Boltzmann (PB) equation. The approximate results for parallel plates with high surface potential in the case of constant potential have been presented in detail by authors [1,2]. In a previous paper on the basis of Langmuir’s [3] suggestion we simplify the Poisson–Boltzmann equation and derive the relation of surface potential, potential midway, and the plate distance. Thus we used the force equation derived by Langmuir [3] to calculate the interaction energy. This method works well for the case in which the high surface potential remains constant during interaction. In the present paper we apply this method to the cases of constant surface charge density. Figure 1 gives a schematic presentation of the system under consideration. Two parallel plates are h apart in a symmetrical electrolyte of valence v. An x-axis is perpendicular to the given plates and one plate is located at its origin.
* Corresponding author.
E-mail address: [email protected] (H. Wang). 0021-9797/03/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0021-9797(03)00323-0
The PB equation for potential relative to the bulk solution is 2ven veψ d 2ψ , = sinh (1) 2 dx ε0 εr kT where e is the elementary electric charge, n is the electrolyte concentration, ε0 the permittivity of a vacuum, εr the relative permittivity of the solution, v the valence of the ions for a symmetric electrolyte, k the Boltzmann constant, and T the absolute temperature. We can arrange that for the surface potentials ψ1 and ψ2 of plates 1 and 2, respectively, ψ2 always is greater than ψ1 , which has the same sign with ψ2 . Equation (1) may be simplified by introducing dimensionless parameters y and ξ : ve ψi . ξ = κx, yi = (2) kT Substitution of those parameters in Eq. (1) shows that d 2y = sinh y. dξ 2 In this case boundary conditions of Eq. (3) are dy ... = − σ 01 at ξ = 0, dξ dy ...0 = σ 2 at ξ = kh, dξ
(3)
(4)
H. Wang et al. / Journal of Colloid and Interface Science 263 (2003) 684–687
685
Equation (9) is valid for all h, so that for h → ∞ A = cosh(y∞,i ) − 1,
i = 1 or 2.
(10)
Thus, we obtain cosh(yh,i ) − cosh(u) = cosh(y∞,i ) − 1,
i = 1 or 2. (11)
For high surface charge density cosh(yh,i ) − cosh(u) = cosh(y∞,i ),
i = 1 or 2,
(12)
or eyh,i − eu = ey∞,i ,
Fig. 1. Schematic representation of dimensionless potential profile between two dissimilar plates with high constant surface charge density; it shows that dimensionless potentials will change with separation h.
where ... 0 0 4πev σ 1 = σ1 , κεkT
...0 0 4πev σ 2 = σ2 κεkT
(5)
are dimensionless surface charges at both plates and σ10 , σ20 are the surface charge densities at the plates. Equation (4), which is referred often to as the constant charge boundary condition, implies that the charge at each plate remains fixed, irrespective of their separation distance, and the surface potentials are changed with their separation distance h. If the separation between two parallel flat plates is h, then yh,1 , yh,2 are dimensionless potentials at interfaces. Figure 1 gives a schematic presentation of the case, and y∞,1 , y∞,2 is the surface potential in the absence of an interaction at h → ∞, which is called the unperturbed surface potential. The solution of Eq. (3) to satisfy the boundary condition (4) is similar to the case of constant surface potential (see Ref. [1] or [2]); thus we obtain 2π . u = 2 ln (6) κh + 2(e−yh,1 /2 + e−yh,2 /2 ) The surface charge densities σ10 and σ20 are related to the unperturbed surface potentials at both plates by y∞,i 4nve sinh , i = 1 or 2. σi0 = (7) κ 2 For two interacting double layers with a distance h the surface charge densities are related to yh,1 , yh,2 and the potential in minimum, u, by 1/2 nεkT 0 , i = 1 or 2. 2 cosh(yh,i ) − 2 cosh(u) σi = 2π (8) The surface charge densities are constant, independent of h; therefore cosh(yh,i ) − cosh(u) = A, where A is constant.
i = 1 or 2,
(9)
i = 1 or 2.
(13)
We make an attempt to obtain a simple expression to calculate the interaction energy in the case of constant charge from Eqs. (6) and (13). Unfortunately, an approximate expression about constant charge does not appear feasible, since it involves a complex tedious expression. Therefore, the interaction energy in the case of constant charge may only be calculated by the formulae derived from Frens and Overbeek [4] or McCormack et al. [5] and banding with Eqs. (6) and (12) derived by us. ψ
VRσ = VR (yh,1 ; yh,2) +
2nkT κ
y∞,1 × 2(yh,1 − y∞,1 ) sinh 2 y∞,2 + 2(yh,2 − y∞,2 ) sinh 2 y∞,1 yh,1 − cosh − 4 cosh 2 2
yh,2 y∞,2 − 4 cosh − cosh , 2 2
(14)
ψ
where VR (yh,1 ; yh,2 ) is Eq. (14a), which is the interaction energy at constant potential but y1 , y2 have changed into yh,1 and yh,2 , and was derived by Refs. [1,2]. Then yh,1 , yh,2 , and κh will be obtained by solving simultaneously Eqs. (6) and (12). 1 2nkT 1 ψ 2 2π − VR = κ κh + b 2π + b 1 3 3 (κh + b) − (2π + b) + (κh − 2π) , − 24π 2 b = 2(e−yh,1 + e−yh,2 ).
(14a)
For similar parallel plates, however, using Eqs. (6) and (13), an approximate expression about yh , y∞ , and κh can be derived. From Eq. (13) we obtain eyh (1 − ey∞ −yh ) = eu
(15)
or eyh /2 (1 − ey∞ −yh )1/2 = eu/2 .
(16)
686
H. Wang et al. / Journal of Colloid and Interface Science 263 (2003) 684–687
For the particles of constant charge yh > y∞ , and especially for small plate separations h, yh y∞ . We expand (1 − ey∞ −yh )1/2 in a power series and obtain 1 2π . eyh /2 − ey∞ −yh /2 = 2 κh + 4e−yh /2 Simplifying, the cubic equation gives
κh,
(19)
σ =2
ψh ψh dσ = −2
0
σ (ψh ) dψh + 2σ ψh .
(20)
0
Unfortunately, Eq. (19) is a complex tedious expression so that an analytic solution of Eq. (20) cannot be obtained. Thus, the interaction energy in the case of a constant charge for similar parallel plates can be calculated from Eqs. (14) and (19). The repulsive force per unit area (or disjoining pressure), P , is given by P = 2nkT (cosh u − 1) = 2nkT (cosh yh − cosh y∞ ) κh + b 2 2π − = nkT , κh + b 2π
(21)
where b = 4e−yh/2 for the similar parallel plates, and for the dissimilar plates b = 2(e−yh,1 /2 + e−yh,2 /2 ). Thereinafter, the numerical results for the variations of interaction free energy and force per unit area with plate separation will be presented. In Table 1 we show the comparison of the present results, Eqs. (6) and (11), with the results of van Olphen [7], who calculated tables of the potential midway between two flat plates as a function of κh at various values of the surface charge, but he did not calculate the energy of interaction. In ... Table 1 we list only the values of σ 0 = 28 (unperturbed sur... 0 face potential y∞ = 6.638) and σ = 280 (unperturbed surface potential y∞ = 11.22) which show when κh → 0 our results become less accurate. The related errors are ∼ 10% ... ... at κh ∼ 0.2 for σ 0 = 28 and κh ∼ 0.02 for σ 0 = 280. The reason is similar to that of the case of constant potential. The interaction energy and force of dissimilar parallel plates with constant surface charge density as a function
... σ 0 = 28
potentials (u)
where (m = 4 − 2π) is a constant. Once yh is found, the double layer interaction energy per area of two interacting parallel similar plates with constant surface charge density σ at separation h is given by Ref. [6]: VRσ
κh/2
Midway
(17)
1 κhe3yh/2 + (4 − 2π)eyh − κhey∞ eyh /2 − 2ey∞ = 0. (18) 2 Abandoning the illogical root we obtain 1/2 1 1 eyh /2 = 2 κh2 ey∞ + m2 /9 cosh 6 3 2 y∞ κh e (1 − m/12) − m3 /27 × arccos h 1 2 y∞ + m2 /9 3/2 6 κh e − m/3
Table 1 The comparisons of present approximate results with the exact results of van Olphen
3 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0
Present results 0.6286 0.4735 0.3528 0.2587 0.1855 0.1285 0.0840 0.0494 0.0225 0.0015
... σ 0 = 280
Exact values
Present results
Exact values
0.6306 0.4756 0.3554 0.2623 0.1905 0.1357 0.09445 0.06409 0.04329 0.02739
0.6937 0.5386 0.4178 0.3238 0.2506 0.1935 0.1491 0.1145 0.0875 0.0666 0.0502 0.0375 0.0276 0.01999 0.01385 0.009163 0.00552 0.00268 0.00465
0.6943 0.5389 0.4181 0.3240 0.2507 0.1937 0.1493 0.1147 0.08776 0.06679 0.05049 0.03781 0.02799 0.02041 0.01460 0.01021 0.00697 0.00463 0.00301
of κh are shown in Fig. 2 for the scaled unperturbed surface potential y∞,i = 5,10. Exact values are calculated by the method in Ref. [5] and approximate results are obtained from Eq. (14) combined with Eqs. (6) and (11). Similar to the case of constant potential, the relative errors of the interaction energy are less than that of the interaction force. At a range of κh 4 approximate results obtained from the equations are accurate except near κh → 0. However, comparing constant surface charge with constant surface potential, we found an applicable range of the present approximation for a constant charge to be superior to constant potential as κh → 0. For the case of constant surface potential an applicable upper limit is [6] 2π − 2(1 + e(y∞,1−y∞ ,2)/2) ≈ 2(π − 1)e−y∞ ,1/2. ey∞ ,1/2 (22) However, for the case of constant charge it is ψ
κhmin =
κhσmin = 2(π − 1)e−yh,1 /2 .
(23)
As κh → 0 yh,1 will exceed greatly y∞,1 , then κhmin in constant charge will be less than in constant potential. Numerical tests reveal (Table 1 and Fig. 2) that the present approximate analytic expressions are surely applicable to the case of constant surface charge density. However, Eq. (6) is divergent at κh → ∞ and zero point at κh ≈ 2π . This means that they only can be used at κh < 2π and an accurate location is at ∼ κh 4. However, when y∞ 5 and κh > 0.6 the interaction-free energy between constant charge and constant potential will tend to superposition except near κh → 0. Thus when κh > 0.6 the calculating
H. Wang et al. / Journal of Colloid and Interface Science 263 (2003) 684–687
687
Acknowledgment The National Science Foundation of China (No. 20273028) supports this work.
References
Fig. 2. Relative errors of the interaction energy and the interaction force at y∞,1 = 5, y∞,2 = 10. Relative errors ε = vi − v/v. vi is the value calculated with Eqs. (14), (6), and (11), and v is the exact values calculated with the method in Ref. [5].
method is the same as the case of the constant surface potential [1,2].
[1] G. Luo, H. Wang, J. Jin, Langmuir 17 (2001) 2167. [2] G. Luo, R. Feng, J. Jin, H. Wang, J. Colloid Interface Sci. 241 (2001) 81. [3] I. Langmuir, J. Chem. Phys. 6 (1938) 893. [4] G. Frens, J.Th.G. Overbeek, J. Colloid Interface Sci. 38 (1972) 376. [5] D. McCormack, S.L. Carnie, D.Y.C. Chan, J. Colloid Interface Sci. 169 (1995) 177. [6] E.J.W. Verwey, J.Th.G. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. [7] H. van Olphen, An Introduction to Clay Colloid Chemistry, 2nd ed., Wiley–Interscience, New York, 1977.
Note
The interaction energy between two parallel plates with constant surface charge density Haoping Wang,a,∗ Chuangye Hou,b and Jun Jin c a Department of Chemistry, Liaoning University of Petroleum and Chemical Technology, Fushun, Liaoning, 113001 PR China b Shallow-Sea Field Branch of Liaohe Field Branch of CNPC, Panjin, Liaoning, 124010 PR China c Beijing East Heavy oil Technical Development Ltd., Beijing, 100081 PR China
Received 1 May 2002; accepted 19 March 2003
Abstract On the basis of Langmuir’s suggestion we simplify the Poisson–Boltzmann equation and derive the relation of surface potential, potential midway, and the plate distance. Thus we obtain the interaction force and energy equations between two dissimilar plates in the case of constant surface charge density. Agreement with the exact numerical values of the interaction of dissimilar plates is good. This method may not only apply to the cases of high constant potential but to the case of high constant charge density. 2003 Elsevier Inc. All rights reserved. Keywords: Electric double layer interaction; Poisson–Boltzmann equation; Langmuir’s method
The subject of this note is the interaction energy and force between two dissimilar plates of highly charged colloidal particles dispersed in aqueous salt solution. The particles have a Gouy double layer, that is, a uniform surface charge neutralized by a diffuse atmosphere of small ions obeying the Poisson–Boltzmann (PB) equation. The approximate results for parallel plates with high surface potential in the case of constant potential have been presented in detail by authors [1,2]. In a previous paper on the basis of Langmuir’s [3] suggestion we simplify the Poisson–Boltzmann equation and derive the relation of surface potential, potential midway, and the plate distance. Thus we used the force equation derived by Langmuir [3] to calculate the interaction energy. This method works well for the case in which the high surface potential remains constant during interaction. In the present paper we apply this method to the cases of constant surface charge density. Figure 1 gives a schematic presentation of the system under consideration. Two parallel plates are h apart in a symmetrical electrolyte of valence v. An x-axis is perpendicular to the given plates and one plate is located at its origin.
* Corresponding author.
E-mail address: [email protected] (H. Wang). 0021-9797/03/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0021-9797(03)00323-0
The PB equation for potential relative to the bulk solution is 2ven veψ d 2ψ , = sinh (1) 2 dx ε0 εr kT where e is the elementary electric charge, n is the electrolyte concentration, ε0 the permittivity of a vacuum, εr the relative permittivity of the solution, v the valence of the ions for a symmetric electrolyte, k the Boltzmann constant, and T the absolute temperature. We can arrange that for the surface potentials ψ1 and ψ2 of plates 1 and 2, respectively, ψ2 always is greater than ψ1 , which has the same sign with ψ2 . Equation (1) may be simplified by introducing dimensionless parameters y and ξ : ve ψi . ξ = κx, yi = (2) kT Substitution of those parameters in Eq. (1) shows that d 2y = sinh y. dξ 2 In this case boundary conditions of Eq. (3) are dy ... = − σ 01 at ξ = 0, dξ dy ...0 = σ 2 at ξ = kh, dξ
(3)
(4)
H. Wang et al. / Journal of Colloid and Interface Science 263 (2003) 684–687
685
Equation (9) is valid for all h, so that for h → ∞ A = cosh(y∞,i ) − 1,
i = 1 or 2.
(10)
Thus, we obtain cosh(yh,i ) − cosh(u) = cosh(y∞,i ) − 1,
i = 1 or 2. (11)
For high surface charge density cosh(yh,i ) − cosh(u) = cosh(y∞,i ),
i = 1 or 2,
(12)
or eyh,i − eu = ey∞,i ,
Fig. 1. Schematic representation of dimensionless potential profile between two dissimilar plates with high constant surface charge density; it shows that dimensionless potentials will change with separation h.
where ... 0 0 4πev σ 1 = σ1 , κεkT
...0 0 4πev σ 2 = σ2 κεkT
(5)
are dimensionless surface charges at both plates and σ10 , σ20 are the surface charge densities at the plates. Equation (4), which is referred often to as the constant charge boundary condition, implies that the charge at each plate remains fixed, irrespective of their separation distance, and the surface potentials are changed with their separation distance h. If the separation between two parallel flat plates is h, then yh,1 , yh,2 are dimensionless potentials at interfaces. Figure 1 gives a schematic presentation of the case, and y∞,1 , y∞,2 is the surface potential in the absence of an interaction at h → ∞, which is called the unperturbed surface potential. The solution of Eq. (3) to satisfy the boundary condition (4) is similar to the case of constant surface potential (see Ref. [1] or [2]); thus we obtain 2π . u = 2 ln (6) κh + 2(e−yh,1 /2 + e−yh,2 /2 ) The surface charge densities σ10 and σ20 are related to the unperturbed surface potentials at both plates by y∞,i 4nve sinh , i = 1 or 2. σi0 = (7) κ 2 For two interacting double layers with a distance h the surface charge densities are related to yh,1 , yh,2 and the potential in minimum, u, by 1/2 nεkT 0 , i = 1 or 2. 2 cosh(yh,i ) − 2 cosh(u) σi = 2π (8) The surface charge densities are constant, independent of h; therefore cosh(yh,i ) − cosh(u) = A, where A is constant.
i = 1 or 2,
(9)
i = 1 or 2.
(13)
We make an attempt to obtain a simple expression to calculate the interaction energy in the case of constant charge from Eqs. (6) and (13). Unfortunately, an approximate expression about constant charge does not appear feasible, since it involves a complex tedious expression. Therefore, the interaction energy in the case of constant charge may only be calculated by the formulae derived from Frens and Overbeek [4] or McCormack et al. [5] and banding with Eqs. (6) and (12) derived by us. ψ
VRσ = VR (yh,1 ; yh,2) +
2nkT κ
y∞,1 × 2(yh,1 − y∞,1 ) sinh 2 y∞,2 + 2(yh,2 − y∞,2 ) sinh 2 y∞,1 yh,1 − cosh − 4 cosh 2 2
yh,2 y∞,2 − 4 cosh − cosh , 2 2
(14)
ψ
where VR (yh,1 ; yh,2 ) is Eq. (14a), which is the interaction energy at constant potential but y1 , y2 have changed into yh,1 and yh,2 , and was derived by Refs. [1,2]. Then yh,1 , yh,2 , and κh will be obtained by solving simultaneously Eqs. (6) and (12). 1 2nkT 1 ψ 2 2π − VR = κ κh + b 2π + b 1 3 3 (κh + b) − (2π + b) + (κh − 2π) , − 24π 2 b = 2(e−yh,1 + e−yh,2 ).
(14a)
For similar parallel plates, however, using Eqs. (6) and (13), an approximate expression about yh , y∞ , and κh can be derived. From Eq. (13) we obtain eyh (1 − ey∞ −yh ) = eu
(15)
or eyh /2 (1 − ey∞ −yh )1/2 = eu/2 .
(16)
686
H. Wang et al. / Journal of Colloid and Interface Science 263 (2003) 684–687
For the particles of constant charge yh > y∞ , and especially for small plate separations h, yh y∞ . We expand (1 − ey∞ −yh )1/2 in a power series and obtain 1 2π . eyh /2 − ey∞ −yh /2 = 2 κh + 4e−yh /2 Simplifying, the cubic equation gives
κh,
(19)
σ =2
ψh ψh dσ = −2
0
σ (ψh ) dψh + 2σ ψh .
(20)
0
Unfortunately, Eq. (19) is a complex tedious expression so that an analytic solution of Eq. (20) cannot be obtained. Thus, the interaction energy in the case of a constant charge for similar parallel plates can be calculated from Eqs. (14) and (19). The repulsive force per unit area (or disjoining pressure), P , is given by P = 2nkT (cosh u − 1) = 2nkT (cosh yh − cosh y∞ ) κh + b 2 2π − = nkT , κh + b 2π
(21)
where b = 4e−yh/2 for the similar parallel plates, and for the dissimilar plates b = 2(e−yh,1 /2 + e−yh,2 /2 ). Thereinafter, the numerical results for the variations of interaction free energy and force per unit area with plate separation will be presented. In Table 1 we show the comparison of the present results, Eqs. (6) and (11), with the results of van Olphen [7], who calculated tables of the potential midway between two flat plates as a function of κh at various values of the surface charge, but he did not calculate the energy of interaction. In ... Table 1 we list only the values of σ 0 = 28 (unperturbed sur... 0 face potential y∞ = 6.638) and σ = 280 (unperturbed surface potential y∞ = 11.22) which show when κh → 0 our results become less accurate. The related errors are ∼ 10% ... ... at κh ∼ 0.2 for σ 0 = 28 and κh ∼ 0.02 for σ 0 = 280. The reason is similar to that of the case of constant potential. The interaction energy and force of dissimilar parallel plates with constant surface charge density as a function
... σ 0 = 28
potentials (u)
where (m = 4 − 2π) is a constant. Once yh is found, the double layer interaction energy per area of two interacting parallel similar plates with constant surface charge density σ at separation h is given by Ref. [6]: VRσ
κh/2
Midway
(17)
1 κhe3yh/2 + (4 − 2π)eyh − κhey∞ eyh /2 − 2ey∞ = 0. (18) 2 Abandoning the illogical root we obtain 1/2 1 1 eyh /2 = 2 κh2 ey∞ + m2 /9 cosh 6 3 2 y∞ κh e (1 − m/12) − m3 /27 × arccos h 1 2 y∞ + m2 /9 3/2 6 κh e − m/3
Table 1 The comparisons of present approximate results with the exact results of van Olphen
3 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0
Present results 0.6286 0.4735 0.3528 0.2587 0.1855 0.1285 0.0840 0.0494 0.0225 0.0015
... σ 0 = 280
Exact values
Present results
Exact values
0.6306 0.4756 0.3554 0.2623 0.1905 0.1357 0.09445 0.06409 0.04329 0.02739
0.6937 0.5386 0.4178 0.3238 0.2506 0.1935 0.1491 0.1145 0.0875 0.0666 0.0502 0.0375 0.0276 0.01999 0.01385 0.009163 0.00552 0.00268 0.00465
0.6943 0.5389 0.4181 0.3240 0.2507 0.1937 0.1493 0.1147 0.08776 0.06679 0.05049 0.03781 0.02799 0.02041 0.01460 0.01021 0.00697 0.00463 0.00301
of κh are shown in Fig. 2 for the scaled unperturbed surface potential y∞,i = 5,10. Exact values are calculated by the method in Ref. [5] and approximate results are obtained from Eq. (14) combined with Eqs. (6) and (11). Similar to the case of constant potential, the relative errors of the interaction energy are less than that of the interaction force. At a range of κh 4 approximate results obtained from the equations are accurate except near κh → 0. However, comparing constant surface charge with constant surface potential, we found an applicable range of the present approximation for a constant charge to be superior to constant potential as κh → 0. For the case of constant surface potential an applicable upper limit is [6] 2π − 2(1 + e(y∞,1−y∞ ,2)/2) ≈ 2(π − 1)e−y∞ ,1/2. ey∞ ,1/2 (22) However, for the case of constant charge it is ψ
κhmin =
κhσmin = 2(π − 1)e−yh,1 /2 .
(23)
As κh → 0 yh,1 will exceed greatly y∞,1 , then κhmin in constant charge will be less than in constant potential. Numerical tests reveal (Table 1 and Fig. 2) that the present approximate analytic expressions are surely applicable to the case of constant surface charge density. However, Eq. (6) is divergent at κh → ∞ and zero point at κh ≈ 2π . This means that they only can be used at κh < 2π and an accurate location is at ∼ κh 4. However, when y∞ 5 and κh > 0.6 the interaction-free energy between constant charge and constant potential will tend to superposition except near κh → 0. Thus when κh > 0.6 the calculating
H. Wang et al. / Journal of Colloid and Interface Science 263 (2003) 684–687
687
Acknowledgment The National Science Foundation of China (No. 20273028) supports this work.
References
Fig. 2. Relative errors of the interaction energy and the interaction force at y∞,1 = 5, y∞,2 = 10. Relative errors ε = vi − v/v. vi is the value calculated with Eqs. (14), (6), and (11), and v is the exact values calculated with the method in Ref. [5].
method is the same as the case of the constant surface potential [1,2].
[1] G. Luo, H. Wang, J. Jin, Langmuir 17 (2001) 2167. [2] G. Luo, R. Feng, J. Jin, H. Wang, J. Colloid Interface Sci. 241 (2001) 81. [3] I. Langmuir, J. Chem. Phys. 6 (1938) 893. [4] G. Frens, J.Th.G. Overbeek, J. Colloid Interface Sci. 38 (1972) 376. [5] D. McCormack, S.L. Carnie, D.Y.C. Chan, J. Colloid Interface Sci. 169 (1995) 177. [6] E.J.W. Verwey, J.Th.G. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. [7] H. van Olphen, An Introduction to Clay Colloid Chemistry, 2nd ed., Wiley–Interscience, New York, 1977.