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Electrical potentials of two identical particles with fixed surface charge density in a salt-free medium
Journal of Colloid and Interface Science 356 (2011) 550–556
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Electrical potentials of two identical particles with fixed surface charge density in a salt-free medium Jyh-Ping Hsu a,⇑, Chih-Hua Huang a, Shiojenn Tseng b a b
Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan Department of Mathematics, Tamkang University, Tamsui, Taipei 25137, Taiwan
a r t i c l e
i n f o
Article history: Received 24 November 2010 Accepted 11 January 2011 Available online 15 January 2011 Keywords: Electrical potential Constant surface charge Two planar, cylindrical, and spherical particles Salt-free medium Perturbation solution
a b s t r a c t The electrical potentials of two identical planar, cylindrical, and spherical particles immersed in a saltfree dispersion are solved analytically by a perturbation approach for the case of constant surface charge density. The system under consideration simulates, for example, micelles, where the ionic species in the liquid phase come mainly from the dissociation of the functional groups on the droplet surface. We show that for planar particles, the present zero-order perturbation solution is exact, and for cylindrical and spherical particles, the first-order perturbation solution provides sufficiently accurate results, with an averaged percentage deviation on the order of 1% under typical conditions. In general, the higher the surface charge density, the higher the valence of counterions, the smaller the separation distance between two particles, and the smaller the curvature of particle surface, the better the performance of the perturbation solution. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Salt-free dispersion is defined as a colloidal dispersion where the ionic species in its dispersion medium come mainly from the counterions dissociated from the dispersed phase. Although this kind of dispersion is a special case of a regular dispersion, where both counterions and coions are present with an appreciable concentration, relevant analyses on the former are not necessarily simpler than those on the latter [1]. Salt-free dispersion has been applied in many areas, including, for example, estimating the diffusion coefficient of sodium ions [2], understanding the structural and thermodynamic properties of moderately or highly concentrated solutions [3], quantifying the attraction between equally charged polyelectrolytes [4], studying the characteristics of amphoteric, amphiphilic polyurethanes [5], getting information on transition from a neutral to charged macromolecules [6], resurrecting abandoned proteins [7], investigating the phase behavior and rheological properties of cationic and anionic surfactant mixtures [8], and determining extremely low surface charge density associated with low-permittivity media through electroosmosis [9]. Many theoretical attempts have been made recently on the analyses of the properties and the electrokinetic phenomena of a salt-free dispersion. These include, for example, electrical potential distribution [10–14], electrophoresis [15–19], electrical double
⇑ Corresponding author. Fax: +886 2 23623040. E-mail address: [email protected] (J.-P. Hsu). 0021-9797/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2011.01.037
layer [20,21], equilibrium sedimentation [22], and critical coagulation concentration or stability [23–26]. In a recent study, we derived an approximate perturbation solution for the electrical potential of two identical planar, cylindrical, and spherical particles in a salt-free dispersion [10]. The derivation was based on a constant surface potential model; that is, the surface potential of a particle is uninfluenced by the neighboring environment such as the presence of another particle or surface. If the surface charge comes from the dissociation/association reactions of functional groups, this corresponds to the case where the rates of those reactions are infinitely fast [27]. Although the constant surface potential model is widely adopted in the assessment of the stability of a colloidal dispersion and in the description of various types of electrokinetic phenomenon, the condition of infinitely fast surface reactions can be unrealistic in practice. The other extreme, where the rates of the surface reactions are infinitely slow, corresponds to the constant surface charge model [28]. This model was often adopted to describe salt-free systems (e.g., [29,30]). Constant surface potential and constant surface charge are idealized models, yet they provide the upper and the lower limits for the actual charged conditions on a surface and, therefore, deserve detailed study. In this study, the perturbation method adopted previously for the resolution of the electrical potential of two identical planes, cylinders, and spheres maintained at constant surface potentials [10] is applied to solve that at constant surface charge densities. Mathematically, the former belongs to a Dirichlet-type problem, and the latter is a Neumann type. The influences of the curvature of the particle surface, the level of the surface charge density, the
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line segment connecting their centers. Because the present problem is symmetric about x = H/2, only the electrical potential in the domain 0 6 x 6 (H/2) needs be considered. Since the particle surface is maintained at a constant surface charge density, the boundary conditions associated with Eq. (1) can be expressed as follows:
valence of counterions, and the separation distance between two particles on the performance of the perturbation solution derived are discussed. 2. Theory Let us consider the case where the surface of a particle is maintained at a constant charge density. Without loss of generality, we assume the particle surface is positively charged. If we let b, C 0b ; e, F, R, T, and w be the valence of counterions, a reference molar concentration of counterions, the permittivity of the medium, Faraday constant, gas constant, the absolute temperature, and the electrical potential, respectively, then the scaled electrical potential y (=Fw/ RT) can be described by [11] 2
x
d y
dy 1 by ¼ e ; þ 2 dx b x þ ðh=kÞ dx
@y ¼ bR; @x
@y ¼ 0; @x
H : 2
ð3Þ
x¼
H : 2
ð4Þ
Assuming k < 1, that is, the double layer is thinner than the radius of a nonplanar particle, a perturbation solution to Eq. (1) can be expressed as [10]
y¼
ð1aÞ
1 X
ki yi :
ð5Þ
i¼0
Here k ¼ 1=ja0 with j = (2IF2/eRT)1/2 and a0 being the reciprocal Debye length and the radius of a cylindrical or a spherical particle, 2 respectively, and I ¼ C 0b b =2 being the ionic strength. For a planar particle, r is the distance from its surface, and is the distance from the center of a cylindrical or a spherical particle. x takes the values of 0, 1, and 2 for planar, cylindrical, and spherical particles, respectively; h takes the values of 0 and 1 for planar particles and cylindrical or spherical particles, respectively. Note that k is a measure for the curvature effect of the surface of a nonplanar particle. A first-order perturbation solution for a single planar, cylindrical, or spherical particle is derived in Appendix A. For two identical particles, as shown in Fig. 1, we let H (=jh) be the scaled separation distance between two planar particles or the scaled shortest surface-to-surface distance between two cylindrical or two spherical particles with h being the corresponding separation distance. x, R (=Fr/eRT), and ym (=Fwm/RT) are the scaled distance from the surface of the left particle, the scaled surface charge density, and the scaled potential at x = H/2, respectively, with r and wm being the surface charge density and the electrical potential at x = H/2, respectively. Note that for cylindrical and spherical particles, Eq. (1) describes the electrical potential outside them and along the
(a)
x¼
In addition, the symmetric condition below needs to be satisfied:
ð1Þ
r h : ka0 k
ð2Þ
@y ¼ 0; @x
y ¼ ym ;
where
x¼
x ¼ 0;
y
Substituting this expression into Eqs. (1)–(3) and collecting terms of the same order in k, we obtain 2
d y0 2
dx
¼
1 by0 e ; b
2
d y1 2
dx 2
d y0 2
dx
x
dy0 1 by0 þ xe ; dx b
n¼1
@y0 ¼ bR and @x
ym x
ð7Þ
2
1 d yn1 dy x y0n ¼ x x n1 þ y0n1 ; 2 b b dx dx dx 2
@yn ¼ 0; @x
n P 1;
nP2
x¼0
ð8Þ
ð9Þ
@y0 ¼ 0; @x
x¼
H ; 2
n¼0
ð10Þ
@y1 ¼ 0; @x
x¼
H ; 2
n¼1
ð11Þ
(b)
H
ð6Þ
2
y1 eby0 ¼ x
d yn
∂y = −bΣ ∂x x =0
H/2
n¼0
∂y
= −bΣ
++ y ∂x x =0 + + + + + y + m + + + + + H/2 + + + + + +
H
++ + + + + + + + + x + + + + + + + + +
Fig. 1. Schematic representation of the problem considered for the case of two identical, positively charged planar particles (a) and two identical, positively charged cylindrical or spherical particles (b), where y, R, and b are the scaled potential, the scaled charge density, and the valence of counterions, respectively; H is the scaled separation distance between two planar particles, and the scaled shortest surface-to-surface distance between two cylindrical or spherical particles; x is the scaled distance from the surface of the left particle; ym is the value of y, at x = H/2.
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" # n @ X i k yi ¼ 0; @x i¼0
H x¼ ; 2
n P 2;
ð12Þ
where y0n ¼ ð@ n eby =@kn Þk¼0 =n!; n P 2. In addition, we define
y0 ¼ y0m
and yn ¼ ynm ;
n P 1;
x¼
H ; 2
The scaled potential at x = H/2 can be approximated by
qffiffiffiffiffiffiffiffiffi by b2 R ffi y0 ¼ 1b ln eby0m sec2 e 20m x þ tan1 pffiffiffiffiffiffiffiffiffiffi by 0m
;
¼ yplanars
06x6
H : 2
For two cylindrical or spherical particles, yn, n P 1, need also to be solved. As pointed out by Hsu and Huang [10], if k is sufficiently small, then using the first-order approximation is satisfactory. That is, yn, n P 2 can be neglected, yielding
ð17Þ
Solving Eq. (7) subject to Eqs. (9) and (11) gives
1 2x þ x tanðPx þ Q Þ x P b Px 2 2x x tanðPx þ QÞ þ tanðPx þ QÞ; b bP H 06x6 2
2x tan W; bP
which is obtained from Eq. (18) by letting x = H/2. Knowing the value of ym is necessary in the estimation of the electrostatic energy between two nonplanar particles [1].
The performance of the present perturbation solution is examined by comparing the results based on that solution with the corresponding exact values obtained by solving Eq. (1) numerically. Here, we focused mainly on the case of two particles and, since the result for two planar particles, Eq. (14), is exact, only that of two cylindrical or spherical particles, Eq. (19), is examined. Fig. 2 shows the simulated distributions of the normalized scaled electrical potential (y/ys), at various values of the scaled surface charge density, R, for both two cylindrical particles and two spherical particles. Both the results based on the present perturbation solution, Eq. (19), and the corresponding exact numerical values are presented. As can be seen in Fig. 2, the higher the surface charge density the better the performance of Eq. (19), in general. To measure the accuracy of the present perturbation solution, we define the averaged percentage deviation in the scaled electrical potential, SEP, and the percentage deviation of the scaled central electrical potential, SCEP, as follows:
R
y1 ¼ M 1 tanðPx þ Q Þ M 2
rffiffiffiffiffiffiffiffiffiffi eby0m 2
P¼
ð18Þ
ð18aÞ 2
b R Q ¼ tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2eby0m
M1 ¼
2x b
ð18bÞ
SEP ¼ 100%
jy yexact jdx R jyexact jdx
ð21Þ
SCEP ¼ 100%
jym ym;exact j : ym;exact
ð22Þ
In these expressions, yexact is the value of y obtained by solving Eq. (1) numerically and ym,exact = yexact(x = H/2). Here, SCEP is calculated by two ways. First, Eq. (15) is solved numerically for y0m, substitute the result obtained into Eqs. (18a)–(18e) and (20a) to get y1m, use Eq. (20) to evaluate ym, and then substitute it into Eq. (22). The SCEP thus obtained is denoted as SCEP1. Second, the only difference is that Eq. (16) is used directly to estimate y1m, and the SCEP thus ob-
2
tan Q tan W þ xbHP tan Q sec2 W þ 2bx þ 2Pbx ðH=2Þ tan W þ P bx ðH=2Þ2 sec2 W 2bx sec2 W
M2 ¼ M1
ð20aÞ
3. Results and discussion
ð16Þ
y ffi y0 þ ky1 :
1 H xH P xH 2 þ tan W tan W P 2 b 4b
ð15Þ
Eqs. (14) and (15) are consistent with the results of Liu and Hsu [25] for the case of two planar particles.pffiffiffiffiffiffiffiffiffiffiffiffi Noteffi that if 4 2 2 1 by0m Þ ffi p=2 expðby Þ b R =2, then because tan ðb R = 2e 0m pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2eby0m =b R, Eq. (15) can be solved directly to give
pffiffiffi 2 ! 2p b R 2 : ln 2 b Hb R þ 4
ð20Þ
where
þ
where yplanars denotes the scaled potential for two planar surfaces. In this expression y0m can be determined by substituting Eq. (14) into Eq. (10) to yield
! rffiffiffiffiffiffiffiffiffiffi 2 eby0m H b R 1 p ffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ 0: þ tan 2 2 2eby0m
ym ffi y0m þ ky1m ;
y1m ¼ M 1 tan W M2
ð14Þ
y0m ¼
ð19Þ
ð13Þ
where ynm denotes the value of yn at x = H/2. For two planar surfaces, since k ¼ 0 Eq. (1) implies that only Eq. (6) needs to be solved. Solving this equation subject to Eqs. (9) and (10) and applying Eq. (13), we obtain
2e
1 y ffi y0 þ k M1 tanðPx þ Q Þ M 2 þ x tanðPx þ QÞ P 2x Px 2 2x H x x tanðPx þ QÞ þ tanðPx þ QÞ ; 0 6 x 6 : 2 b b bP
P sec2 W P tan Q tan W P2 ðH=2Þ sec2 W tan Q
P sec2 Q 2x þ : tan Q tan Q b
W ¼ PðH=2Þ þ Q Therefore, for two cylindrical or spherical particles,
ð18dÞ ð18eÞ
ð18cÞ
tained is denoted as SCEP2. Table 1a summarizes the SEP, SCEP1, and SCEP2 for the case of Fig. 2. This table reveals that the higher the surface charge density (or surface potential) the better the performance of the present perturbation result. At the assumed value of k, regardless of the level of the surface charge density, the performance of Eq. (19) is satisfactory; both SEP and SCEP1 are below
J.-P. Hsu et al. / Journal of Colloid and Interface Science 356 (2011) 550–556
553
Fig. 2. Distributions of the normalized scaled electrical potential, (y/ys), at various values of R for two cylindrical particles (a) and two spherical particles (b) at b = 1, H = 3, and k ¼ 0:1; solid curves, present perturbation results based on Eq. (19); discrete symbols, exact values.
1%. The deviation of SCEP2 at small values of R, however, can be very large, which is expected because the condition of 4 expðby0m Þ b R2 =2 is not satisfied and, therefore, the approximate expression, Eq. (16), is inaccurate. The simulated distributions of the scaled electrical potential y at various values of b for both two cylindrical particles and two spherical particles are presented in Fig. 3, and the corresponding SEP and SCEP are summarized in Table 1b. Fig. 3 and Table 1b reveal that the larger the b the more accurate the present perturbation solution, Eq. (19). Note that the scaled surface potential ys does not increase monotonically with b, nor is the approximate scaled midpoint scaled potential ym. According to Eqs. (15) and (20), this is because R (and therefore, ys), ym, b, and H are all related to each other. SEP, SCEP1, and SCEP2, on the other hand, all decrease monotonically with increasing b. The influence of the separation distance between two particles on the performance of the present perturbation solution, Eq. (19), is illustrated in Fig. 4 and Table 1c. As seen in Table 1c, the smaller the separation distance between two particles the higher the surface potential and the smaller the SEP and SCEP1, implying the better the performance of Eq. (19). Note that SCEP2 does not vary monotonically with H. The influence of the perturbation parameter k, which measures the curvature effect of the surface of a nonplanar particle, on the performance of Eq. (19) is illustrated in Fig. 5 and Table 1d. As expected, the smaller the k the more accurate Eq. (19) is. Note that the scaled surface potential ys decreases with increasing k. Surprisingly, as seen in Table 1d, Eq. (19) remains accurate even at k ¼ 1. As seen in Table 1d, although SCEP1 remains small for k up to unity, SCEP2 is appreciable. A comparison between the present results and those of Hsu and Huang [10] reveals that the performance of the perturbation method adopted for the case of constant surface charge density is better than that for the case of constant surface potential.
ditions: higher surface charge density, higher valance of counterions, smaller separation distance between two particles, and smaller curvature of particle surface. The performance of the present perturbation method for the case of constant surface charge density is better than that for the case of constant surface potential. Acknowledgment This work is supported by the National Science Council of the Republic of China. Appendix A. Electrical potential for a single particle For convenience, Eq. (1) in the text is rewritten 2
d y 2
dx
þ
x dy 1 by ¼ e : x þ ðh=kÞ dx b
ðA1Þ
If the particle surface is maintained at a constant charge density
r, then the boundary conditions associated with Eq. (A1) on the particle surface can be expressed as
@y ¼ bR; @x
x¼0
ðA2Þ
where R = Fr/eRT is the scaled surface charge density. At a point far away from the particle surface, the following two types of boundary condition can be assigned:
y ¼ 0 as x ! 1
ðA3Þ
@y ¼ 0 as x ! 1: @x
ðA4Þ
A.1. Case I y = 0 as x ? 1 4. Summary Under the condition of constant surface charge density, we derived a perturbation solution for the electrical potential of two identical planar, cylindrical, and spherical particles immersed in a salt-free dispersion. For planar particles, the zero-order perturbation solution is exact, and for cylindrical and spherical particles, the first-order perturbation solution provides sufficiently accurate results under typical conditions. The performance of the present perturbation solution is more satisfactory under the following con-
For a planar particle, substituting x = 0 and h = 0 into Eq. (A1) and solving the resultant equation subject to Eqs. (A2) and (A3), we obtain the following scaled electrical potential for a planar particle, yplanar:
yplanar ¼
" ! !# pffiffiffi 2 1 2 b R : ln sec2 x þ tan1 pffiffiffi b 2 2
ðA5Þ
It can be shown that R and the scaled surface potential ys = Fws/ RT with ws being the corresponding surface potential are related by
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Table 1 Averaged percentage deviation (%) of the scaled electrical potential of two particles (SEP) and the percentage deviation (%) of the scaled central electrical potential (SCEP1, SCEP2) based on the present perturbation solution for various values of R at b = 1, k ¼ 0:1, and H = 1 (a), those for various values of b at R = 5, k ¼ 0:1, and H = 3 (b), those for various values of H at R = 5, k ¼ 0:1, and b = 1 (c), and those for various values of k at R = 5, b = 1, and H = 1 (d); ys,c and ys,s denote the surface potential of cylindrical particles and that of spherical particles, respectively. (a) ys,c/ys,s
1 0.827/0.803
2 1.665/1.641
5 2.945/2.922
8 3.701/3.681
Cylindrical particles SEP SCEP1 SCEP2
0.0574 0.0463 129.3399
0.0332 0.0206 33.3599
0.0272 0.0112 6.5060
0.0167 0.0093 2.5177
Spherical particles SEP SCEP1 SCEP2
0.0680 0.0578 124.4701
0.0349 0.0228 35.4267
0.0254 0.0112 6.7548
0.0146 0.0099 2.5955
R
(b) b ys,c/ys,s
1 2.585/2.53
2 2.645/2.63
3 2.305/2.302
Cylindrical particles SEP SCEP1 SCEP2
0.1744 0.6067 5.9172
0.0765 0.2948 0.3613
0.0350 0.2669 0.2727
Spherical particles SEP SCEP1 SCEP2
0.1518 0.9024 7.3733
0.1095 0.5032 0.5761
0.0710 0.4648 0.4711
(c) H ys,c/ys,s
1 2.945/2.922
2 2.679/2.641
3 2.585/2.537
Cylindrical particles SEP SCEP1 SCEP2
0.0272 0.0112 6.5060
0.0717 0.0855 4.0142
0.1744 0.6067 5.9172
Spherical particles SEP SCEP1 SCEP2
0.0254 0.0112 6.7548
0.0592 0.1000 4.3278
0.1518 0.9024 7.3733
(d) k ys,c/ys,s
0.1 2.945/2.922
0.3 2.901/2.834
0.5 2.859/2.745
Cylindrical particles SEP SCEP1 SCEP2
0.0272 0.0112 6.5060
0.0967 0.0982 7.0892
0.2327 0.2666 7.7664
0.5552 0.6620 8.9486
0.8490 1.0297 11.3344
Spherical particles SEP SCEP1 SCEP2
0.0254 0.0112 6.7548
0.0831 0.1005 7.8811
0.2040 0.2809 9.1738
0.5281 0.7321 11.4499
0.8538 1.1635 13.2137
R¼
pffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ebys 1: 2 b
2
ðA6Þ
As pointed out by Hsu and Huang [10], because Eq. (A6) is of a periodic nature, a characteristic scaled distance D is defined, which is the shortest scaled distance at which yplanar ffi 0. Therefore, the solution to Eq. (A1) is y = yplanar for x 6 D, and y = 0 for x > D. For a single cylindrical and a spherical particle, we assume that the double layer is sufficiently thin compared with its radius; that is, k ¼ 1=ja0 < 1. Under this condition, y is expanded as
y¼
1 X
ki yi :
ðA7Þ
d y0 2
dx
¼
1 by0 e ; b
0.8 2.798/2.614
;n ¼ 0
2
d y1 2
dx
1.0 2.758/2.527
ðA8Þ
2
y1 eby0 ¼ x
2
d yn
d y0 2
dx
x
dy0 1 by0 þ xe ; dx b
n¼1
ðA9Þ
2
1 d yn1 dy x y0n ¼ x x n1 þ y0n1 ; 2 b b dx dx dx 2
n P 2;
ðA10Þ
where y0n ¼ ð@ n eby =@kn Þk¼0 =n!. The corresponding boundary conditions are
i¼0
Substituting this expression into Eq. (A1) and collecting terms of the same order in k, we obtain
@y0 ¼ bR and @x
@yn ¼ 0; @x
n P 1;
x¼0
ðA11Þ
J.-P. Hsu et al. / Journal of Colloid and Interface Science 356 (2011) 550–556
555
Fig. 3. Distributions of the scaled electrical potential y at various values of b for two cylindrical particles (a) and two spherical particles (b) at H = 3, k ¼ 0:1, and R = 5; solid curves, present perturbation results based on Eq. (19); discrete symbols, exact values.
Fig. 4. Distributions of the scaled electrical potential y at various values of H for two cylindrical particles (a) and two spherical particles (b) at b = 1, k ¼ 0:1, and R = 5; solid curves, present perturbation results based on Eq. (19); discrete symbols, exact values.
Fig. 5. Distributions of the scaled electrical potential y at various values of k for two cylindrical particles (a) and two spherical particles (b) at b = 1, H = 1, and R = 5; solid curves, present perturbation results based on Eq. (19); discrete symbols, exact values.
yn ¼ 0;
n P 0;
x P D:
ðA12Þ
Solving Eq. (A8) subject to Eqs. (A11) and (A12), we obtain
" ! !# pffiffiffi 2 1 2 b R 2 1 pffiffiffi ¼ yplanar x þ tan y0 ¼ ln sec b 2 2
ðA13Þ
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That is, the zero-order perturbation solution of a single cylindrical or a spherical particle is the exact solution of a single planar particle. Substituting Eq. (A13) into Eq. (A9) and solving the resultant expression subject to Eqs. (A11) and (A12) for n = 1 yields
1 2x y1 ¼ C 1 tanðAx þ BÞ þ C 2 x tanðAx þ BÞ x A b
Ax 2 2x x tanðAx þ BÞ þ tanðAx þ BÞ; b bA
Substituting Eq. (A5) into Eq. (A9) and solving the resultant expression subject to Eqs. (A16) and (A17) for n = 1, we obtain Eq. (A14) with
pffiffiffiffiffiffiffiffi A ¼ 1=2 2
B ¼ tan
C1 ¼ C2D þ
Ax 2 D b
ðA14Þ
ðA14aÞ 2
B ¼ tan1
b R pffiffiffi 2
! ðA14bÞ
2xAR tan B 2x tan2 B C1 ¼ bA sec2 B C2 ¼
ðA14cÞ
2xAR : b
ðA14dÞ
The higher-order solutions, yn, n P 2, can be obtained from Eqs. (A10)–(A12) by following a similar procedure as that employed in the derivation of y1. This, however, becomes nontrivial, especially as n gets large. As concluded by Hsu and Huang [10], if k is sufficiently small, then using the approximate expression below is usually satisfactory:
y ffi y0 þ ky1 :
ðA15Þ
A.2. Case II oy/ox = 0 as x ? 1 For a planar particle, solving Eq. (A1) with x = 0 and h = 0 subject to Eqs. (A2) and (A4) also leads to (A5). In the present case, the boundary conditions associated with Eq. (A1) for a cylindrical or a spherical particle become
@y0 ¼ bR and @x
@yn ¼ 0; @x
1 X @ i ðk yi Þ ¼ 0; @x i¼0
x ! 1:
n P 1;
x¼0
ðA16Þ
ðA17Þ
!
b R pffiffiffi 2
1
where
pffiffiffiffiffiffiffiffi A ¼ 1=2
ðA18Þ
C2 ¼
A2 xD2 b
sec2 B þ 2bx tan2 B : AD sec2 B tanB
ðA18aÞ
ðA18bÞ
ðA18cÞ
References [1] R.J. Hunter, Foundations of Colloid Science, vol. 1, Oxford Univ. Press, New York, 1992. [2] P. Ander, W. Lubas, Macromolecules 14 (1981) 1058. [3] B. Sjoberg, K. Mortensen, Biophys. Chem. 65 (1997) 75. [4] F. Gro1hn, M. Antonietti, Macromolecules 33 (2000) 5938. [5] A. Dong, G. Hou, Y. Wang, D. Sun, J. Polym. Sci. Part B: Polym. Phys. 40 (2002) 972. [6] M. Nagy, Colloids Surf. A 250 (2004) 467. [7] M. Li, J. Liu, X. Ran, M. Fang, J. Shi, H. Qin, J.M. Goh, J. Song, Biophys. J. 91 (2006) 4201. [8] H.G. Li, J.C. Hao, J. Phys. Chem. B 112 (2008) 10497. [9] F.M. Chang, Y.W. Chang, Y.J. Sheng, H.K. Tsao, Appl. Phys. Lett. 97 (2010) 164101. [10] J.P. Hsu, C.H. Huang, J. Colloid Interface Sci. 348 (2010) 402. [11] J.P. Hsu, H.Y. Yu, S. Tseng, J. Phys. Chem. B 110 (2006) 25007. [12] S. Tseng, N.B. Wong, P.C. Liu, J.P. Hsu, Langmuir 23 (2007) 10448. [13] H. Ohshima, J. Colloid Interface Sci. 260 (2003) 339. [14] H. Ohshima, J. Colloid Interface Sci. 268 (2003) 429. [15] H. Ohshima, J. Colloid Interface Sci. 248 (2002) 499. [16] H. Ohshima, Colloids Surf. A 222 (2003) 207. [17] H. Ohshima, Chem. Eng. Sci. 61 (2006) 2104. [18] C.P. Chiang, E. Lee, Y.Y. He, J.P. Hsu, J. Phys. Chem. B 110 (2006) 1490. [19] H. Ohshima, Colloid Polym. Sci. 285 (2007) 1411. [20] R. Ni, D. Cao, W. Wang, J. Phys. Chem. B 110 (2006) 26232. [21] E. Ruiz-Reina, F. Carrique, J. Phys. Chem. B 112 (2008) 11960. [22] T.Y. Wang, H.T. Li, Y.J. Sheng, H.K. Tsao, J. Chem. Phys. 129 (2008) 204504. [23] A.K. Mukherjee, K.S. Schmitz, L.B. Bhuiyan, Langmuir 18 (2002) 4210. [24] J.P. Hsu, H.Y. Yu, S. Tseng, J. Phys. Chem. B 110 (2006) 7600. [25] B.T. Liu, J.P. Hsu, J. Chem. Phys. 128 (2008) 104509. [26] B.T. Liu, J.P. Hsu, Langmuir 25 (2009) 9045. [27] J.E. Jones, S. Levine, J. Colloid Interface Sci. 30 (1969) 241. [28] G. Frens, D.J.C. Engel, J. Th, G. Overbeek, Trans. Faraday Soc. 63 (1967) 418. [29] F. Gröhn, M. Antonietti, Macromolecules 33 (2000) 5938. [30] P. Ander, W. Lubas, Am. Chem. Soc. 14 (1981) 1058.
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Electrical potentials of two identical particles with fixed surface charge density in a salt-free medium Jyh-Ping Hsu a,⇑, Chih-Hua Huang a, Shiojenn Tseng b a b
Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan Department of Mathematics, Tamkang University, Tamsui, Taipei 25137, Taiwan
a r t i c l e
i n f o
Article history: Received 24 November 2010 Accepted 11 January 2011 Available online 15 January 2011 Keywords: Electrical potential Constant surface charge Two planar, cylindrical, and spherical particles Salt-free medium Perturbation solution
a b s t r a c t The electrical potentials of two identical planar, cylindrical, and spherical particles immersed in a saltfree dispersion are solved analytically by a perturbation approach for the case of constant surface charge density. The system under consideration simulates, for example, micelles, where the ionic species in the liquid phase come mainly from the dissociation of the functional groups on the droplet surface. We show that for planar particles, the present zero-order perturbation solution is exact, and for cylindrical and spherical particles, the first-order perturbation solution provides sufficiently accurate results, with an averaged percentage deviation on the order of 1% under typical conditions. In general, the higher the surface charge density, the higher the valence of counterions, the smaller the separation distance between two particles, and the smaller the curvature of particle surface, the better the performance of the perturbation solution. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Salt-free dispersion is defined as a colloidal dispersion where the ionic species in its dispersion medium come mainly from the counterions dissociated from the dispersed phase. Although this kind of dispersion is a special case of a regular dispersion, where both counterions and coions are present with an appreciable concentration, relevant analyses on the former are not necessarily simpler than those on the latter [1]. Salt-free dispersion has been applied in many areas, including, for example, estimating the diffusion coefficient of sodium ions [2], understanding the structural and thermodynamic properties of moderately or highly concentrated solutions [3], quantifying the attraction between equally charged polyelectrolytes [4], studying the characteristics of amphoteric, amphiphilic polyurethanes [5], getting information on transition from a neutral to charged macromolecules [6], resurrecting abandoned proteins [7], investigating the phase behavior and rheological properties of cationic and anionic surfactant mixtures [8], and determining extremely low surface charge density associated with low-permittivity media through electroosmosis [9]. Many theoretical attempts have been made recently on the analyses of the properties and the electrokinetic phenomena of a salt-free dispersion. These include, for example, electrical potential distribution [10–14], electrophoresis [15–19], electrical double
⇑ Corresponding author. Fax: +886 2 23623040. E-mail address: [email protected] (J.-P. Hsu). 0021-9797/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2011.01.037
layer [20,21], equilibrium sedimentation [22], and critical coagulation concentration or stability [23–26]. In a recent study, we derived an approximate perturbation solution for the electrical potential of two identical planar, cylindrical, and spherical particles in a salt-free dispersion [10]. The derivation was based on a constant surface potential model; that is, the surface potential of a particle is uninfluenced by the neighboring environment such as the presence of another particle or surface. If the surface charge comes from the dissociation/association reactions of functional groups, this corresponds to the case where the rates of those reactions are infinitely fast [27]. Although the constant surface potential model is widely adopted in the assessment of the stability of a colloidal dispersion and in the description of various types of electrokinetic phenomenon, the condition of infinitely fast surface reactions can be unrealistic in practice. The other extreme, where the rates of the surface reactions are infinitely slow, corresponds to the constant surface charge model [28]. This model was often adopted to describe salt-free systems (e.g., [29,30]). Constant surface potential and constant surface charge are idealized models, yet they provide the upper and the lower limits for the actual charged conditions on a surface and, therefore, deserve detailed study. In this study, the perturbation method adopted previously for the resolution of the electrical potential of two identical planes, cylinders, and spheres maintained at constant surface potentials [10] is applied to solve that at constant surface charge densities. Mathematically, the former belongs to a Dirichlet-type problem, and the latter is a Neumann type. The influences of the curvature of the particle surface, the level of the surface charge density, the
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line segment connecting their centers. Because the present problem is symmetric about x = H/2, only the electrical potential in the domain 0 6 x 6 (H/2) needs be considered. Since the particle surface is maintained at a constant surface charge density, the boundary conditions associated with Eq. (1) can be expressed as follows:
valence of counterions, and the separation distance between two particles on the performance of the perturbation solution derived are discussed. 2. Theory Let us consider the case where the surface of a particle is maintained at a constant charge density. Without loss of generality, we assume the particle surface is positively charged. If we let b, C 0b ; e, F, R, T, and w be the valence of counterions, a reference molar concentration of counterions, the permittivity of the medium, Faraday constant, gas constant, the absolute temperature, and the electrical potential, respectively, then the scaled electrical potential y (=Fw/ RT) can be described by [11] 2
x
d y
dy 1 by ¼ e ; þ 2 dx b x þ ðh=kÞ dx
@y ¼ bR; @x
@y ¼ 0; @x
H : 2
ð3Þ
x¼
H : 2
ð4Þ
Assuming k < 1, that is, the double layer is thinner than the radius of a nonplanar particle, a perturbation solution to Eq. (1) can be expressed as [10]
y¼
ð1aÞ
1 X
ki yi :
ð5Þ
i¼0
Here k ¼ 1=ja0 with j = (2IF2/eRT)1/2 and a0 being the reciprocal Debye length and the radius of a cylindrical or a spherical particle, 2 respectively, and I ¼ C 0b b =2 being the ionic strength. For a planar particle, r is the distance from its surface, and is the distance from the center of a cylindrical or a spherical particle. x takes the values of 0, 1, and 2 for planar, cylindrical, and spherical particles, respectively; h takes the values of 0 and 1 for planar particles and cylindrical or spherical particles, respectively. Note that k is a measure for the curvature effect of the surface of a nonplanar particle. A first-order perturbation solution for a single planar, cylindrical, or spherical particle is derived in Appendix A. For two identical particles, as shown in Fig. 1, we let H (=jh) be the scaled separation distance between two planar particles or the scaled shortest surface-to-surface distance between two cylindrical or two spherical particles with h being the corresponding separation distance. x, R (=Fr/eRT), and ym (=Fwm/RT) are the scaled distance from the surface of the left particle, the scaled surface charge density, and the scaled potential at x = H/2, respectively, with r and wm being the surface charge density and the electrical potential at x = H/2, respectively. Note that for cylindrical and spherical particles, Eq. (1) describes the electrical potential outside them and along the
(a)
x¼
In addition, the symmetric condition below needs to be satisfied:
ð1Þ
r h : ka0 k
ð2Þ
@y ¼ 0; @x
y ¼ ym ;
where
x¼
x ¼ 0;
y
Substituting this expression into Eqs. (1)–(3) and collecting terms of the same order in k, we obtain 2
d y0 2
dx
¼
1 by0 e ; b
2
d y1 2
dx 2
d y0 2
dx
x
dy0 1 by0 þ xe ; dx b
n¼1
@y0 ¼ bR and @x
ym x
ð7Þ
2
1 d yn1 dy x y0n ¼ x x n1 þ y0n1 ; 2 b b dx dx dx 2
@yn ¼ 0; @x
n P 1;
nP2
x¼0
ð8Þ
ð9Þ
@y0 ¼ 0; @x
x¼
H ; 2
n¼0
ð10Þ
@y1 ¼ 0; @x
x¼
H ; 2
n¼1
ð11Þ
(b)
H
ð6Þ
2
y1 eby0 ¼ x
d yn
∂y = −bΣ ∂x x =0
H/2
n¼0
∂y
= −bΣ
++ y ∂x x =0 + + + + + y + m + + + + + H/2 + + + + + +
H
++ + + + + + + + + x + + + + + + + + +
Fig. 1. Schematic representation of the problem considered for the case of two identical, positively charged planar particles (a) and two identical, positively charged cylindrical or spherical particles (b), where y, R, and b are the scaled potential, the scaled charge density, and the valence of counterions, respectively; H is the scaled separation distance between two planar particles, and the scaled shortest surface-to-surface distance between two cylindrical or spherical particles; x is the scaled distance from the surface of the left particle; ym is the value of y, at x = H/2.
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" # n @ X i k yi ¼ 0; @x i¼0
H x¼ ; 2
n P 2;
ð12Þ
where y0n ¼ ð@ n eby =@kn Þk¼0 =n!; n P 2. In addition, we define
y0 ¼ y0m
and yn ¼ ynm ;
n P 1;
x¼
H ; 2
The scaled potential at x = H/2 can be approximated by
qffiffiffiffiffiffiffiffiffi by b2 R ffi y0 ¼ 1b ln eby0m sec2 e 20m x þ tan1 pffiffiffiffiffiffiffiffiffiffi by 0m
;
¼ yplanars
06x6
H : 2
For two cylindrical or spherical particles, yn, n P 1, need also to be solved. As pointed out by Hsu and Huang [10], if k is sufficiently small, then using the first-order approximation is satisfactory. That is, yn, n P 2 can be neglected, yielding
ð17Þ
Solving Eq. (7) subject to Eqs. (9) and (11) gives
1 2x þ x tanðPx þ Q Þ x P b Px 2 2x x tanðPx þ QÞ þ tanðPx þ QÞ; b bP H 06x6 2
2x tan W; bP
which is obtained from Eq. (18) by letting x = H/2. Knowing the value of ym is necessary in the estimation of the electrostatic energy between two nonplanar particles [1].
The performance of the present perturbation solution is examined by comparing the results based on that solution with the corresponding exact values obtained by solving Eq. (1) numerically. Here, we focused mainly on the case of two particles and, since the result for two planar particles, Eq. (14), is exact, only that of two cylindrical or spherical particles, Eq. (19), is examined. Fig. 2 shows the simulated distributions of the normalized scaled electrical potential (y/ys), at various values of the scaled surface charge density, R, for both two cylindrical particles and two spherical particles. Both the results based on the present perturbation solution, Eq. (19), and the corresponding exact numerical values are presented. As can be seen in Fig. 2, the higher the surface charge density the better the performance of Eq. (19), in general. To measure the accuracy of the present perturbation solution, we define the averaged percentage deviation in the scaled electrical potential, SEP, and the percentage deviation of the scaled central electrical potential, SCEP, as follows:
R
y1 ¼ M 1 tanðPx þ Q Þ M 2
rffiffiffiffiffiffiffiffiffiffi eby0m 2
P¼
ð18Þ
ð18aÞ 2
b R Q ¼ tan1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2eby0m
M1 ¼
2x b
ð18bÞ
SEP ¼ 100%
jy yexact jdx R jyexact jdx
ð21Þ
SCEP ¼ 100%
jym ym;exact j : ym;exact
ð22Þ
In these expressions, yexact is the value of y obtained by solving Eq. (1) numerically and ym,exact = yexact(x = H/2). Here, SCEP is calculated by two ways. First, Eq. (15) is solved numerically for y0m, substitute the result obtained into Eqs. (18a)–(18e) and (20a) to get y1m, use Eq. (20) to evaluate ym, and then substitute it into Eq. (22). The SCEP thus obtained is denoted as SCEP1. Second, the only difference is that Eq. (16) is used directly to estimate y1m, and the SCEP thus ob-
2
tan Q tan W þ xbHP tan Q sec2 W þ 2bx þ 2Pbx ðH=2Þ tan W þ P bx ðH=2Þ2 sec2 W 2bx sec2 W
M2 ¼ M1
ð20aÞ
3. Results and discussion
ð16Þ
y ffi y0 þ ky1 :
1 H xH P xH 2 þ tan W tan W P 2 b 4b
ð15Þ
Eqs. (14) and (15) are consistent with the results of Liu and Hsu [25] for the case of two planar particles.pffiffiffiffiffiffiffiffiffiffiffiffi Noteffi that if 4 2 2 1 by0m Þ ffi p=2 expðby Þ b R =2, then because tan ðb R = 2e 0m pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2eby0m =b R, Eq. (15) can be solved directly to give
pffiffiffi 2 ! 2p b R 2 : ln 2 b Hb R þ 4
ð20Þ
where
þ
where yplanars denotes the scaled potential for two planar surfaces. In this expression y0m can be determined by substituting Eq. (14) into Eq. (10) to yield
! rffiffiffiffiffiffiffiffiffiffi 2 eby0m H b R 1 p ffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ 0: þ tan 2 2 2eby0m
ym ffi y0m þ ky1m ;
y1m ¼ M 1 tan W M2
ð14Þ
y0m ¼
ð19Þ
ð13Þ
where ynm denotes the value of yn at x = H/2. For two planar surfaces, since k ¼ 0 Eq. (1) implies that only Eq. (6) needs to be solved. Solving this equation subject to Eqs. (9) and (10) and applying Eq. (13), we obtain
2e
1 y ffi y0 þ k M1 tanðPx þ Q Þ M 2 þ x tanðPx þ QÞ P 2x Px 2 2x H x x tanðPx þ QÞ þ tanðPx þ QÞ ; 0 6 x 6 : 2 b b bP
P sec2 W P tan Q tan W P2 ðH=2Þ sec2 W tan Q
P sec2 Q 2x þ : tan Q tan Q b
W ¼ PðH=2Þ þ Q Therefore, for two cylindrical or spherical particles,
ð18dÞ ð18eÞ
ð18cÞ
tained is denoted as SCEP2. Table 1a summarizes the SEP, SCEP1, and SCEP2 for the case of Fig. 2. This table reveals that the higher the surface charge density (or surface potential) the better the performance of the present perturbation result. At the assumed value of k, regardless of the level of the surface charge density, the performance of Eq. (19) is satisfactory; both SEP and SCEP1 are below
J.-P. Hsu et al. / Journal of Colloid and Interface Science 356 (2011) 550–556
553
Fig. 2. Distributions of the normalized scaled electrical potential, (y/ys), at various values of R for two cylindrical particles (a) and two spherical particles (b) at b = 1, H = 3, and k ¼ 0:1; solid curves, present perturbation results based on Eq. (19); discrete symbols, exact values.
1%. The deviation of SCEP2 at small values of R, however, can be very large, which is expected because the condition of 4 expðby0m Þ b R2 =2 is not satisfied and, therefore, the approximate expression, Eq. (16), is inaccurate. The simulated distributions of the scaled electrical potential y at various values of b for both two cylindrical particles and two spherical particles are presented in Fig. 3, and the corresponding SEP and SCEP are summarized in Table 1b. Fig. 3 and Table 1b reveal that the larger the b the more accurate the present perturbation solution, Eq. (19). Note that the scaled surface potential ys does not increase monotonically with b, nor is the approximate scaled midpoint scaled potential ym. According to Eqs. (15) and (20), this is because R (and therefore, ys), ym, b, and H are all related to each other. SEP, SCEP1, and SCEP2, on the other hand, all decrease monotonically with increasing b. The influence of the separation distance between two particles on the performance of the present perturbation solution, Eq. (19), is illustrated in Fig. 4 and Table 1c. As seen in Table 1c, the smaller the separation distance between two particles the higher the surface potential and the smaller the SEP and SCEP1, implying the better the performance of Eq. (19). Note that SCEP2 does not vary monotonically with H. The influence of the perturbation parameter k, which measures the curvature effect of the surface of a nonplanar particle, on the performance of Eq. (19) is illustrated in Fig. 5 and Table 1d. As expected, the smaller the k the more accurate Eq. (19) is. Note that the scaled surface potential ys decreases with increasing k. Surprisingly, as seen in Table 1d, Eq. (19) remains accurate even at k ¼ 1. As seen in Table 1d, although SCEP1 remains small for k up to unity, SCEP2 is appreciable. A comparison between the present results and those of Hsu and Huang [10] reveals that the performance of the perturbation method adopted for the case of constant surface charge density is better than that for the case of constant surface potential.
ditions: higher surface charge density, higher valance of counterions, smaller separation distance between two particles, and smaller curvature of particle surface. The performance of the present perturbation method for the case of constant surface charge density is better than that for the case of constant surface potential. Acknowledgment This work is supported by the National Science Council of the Republic of China. Appendix A. Electrical potential for a single particle For convenience, Eq. (1) in the text is rewritten 2
d y 2
dx
þ
x dy 1 by ¼ e : x þ ðh=kÞ dx b
ðA1Þ
If the particle surface is maintained at a constant charge density
r, then the boundary conditions associated with Eq. (A1) on the particle surface can be expressed as
@y ¼ bR; @x
x¼0
ðA2Þ
where R = Fr/eRT is the scaled surface charge density. At a point far away from the particle surface, the following two types of boundary condition can be assigned:
y ¼ 0 as x ! 1
ðA3Þ
@y ¼ 0 as x ! 1: @x
ðA4Þ
A.1. Case I y = 0 as x ? 1 4. Summary Under the condition of constant surface charge density, we derived a perturbation solution for the electrical potential of two identical planar, cylindrical, and spherical particles immersed in a salt-free dispersion. For planar particles, the zero-order perturbation solution is exact, and for cylindrical and spherical particles, the first-order perturbation solution provides sufficiently accurate results under typical conditions. The performance of the present perturbation solution is more satisfactory under the following con-
For a planar particle, substituting x = 0 and h = 0 into Eq. (A1) and solving the resultant equation subject to Eqs. (A2) and (A3), we obtain the following scaled electrical potential for a planar particle, yplanar:
yplanar ¼
" ! !# pffiffiffi 2 1 2 b R : ln sec2 x þ tan1 pffiffiffi b 2 2
ðA5Þ
It can be shown that R and the scaled surface potential ys = Fws/ RT with ws being the corresponding surface potential are related by
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Table 1 Averaged percentage deviation (%) of the scaled electrical potential of two particles (SEP) and the percentage deviation (%) of the scaled central electrical potential (SCEP1, SCEP2) based on the present perturbation solution for various values of R at b = 1, k ¼ 0:1, and H = 1 (a), those for various values of b at R = 5, k ¼ 0:1, and H = 3 (b), those for various values of H at R = 5, k ¼ 0:1, and b = 1 (c), and those for various values of k at R = 5, b = 1, and H = 1 (d); ys,c and ys,s denote the surface potential of cylindrical particles and that of spherical particles, respectively. (a) ys,c/ys,s
1 0.827/0.803
2 1.665/1.641
5 2.945/2.922
8 3.701/3.681
Cylindrical particles SEP SCEP1 SCEP2
0.0574 0.0463 129.3399
0.0332 0.0206 33.3599
0.0272 0.0112 6.5060
0.0167 0.0093 2.5177
Spherical particles SEP SCEP1 SCEP2
0.0680 0.0578 124.4701
0.0349 0.0228 35.4267
0.0254 0.0112 6.7548
0.0146 0.0099 2.5955
R
(b) b ys,c/ys,s
1 2.585/2.53
2 2.645/2.63
3 2.305/2.302
Cylindrical particles SEP SCEP1 SCEP2
0.1744 0.6067 5.9172
0.0765 0.2948 0.3613
0.0350 0.2669 0.2727
Spherical particles SEP SCEP1 SCEP2
0.1518 0.9024 7.3733
0.1095 0.5032 0.5761
0.0710 0.4648 0.4711
(c) H ys,c/ys,s
1 2.945/2.922
2 2.679/2.641
3 2.585/2.537
Cylindrical particles SEP SCEP1 SCEP2
0.0272 0.0112 6.5060
0.0717 0.0855 4.0142
0.1744 0.6067 5.9172
Spherical particles SEP SCEP1 SCEP2
0.0254 0.0112 6.7548
0.0592 0.1000 4.3278
0.1518 0.9024 7.3733
(d) k ys,c/ys,s
0.1 2.945/2.922
0.3 2.901/2.834
0.5 2.859/2.745
Cylindrical particles SEP SCEP1 SCEP2
0.0272 0.0112 6.5060
0.0967 0.0982 7.0892
0.2327 0.2666 7.7664
0.5552 0.6620 8.9486
0.8490 1.0297 11.3344
Spherical particles SEP SCEP1 SCEP2
0.0254 0.0112 6.7548
0.0831 0.1005 7.8811
0.2040 0.2809 9.1738
0.5281 0.7321 11.4499
0.8538 1.1635 13.2137
R¼
pffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ebys 1: 2 b
2
ðA6Þ
As pointed out by Hsu and Huang [10], because Eq. (A6) is of a periodic nature, a characteristic scaled distance D is defined, which is the shortest scaled distance at which yplanar ffi 0. Therefore, the solution to Eq. (A1) is y = yplanar for x 6 D, and y = 0 for x > D. For a single cylindrical and a spherical particle, we assume that the double layer is sufficiently thin compared with its radius; that is, k ¼ 1=ja0 < 1. Under this condition, y is expanded as
y¼
1 X
ki yi :
ðA7Þ
d y0 2
dx
¼
1 by0 e ; b
0.8 2.798/2.614
;n ¼ 0
2
d y1 2
dx
1.0 2.758/2.527
ðA8Þ
2
y1 eby0 ¼ x
2
d yn
d y0 2
dx
x
dy0 1 by0 þ xe ; dx b
n¼1
ðA9Þ
2
1 d yn1 dy x y0n ¼ x x n1 þ y0n1 ; 2 b b dx dx dx 2
n P 2;
ðA10Þ
where y0n ¼ ð@ n eby =@kn Þk¼0 =n!. The corresponding boundary conditions are
i¼0
Substituting this expression into Eq. (A1) and collecting terms of the same order in k, we obtain
@y0 ¼ bR and @x
@yn ¼ 0; @x
n P 1;
x¼0
ðA11Þ
J.-P. Hsu et al. / Journal of Colloid and Interface Science 356 (2011) 550–556
555
Fig. 3. Distributions of the scaled electrical potential y at various values of b for two cylindrical particles (a) and two spherical particles (b) at H = 3, k ¼ 0:1, and R = 5; solid curves, present perturbation results based on Eq. (19); discrete symbols, exact values.
Fig. 4. Distributions of the scaled electrical potential y at various values of H for two cylindrical particles (a) and two spherical particles (b) at b = 1, k ¼ 0:1, and R = 5; solid curves, present perturbation results based on Eq. (19); discrete symbols, exact values.
Fig. 5. Distributions of the scaled electrical potential y at various values of k for two cylindrical particles (a) and two spherical particles (b) at b = 1, H = 1, and R = 5; solid curves, present perturbation results based on Eq. (19); discrete symbols, exact values.
yn ¼ 0;
n P 0;
x P D:
ðA12Þ
Solving Eq. (A8) subject to Eqs. (A11) and (A12), we obtain
" ! !# pffiffiffi 2 1 2 b R 2 1 pffiffiffi ¼ yplanar x þ tan y0 ¼ ln sec b 2 2
ðA13Þ
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That is, the zero-order perturbation solution of a single cylindrical or a spherical particle is the exact solution of a single planar particle. Substituting Eq. (A13) into Eq. (A9) and solving the resultant expression subject to Eqs. (A11) and (A12) for n = 1 yields
1 2x y1 ¼ C 1 tanðAx þ BÞ þ C 2 x tanðAx þ BÞ x A b
Ax 2 2x x tanðAx þ BÞ þ tanðAx þ BÞ; b bA
Substituting Eq. (A5) into Eq. (A9) and solving the resultant expression subject to Eqs. (A16) and (A17) for n = 1, we obtain Eq. (A14) with
pffiffiffiffiffiffiffiffi A ¼ 1=2 2
B ¼ tan
C1 ¼ C2D þ
Ax 2 D b
ðA14Þ
ðA14aÞ 2
B ¼ tan1
b R pffiffiffi 2
! ðA14bÞ
2xAR tan B 2x tan2 B C1 ¼ bA sec2 B C2 ¼
ðA14cÞ
2xAR : b
ðA14dÞ
The higher-order solutions, yn, n P 2, can be obtained from Eqs. (A10)–(A12) by following a similar procedure as that employed in the derivation of y1. This, however, becomes nontrivial, especially as n gets large. As concluded by Hsu and Huang [10], if k is sufficiently small, then using the approximate expression below is usually satisfactory:
y ffi y0 þ ky1 :
ðA15Þ
A.2. Case II oy/ox = 0 as x ? 1 For a planar particle, solving Eq. (A1) with x = 0 and h = 0 subject to Eqs. (A2) and (A4) also leads to (A5). In the present case, the boundary conditions associated with Eq. (A1) for a cylindrical or a spherical particle become
@y0 ¼ bR and @x
@yn ¼ 0; @x
1 X @ i ðk yi Þ ¼ 0; @x i¼0
x ! 1:
n P 1;
x¼0
ðA16Þ
ðA17Þ
!
b R pffiffiffi 2
1
where
pffiffiffiffiffiffiffiffi A ¼ 1=2
ðA18Þ
C2 ¼
A2 xD2 b
sec2 B þ 2bx tan2 B : AD sec2 B tanB
ðA18aÞ
ðA18bÞ
ðA18cÞ
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