Langmuir 1999, 15, 5219-5226
5219
Stability of Colloidal Dispersions: Charge Regulation/ Adsorption Model Jyh-Ping Hsu* and Bo-Tau Liu Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617 Republic of China Received August 18, 1998. In Final Form: April 22, 1999 The modified stability ratio, W′, and the critical coagulation concentration, CCC, of a colloidal dispersion containing spherical particles are analyzed theoretically. We consider the case where the surface of a particle is capable of regulating its charged condition through the dissociation of functional groups, the exchange of ions with the liquid phase, or both. An amphoteric surface is assumed for the former. The analysis leads to an approximate analytic expression for the electrical interaction energy between two particles. We show that the smaller the (pK- - pK+), K- and K+ being the dissociation constants, the greater the interaction energy. It is found that a system may have two different CCC’s and two different electrolyte concentrations at which W′ ) 1. However, the greater the (pK- + pK+), or the greater the (pK- pK+), the less the significance of the second CCC and the electrolyte concentration at which W′ ) 1. The number of CCC and that of electrolyte concentration at which W′ ) 1 may not be the same, and we show that W′ provides a more realistic description of the qualitative behavior of a colloidal dispersion than does CCC.
1. Introduction The stability ratio and the critical coagulation concentration are two of the basic properties of a colloidal dispersion. These properties are closely related to the interactions between two charged entities which include the electrical and the van der Waals interactions. The electrical interaction between two charged surfaces or particles has been investigated extensively in the literature. Often, it is assumed that the two interacting entities are maintained at constant and uniform surface conditions for a simpler mathematical treatment.1 In particular, constant surface potential or charge density models are considered in most of the relevant studies. Although this may be adequate under certain conditions, it can be unrealistic for surfaces bearing dissociable functional groups or capable of exchanging ions with the surrounding liquid phase. The so-called charge-regulation phenomenon is mainly due to the fact that two interacting surfaces tend to minimize the electrical interaction energy through adjusting the amount of potential determining ion carried by them.2,3 Ninham and Parsegian4 pointed out that if two identical biological surfaces approaching each other are in ionic equilibrium with the surrounding liquid phase, neither the surface potential nor the surface charge density remain constant. A similar problem which considered two planar surfaces bearing multiple ionizable functional groups was discussed by Prieve and Ruckenstein.5 As suggested by Chan et al.,6 the constant surface potential and constant surface charge models can be viewed as two extreme cases. In the former, the exchange of the ions * To whom correspondence should be addressed. Fax: 886-223623040. E-mail:
[email protected]. (1) Hunter, R. J. Foundations of Colloid Science; Oxford University: London, 1989; Vol. 1. (2) Healy, T. W.; Chan, D.; White, L. R. Pure Appl. Chem. 1980, 52, 1207. (3) Chan, D.; Perram, J. W.; White, L. R.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1046. (4) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405. (5) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 205. (6) Chan, D.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 1 1976, 72, 2844.
between the functional groups of a surface and the liquid phase is fast, and it is slow in the latter. Compared with the results for the electrical interaction energy between two surfaces having a constant surface potential (or charge density), the reported analyses for charge-regulated surfaces are limited. Carnie and Chan7 derived the electrical double-layer interaction energy between two planar, parallel charged surfaces for the case in which the boundary condition is linearized. The case of charge-regulated surfaces was also discussed. Pujar and Zydney8 adopted the approach of Smith and Deen9 to analyze the problem of a spherical particle in a cylindrical pore taking the effect of charge regulation into account. The electrical potential distribution derived involves a harmonic function expansion, the coefficients of which needed to be determined through a complicated procedure. An approximate expression for the interaction energy was also derived. Hsu and Kuo10 simulated the adsorption of a charge-regulated particle to a rigid, charged surface by adopting the Deryaguin’s approximation. In the present study, the interactions between two spherical particles are discussed under the condition that the surface of a particle contains dissociable functional groups, is capable of exchanging ions with the surrounding liquid phase, or both. An attempt is made to derive a simple approximate analytic expression for the electrical interaction energy between two particles. Both the stability ratio and the critical coagulation concentration of the system under consideration are estimated. 2. Analysis By referring to Figure 1, we consider two spherical particles in an electrolyte solution. The particles may have different sizes. Let R be the distance between the centers of particles 1 and 2, r21 the distance measured from the center of particle 1 to the surface of particle 2, and a1 and (7) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 161, 206. (8) Pujar, N. S.; Zydney, A. L. J. Colloid Interface Sci. 1997, 192, 338. (9) Smith, F. G.; Deen, W. M. J. Colloid Interface Sci. 1980, 78, 444. (10) Hsu, J. P.; Kuo, Y. C. Langmuir 1997, 13, 4372.
10.1021/la981053l CCC: $18.00 © 1999 American Chemical Society Published on Web 06/26/1999
5220 Langmuir, Vol. 15, No. 16, 1999
Hsu and Liu
Ξ ) σ0 Γ)
δ sinh(ψN/ψ0) 1 + δ cosh(ψN/ψ0)
δ cosh(ψN/ψ0) + δ2 σ0 [1 + δ cosh(ψ /ψ )]2 ψ0 N
Figure 1. Schematic representation of the system under consideration. R is the center-to-center distance between particles 1 and 2, r21 is the scaled distance between the center of particle 1 to a point on the surface of particle 2, and a1 and a2 are the scaled radii of particles 1 and 2.
AB T A
Z-
(1a)
Z+
+B
(1b)
where AB denotes the functional group, and BZ+ the potential determining ion (PDI), with Z being the valence. For the case of an amphoteric surface, the PDI is usually the hydrogen ion,11 and the corresponding dissociation constants for eq 1a,b are denoted by K+ and K-, respectively. For some other cases, the electrolyte, C, in the bulk solution may adsorb to the surface, and charge inversion may occur. For simplicity, we assume B ) C. Therefore, two cases are considered in the following discussions: If B * C, the electrolyte in the bulk liquid phase contains indifferent ions (IDI), and if B ) C, it contains PDI. The net surface charge density can be evaluated by6
(
)
ψN - ψ S δ sinh ψ0 σ ) σ0 ψN - ψS 1 + δ cosh ψ0
(
)
(2)
σ ) σ0
δ sinh(ψN/ψ0) 1 + δ cosh(ψN/ψ0)
-
δ cosh(ψN/ψ0) + δ2 σ0 ψS [1 + cosh(ψ /ψ )]2 ψ0
) Ξ - ΓψS
N
∆ψ ) κ2ψ 2
2
0
(3)
(4)
2
κ ) 2F Z C/RT
(4a)
where ∆ denotes the Laplace operator, κ is the reciprocal Debye length, F and R are, respectively, the Faraday constant and the gas constant, and is the dielectric constant. It can be shown that a particular solution to eq 4 is
G(r b) )
exp(-κr b) 4πr b
(5)
where b r is the scaled position vector from a fixed point charge. G is known as the free Green’s function.13 Applying the Green’s formula, the electrical potential can be expressed as 2
ψ)
∫S ∑ j)1
[
]
G
i
σi - ψSiχi dS
(6)
with
χi )
where ψS is the surface potential, σ0 ) eZNS, ψ0 ) kBT/e, δ ) 2 × 10-∆pK/2, ψN ) 2.303(pK0 - pB)ψ0, pK0 ) (pK+ + pK-)/2, and ∆pK ) pK- - pK+. In these expressions, e and NS are, respectively, the elementary charge and the number density of the surface groups, and kB and T are the Boltzmann constant and the absolute temperature, respectively, ψN is the Nernst potential; it is related to the point of zero charge when pK0 ) pB. Ettelaie and Buscall12 pointed out that a surface with a low ∆pK is capable of regulating its charged condition efficiently, such that its surface potential is more or less constant as the separation distance between two surfaces varies. In contrast, a surface with a high ∆pK is considered to have properties close to those with a constant surface charge. If ψS is low, eq 2 can be approximated by
(3b)
0
Ξ and Γ can also be interpreted in another way.7 Suppose that the electrical potential of the system, ψ is low, and its spatial variation at equilibrium can be described approximately by the linearized PoissonBoltzmann equation
a2 the radii of particles 1 and 2, respectively. Suppose that the following dissociation reactions occur at the surface of a particle: Z+ ABZ+ 2 T AB + B
(3a)
∂G ∂ni
(6a)
In these expressions, Si denotes the surface of particle i, ni is the outward normal vector of Si, and ψSi and σi are the surface potential and surface charge of particle i, respectively. We have assumed implicitly that the dielectric constant of the solid phase is far smaller than that of the liquid phase, which is usually satisfied under conditions of practical significance. Similar to the treatment in the potential theory,14 for a point on the surface of particle j, Sj, we have
1 2
2
ψSj )
∫S ∑ i)1
i
[
G
]
σi - ψSiχi dS
(7)
Suppose that ψSi can be decomposed as
ψSi ) ψS0 i + ∆ψSi
(8)
where
where ψS0 i denotes the unperturbed surface potential of isolated particle i, and ∆ψSi a perturbed surface potential
(11) Hunter, R. J. Foundations of Colloid Science; Oxford University: London, 1989; Vol. 2. (12) Ettelaie, R.; Buscall, R. Adv. Colloid Interface Sci. 1995, 61, 131.
(13) Greenberg, M. D. Application of Green’s Functions in Science and Engineering; Prentice Hall: Englewood Cliffs, NJ, 1971. (14) Jeffreys, H.; Jeffreys, B. S. Methods of Mathematical Physics; Cambridge University Press: Cambridge, 1956.
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Langmuir, Vol. 15, No. 16, 1999 5221
due to the presence of the other particle. ψS0 i can be calculated by substituting eq 3 into eq 7
Ξi β Ξ iai ii ψ0Si ) ) Γiai + (κai + 1) 1 Γi + βii - γii 2
(9)
γii ) -
[ (
[
ℵ ℵ ℵ ) ℵ1,1 ℵ1,2 2,1 2,2
(
)
]
(9b)
Substituting eqs 8 and 9 into eq 7 and applying eq 3, we obtain
)
χ2
1 ∆ψS2 ) 2
)
χ2
(
)
1
(
(
)
)
∫
(
-Γ1 -Γ2 G + χ1 ∆ψS1 dS + S2 G+ -Γ1 Ξ2 β21 + γ21 ψS0 1 + β21 (10b) ∆ψS2 dS +
∫S
1
(
)
∫
where
βji ) aiβii
e-κ(rji-ai) rji -κ(rji-ai)
1 e γji ) ai + γii 2 rji
(
)
(10c) (10d)
The subscript j in these expressions denotes that a point on Sj is considered. Equations 10a and b are implicit expressions for ∆ψS1 and ∆ψS2 which have the form of an integral equation with an inseparable kernel. To solve eqs 10a and b for ∆ψS1 and ∆ψS2, we assume that the surface integral of a smooth function F can be approximated by
∫
M
F dS ≈
wiFi ∑ i)1
(11)
where wi is a weighting factor and Fi is the value of F at point i. The values of wi and M depend on the integration scheme adopted. On the basis of eqs 11, 10a and b can be expressed as
χ∆ψ ) B where
(12)
γ12,1 +
ψS0 2 γ12,M + ψS0 1 γ21,1 +
(
-Γ1 -Γ2 G + χ1 ∆ψS1 dS + S2 G+ Ξ2 -Γ2 β12 + γ12 ψS0 2 + β12 (10a) ∆ψS2 dS +
∫S
)
[ ] (
ψS0 2
B)
1 ∆ψS1 ) 2
(12b) (12c)
-Γk l,i Gk,j + χl,i i, j ) 1, ..., M l, k ) 1, 2 k,j -Γ 1 k l,i ) -wj Gk,j + χl,i k,j 2 i ) j and l ) k, i, j ) 1, ..., M l, k ) 1, 2 (12d)
(9a)
1 1 1 - 1+ exp(-2κai) 2 κai κai
]
ℵl,k ) [ℵi,j,l,k] i, j)1,...,M l,k ) 1, 2 ℵi,j,l,k ) -wj
where
sinh(κai) exp(-κai) βii ) κ
∆ψ ) [∆ψ1,1, ∆ψ1,2, ... ∆ψ1,M, ∆ψ2,1, ..., ∆ψ2,M]t (12a)
ψS0 1 γ21,M +
)
-Γ2ψ0S2 + Ξ2
β12,1
l -Γ2ψ0S2 + Ξ2 -Γ1ψ
0
S1
β12,M
+ Ξ1
(12e)
β21,1
l -Γ1ψ0S1 + Ξ1
β21,M
In these expressions, superscript t denotes the transpose l,i l,i of a matrix, Gk,j and χk,j are, respectively, the values of G and χ in which the distance r is measured from the point j of particle k to the point i of particle l and the differentiation is conducted on Sk, γji,k and βji,k denote, respectively, the values of γji and βji evaluated at the point l,i l,i and χk,j become k of particle j. If l ) k and i ) j, then Gk,j singular. In this case, we suggest using the following approach to evaluate their values. According to eqs 9a l,i k,j and χk,j can be evaluated by and b, the values of Gk,j l,i l,i and χk,j from subtracting the contribution of all other Gk,j Gkk and γkk, that is,
ℵj,j,k,k )
1
-
2
(
-Γk
)
βkk + γkk +
M
∑ Wi i)1 i*j
(
-Γk
k,j k,j Gk,i + χk,i
)
(13)
For the present linear system, the interaction energy between two particles can be evaluated by7
VDL )
∫S
1 Ξ 2 1
1
∆ψS1 dS +
∫S
1 Ξ 2 2
2
∆ψS2 dS (14)
Under typical conditions, ∆ψSi varies smoothly over surface Si. This implies that the dimensions of the matrixes in eq 12 need not be large, and it can be solved efficiently. Substituting the solution of eq 12 into eq 14, the electrical interaction energy between two particles can be calculated. Although a numerical scheme can be implemented without too much difficulty, an approximately analytic expression for the electrical interaction energy is highly desirable for practical considerations. This can achieved by adopting the method proposed by McCartney and Levine15 and Bell l,i l,i and χk,j are large when l ) k and i ) j, et al.16 Since Gk,j (i.e., at a singular point), ∆ψSi on the left-hand sides of eqs (15) McCartney, L. N.; Levine, S. J. Colloid Interface Sci. 1969, 30, 345. (16) Bell, G. M.; Levine, S.; McCartney, L. N. J. Colloid Interface Sci. 1970, 33, 335.
5222 Langmuir, Vol. 15, No. 16, 1999
Hsu and Liu
10a and b can be moved out from the integral sign, and we have
(
)
(
)
-Γ1 -Γ2 1 β + γ11 β + γ12 ∆ψS2 ) ∆ψS1 + 11 2 12 a2e-κ(r12-a2) (15a) -ψS0 2 r12
(
)
(
)
-Γ1 -Γ2 1 β21 + γ21 ∆ψS1 + β + γ22 ∆ψS2 ) 22 2 a1e-κ(r21-a1) (15b) -ψS0 1 r21
{
VDL )
πa1a2 × R
f2ψS0 12 + f1ψS0 22
ln(1 - f1f2e-2κ(R-a1-a2)) f1f2 4ψS0 1 ψS0 2 tan-1 ( |f f |e-κ(R-a1-a2)) x 12 + -1 (x|f1f2|e-κ(R-a1-a2)) tanh x|f1f2|
[
(
1-
(
Ξ1β11
-1
)(
ψS0 2
Ξ2β22
∆ψS2 ) ψS0 1
ψS0 2
Ξ2β22
-1
)
(
(ψS0 2)2
)
a1e-κ(r21-a1) a2e-κ(r12-a2) r21 r12 (16a)
)
ψS0 1
a1e-κ(r21-a1) a1e-κ(r21-a1) a2e-κ(r12-a2) + -1 r21 Ξ2β22 Ξ1β11 r21 r12
1-
(
ψS0 1
Ξ1β11
-1
)(
)
ψS0 2
a1e-κ(r21-a1) a2e-κ(r12-a2) -1 Ξ2β22 r21 r12 (16b)
If we consider only the points which make a small angle between rji and R, then eqs 16a and 16b can be approximated by17
∆ψS1 ≈ ψS0 1
(
1-
ψS0 2
Ξ1β11 ψS0 1
Ξ1β11
+
-1
(ψS0 1)2
Ξ1β11 Ξ2β22
)(
ψS0 2
Ξ2β22
∆ψS2 ≈ ψS0 1
1-
(
ψS0 2
Ξ2β22 ψS0 1
Ξ1β11
+
-1
(
ψS0 2
(ψS0 2)2
)
)(
(
Ξ2β22
)
- 1 e-2κ(r12-a2)
ψS0 1
Ξ2β22 Ξ1β11 ψS0 2
-1
)
a2e-κ(r12-a2) r12 (17a)
-1
)
- 1 e-2κ(r21-a1)
Γ11 1 + γ11 β 2 11 f1 ) Γ11 1 - γ11 + β 2 11
(19a)
Γ22 1 + γ22 β 2 22 f2 ) Γ22 1 - γ22 + β 2 22
(19b)
If the surface potential is constant, then eq 19 becomes
VDL ) π
a1a2 × R (ψS1 + ψS2)2 ln[1 + e-κ(R-a1-a2)]
{
+ (ψS1 - ψS2)2 ln[1 - e-κ(R-a1-a2)]
VDL ) 4πa1a2 ψS0 1 ψS0 2
ln(1 + ab) ≈ b ln(1 + a) a, b f 0
e-κ(R-a1-a2) R
(20a)
VT/kBT
∫2a∞ e R2
dR
(21)
where VT is the total interaction energy between two particles, which is the sum of the electrical energy and the van der Waals energy, VVDW, that is,
VT ) VVDW + VDL
(22)
For two identical spheres, VVDW can be evaluated by1
VVDW ) (18)
(20)
This expression can also be derived by neglecting the second term on the left-hand side of eq 15a and the first term on the left-hand side of eq 15b and substituting the resultant expressions into eq 14. Equation 20a can also be derived on the basis of linear superposition. This is expected because if the double layer is thin, particles can be treated as isolated entities. 2.1. Stability Ratio. The stability ratio for sphere particles of radius a, W, can be defined as18
(17b)
Substituting these expressions into eq 14, and applying the relation
}
This is the same expression as that derived by Sader et al.17 For thin double layers, eq 19 can further be simplified as
W ) 2a a1e-κ(r21-a1) r21
}
where
ψS0 1 ψS0 2 a2e-κ(r12-a2) (ψS0 1)2 ψS0 2 a1e-κ(r21-a1) a2e-κ(r12-a2) + -1 Ξ1β11 r12 Ξ1β11 Ξ2β22 r21 r12 ψS0 1
]
(19)
Solving these equation for ∆ψS1 and ∆ψS2 yields ∆ψS1 )
f1f2 < 0 f1f2 > 0
[
(
)]
A132 1 1 1 + + 2 ln 1 - 2 12 x2 - 1 x2 x
(23)
we obtain
where A132 denotes the Hamaker constant and x ) R/2a. According to its definition, W is interpreted as the ratio
(17) Sader, J. E.; Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1995, 171, 46.
(18) Hiemenz, P. C. Principles of Colloid and Surface Chemistry, 2nd ed.; Dekker: New York, 1986.
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Langmuir, Vol. 15, No. 16, 1999 5223
[(rate of coagulation when VT ) 0)/(rate of coagulation when VT * 0)]. Since it is possible that VT does not vanish, a modified stability ratio, W′, is defined as below:19
W)
∫2a∞
exp(VT/kBT) R2
dR (24)
eVVDW/kBT dR 2a R2
∫
∞
Note that W′ approaches unity as the concentration of electrolyte increases, a desirable property. 2.2. Critical Coagulation Concentration. Suppose that the radius of a particle is much larger than the surfaceto-surface distance between two particles, H, and κa is large. For two identical particles, a1 ) a2 ) a, ψS0 1 ) ψS0 2 ) ψ0S, and eq 22 reduces to
VT ) 2πa(ψ0S)2 e-κH -
A132 1 24 x - 1
(25)
Applying the conditions that both the total interaction energy and its derivative with respect to the separation distance between two particles vanish at the critical coagulation concentration, we obtain
[
Ξa 24π A132 exp(1) Γa + (κCa + 1)
κC )
]
2
(26)
where κC denotes the value of κ at the critical coagulation concentration. Note that if both Ξ and Γ approach to infinity, eq 3 implies that the present charge-regulation condition reduces to a constant potential model. In this case, Ξ/Γ ) ψS, ψS being the surface potential of a particle, and eq 26 becomes
κC )
24π ψ2 A132 exp(1) S
(27)
This is consistent with the result of DLVO theory. On the other hand, if Γ f 0, the surface of a particle becomes constant charge. In this case, Ξ ) σ, σ being the surface charge density of a particle, and eq 26 becomes
κC )
[
σa 24π A132 exp(1) (κCa + 1)
]
2
(28)
If the dissociation of the electrolyte in the liquid phase yields indifferent ions, eq 26 can be expressed explicitly as
κC )
[x
(Γ/ + 1/a)3 6πΞ2 + + 27 A132 exp(1)
[x
Ξ
]
2/3
x
6π A132 exp(1)
+
(Γ/ + 1/a)3 6πΞ2 + 27 A132 exp(1) Ξ
x
] ( )
6π A132 exp(1)
2/3
-
2 Γ 1 + (29) 3 a
Otherwise, κC needs to be determined numerically. (19) McGown, D. N.; Parfitt, G. D. J. Phys. Chem. 1967, 71, 449.
Figure 2. Variation of the scaled electrical interaction energy, VDL/aψ20, as a function of the concentration of IDI, C. Key: s, exact solution based on eqs 12 and 14; - - -, result based on eq 19. (a) 1, ∆pK ) -3; 2, ∆pK ) 0; (b) 3, ∆pK ) 2; 4, ∆pK ) 3. Parameters used are a1 ) a2 ) a ) 10-7m, R ) 2.1 × 10-7 m, NS1 ) NS2 ) 1012 sites/cm2, Z ) 1, T ) 298 K, pK0 ) 4, and pB ) 5.
3. Discussion The variations of VDL as a function of the concentration of PDI and that of IDI are shown in Figures 2 and 3, respectively. For comparison, both the exact numerical results calculated by eqs 12 and 14 and those based on the approximate expression, eq 19, are illustrated in these figures. The values of the parameters assumed in the numerical calculations are on the same order of magnitudes as those adopted in the literature.3,7,8,12 Figures 2 and 3 reveal that for a fixed concentration of ions the greater the ∆pK, the lower the VDL, for both PDI and IDI. For the case of IDI, VDL decreases with the increase in C. This is because that the higher the C, the thinner the double layer. The variation of VDL as a function of C has a different nature for PDI. As can be seen from Figure 3, it decreases first with the increase in C, rises to a maximal value, and then decreases again for a further increase in C. This is because that the adsorption of PDI onto the
5224 Langmuir, Vol. 15, No. 16, 1999
Figure 3. Variation of the scaled electrical interaction energy, VDL/aψ20, as a function of the concentration of PDI, C, for the case of Figure 2 except that B ) C.
surface of a particle has the effect of lessening its net charges, and, therefore, reduces VDL. If C is higher than a certain value, the adsorption of PDI leads to charge reversal on particle surface. A further increase in C results in accumulation of inverse charge on particle surface, and therefore, VDL is increased. If C is increased again, the effect of double layer compression becomes significant, and VDL is reduced. Figures 2 and 3 reveal that the performance of the approximate expression eq 19 is satisfactory. The higher the concentration of ions, the better the performance, in general. The approximate expression for electrical interaction energy is used thereinafter. The variations of the modified stability ratio W′ as a function of the concentration of ions under two different conditions are shown in Figures 4 and 5. As can be seen from these figures, for the case of PDI, there exist two values of C at which W′ is unity, but only one such C is present for the case of IDI. The behavior was observed experimentally by Ketelson et al.20 for the dispersion of Stober silica in an acetone-water solution. This implies that the difference between PDI and IDI is significant. A
Hsu and Liu
Figure 4. Variation of the modified stability ratio W′ as a function of the concentration of ions of C for the case A132 ) 10-19 J. Parameters used are the same as those of Figure 2 except that B ) 4 × 10-5 M for IDI. Key: s, PDI; - - -, IDI. (a) 1, ∆pK ) -3 and 2, ∆pK ) 0; (b) 3, ∆pK ) 2 and 4, ∆pK ) 3.
comparison between Figures 4 and 5 reveals that the greater the pK0, the less appreciable the second C at which W′ ) 1. This is because that if pK0 is high, the concentration of counterions in the liquid-phase necessary to inverse the charges on the surface of a particle is also high. Therefore, if the rate of variation of the amount of the inverse charge on the particle surface (or ψN) as the concentration of counterions varies is slower than that of the variation of the thickness of the double layer as the concentration of counterions varies, the second C at which W′ ) 1 disappears. Figures 4 and 5 also reveal that the larger the ∆pK, the less appreciable the second C at which W′ ) 1. Figure 6 shows the variation of κC as a function of ∆pK for the case of IDI. This figure reveals that κC decreases with the increase in ∆pK. If ∆pK is negative, κC is roughly constant. (20) Ketelson, H. A.; Pelton, R.; Brook, M. A. Langmuir 1996, 12, 1134.
Stability of Colloidal Dispersions
Langmuir, Vol. 15, No. 16, 1999 5225
Figure 6. Variation of κC as a function of ∆pK for the case of IDI. Parameters used are the same as those of Figure 2. Curve 1: pK0 ) 4 and B ) 4 × 10-5 M. Curve 2: pK0 ) 3 and B ) 5 × 10-4 M.
Figure 5. Variation of the modified stability ratio W′ as a function of the concentration of ions, C, for the case of Figure 4 except that pK0 ) 3 and B ) 5 × 10-4 M. Key: same as that of Figure 4.
Figure 7 illustrates the variation of κC as a function of ∆pK for the case of PDI. As can be seen from this figure, if ∆pK is small, there may exist two critical coagulation concentrations, and if it is large, only one exists. This behavior can be elaborated as follows. Let us define the function f as
f)
(
Ξa 24π A132 exp(1) -Γa + (κCa + 1)
)
2
- κC
(30)
According to eq 26, if f vanishes, the corresponding electrolyte concentration is the critical coagulation concentration. Figure 8 shows the variation of f as a function of κC at two different ∆pK for the case of PDI. This figure suggests that if ∆pK is large, only one critical coagulation concentration exists, as predicted by Figure 7. On the other hand, if ∆pK is small, there exists three κC’s at which f vanishes, and therefore, three critical coagulation concentrations. It should be pointed out, however, that the critical coagulation concentration corresponding to the middle κC has no physical meaning. By referring to
Figure 7. Variation of κC as a function of ∆pK for the case of PDI. Parameters used are the same as those of Figure 3.
Figure 9a, as the concentration of electrolyte increases, the total interaction energy between two particles VT decreases. The electrolyte concentration at which both VT and its derivative with respect to the closest surface-tosurface distance between two particles, (dVT/dH), vanish (curve 2) is defined as the first critical coagulation concentration, CCC1; the corresponding κCis denoted as κC1. When the electrolyte concentration becomes higher than CCC1, charge reversal occurs on the surfaces of the particles, and VT increases. As shown in Figure 9a, at a certain electrolyte concentration, CCC2, both VT and (dVT/ dH) vanish again (curve 4). The corresponding κC at this electrolyte concentration is denoted as κC2. Apparently, CCC2 should not be viewed as a critical coagulation concentration. Therefore κC2 is of no practical significance. When the electrolyte concentration further increases, VT is lowered again, and at a certain electrolyte concentration, CCC3, both VT and (dVT/dH) vanish (curve 6). The value of κ at CCC3 is denoted as κC3. Here, the role played by
5226 Langmuir, Vol. 15, No. 16, 1999
Hsu and Liu
Figure 8. Variation of the function f defined in eq 30 as a function of κC for the case of PDI. Parameters used are the same as those of Figure 2. Curve 1: ∆pK ) -10. Curve 2: ∆pK ) 10.
CCC3 is similar to that by CCC1, except that the sign of the surface charge in the former is different from that of the latter. It should be pointed out that the behavior of a colloidal dispersion described by the stability ratio and that by the critical coagulation concentration may be different. Figure 7, for example, suggests that if ∆pK is positive, only one critical coagulation concentration exists. On the other hand, Figure 4b suggests that there exist two C’s at which W′ ) 1; i.e., a dispersed system is unstable. The inconsistency between Figures 4b and 7 can be explained by the variation of VT as a function of H at various electrolyte concentrations C shown in Figure 9b. Starting from curve 1 of Figure 9b, VT decreases with the increase in C due to the decrease in the electrical interaction energy between two particles VDL. At a certain level of C, CCC1′, VT becomes all negative (curve 3), and W′ ) 1. This corresponds to the lower C in Figure 4b at which log(W′) vanishes. If C is further increased, the inversion of surface charge leads to an increase in VDL, and therefore, VT may vary from curves 3 to 4 in Figure 9b. A further increase in C to CCC2′ may lead to curve 6. This corresponds to the higher C in Figure 4b at which log(W′) vanishes. Note that in the case of Figure 9a there are two CCC’s and two C’s at which W′ ) 1; that is, the qualitative behavior of a colloidal dispersion described by CCC and that by W′ are the same. However, in the case of Figure 9b, there is only one CCC (curve 2) but two C’s at which W′ ) 1; i.e., the qualitative behavior of a colloidal dispersion described by CCC and that described by W′ are different. According to the variation of VT as a function of H at various electrolyte
Figure 9. Schematic representation of the variation of the total interaction energy between two particles VT as a function of the closest surface-to-surface distance between two particles, H, at various electrolyte concentrations C. (a) CCC1 and CCC3 are critical coagulation concentrations; CCC2 has no physical meaning. (b) CCC1′ and CCC2′ are the values of C at which W′ ) 1.
concentrations shown in Figure 9b, the qualitative behavior described by W′ is more realistic than that described by CCC. In summary, the stability behavior of a system containing spherical particles which are capable of modifying its charged properties through the dissociation of functional groups or exchange ions with the liquid phase is discussed theoretically. A systematic approach, which yields an approximate analytical expression for the electrical interaction energy between two particles under a general surface condition, is presented; the available results in the literature can be recovered as the special cases of the present study. We show that multiple instability points may exist, and the stability ratio is found to be a more realistic description about the qualitative behavior of a colloidal dispersion than the critical coagulation concentration. Acknowledgment. This work is partially financially supported by the National Science Council of the Republic of China. LA981053L
