Langmuir 2003, 19, 3049-3055
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Colloid Restabilization at High Electrolyte Concentrations: Effect of Ion Valency Eli Ruckenstein* and Haohao Huang Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received October 22, 2002. In Final Form: January 3, 2003 The scope of this paper is to explain the restabilization of colloidal dispersions at sufficiently high electrolyte concentrations. This restabilization could not be explained in the framework of the traditional double layer theory. An improved theory which accounts for the field generated in water by neighboring dipoles (the polarization gradient) as well as the field induced by surface dipoles could, however, explain the experimental results. The latter dipoles are generated by the adsorption of the ions of the electrolyte upon the surface charges with opposite sign formed through the dissociation of acidic and basic sites. This adsorption changes not only the surface charge but also generates ion pairs (dipoles), which induce a field in the neighboring water molecules that propagates further in the liquid. A higher valency has an important effect upon the restabilization because it makes the field induced by the neighboring water molecules stronger and the change of the surface charge with electrolyte concentration greater. Experimental results from literature regarding the stability of a dispersion of protein-covered latexes are explained on the basis of the new theory.
1. Introduction Experiment revealed that, in most cases, the stability of a colloidal dispersion is decreased as the electrolyte concentration increases. The DLVO theory could explain, sometimes quantitatively, these experimental findings. However, many decades ago, Voet1 prepared stable sols of metals (Pt, Pd) and salts (sulfides, halides) in highly concentrated solutions of H2SO4, H3PO4, and CaCl2. Their dilution with water induced coagulation. Similar results have been reported more recently by Healy et al.,2 who observed that amphoteric latex particles did not coagulate at high ionic strength of LiNO3 (>1 M). Finally, MolinaBolivar and Ortega-Vinuesa3 have observed that the stability ratio of a dispersion of polystyrene particles stabilized with adsorbed protein molecules (Ig-G) decreased with increasing electrolyte concentration, passed through a minimum, after which it increased. Consequently, at sufficiently large electrolyte concentrations, the dispersion was restabilized. The traditional theory could not explain the above observations. In recent papers concerned with monovalent electrolytes,4,5 the restabilization was attributed to the following two effects: (i) the reassociation of charges on the surface and their replacement by ion pairs (surface dipoles), and (ii) the fields generated by the surface dipoles and in the bulk by neighboring dipoles. While the surface charge density formed through the dissociation of acidic and basic sites is decreased by the adsorption of counterions, the charges are replaced by ion pairs (dipoles) which polarize the water molecules nearby. This polar* To whom correspondence should be addressed: Telephone: (716) 645-2911 ext. 2214. Fax: (716) 645-3822. E-mail: feaeliru@ acsu.buffalo.edu (1) Voet, A. Thesis, University of Amsterdam, 1935; see also: Kruyt, H. R. Colloid Science; Elsevier: New York, 1952. (2) Healy, T. W.; Homola, A.; James, R. O.; Hunter, R. J. Faraday Discuss. Chem. Soc. 1978, 65, 156. (3) Molina-Bolivar, J. A.; Ortega-Vinuesa, J. L. Langmuir 1999, 15, 2644. (4) Ruckenstein, E.; Manciu, M. Langmuir 2002, 18, 7584. (5) Huang, H.; Manciu, M.; Ruckenstein, E. J. Colloid Interface Sci., in press.
ization propagates, and the overlap of the polarization layers generates a repulsive force (the hydration force) which becomes dominant at high electrolyte concentrations and is responsible for the restabilization. The previously developed theory6 assumed additivity between the traditional double layer force and the hydration force generated by the surface dipoles. Because both forces depend on polarization, a new unitary theory was formulated in which they were coupled in a single repulsive force4 in which the two effects were no longer additive. It is well-known that the valency of the counterion ν has a strong effect on colloidal stability,7 since the critical coagulation concentration is proportional to ν-6. The theoretical confirmation of this Schulze-Hardy rule was one of the major successes of the DLVO theory.8 The scope of the present paper is to examine the effect of the valency of the electrolyte ions on colloidal restabilization. The valency is expected to be important for at least two reasons: (i) the strong effect which it has on the screening of the electric field; (ii) the adsorption of multivalent ions on surface charges of opposite sign which can easily cause a charge inversion on the surface. The presentation is organized as follows: First, an expression for the surface charge density will be derived by assuming that the surface charge is generated by the dissociation of acidic and basic groups and the adsorption of cations and anions of the electrolyte upon the surface charges of opposite sign. Because intuition suggests that a charge reversal should occur with increasing electrolyte concentration and this reversal is compatible with a minimum in the stability ratio, the traditional double layer theory was first employed to verify if it can predict restabilization. The conclusion was that the traditional theory cannot predict restabilization when the electrolyte concentration becomes very large. Further, it will be shown that a unitary theory, which (6) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7582. (7) Overbeek, J. Th. G. In Colloid Science; Kruyt, H. R. Eds.; Elsevier: New York, 1952; Vol, 1. (8) Verwey, E. J.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948.
10.1021/la026729y CCC: $25.00 © 2003 American Chemical Society Published on Web 02/27/2003
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involves the charge of the surface, the surface dipole density, and the fields caused by the surface dipoles and by the neighboring water dipoles in the water layer, can provide reasonable results. A comparison with experiment will conclude the paper.
ions and anions N-, and [H+]S, [Mν+]S, [OH-]S, and [N-]S are the concentrations of the corresponding ions near the surface. The surface charge density is obtained by subtracting the negative charge density from the positive one
2. Dissociation and Adsorption of Cations and Anions on Protein Stabilized Latexes
σ ) e[NB(1 - θOH - θN) + (ν - 1)NAθM NA(1 - θH - θM)] ) e[NB(1 - θOH - θN) NA(1 - θH - νθM)] (3)
Molina-Bolivar and Ortega-Vinuesa examined experimentally the stability of a dispersion of polystyrene latexes with chemisorbed protein (Ig-G) molecules upon their surface.3 A latex suspension was introduced into a buffer containing protein molecules and the latter molecules reacted with the chloromethyl groups of the latex surface. After the protein binding, the stability ratio W, the ratio of the rate constants of rapid and slow coagulations, of the latexes in various electrolyte solutions was determined using the low-angle light-scattering technique. Let us first derive an expression for the surface charge generated through the dissociation of the acidic and basic groups of the protein and adsorption of the ions of the electrolyte on the opposite charges of the surface.9 The dissociations of the protein can be represented by the equilibria
where NB and NA represent the numbers of basic and acidic sites per unit area, respectively, and e is the protonic charge. Combined with eq 2a-d and Boltzmann distributions, eq 3 becomes
(
σ)
SA T SA- + H+
(1a)
SB T SB+ + OH-
(1b)
NBKOHKN + eΨs eΨs KOHKN + [OH-] exp KN + [N-] exp KOH kT kT νeΨs ν+ NA(ν - 1)KH[M ] exp - NAKHKM kT eΨs νeΨs KHKM + KM[H+] exp + KH[Mν+] exp kT kT (4)
where SA is an acidic site, SA- is a dissociated acidic site, SB is a basic site, and SB+ is a dissociated basic site. Since the cations and anions of the electrolyte can be adsorbed upon the dissociated acidic and, respectively, basic sites, the following adsorption equilibria should be included
where [H+], [OH-], [Mν+], and [N-] are concentrations in the bulk, Ψs is the surface potential, k is the Boltzmann constant, and T is the temperature in K If the adsorption of anions is neglected (such an approximation is reasonable for a negatively charged surface or a sufficiently large KN), eq 4 reduces to
SA- + Mν+ T (SA-Mν-1)
(1c)
SB+ + N- T SB-N
(1d)
-Mν-1)
ν+
is an acidic where M is a cation of valency ν, (SA site occupied by cation Mν+ and SB-N is a basic site occupied by anion N-. Equilibrium 1c implies that a cation of valency ν is adsorbed on a single monovalent site. Neglecting the interactions between the adsorbed species, one can write the following equilibria
KH ≡
(1 - θM - θH)[H+]S θH
(2a)
KM ≡
(1 - θM - θH)[Mν+]S θM
(2b)
(1 - θOH - θN)[OH-]S KOH ≡ θOH KN ≡
(1 - θOH - θN)[N-]S θN
(2c)
(2d)
where KH, KOH, KM, and KN are equilibrium constants for H, OH, and the cation and anion of the electrolyte, respectively, θH and θM denote the fractions of acidic sites occupied by hydrogen ions and cations Mν+, θOH and θN denote the fractions of basic sites occupied by hydroxide (9) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 205.
e
σ)e
( ) ( ( )
(
NBKOH
( ) ( ( )
KOH + [OH-] exp
eΨs kT
)
( ) (
)
)
-
)
)
νeΨs - NAKHKM kT eΨs νeΨs KHKM + KM[H+] exp + KH[Mν+] exp kT kT (5) NA(ν - 1)KH[Mν+] exp -
(
)
3. Calculation of the Stability Ratio on the Basis of the Traditional Double Layer Theory The free energy of the system due to repulsion was calculated using the Derjaguin approximation, hence by expressing the repulsive free energy between two spherical particles in terms of that between two parallel plates. The Poisson-Boltzmann equation in one dimension has the form
d2Ψ 2
dz
)-
1
∑i ni0νie exp(-νieΨ/kT)
(6)
0
where Ψ is the electrical potential, z the distance from the middle distance between plates, ni0 the number of ions of species i per unit volume in the bulk solution, the dielectric constant of the medium, and 0 the vacuum permittivity.
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The electroneutrality provides the boundary conditions
|
|
dΨ dΨ σ ))dz z)-d dz z)d 0
(7)
where 2d is the distance between the two plates. The repulsive free energy of the system is the sum of three contributions10
F ) Fele + Fentropy+ Fchem
(8)
where Fele, Fentropy, and Fchem are the electrostatic, entropic, and chemical contributions to the free energy, respectively. The quantity needed in the calculation of the stability ratio is the repulsive free energy per unit area and hence the work per unit area needed to bring the two plates from infinity to the distance 2d
∆F ) F(2d) - F(2d)∞)
(9)
The electrostatic and entropic contributions to the free energy can be calculated using the following expressions10
∫-dd (dΨ dz )
1 Fele ) 0 2 Fentropy ) kT
∑i ∫-d d
2
( () ni ln
ni
0
ni
dz
(10)
)
- ni + ni0 dz (11)
where ni is the number ions of species i per unit volume. The chemical contribution to the free energy can be calculated by decomposing the pathway from infinity to 2d into infinitesimal steps at constant charge, for which the chemical contributions are zero,8 each followed by an infinitesimal step at constant potential, for which the chemical contributions are -2Ψs∆σ.8 The summation over the entire path leads to11 σ(2d) Ψs dσ ∫σ(∞)
∆Fchem ) -2
(12)
The free energy of the surface layer formed of the surface dipoles and the water molecules between them is considered independent of the distance 2d. If the radius of the spherical colloidal particles a is much larger than the shortest distance between the surfaces of two particles, the repulsive free energy between two identical spherical particles at a distance H0 of shortest approach is given in the Derjaguin approximation by12
VR ) πa
∫H∞ ∆F dH
(13)
0
where H ) 2d. The van der Waals attractive energy between two spheres of radius a has the form8
[
AH 2a2 2a2 + + VA ) 6 H0(4a + H0) (2a + H )2 0 H0(4a + H0) (14) ln (2a + H0)2
Figure 1. (a) Stability ratio W against electrolyte concentration C (in M). The calculations were carried out with the traditional theory. NA ) NB ) 2 × 1018 m-2, KH ) 6 × 10-10 M, AH ) 1 × 10-20J, pH ) 8.0, a ) 102 nm, and KN ) 6 × 10-5 M. Key: (1) 3:1 electrolyte with KM ) 0.001 M; (2) 2:1 electrolyte with KM ) 0.001 M; (3) 3:1 electrolyte with KM ) 0.01 M; (4) 3:1 electrolyte with KM ) 0.1 M; (5) 2:1 electrolyte with KM ) 0.01 M; (6) 2:1 electrolyte with KM ) 0.1 M. (b) Stability ratio W against electrolyte concentration C (in M). The calculations were carried out with the traditional theory. The adsorption of anions is neglected. NA ) NB ) 2 × 1018 m-2, KH ) 6 × 10-10 M, AH ) 1 × 10-20J, pH ) 8.0, a ) 102 nm, and KN ) ∞. Key: (1) 3:1 electrolyte with KM ) 0.001 M; (2) 2:1 electrolyte with KM ) 0.001 M; (3) 3:1 electrolyte with KM ) 0.01 M; (4) 3:1 electrolyte with KM ) 0.1 M; (5) 2:1 electrolyte with KM ) 0.01 M; (6) 2:1 electrolyte with KM ) 0.1 M.
The colloidal stability can be expressed through the stability ratio W, which is the ratio between the rate constants of rapid and slow coagulations13,14
∫0∞ (u + 2)2 exp(kTT) du β(u)
W)
∫0
∞
V
( )
VA exp du 2 kT (u + 2) β(u)
(15)
where AH is the Hamaker constant.
where β is a hydrodynamic correction factor, u ) H0/a, VT is the sum of the repulsive VR and the van der Waals attractive VA free energies between two spherical particles. The hydrodynamic correction factor β was taken as unity. Parts a and b of Figure 1 examine the effect of the valency on the stability ratio in the framework of the
(10) Overbeek, J. Th. G. Colloids Surf. 1990, 51, 61. (11) Manciu, M.; Ruckenstein, E, Langmuir, in press. (12) Hunter, R. J. Foundations of Colloid Science;Clarendon Press: Oxford, England, 1986; Vol. 1.
(13) Honig, E. P.; Roebersen, G. J.; Wiersema, P. H. J. Colloid Interface Sci. 1971, 36, 97. (14) Derjaguin, B. V.; Muller, V. M. Dokl. Akad. Nauk SSSR (Engl. Transl.) 1967, 176, 738.
]
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important for monovalent than multivalent ions. Figures 2 and 3, in which the surface potential and the surface charge density are plotted against the electrolyte concentration, confirm the above interpretation. When the cation adsorption is weak, the screening effect of the electrolyte ions is dominant at all electrolyte concentrations. The above calculations indicate that at sufficiently high ionic strengths the traditional double layer theory predicts the screening of the repulsive force. In contrast, experiment appears to indicate that at high ionic strengths a restabilization should occur. 4. The Unitary Theory and Its Application to the Calculation of the Stability Ratio
Figure 2. Reversal of the surface potential for a single plate. The calculations were carried out with the traditional theory. Key: (1) 2:1 electrolyte withKM ) 0.001 M; (2) 2:1 electrolyte with KM ) 0.01 M; (3) 2:1 electrolyte with KM ) 0.1 M. The other parameters are as in Figure 1a.
It is well-known that the traditional double layer theory is valid in a limited range of concentrations for monovalent electrolytes, but is much less valid for higher valency electrolytes.15 The traditional theory starts from the Poisson equation
0
dP dE )Fdz dz
(16)
where F is the charge density, E is the macroscopic field, and P is the polarization. The main assumption in that theory is that
P ) χE
Figure 3. Reversal of the surface charge for a single plate. The calculations were carried out with the traditional theory. Key: (1) 2:1 electrolyte with KM ) 0.001 M; (2) 2:1 electrolyte with KM ) 0.01 M; (3) 2:1 electrolyte with KM ) 0.1 M. The other parameters are as in Figure 1a.
classical theory. In Figure 1a, the anion adsorption equilibrium constant KN ) 6 × 10-5 M, whereas in Figure 1b, KN ) ∞. There is a small difference between the two figures as a result of a low anion adsorption in Figure 1a and zero anion adsorption in Figure 1b. For strong cation adsorption (KM ) 0.001 M), the stability ratio passes through a minimum, followed by a maximum, after which it decreases with increasing electrolyte concentration. However, for low cation adsorptions (KM ) 0.01 or 0.1 M), the stability ratio decreases monotonically with increasing electrolyte concentration. The above behavior is a result of two competing factors: one of them is the screening effect of the electrolyte ions, and the other is the change in the sign of the surface charge. At relatively low ionic strengths, the stability decreases due to ion screening. When the electrolyte concentration becomes sufficiently large, the multivalency and strong adsorption easily generate a change in the surface charge from negative to positive, and the system is restabilized. The stability decreases, however, at sufficiently large electrolyte concentrations because the screening becomes again dominant. It is important to emphasize that for monovalent ions W decays monotonically as the electrolyte concentration decreases. This occurs because the effect of charge inversion is much less
(17)
where χ is the electrical susceptibility, which is considered constant. Equation 17 is however valid only for a uniform field and hence at most when the gradient of the field is sufficiently small. Because, particularly for high valency electrolytes, the variation of the field with the distance is relatively steep, the above expression has to be modified by including the field Ep induced upon a molecule of water by the neighboring water dipoles. In the traditional Lorentz approach, on which the traditional double layer theory is based, the local field that acts upon a dipole is given by E + (P/30).16 In the present case the local field has to contain the additional field Ep induced by the neighboring dipoles
Elocal ) E + (P/30) + Ep
(18)
Ep(zi) ) C1mi-1 + C0mi + C1mi+1
(19)
where
In eq 19, the field is calculated by assuming that the water molecules are organized in icelike layers, mi being the average dipole moment of water in layer i and C0 and C1 being interaction coefficients given by6
3.766 1.827 ; C1 ) C0 ) 3 4π0′′l 4π0′′l3
(20)
In eq 20, l is the distance between the centers of two adjacent water molecules, and ′′ is the dielectric constant for the interaction between two neighboring molecules. Only the neighboring water molecules are assumed to induce a field upon a water molecule of layer i, because the effect of the more distant molecules is screened by the intervening ones (the dielectric constant ′′ ) 1 for neighboring molecules and around 80 for the more distant ones, at room temperature). (15) Levine, S.; Bell, G. M. Discuss. Faraday Soc. 1966, 42, 69. (16) Frankl, D. R. Electromagnetic Theory; Prentice Hall: Englewood Cliffs, NJ, 1986.
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Expanding eq 19 in series, one obtains
Ep ≈ (C0 + 2C1)m + C1∆2
2 ∂2m 2 ∂ m ≈ C ∆ (21) 1 ∂z2 ∂z2
where ∆ is the distance between the centers of two adjacent water layers. The average dipole moment of a water molecule is given by
m ) γElocal
(22)
where γ is the polarizability for which, as in the traditional double layer theory, the Clausius-Mossotti equation
γ ) 30v0( - 1)/( + 2)
(23)
where v0 is the volume of a water molecule, will be employed. Combining eqs 18, 21, 22, and 23, yields
∂2P P(z) ) 0( - 1)E(z) + 0v0( - 1)C1∆2 2 ∂z
(24)
Using Boltzmann expressions for the ion distributions, the Poisson-Boltzmann equation becomes
d2Ψ
)-
dz2
1 dm(z)
1
∑ni0νie exp(-νieΨ/kT) + v i 0
0 0
dz
KMKH + KM[H + ]s + KH[Mν+]s 1 ) NAθM K N [Mν+] H
A
m|z)(d ) 0 v0( - 1)(E + Es + Ep)|z)(d
∫-dd F dz ) 21∫-dd
1 2
σ))
[( |
(
)
∂2Ψ 1 ∂m 0 2 dz v ∂z 0 ∂z
| ) |
∂Ψ 1 ∂Ψ 1 0 (m|z)d - m|z)-d) z)d z)-d 2 ∂z ∂z v0
)e
|
]
1 ∂Ψ ) 0 - m ∂z z)d v0 z)d
(
NBKOH
( ) ( ( )
KOH + [OH-] exp
eΨs kT
-
)
)
νeΨs - NAKHKM kT eΨs νeΨs KHKM + KM[H+] exp + KH[Mν+] exp kT kT (30) NA(ν - 1)KH[Mν+] exp -
(
)
In addition, due to the symmetry of the system
s
) {KMKH + KM[H+] exp(-eΨs/kT) + KH[Mν+] ×
m ) 0 and
ν+
exp(-νeΨs/kT)}/{KHNA[M ] exp(-νeΨs/kT)} (27) By inclusion of the field induced by the surface dipole into the local field that is acting upon the first layer of water molecules, the dipole moment of the first layer of water molecules near the surface is given by
m|z)(d ) γ Elocal ) γ(E + Es + Ep + P/30)|z)(d (28)
(19a)
dΨ ) 0 for z ) 0 dz
(31)
Using eqs 29-31 as boundary conditions, eqs 24 and 25 can be solved to provide the profiles of the electric potential Ψ and the dipole moment m ) Pv0. Because of the interactions with neighboring dipoles, which generate the field Ep, the electrical contribution to the free energy contains an additional term when compared to the traditional theory
where
Ep|z)(d ) -(C0m1 + C1m2)
(29)
The electroneutrality condition provides a second boundary condition
(26)
where p is the normal component of the dipole moment of a surface dipole, ′ is the local dielectric constant near the surface, A is the surface area per surface dipole (1/A is the surface dipole density), and ∆′ is the distance between a surface dipole and the center of the first layer of water molecules. The surface area per dipole, and hence the number of dipoles per unit area, is related to the surface potential and electrolyte concentration through the following equation
A)
Equation 19a is obtained from eq 19 by taking into account that for the first layer of water near the surface the layer i - 1 is missing. Consequently, since P ) m/v0
(25)
Let us now derive the boundary conditions. The adsorption of the ions of the electrolyte changes the charge of the surface and generates simultaneously surface ion pairs (dipoles). Let us first assume that only the cation is adsorbed. The electric field induced by a surface dipole on the first layer of water molecules is given by6
Es ) (p/′)/2π0 (A/π + ∆′2)3/2
Figure 4. Stability ratio W against electrolyte concentration C (in M). The calculations were carried out with the new theory. The adsorption of anions is neglected. NA ) NB ) 2 × 1018 m-2, KH ) 6 × 10-10 M, pH ) 8.0, AH ) 1 × 10-20J, and a ) 102 nm. Key: (1) 3:1 electrolyte with KM ) 0.1 M; (2) 2:1 electrolyte with KM ) 0.1 M; (3) 1:1 electrolyte with KM ) 0.1 M.
Fele )
∫-dd
1 2
((
0E(z) +
)
)
m(z) m(z) E(z) E (z) dz (32) v0 v0 p
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Figure 4, in which the stability ratio is plotted against the electrolyte concentration, contains three domains: the DLVO domain, the rapid aggregation domain, and the restabilization domain. The calculations revealed that the surface dipoles and the additional contribution to the local field generated by the neighboring dipoles in the water layers can account for the restabilization. With increasing electrolyte concentration, the density of surface dipoles increases and the colloidal dispersion becomes increasingly more stable for electrolyte concentrations higher than the critical stabilization concentration. Because of the surface charge reversal as the electrolyte concentration increases, it is more realistic to also include the adsorption of anions among the adsorption equilibria, since they can have a relatively high concentration near a positively charged surface. The adsorption of anions generates surface dipoles with a sign opposite to that generated by the adsorption of cations. The total electric field induced by the surface dipoles is obtained by adding the electric fields induced by the two kinds of ion pairs (surface dipoles)
Es ) (pa/′)/2π0 (Aa/π + ∆′2)3/2 +
Figure 5. Stability ratio W against electrolyte concentration C(in M). The calculations were carried out with the new theory. The adsorption of anions is taken into account. NA ) NB ) 2 × 1018 m-2, KH ) 8 × 10-10 M, pH ) 8.0, AH ) 1 × 10-20J, a ) 102 nm, and KN ) 6 × 10-5 M. Key: (1) 1:1 electrolyte with KM ) 0.001 M; (2) 2:1 electrolyte with KM ) 0.001 M; (3) 3:1 electrolyte with KM ) 0.001 M.
(pb/′)/2π0 (Ab/π + ∆′2)3/2 (33) where pa, pb are the normal components of the surface dipole moments generated by the adsorptions of anions and cations, respectively, and the surface area per dipole Aa and Ab can be calculated using the following expressions
KMKH + KM[H+]s + KH[Mν+]s 1 ) NAθM N K [Mν+]
Aa )
A
H
s
+
) {KMKH + KM[H ] exp(-eΨs/kT) + KH[Mν+] × exp(-νeΨs/kT)}/{NAKH[Mν+] exp(-νeΨs/kT)} (34) KNKOH + KN[OH-]s + KOH[N-]s 1 ) NBθN N K [N-]
Ab )
B
OH
Figure 6. Relationship between the critical coagulation concentration and valency for the parameters of Figure 5.
s
KNKOH exp(-eΨs/kT) + KN[OH-] + KOH[N-]
)
NBKOH[N-]
m|z)-d ) 0 v0( - 1)(E + Ep + (pa/′)/2π0 (Aa/π +
(35)
where [N-]s and [N-] are the concentrations of N- near the surface and in the bulk. In this case the boundary conditions acquire the forms
σ)-
(
∫-dd Fdz ) 21∫-dd
1 2
(
)
1 ∂m ∂2Ψ 0 2 dz ) v ∂z 0 ∂z ∂Ψ 1 - m 0 ∂z z)d v0
|
|
z)d
NBKOHKN + eΨs eΨs KOHKN + [OH-]exp KN + [N-]exp KOH kT kT νeΨs ν+ NA(ν - 1)KH[M ] exp - NAKHKM kT eΨs νeΨs KHKM + KM[H+] exp + KH[Mν+] exp kT kT
)e
( ) ( ( )
)
( ) (
)
)
(36) and
∆′2)3/2 + (pb/′)/2π0 (Ab/π + ∆′2)3/2)|z)-d (37) Figure 5, in which the stability ratio is plotted against electrolyte concentration, for cations of valencies 1, 2, and 3 and monovalent anions, shows that the critical coagulation concentration increases with decreasing cation valency (Figure 6). This rule is not valid for the critical stabilization concentration. 5. Comparison with Experiment The experiments carried out by Molina-Bolivar and Ortega-Venuesa with a monovalent cation NaCl and a divalent cation CaCl2 electrolytes, described in the Introduction, are compared with calculated results in Figure 7. The adsorption equilibrium constant KOH was calculated in term of KH using the expression
KOH )
10-7 10-7NB NA - NB KHNA NA
(38)
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Langmuir, Vol. 19, No. 7, 2003 3055
which has the right order of magnitude, was selected for the Hamaker constant.12 The values of the other parameters employed in the calculations were: KCl ) 6 × 10-5 M, pa/′ ) 3 D for Ca2+, pa/′ ) 1.8 D for Na+, pb/′ ) -1 D. In these experiments, the diameter of the polystyrene particles was 204 nm, and the pH of the solution was 8.0. The calculations demonstrate that the surface dipoles and the field caused by neighboring molecules in water play important roles in colloid stability and can explain the restabilization.
Figure 7. Comparison between experimental data and calculations for the dependence of the stability ratio W on the electrolyte concentration C (in M). The calculations were carried out with the new theory. NA ) NB ) 2 × 1018 m-2, KH ) 8 × 10-10 M, pH ) 8.0, AH ) 1 × 10-20J, D ) 2a ) 204 nm, and KCl ) 6 × 10-5 M. Key: (1) NaCl with KNa ) 1 M; (2) CaCl2 with KCa ) 0.02 M. 9 and 4 are experimental data for NaCl and CaCl2 taken from Figure 6 of ref 3.
which was obtained from eq 4 at the isoelectric point and in the absence of electrolyte. The value of the pH at the isoelectric point was selected 7.0, on the basis of the experimental values, which are in the range 6.0-8.0. Selecting a distance of 7 Å between two acidic or basic sites, one obtains NA ) NB ) 2 × 1018 m-2. The pKH of amino acid residues of globular proteins are in the range of 1.95-9.5,17 and, for this reason, a value of 8 × 10-10 M was selected for KH (pKH ) 9.1). The value of 0.02 M was selected for the KCa of Ig-G. The value AH ) 1 × 10-20J, (17) Timasheff, S. N. Biological Polyelectrolyte; Veis, A., Ed.; Marcel Dekker: New York, 1970.
6. Conclusions In the framework of the traditional double layer theory, when there is a strong adsorption of a multivalent cation of the electrolyte, the stability ratio of a colloidal dispersion first decreases, passes through a minimum, followed by a maximum, after which it decreases as the concentration of the (multivalent) electrolyte increases. When the adsorption is weak, the stability ratio decreases monotonically with increasing electrolyte concentration. For a monovalent electrolyte, the stability ratio calculated in the framework of the classical double layer theory decreases monotonically with increasing electrolyte concentration. Consequently, in the traditional theory, the ion screening becomes dominant at high ionic strength, whereas experiment appears to show that the colloidal dispersion is restabilized. A new theory was proposed in which the adsorption of the ions of the electrolyte generates surface ion pairs (dipoles), which induce an electric field. In addition, the neighboring dipoles in water also generate a field in water. The restabilization of the colloidal dispersion at sufficiently high ionic strengths appears to be an effect of these fields. The charge inversion produced by the adsorption of the ions of the electrolyte plays also a role particularly for the high valency ions. LA026729Y
