Langmuir 1996, 12, 3437-3441
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Particle Charges in Nonaqueous Colloidal Dispersions Inan Chen Wilson Center for Research and Technology, Xerox Corporation, Webster, New York 14580 Received November 29, 1995. In Final Form: May 2, 1996X Numerical solutions of the Poisson-Boltzmann equation for potentials in the electrical double layer surrounding a particle are used to derive a new relationship between the particle charge q and the surface potential ζ. Unlike the linear ζ-q relation for a particle in charge-free media, the new relation shows that as the particle charge increases, the initial linear increase of the potential slows down and asymptotes to a finite value. The asymptotic values of the potential at high particle charges are dependent on the Debye length (or the ionic charge density) of the media and are of the order of a few hundred millivolts for particles in typical nonaqueous dispersions, e.g., those used as liquid developers for electrographic images. Thus, with this relationship, the reported values of charge and the electrophoretic mobility determined experimentally for these dispersions correspond to physically reasonable values of ζ potentials, which are smaller than that expected from the linear relation by more than an order of magnitude. In addition, the variations of particle charges with ionic charge densities and particle concentrations are examined.
I. Introduction The charge q and the mobility µ of colloidal particles are two important figures of merit in electrophoretic applications of nonaqueous colloidal dispersions, an example of which is the liquid developer for electrographic images.1-4 The two quantities are related by the Stokes’ law5
µ ) q/6πηR
(1)
where R is the particle radius and η is the coefficient of viscosity of the fluid. Denoting the mass density of particle by Fm and the mass of particle by m, the above relation can be rewritten in terms of the “charge-to-mass ratio”, q/m as
µ ) 2(q/m) FmR2/9η
(2)
With Fm ) 1 g/cm3, R ) 1 µm, and η ) 3 cP, a value of q/m ) 200 µC/g corresponds to a mobility of µ ) 1.5 × 10-4 cm2/(V s). The mobility is also related to the surface potential, or “zeta potential” ζ, by the Hueckel equation6
ζ ) 3ηµ/2
(3)
where is the permittivity of the fluid. With ) 2 × 10-13 F/cm, a typical value for hydrocarbon liquids, the above value of µ (or q/m) gives a zeta potential of ζ ) 3.4 V! A possible cause leading to this counterintuitively large value can be suggested as follows. Eliminating µ from eqs 1 and 3, or eqs 2 and 3, ζ is related to q, or q/m by
ζ ) q/4πR ) (q/m)FmR2/3
(4)
This can be recognized as the expression for the surface potential of a spherical particle with charge q in a chargeX
Abstract published in Advance ACS Abstracts, July 1, 1996.
(1) Gibson, G. A.; Luebbe, R. H. J. Imaging Technol. 1991, 17, 207. (2) Larson, J. R.; Lane, G. A.; Swanson, J. R.; Trout, T. J.; El-Sayed, L. J. Imaging Technol. 1991, 17, 210. (3) Larson, J. R.; Morrison, I. D.; Robinson, T. S. IS&T’s Int Congr. Adv. Non-Impact Print. Technol., Final Program Proc., 8th 1992, 193. (4) Morrison, I. D.; Tarnawskyj, C. J. Langmuir 1991, 7, 2358. (5) Wiersema, P. H. On the Theory of Electrophoresis; Pasmans: The Hague, 1964. (6) Ross, S.; Morrison, I. D. Colloidal Systems and Interfaces; John Wiley & Sons: New York, 1988; p 346.
S0743-7463(95)01091-2 CCC: $12.00
free medium.4 In colloidal dispersions, the charged particles are dispersed, not in a charge-free medium, but in a medium containing surfactants ions of both polarities. The charged particle attracts ions of opposite polarity (counterions), and repels those of the same polarity (coions), forming an electrical double layer near the surface. The counterion and co-ion distributions around the particle are determined by the competition between the Coulomb force and the thermal motion of ions, resulting in a Boltzmann distribution at equilibrium (Gouy-Chapman model).7 The ζ-q relation for such a particle can be expected to differ from eq 4. The primary objective of this work is to derive the surface potential-charge (ζ-q) relation for particles surrounded by an electric double layer, from the solutions of PoissonBoltzmann equations.8 It will be shown that with the new relationship, the experimentally determined q/m (∼100 µC/g) or µ values (∼10-4 cm2/(V s))9 correspond to more reasonable values of zeta potential (e400 mV). The mathematical procedure for the evaluation of particle charge is described in the next section (section II). The resultant ζ vs q/m relations and the nature of particle and ion charge distributions associated with the electrical double layer are presented and discussed in section III. The Appendix describes the features of the potential distributions which are used to calculate the charges. II. Mathematical Procedure The Poisson-Boltzmann equation for the electrical potential ψ in spherical symmetry, with the origin at the center of particle, is given by
(1/r2)(d/dr)[r2dψ/dr] ) -F/
(5)
with the net space charge density F given in terms of the potential as
F ) F+ + F- ) zen0[exp(-zeψ/kT) - exp(zeψ/kT)] (6) where k is the Boltzmann constant, T is the absolute (7) Gouy, G. J. Phys. Theor. Appl. 1910, 9, 457. Chapman, D. L. Philos. Mag. 1913, 25, 475. Reference 6, p 232. (8) Loeb, A. L.; Overbeek, J. Th. G.; Wiersema, P. H. The electrical double layer around a spherical colloid particle; MIT Press: Cambridge, MA, 1961. (9) Chen, I.; Mort, J.; Machonkin, M. A.; Larson, J. R. Program & Abstract, 69th Colloid & Surface Science Symposium, Salt Lake City, Utah, 1995; p 80.
© 1996 American Chemical Society
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Chen
temperature, (ze is the charge on each positive/negative ion, and n0 is the number density of surfactant ions of each polarity in the absence of (or at a large distance from) the particle. This “nominal ionic charge density”, zen0, is related to the conductivity of the surfactant dispersion by σ ) 2(zen0)µi, where µi is the ion mobility (assumed to be identical for anions and cations). In terms of the nominal ionic charge density, a “Debye length” D can be defined as
D ) [(kT/ze)/(zen0)]1/2
(7)
D can be chosen over zen0 as an input parameter for the Poisson-Boltzmann equation, because its dimension enables direct comparison with the particle radius R. The potential value at the particle surface, commonly denoted as the zeta potential is determined by the difference in the electrochemical properties (such as chemical potentials or electron affinities) of the particles and the surfactant ions. The zeta potential ζ serves as one of the boundary conditions, at r ) R, for the PoissonBoltzmann equation
Another boundary condition at a large distance is specified as follows. Let f be the fractional volume of particles in the dispersion, then the number of particles in a unit volume of dispersion is 3
(9)
Defining the “particle space” as the average spherical space per particle of volume 1/N ) 4πR3/3f, then the “radius of particle space” S is given by 1/3
(10)
S ) R/f
For example, with the particle concentration of 2%, i.e., f ) 0.02, the radius of particle space S is 3.68 times the particle radius R, and for f ) 0.2 (20%), S ) 1.71R. The boundary condition at a large distance can now be stated as the vanishing of the potential gradient at r ) S, due to the symmetry
dψ/dr ) 0 at r ) S
(11)
From the numerical solutions, ψ(r), of the PoissonBoltzmann equation, various charge quantities can be calculated as follows. The charge on a particle is equal and opposite in sign to the total net space charge surrounding the particle. Therefore, the charge per particle q is given by
∫R Fr2 dr ) 4π∫R (d/dr)[r2dψ/dr] dr )
q ) -4π
S
Furthermore, multiplying q of eq 12 by the number of particles per unit volume N, eq 9, the particle charge density, Q, is given by
(8)
ψ(R) ) ζ
N ) 3f/4πR
Figure 1. Calculated ζ vs q/m relations for a 2% solid content dispersion with three values of Debye length D (solid curves) and the linear relation in charge-free medium, eq 4 (dashed). The normalized unit for the potential is kT/ze (≈25 mV) and that for q/m is kT/zeR2Fm (≈0.5 µC/g). See section III for the details of the normalized units.
S
-4πR2[dψ/dr]R (12) where the space charge density F is expressed in terms of the derivative of potential ψ using eq 5, and the boundary condition, eq 11, is applied. Dividing this expression of q by the particle mass m ) 4πR3Fm/3, the charge/mass ratio is given by
q/m ) -(3/RFm)[dψ/dr]R
(13)
The gradient of ψ can be calculated from the solution of the Poisson-Boltzmann equation with a given value of ζ as the boundsary condition, eq 8. Thus, eq 13 represents implicitkly the q/m vs ζ relation.
Q ) -(3f/R)[dψ/dr]R
(14)
Similarly, the positive and the negative ion charges (qp and qn) surrounding the particle can be expressed as
∫RSexp(-zeψ/kT)r2 dr ≡ 4π(zen0)I+
qp ) 4π(zen0)
qn ) -4π(zen0)
(15)
∫RSexp(zeψ/kT)r2 dr ≡ -4π(zen0)I-
(16)
where I( denote the integrals that can be calculated from the potential ψ(r). Multiplying these quantities by the number of particles per unit volume N, the charge densities of (positive) co-ions Qp and (negative) counterions Qn are
Qp, Qn ) (4πN(zen0)I( ) (3(f/R3)(zen0)I( (17) The variations of Q, Qp, and Qn with the nominal ionic charge density, the zeta potential, and the particle concentration are shown with numerical examples in the next section. The accuracy of numerical results can be verified by the charge neutrality condition among the three charge densities, Q, Qp, and Qn,
Q + Qp + Qn ) 0
(18)
III. Results and Discussion All numerical examples in this section and the Appendix are given in normalized units defined below. Typical values of practical interest for the units are also given: length (R) ≈1 µm; potential (kT/ze) ≈25 mV (with z ) 1 at room temperature); charge density (kT/zeR2) ≈0.5 µC/ cm3 (with ) 2 × 10-13 F/cm and R ) 1 µm); charge/mass ratio (kT/zeR2Fm) ≈0.5 µC/g (with Fm ) 1 g/cm3 and above and R). III. 1. Potential-Charge Relation. The surface potential vs the charge/mass ratio relation (ζ vs q/m), calculated by the procedure described in the previous section, in particular, eq 13, is shown in Figure 1 for the case of 2% solid content (f ) 0.02). The three solid curves differ in the values of the Debye lengths D, given in units of R, the particle radius. With the typical values of parameters given above, the three cases with D ) 3R, R, and 0.3R correspond to the nominal ionic charge density of zen0 ) 0.05, 0.5, and 5 µC/cm3, respectively. The dashed
Particle Charges in Nonaqueous Colloidal Dispersions
Langmuir, Vol. 12, No. 14, 1996 3439
Figure 2. Particle charge density Q, co-ion charge density Qp, and counterion charge density Qn, as functions of zeta potential. The normalized units are kT/zeR2 (≈0.5 µC/cm3) for charge densities and kT/ze (≈25 mV) for zeta potential.
curve shows the linear ζ vs q/m relation in charge-free medium, of eq 4 (equivalent to D ) ∞ or zen0 ) 0). In the presence of electrical double layer, the surface potential ζ is seen to increase linearly at small values of q/m (e a few tens of µC/g) as in the charge-free medium, but asymptotes to a finite value at higher values of q/m (g50 µC/g). The asymptotic value, although dependent on the Debye length D (or zen0), is only slightly larger than 10(kT/ze) (≈250 mV). For example, with the above cited parameter values of practical interest, a q/m ) 100 µC/g corresponds to ζ ≈ 240, 190, and 125 mV for Debye lengths of D ) 3, 1, and 0.3 µm (or zen0 ) 0.05, 0.5, and 5 µC/cm3), respectively. This is a reduction by about an order of magnitude from the values expected from the linear relation, eq 4. The reduction is more significant for a larger value of q/m. It should be noted that in this discussion the ζ values refer to the potentials at the physical surface r ) R of the particles. The decrease of potential near the particle surface is faster for the larger ζ, as shown in Figure 5 of the Appendix. Thus, if the particle size includes the inner localized portion of the double layer (Stern layer),10 the observed ζ potential in reality can be somewhat smaller than that expected from the above calculations. III. 2. Particle and Ion Charges. The variations of the particle charge density Q, eq 14, and the co-ion and counterion charge densities, Qp and Qn, eqs 17 and 18, with the ζ potential are shown in Figure 2 with an example for particle concentration f ) 0.02 and Debye length D ) R. As the ζ potential increases from 4 to 16 kT/ze (i.e. ≈100 to 400 mV), both the particle charge density Q and the counterion charge density Qn increase by about 4 orders of magnitude, while the co-ion density Qp decreases slightly from the nominal value zen0. An efficient charging of particles by the surfactants is represented by the particle charge density Q exceeding the co-ion charge density Qp, which is seen to occur for larger ζ values (>4kT/ze for this example). Similar features are observed for other values of D within the range of practical interest. The dependence of particle charge Q on the Debye length or equivalently, on the nominal ionic charge density, zen0, is shown in Figure 3A, for the case of particle concentration f ) 0.02 and various ζ values. Q is seen to vary as the square-root of zen0 at smaller values of ζ (
