4090
Langmuir 2000, 16, 4090-4093
Experimental Test of the Ion Condensation A. Ferna´ndez-Nieves, A. Ferna´ndez-Barbero, and F. J. de las Nieves* Group of Complex Fluids Physics, Department of Applied Physics, University of Almerı´a, Almerı´a 04120, Spain Received July 12, 1999. In Final Form: January 14, 2000 The objective of this work is to test the very recent Levin’s theory for charged colloids (Levin, Y.; Barbosa, M. C.; Tamashiro, M. N. Europhys. Lett. 1998, 41 (2), 123). To accomplish this, the fraction of condensed ions has been experimentally obtained from electrophoretic mobility data and surface charge values. Latexes with different functionalities, surface charges, and sizes were selected in order to support the universality of the ion condensation effect. The good agreement between charge and fraction of condensed ions confirms Levin’s theory and makes ion condensation a clear candidate for explaining the observed insensitivity of the electrophoretic mobility to surface charge variations, for sufficiently large charge.
1. Introduction Charged colloidal suspensions represent a severe challenge due to the high asymmetry between the charge on a polyion and a counterion, making the use of traditional methods of liquid-state theory unfeasible. This topic was firstly tackled by Manning,1,2 whose works represent already a classical starting point for studying charge asymmetry phenomena. Very recently, an extension of the Debye-Hu¨ckelBjerrum theory to the fluid state of a highly asymmetric charged colloid has been published.3,4 The full free energy of a polyelectrolyte solution was calculated, finding that counterions condense onto polyions, forming clusters composed of one polyion and n counterions. A direct consequence of this ion condensation is the apparence of an effective particle charge density, lower than the surface one, which must govern the colloidal features. From an experimental point of view, an electrophoretic mobility insensitivity to surface charge variations, for sufficiently large charge, has been widely observed in many and different colloidal systems. An example of such unexpected phenomenon is shown in Figure 1, where the electrophoretic mobilities of three different sulfonate latex particles have been plotted versus pH.5 Similar values were obtained, independently of the particle surface charge density. This is an anomalous behavior since the particle velocity should increase as its surface charge density rises, as predicted by basic theory. This fact, far from being system dependent, can be found in the literature with independence of the particle size, the sign of the charge, or the surface end group. Native DNA, polyphosphate, and several polymers in solution are also systems where such behavior becomes apparent.6 From another perspective, the long-range structure in dense colloidal systems also seems to be governed by an effective charge, much lower than the titrated one.7 Charge * To whom correspondence should be addressed. E-mail:
[email protected]. (1) Manning, G. S. Q. Rev. Biophys. 1978, 11, 179. (2) Manning, G. S. Ber. Busen-ges. Phys. Chem. 1996, 100 (6), 909. (3) Tamashiro, M. N.; Levin, Y.; Barbosa, M. C. Physica A 1998, 258, 341. (4) Levin, Y.; Barbosa, M. C.; Tamashiro, M. N. Europhys. Lett. 1998, 41 (2), 123. (5) de las Nieves, F. J.; Daniels, E. S.; El-Aasser, M. S. Colloids Surf. 1991, 60, 107. (6) Manning, G. S. J. Phys. Chem. 1981, 85, 1506.
Figure 1. Electrophoretic mobility of three sulfonate latexes with different surface charge densities as a function of pH at 0.01 M NaCl. As expected, µ is pH-independent due to the strong acid nature of the surface end groups (for details see ref 5). However, µ presents similar values, independently of the surface charge density.
renormalization procedures8 have been successfully employed to account for this effect, the effective charge being a consequence of the interaction between particles, causing a potential screening in relation to that of single particles. This universal kind of charge saturation, could be connected to the ion condensation, described in the DebyeHu¨ckel-Bjerrum theory. The aim of this work is to determine the fraction of condensed ions from electrophoretic mobility and surface charge data, for a subsequent comparison with the prediction of Levin’s theory for charged colloids.3,4 The comparison will be extended to colloidal systems of different sizes, functionalities, and surface charges in order to support the universal nature of the effect. The theoretical insensibility of the fraction of condensed ions with respect to salt concentration will also be experimentally tested. (7) Quesada-Pe´rez, M.; Callejas-Ferna´ndez, J.; Hidalgo-A Ä lvarez, R. J. Chem. Phys. 1999, 110, 6025. (8) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P.; Hone, D. J. Chem. Phys. 1984, 80, 5776.
10.1021/la990920z CCC: $19.00 © 2000 American Chemical Society Published on Web 03/28/2000
Experimental Test of the Ion Condensation
Langmuir, Vol. 16, No. 9, 2000 4091
The good agreement arising from the comparison confirms that (i) the fraction of condensed ions is independent of salt concentration and (ii) ion condensation is a clear candidate for explaining the observed insensitivity of the electrophoretic mobility to surface charge variations. 2. Ion Condensation We consider a set of N particles inside a volume V. The particles are idealized as hard spheres of radius a, each carrying Z ionized groups of charge q, uniformly spaced on the surface. Consequently, a total of Z N counterions must be present in order to preserve the overall electroneutrality of the system. The fluid will be composed of bare particles of density F0, free counterions of density Ff, and clusters consisting of a particle and n associated counterions (0 < n e Z). The density of the clusters with n counterions is Fn. The contributions to the Helmholtz free-energy density, f ) -F/V, arise from electrostatic interactions and entropic terms.4 The particle-counterion contribution is obtained using the usual Debye-Hu¨ckel theory applied to an n-cluster of effective charge σn inside the ionic atmosphere. The resultant free energy density is obtained through a Debye charging process, leading to
βf particle-counterion ) -
λB
Z
(Z - n)2Fn ∑ 2a(1 + κa)n)0
(1)
where β-1 ) kT. λB ) e2β/(4π) is the Bjerrum length and κ ) (4πλBFf)1/2 is the inverse of the Debye length. The calculation of the particle-particle contribution to the free energy is based on the usual van der Waals theory, giving rise to the following expression:3
βfparticle-particle ) -
1 + 2κa Ff (1 + κa)2 2
(2)
where the long-ranged interaction between two clusters is screened by the cloud of free counterions, producing an effective short-ranged potential of a DLVO form. The counterion-counterion contribution originating from the interactions between free counterions, is calculated using the one-component plasma theory,3,9 yielding an expression for the free-energy density valid over a wide range of mass densities:
βfcounterion-counterion ) -Ff F(Ff)
(3)
where
F(Ff) )
[
(
)
ω2 + ω + 1 1 2π 1 + 3/2 + ln - ω2 4 3 3
(
)]
2ω + 1 2 tan-1 1/2 3 31/2 ω(Ff) ) {1 + 3[4πλB3Ff]1/2}1/3
(4) (5)
βf mixing )
( )]
[
φs
∑s Fs - Fs ln ζ
(6)
s
with s accounting for bare particles, free counterions, and clusters particle-counterions. φn ) (4/3)πFn3a3 and φf ) (4/3)πFf3d3 are the volume fractions occupied by each species s. d is the distance over which the counterions will keep from approaching one another and is equal to the Bjerrum length in the limit of small densities. ζs are the internal partition functions for an isolated species s. Since the bare polyions and the free (unassociated) counterions do not have internal structure, their internal partition functions are simply given by ζ0 ) ζf ) 1. For an n-cluster the internal partition function is4
ζn )
[
]
Zn - (n2/2) Z! exp a/λB (Z - n)!n!
The total free energy is finally calculated from the entropic and electrostatic contributions as
f ) f mixing + f particle-counterion + f counterion-counterion + f particle-particle (7) The minimization of the free energy under the constraints of a fixed number of polyions and counterions leads to the law of mass action:
µ0 + nµf ) µn
(8)
where the chemical potentials are calculated as µs ) -∂f/ ∂Fs. The relation between bare and effective charge is obtained by solving this set of Z coupled nonlinear algebraic equations (one for each cluster). As demonstrated by Levin et al.,4 the width of the cluster size distribution remains quite narrow, allowing the average cluster size to be replaced by one characteristic size. In this case, the theory becomes extremely simple. Therefore, only one algebraic equation must be solved instead of Z coupled ones. Equation 8 may then be used for determining the fraction of condensed ions n/Z, assuming a single cluster distribution. It is interesting to remark that Levin’s theory is constructed in the absence of salt. The test of this point will be one of the tasks of the present paper. 3. Experimental Determination of the Ion Condensation In this work the effective charge of a colloidal particle is experimentally determined from electrophoretic mobility and surface charge density measurements. In this section, the method yielding the fraction of condensed ions is described. The electrophoretic mobility is related to the electrical surface potential through the Smoluchowski equation:10
µ ) ψ(0) η
(9)
It is interesting to point out that, in the bulk, this contribution is very small. The entropic (mixing) contribution reduces to a sum of ideal gas terms:
which is valid for large particles as compared with the Debye screening length, κ-1. This condition implies that the polarization of the electric double layer is not relevant, and thus, retardation and relaxation effects may be
(9) Penfold, R.; Nordholm, S.; Jo¨nsson, B.; Woodward, C. E. J. Chem. Phys. 1990, 92, 1915.
(10) Hunter, R. J. Foundations in Colloid Science, Vol. I; Oxford University Press, Oxford, U.K., 1995.
4092
Langmuir, Vol. 16, No. 9, 2000
Ferna´ ndez-Nieves et al. Table 1: Latexes Used in the Calculation of the Fraction of Condensed Ionsa latices
functionality
FJN20 FJN21 FJN22 K8 K8 JM1 AD1 DBG1 DBG2 SN8 SN13
sulfonate sulfonate sulfonate amine amine sulfonate sulfate/aldehyde sulfonate sulfonate sulfonate sulfonate
ref surface charge µC/cm2 diam, nm 5 5 5 17 17 18 19 19 20 21 21
-13.9 -16.3 -19.4 +20.0 +8.5 -4.2 -2.9 -6.0 -1.3 -12.3 -17.0
248 263 244 265 265 195 324 161 287 179 178
a The references in which the electrokinetic characterization were overtaken are shown. Errors in surface charge density and particle size are ∼10%.
potential of a sphere with its effective charge:3 Figure 2. Experimental insensitivity of the fraction of condensed ions with salt concentration variations: DBG2 (O), DBG1 (b), AD1 (/), and K8 (3). Error bars have been included for some experimental points in order to give the approximate level of accuracy of the calculations.
neglected.11 Moreover, in this high-salt region the ζ potential should equal the diffuse potential which in the absence of specific adsorption coincides with the surface potential. These facts are taken into account in writing eq 9. In order to relate the surface potential with the surface charge, the Poisson equation must be solved. At this stage, the ion condensation is included through a renormalization of the particle charge. Thus, the effective surface charge density of a cluster with n condensed counterions is
σn )
(Z - n) q 4πa2
Z-n ) σ0 Z
(10)
where σ0 and Z are the surface charge density and number of superficial charges on a colloidal particle, respectively. Outside the particle (r g a), the charge density is Z
F(r) ) -qFf exp[-βqψ(r)] +
∑
(Z - n)qFn +
n)0
σnδ(r - a) (11) where a Boltzmann distribution has been considered for the counterions. Note that only the free counterions are assumed to be polarized; the bare particles and clusters are too massive to be affected by the electrostatic fluctuations and contribute only to the neutralizing background. Substituting the charge density into the Poisson equation, the nonlinear Poisson-Boltzmann equation is obtained. After linearization of the exponential factor in eq 11, the electrostatic potential satisfies the Laplace (r < a) and Helmholtz (r g a) equations:
∇2ψ(r) )
{
r
