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Effective Screened Coulomb Charge of Spherical Colloidal Particles Kenneth S. Schmitz Department of Chemistry, University of MissourisKansas City, Kansas City, Missouri 64110 Received April 28, 1999. In Final Form: October 27, 1999 The application of the well-known DLVO pair potential and its variations in the literature to the evaluation of the charge on macroions routinely results in an “effective” charge considerably smaller than the “bare” surface charge. Various theories have been proposed for “charge renormalization” on the basis of either surface effects (viz., counterion “condensation”) or equivalence of linearized and nonlinearized electrostatic potentials at the surface of a computation cell in the numerical analysis of the Poisson-Boltzmann equation. In the present discourse, a model for the effective charge is proposed that is based on the equivalence of the screened Coulomb macroion-counterion interaction to the thermal energy kBT. The distance at which these interactions are equal, Rtherm, defines a distance partition for the counterions into two populations: those tightly associated with the macroion and contributing to the charge reduction and those “free” in solution and contributing to the properties of the surrounding medium. The characteristics of this model are in good agreement with other theoretical approaches based on more elaborate solutions to the PoissonBoltzmann equation and computer simulations. More important, this relatively simple model is in agreement with the experimentally determined trends relating the effective charge to the titration charge. Limitations of the proposed model, as well as the screened Coulomb potential, are also discussed.
1. Introduction One of the major problems in the study of charged colloidal particles is the magnitude of the charge of the colloidal particle that affects the physical properties of the system. Although at first glance one may suggest that the charge of the colloidal particle be obtained by titration methods, it is not always clear that this value is the “effective” charge that governs the physics of the system. For example, the DLVO potential has been used for several decades as the premier vehicle to interpret physical data on colloidal stability. The DLVO potential is the sum of a Coulomb repulsive part (UC) and an ad hoc addition of a van der Waals attraction part (UvdW) for two colloids separated by a distance R,1
UDLVO ) UC + UvdW )
(1)
[
q2p exp(2κap)
exp(-κR) AH 1 + R 12 x2 - 1 4πro(1 + κap) 2
( )]
1 x2 - 1 + 2ln x2 x2
where x ) r/2ap is the reduced distance of approach normalized by the particle radius ap and charge qp ) Zpqe; here, Zp is the magnitude with sign of the colloidal particle, qe is the magnitude of the electron charge, AH is Hamaker’s constant (10-12 J < AH < 10-19 J), and
κ2 ) 4πλB
∑
j(added)
njZ2j (1 - φp)
+
4πλB|Zp|npZ2c (1 - φp)
) κ2DH + κ2c (2)
defines a generalized screening parameter κ composed of the Debye-Hu¨ckel screening parameter κDH for the added electrolyte and κc for the counterions released from the (1) Verwey, E. J.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier Publishing Co. Inc.: New York, 1948.
macroions. The parameters in eq 2 are as follows: nj ) Nj /VT is the number density of the jth ion of charge magnitude (with sign) Zj; VT is the total volume of the system that one would use when experimental data is used in calculating the microion concentrations; 1 - φp is the volume correction term to obtain the “free” volume for the microions where the macroion volume fraction is φp ) np(4π/3)a3p, where (4π/3)a3p is the volume per colloid particle; and λB ) q2e /4πrokBT is the Bjerrum length, where r is the relative permittivity of the medium, o is the permittivity of the vacuum, kB is the Boltzmann constant, and T is the absolute temperature. The counterion concentration is taken to be nc ) |Zp|np. The symbol κDH is to emphasized that it refers to the added electrolyte in accordance with the well-known Debye-Hu¨ckel expansion of the Boltzmann weighting factor in which the first term is set equal to zero because of electrical neutrality. The separation of κ into its two components, the added salt and the released counterions, is also to emphasize that the original formulation of the DLVO potential was based only on κDH. That is, the macroion concentration was taken in the zero concentration limit such that there was no contribution to the pair potential by other macroions. This “true pair potential” was first emphasized by Beresford-Smith and co-workers2 in their theory for the pair potential, who also distinguished between the contribution of added electrolyte and the counterions, given, respectively, in their notation by κsalt and κcounterions. We make this distinction for clarity in later development for salt-free solutions. In using the potential UC to fit experimental data, the value of Zp has routinely been found to be much smaller than that anticipated on the basis of titration measurements. Attempts to justify the use of the DLVO potential and its variations have led to the concept of “charge renormalization” that generally considers “surface-bound” counterions in the readjustment of the “surface charge”. (2) Beresford-Smith, B.; Chan, D. Y. C.; Mitchell, D. J. J. Colloid Interface Sci. 1985, 105, 216.
10.1021/la990522w CCC: $19.00 © 2000 American Chemical Society Published on Web 01/04/2000
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The screening parameter κ in eq 2 is actually a “farfield” parameter as it is defined in linearized PB theories. The focus of the present study is an “effective screened Coulomb” charge, Zeff, that takes into consideration the structure of the ion cloud about a single macroion. That this counterion structure may have profound influence on the calculation of Zeff is illustrated in the following calculation along one dimension. We may think of a “screened charge” as that which is obtained by moving a test charge toward the macroion of interest with a distribution of surrounding counterions. We compare the unscreened Coulomb interaction of a test charge as it approaches a surface in the absence (Uabs) and presence (Upres) of neutralizing point charges distributed at uniform intervals along a line, viz., xj ) 50 + 2j, where j ) 1-6, and the macroion is assumed to have a radius of ap ) 50 Å. The charges at these locations, moving from the macroion, are arbitrarily given the monotonic decreasing total charge distribution -30, -20, -15, -11, -8, and -6 to neutralize the macroion charge of +90. The location of the test charge is generated by the expression xt ) 51 + 2(t - 1), with t ) 1-7. The ratio Upres/Uabs for this set of calculations is {Upres/Uabs} ) {0.0937, 0.0904, 0.0873, 0.0845, 0.0819, 0.0794, 0.0770}. If we now have a uniform distribution of neutralizing charge, i.e., -15 at all positions, then the set becomes {Upres/Uabs} ) {0.1222, 0.1181, 0.1142, 0.1106, 0.1072, 0.1040, 0.1001}. One may conclude that because the elements of the set {Upres/Uabs} are larger for the uniform distribution than for the monotonic distribution then the effective charge of the macroion is larger for the former distribution than the latter. Discussed in some detail is the analytic expression for the screened Coulomb pair potential for the microionmacroion interaction, and the inherent limitations of this functional form from dilute-to-concentrated macroion concentrations. The concept of “thermodynamically bound” counterions is examined in more detail through this analytic form of a pair potential for the colloidal particle with the surrounding cloud of counterions.
1fb )
1 |Zc|ξOM |Zc|
(4)
where |Zc| is the valency of the counterions and the Oosawa-Manning condensation parameter is ξOM ) λB/ 〈b〉, where 〈b〉 is the average spacing between charged groups along the rod. Condensation of the counterions thus occurs if ξOM > 1. For example, in the case of DNA with 〈b〉 ) 1.7 Å and |Zc| ) 1, the fraction of bound sites is fb ) 0.76. The fraction of “contact” counterions onto planar surfaces was examined through the Poisson-Boltzmann (PB) equation for two parallel plates by Enstro¨m and Wennerstro¨m.5 These authors defined a region ∆ from the surface of one plate as a contact region in which counterions are adsorbed onto the surface. The fraction of condensed counterions fb was obtained by integration of the charge density over the region from the surface to this distance ∆, with the result, in the present notation,
fb ) 1 -
1 ∆σZcqe 12rokBT
(5)
where σ is the surface charge density. In accordance with their calculations for σ ) 0.2 C m-2 and ∆ ) 3 Å, the fraction of surface-bound counterions is in excess of 0.63. Gisler et al.6 presented a different approach to charge renormalization. They solved the linearized and nonlinear PB equation and equated their values and first derivatives at the computation cell boundary. The slope of the somatched linearized PB potential at the particle surface, which is a measure of the effective macroion charge, was found to be smaller than the corresponding quantity for the nonlinear PB potential, which was proportional to the bare charge. One may therefore say that a certain number of counterions “condensed” onto the macroion surface to account for this difference in surface charge.
2. “Condensation” Models and Renormalized Charge
3. Experimental Data and the Condensation Renormalized Charge
Alexander et al.3 formulated a model for charge renormalization to account for this discrepancy in the “model” and “anticipated” charges. In their approach, the counterions are allowed to adsorb onto the surface of the colloidal particle until the configurational entropy is balanced by the electrostatic potential at the surface of the colloidal particle. That is,
Although the Manning condensation theory has enjoyed widespread application to rodlike molecules such as DNA, the use of the charge renormalization model of Alexander et al. is relatively fresh in the literature as a model to explain experimental results. In fact, in a few cases, the predicted renormalized charge Z*p has greatly underestimated the experimental determination of the effective colloidal charge. Ito et al.7 used conductivity methods to determine the fraction of free counterions, ff ) 1 - fb, in solutions of linear polystyrenesulfonate (PSS) and sulfonated polystyrene latex spheres (PLS). They obtained for the PSS the value ff ) 0.37 and for the PLS the range of values 0.03 < ff < 0.16. For their PLS sample #1, they used the expression of Alexander et al.3 and calculated a value of ff ) 0.001, which was considerably smaller than their experimental value of ff ) 0.04. Roberts, et al.8 employed steady-state voltammetry methods for hydrogen reduction to obtain a value of ff for PLS particles. They also obtained an experimental value of ff ) 0.04. These authors proposed an empirical model in which a certain
Z*pλB ) ln(φc) ap
(3)
where Z*p is the renormalized charge of the colloidal particle and φc is the volume fraction of the counterions. The screening parameter is κ ) κc where Z*p is used in place of Zp in eq 2. The philosophical basis of this model finds its parallel with the condensation theory of Manning4 for counterion modification of the charge density of rigid rods. In the Manning model, the surface charge density along the rod is adjusted in such a way as to counterbalance the entropy of mixing of the counterions. The fraction of “bound sites” is given in the Manning theory as (3) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P. J. Chem. Phys. 1984, 80, 5776. (4) Manning, G. S. Q. Rev. Biophys. 1978, 11, 179.
(5) Enstro¨m, S.; Wennerstro¨m, H. J. Phys. Chem. 1978, 82, 2711. (6) Gisler, T.; Schulz, S. F.; Borkovec, M.; Sticher, H.; Schurtenberger, P.; D’Aguanno, B.; Klein, R. J. Chem. Phys. 1994, 101, 1994. (7) Ito, K.; Ise, N.; Okubo, T., J. Chem. Phys. 1985, 82, 5732. (8) Roberts, J. M.; Linse, P.; Osteryoung, J. G. Langmuir 1998, 14, 204.
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fraction of counterions diffused with the PLS particles,
D R+γ ) o γ+1 D
(6)
where D is the measured diffusion coefficient of the counterion (H+) in the presence of PLS and D° in the absence of PLS, γ is the ratio of the concentrations of 1:1 electrolyte to the counterion, and R ) 0.024 was determined from the ratio of slopes of perchloric acid and the latex particles. Although their equation had no adjustable parameters, these authors point out that eq 6 gave a good fit to their data. The solution to the nonlinear PB equation using a cell model indicated that the value of ff could be accounted for if a certain fraction of counterions were bound to the charged sphere.8 4. “Experiment-Defined” Macroion Charges Several years ago, we emphasized a difference between three types of charges that may be obtained from experiment: the titration charge, Ztit; the electrolyte dissipation charge (obtained from the friction factor in nonperturbation methods such as dynamic light scattering), Zedc; and the electrophoretic mobility charge (obtained under perturbation methods), Zemc.9,10 The physical picture that distinguished these three charges is the role of the surrounding cloud of counterions. The titration charge is a surface charge, the electrolyte dissipation charge includes those ions that move with the macroion under “zero shear” conditions, and the electrophoretic mobility charge includes only those counterions that are not “stripped off” under the imposed shear of the hydrodynamic flow field. The partitioning of the charges into these three categories is to emphasize that the value of the effective charge may be dependent upon the method of its determination. Other effects, such as the influence of other macroions, are not considered in the present development. 5. Screened Coulomb “Effective” Charge, Zeff It is assumed that the macroions are sufficiently far apart that the distribution function of the counterions (c) about the macroion of interest (p) can be described in spherical coordinates. Thus, the pair distribution function of counterions about the colloidal particle is denoted by gpc(r), which is related to the pair interaction energy Upc(r) by
gpc(r) ) exp(-βUpc(r))
(9)
(9) Schmitz, K. S. Chem. Phys. 1982, 66, 177. (10) Schmitz, K. S.; Lu, M. Proc. Natl. Acad. Sci. U.S.A. 1983, 80, 425.
[
]
|ZcZp|λB exp(κ(ap - R)) 1 + κap R
(10)
where the approximation assumes point electrolyte ions for the counterion. Let us now differentiate between the thermodynamically bound and “thermodynamically free” counterions as being the separation distance for which β|Upc(R)| ) 1 and identify this distance as Rtherm. Hence, we seek the solution to the transcendental equation,
[
]
|ZcZp|λB exp(κ(ap - Rtherm)) )1 1 + κap Rtherm
(11)
It is not necessary, however, to know the precise form of Rtherm to estimate the apparent charge on the colloidal particle. What we need to know is that there exists a value of Rtherm within the range of values ap ) Rtherm e Rcell. If Rtherm ) ap, then no counterions are thermodynamically bound, and the effective charge is equal to the titration charge. The reason for the inequality Rtherm < Rcell is to preserve the mathematical form of the screened Coulomb potential in eq 11. If Rtherm g Rcell, then the bound counterions also act under the influence of the neighboring colloidal particles. The latter situation must result in nonlinear electrostatic effects and thus the breakdown of the linearized solution to the PB equation from which eq 11 was based. With these limitations in mind on the applicability of eqs 10 and 11, we now make the transformation from a distribution function based on βUpc(R) to one based on Rtherm. Because the DLVO theory is based on the linearized PB equation, we take as our guide for the macroion-microion distribution function the linearized form of gpc(R), viz., gpc(R) = 1 - βUpc(R), where Upc(R) is of the Yukawa form exp(-κR)/R. We thus assume the Yukawa form and write
exp(-R/Rtherm) R
ψpc(R) ) A
(12)
where A is a normalization factor that satisfies the condition
(8)
where it is assumed that Upc(R f ∞) ) 0 and Fpc(∞) represents the uniform distribution of counterions an infinite distance from the parent colloidal particle. If, however, the concentration of colloid particles is finite and the potential does not go to zero before entering the domain of a second macroion, then we may write for a spherical cell surrounding one macroion
Fpc(R) ) Fpc(Rcell) exp(-β∆Upc(R))
β|Upc(R)| )
(7)
where β ) 1/(kBT). We may then write the number density distribution function of counterions about the macroion as
Fpc(R) ) Fpc(∞) exp(-βUpc(R))
where the range of R is ap e R e Rcell and ∆Upc(R) ) Upc(R) - Upc(Rcell). At present, Rcell remains undefined except for the requirement of spherical symmetry so as to develop the relevant expressions without distraction. A discussion regarding Rcell is delayed to the following section when these general equations are expressed in a form to be applied to real systems. The functional form of Upc(R) in the present analysis is based on the screened Coulombic interaction between the counterion and the macroion, viz.,
A
∫aR
cell
p
exp(-R/Rtherm) 2 R dR ) 1 R
(13)
The number distribution function thus obtained is
ψpc(r) )
[
rtherm
exp(1/rtherm) × rcell - 1 (1 + rtherm) - exp (rcell + rtherm) rtherm exp(-r/rtherm) (14) r
(
)
]
where r ) R/ap, rtherm ) Rtherm/ap, and rcell ) Rcell/ap are
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reduced parameters. The fraction of counterions in the region ap to Rtherm, considered to be bound, in reduced parameters is
nc(rtherm) ) fb ) nc
∫1r
ψpc(r)r2 dr )
therm
(
)
rtherm - 1 rtherm
1 + rtherm - 2rthermexp -
(
)
rcell - 1 1 + rtherm - exp (rcell + rtherm) rtherm
(15)
where nc is the bulk concentration of counterions. The fraction charge reduction Zeff/Zp ) fZ is thus calculated to be fZ ) 1 - fb ) exp(-rcell/rtherm)(rcell + rtherm) - 2rthermexp(-1) exp(-rcell/rtherm)(rcell + rtherm) - (1 + rtherm)exp(-1/rtherm)
(16)
6. Zeff in Salt-Free Suspensions The only differences between salt-free and added electrolyte systems is the calculation of the screening parameter κ and the radius of the cell. These parameters may be adjusted independently upon addition of the electrolyte because κ is a function of the electrolyte concentration and the cell radius may be chosen at distances less than rcell if the pair potential is negligible. However, in salt-free solutions, the only source of the ions is the colloidal particle. The screening parameter for saltfree solutions is given in terms of the volume fraction,
κc )
x
4πλBnp|Zp|Z2c x3λBZ2c ) 1 - φp ap
x
x
φp 1 - φp
|Zp| (17) ap
The transcendental equation to solve for the salt-free case is
[
]
exp(y(1 - rtherm)) y2 )1 rtherm 3φp|Zc|(1 + y)
(18)
where
y ) κcap )
x ( )x 3λBZ2c
φp 1 - φp
|Zp| ap
(19)
7. Choice of Computation Cell Boundary It is desirable to express Rcell (or rcell) in eqs 9 and 13-16 in terms of the volume fraction φp. The choice of Rcell should be for convenience as to reflect the computation model used in the comparison. We consider two different schemes, a cubic and a spherical box, that have been used, respectively, in Monte Carlo (MC) simulations for a collection of macroions and in solving the PoissonBoltzmann equation involving only one macroion. In MC or Brownian dynamics (BD, or biased MC) simulations of a system composed of several macroions, the volume associated with each macroion is usually given as a rectangular box.11 For a uniform distribution of macroions in a cubic array, the distance between neigh(11) Allen, M. P.; Tildesley, D. J. Computer Simulations of Liquids; Clarendon Press: Oxford, England, 1993.
boring macroions is B, where B is the length of one side of the cubic box with the macroion at its center. The radial distribution function determined about the macroion of interest should not, therefore, penetrate nearby cells. Thus, one may define Rcell ) B/2 and the volume fraction by φp ) (4π/3)a3p/B3. The desired relationship between Rcell and φp for this model is
Rcell )
( ) π 6φp
1/3
ap
(20)
In this definition, however, one must know the fraction of counterions within the computation volume because the microions are free to roam the entire free volume Vfree ) B3 - (4π/3)a3p. Although this choice of Rcell may be valid for computer simulations, it is not convenient for the use of eqs 12-16 due to the normalization of the distribution function given by eq 13. A second choice is to construct a fictitious sphere with a volume equal to that of the cubic box, i.e., B3 ) (4π/3) R3cell. In this case, φp ) (4π/3)a3p/B3 ) (4π/3)a3p/(4π/3)R3cell and
Rcell ) (1/φp)1/3ap
(21)
Although this choice of the computation cell has the advantage that all of the counterions are within its boundaries, hence eqs 12-16 are valid, it cannot be used to mimic a system of more than one macroion. This is due to the fact that Rcell ) (3/4π)1/3B > B/2. This means that there is interpenetration of the spherical cells of neighboring macroions and therefore charge neutrality of the individual cell is violated. Furthermore, unlike the cubic cell, there are regions in the system that are voids if one has a collection spherical cells. Thus, the advantage of the spherical cell is the simplified mathematics accompanying the spherical symmetry. The disadvantage is that a system of macroions cannot be studied due to overlapping cells and void regions in the system. The focus in the present study is on the distribution of counterions about one macroion with the implied presence of other macroions. The latter applies when comparing the calculations with experimental data. Calculations reveal that the relationship between the volume fraction φp and the cell radius Rcell greatly affects the value of fZ through the screening parameter. Thus, if scheme 1 is employed, the value of κc is overestimated by use of the smaller volume, whereas if scheme 2 is employed, then the value of κc is underestimated due to the neglect of the overlap between neighboring macroion cells. We therefore report parallel calculations using these two schemes. The calculation of Zeff should not be dependent upon the choice of the cell if the counterions are found largely in the vicinity of the macroion or the method of calculation. To this end, we mention in the Discussion section recent BD simulations that employ scheme 1, and thus Rcell as defined by eq 20, and PB calculations that employ scheme 2, and thus Rcell as defined by eq 21. The relationship between the cubic box and the two definitions of Rcell is graphically shown in Figure 1. 8. Calculations of Zeff for Salt-Free Suspensions for the Screened Coulomb Potential All calculations related to schemes 1 and 2 were performed with Mathematica, a means of doing mathematics on a computer. Because β|Upc(r)| - 1 is a monotonic function of r that passes from positive to negative values, the value of rtherm was more easily found by determining
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Figure 2. Range of validity of the macroion-microion screened coulomb interaction: scheme 2. Shown here is the validity range of (|Zp|/ap)1/2 (Å-1/2) with the constraint that κcap < 3 for the three volume fractions: φp ) 0.004 (s), φp ) 0.04 (- - -), and φp ) 0.2 (- - -). Figure 1. Cutaway view of computation cells. Shown here are cutaway views of three computation cells. The cubic cell is convenient for calculations involving more than one macroion in the system due to its stacking properties. The inner spherical cell may be used to calculate the distribution function for the microions about one macroion in a system composed of several macroions. The drawback in using this spherical cell embedded in the cubic cell is that the number of microions within the spherical unit does not neutralize the macroion charge. The outer spherical cell has the same volume as the cubic cell and, hence, contains all of the number of neutralizing ions. However, as shown in this figure, this spherical cell penetrates into any neighboring sphere about the adjacent macroion and thus must necessarily contain counterions from the second macroion. The equal volume spherical cell is thus convenient for studying isolated macroions and is employed in the present study to validate eqs 12-16.
the minimum in its absolute value |β|Upc(r)| - 1| than solving the transcendental eq 20. For this determination, we used the Mathematica package FindMinimum to obtain the value of rtherm, which was then used in the subsequent calculations. According to eqs 18 and 19, there are two independent parameters in salt-free suspensions of colloids: the volume fraction φp and the ratio |Zp|/ap. We chose the volume fractions φp ) 0.004 and 0.04 to represent the extremes of concentrations used in experiments and φp ) 0.2 to test the limits of the screened potential. As pointed out by one reviewer, the screened Coulomb pair interaction is valid for κap < 3. Thus, for the present calculations using κc as defined by eq 17, we use an upper limit of κcap ) 3 to assess the validity of our calculations. In this analysis, we use the value |Zc| ) 1. We first determine the upper limits for (|Zp|/ap)1/2 at the three volume fractions of interest. A plot of κcap versus (|Zp|/ap)1/2 was found to be virtually identical for schemes 1 and 2. Shown in Figure 2 is the set of plots for scheme 2. Hence, we have for both methods the upper limits of (|Zp|/ap)1/2 ) (10, 3, 1) for φp ) (0.004, 0.04, 0.2), respectively. Shown in Figure 3 are the ratios rtherm/rcell, and shown in Figure 4 are the values of the charge ratio Zeff/Zp ) fZ versus (|Zp|/ap)1/2 for these volume fractions. To compare the effect of the choice of the boundary rcell, calculations were performed for the “infinite volume” case by setting all terms containing exp(-rcell/ rtherm) equal to zero in eq 16. These calculations are shown in Figure 5. 9. Discussion It is common to relate the electrostatic interaction energy as a ratio with the thermal energy kBT and then to interpret this ratio in terms of some physical process. In the case of macroions, the usual procedure is to use the
Figure 3. Ratio of the thermodynamic radius to cell radius versus (|Zp|/ap)1/2 for selected volume fractions. The reduced thermodynamic radius rtherm was calculated from eq 20 and the cell radius rcell from eq 19. The ratios rtherm/rcell were calculated over the range of validity of the screened Coulomb potential as given in Figure 1 for the three volume fractions: φp ) 0.004 (s), φp ) 0.04 (- - -), and φp ) 0.2 (- - -). Top: scheme 1. Bottom: scheme 2.
unscreened Coulomb potential for a surface-located calculation. Included in this category are the theories of Alexander et al.3 (cf. eq 3) and Enstro¨m and Wennerstro¨m5 (cf. eq 5). Rouzina and Bloomfield12 also compared the unscreened Coulomb interaction between counterions bound on a surface to the thermal energy. The physical process examined in their study was not condensation of the counterions but rather the conditions under which charged plates of like sign might exhibit an electrostatic attraction. They justified using an unscreened Coulomb (12) Rouzina, I.; Bloomfield, V. A. J. Phys. Chem. 1996, 100, 9977.
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Figure 4. Ratio of effective to surface charge versus (|Zp|/ap)1/2 for selected volume fractions. The ratio of the effective charge Zeff to the surface charge Zp was calculated from eq 17 using the identity fZ ) Zeff/Zp. The three volume fractions φp shown here are φp ) 0.004 (s), φp ) 0.04 (- - -), and φp ) 0.2 (- - -). Top: scheme 1. Bottom: scheme 2
Figure 5. Comparison of calculations of Zeff/Zp for the infinite and finite cell size: scheme 2. Calculations of fZ for the infinite cell radius were performed by setting to zero the terms with exp(-rcell/rtherm) in eq 17. The three volume fractions φp shown here are φp ) 0.004 (s), φp ) 0.04 (- - -), and φp ) 0.2 (- - -).
potential rather than the screened form on the assumption that the local concentration of condensed counterions was much lower than the ion concentration in the bulk. For a one-component plasma gas, Stevens and Robbins13 used the following ratio in their study of attraction between like charges, where we use the present notation,
Γ)
Z2λB 〈R〉
(22)
(13) Stevens, M. J.; Robbins, M. O. Europhys. Lett. 1990, 12, 81.
Clearly, eq 12 reduces to eq 22 for Z ) Zp ) Zc and κ ) 0. In contrast, Manning4 employed a screened Coulomb interaction between surface sites in a rodlike polyion (cf. eq 4). Hence, there is reason to believe that the use of the criterion β|Upc| ) 1 is a reasonable assumption with historical precedence to differentiate between two different “classes” of counterions. We are concerned herein with ions in the vicinity of the colloid particle and their influence on the calculation of an effective charge as one moves away from the colloid surface. One might argue that the Manning theory likewise includes counterions in the diffuse ion cloud as a means of surface charge reduction. However, in the Manning theory, this counterion distribution is uniform, whereas in the present model, the counterion cloud has a structure. As shown in the Introduction, a simple model calculation indicates that the value of Zeff differs for the uniform and monotonic decay distributions. We thus define the distance Rtherm at which the distribution function gpc(R) is equal to exp(1) as a means of differentiating between bound and free counterions. Only the free counterions contribute, for example, to the solution conductivity.7 Under such an assumption, the bound counterions must travel with the macroions. Hence, the unit of the macroion and tightly associated counterions is referred to as the “kinetic unit” in the remainder of this text. This definition of Rtherm is quite general and is based on the assumption that microions statistically remain in the vicinity of the macroion if the pair attraction energy is greater than that of the thermal motion of the counterions. The numerical evaluation of Rtherm is dependent, however, on the form of the counterion distribution function. Because of its analytical simplicity and widespread use of the screened Coulomb potential, we have used the microion-macroion interaction potential given by eq 11 to calculate rtherm to be used in the calculations of fZ. Therefore, our calculations are subject not only to the approximations leading to eq 17 but also the limitations underlying the derivation of eq 11. Hence, the range of values of (|Zp|/ap)1/2 are dependent upon the valid range of values of κap and the volume fraction φp. This range is shown in Figure 2 for the salt-free case used in the present study. Another factor to consider is the relative range of Rtherm and the distance between adjacent macroions. If Rtherm is greater than Rcell, then the counterions are “collectively influenced” by all of the macroions in the cluster. The situation that one obtains is likened to the one-component plasma model in which the macroions are considered to be immersed in a sea of counterions. That is, there is no structure to the counterion distribution in the vicinity of any one macroion. As shown in Figure 3, this situation is approached for φp ) 0.2 where rtherm/rcell ) 0.8 for κcap ) 1. Attention is now focused on the minimum in Figure 4. As a point of comparison, both the models of Alexander et al.3 and Gisler et al.6 indicate that the effective charge Z*p saturates as the magnitude of the bare surface charge Zp is increased. Thus, the ratio Z*p/Zp should show a continuous decrease as Zp is increased. These theories are based on surface charge effects, or what might be called the Stern layer, and solutions to the nonlinear PB equation. In contrast, the present model addresses counterions within a distance rtherm from the macroion surface. The neutralizing counterions are obtained by integration over the volume of a concentric sphere of radius rtherm. The ratio Zeff/Zp exhibits a minimum because the integrand in eq 16, to determine the fraction of tightly associated counterions, is a functional form that exhibits a maximum, i.e., x exp(-x). The presence of the minimum may thus
Charge of Spherical Colloidal Particles
Figure 6. Reduced thermal radius as a function of (Zp/ap)1/2: scheme 2. As shown above, the theoretical limit of rtherm ) 1 as (Zp/ap)1/2 f 0 is violated. This violation is attributed to the uncertainties in the solution of eq 20 in the low-charge regime. Hence, calculations of the ratio Zeff/Zp greater than unity in Figures 3 and 4 are artifacts.
reflect the failure of the screened Coulomb form of the pair interaction. Hence, the validity of the DLVO potential may not only be limited to κap < 3 but also to (Zp/ap)1/2 < 2. However, the minimum may also have physical significance. The following scenario may provide a physical reason for the minimum in Zeff/Zp as embedded in the mathematics. As the particles are charged, the electrostatic interaction may be said to be an unscreened Coulomb interaction. The newly created counterions interact strongly with the charged sphere. Increasing the surface charge Zp results in a decrease in the effective charge Zeff because a greater fraction of the counterions are within the distance rtherm. Because rcell is fixed by the volume fraction, a further increase in Zp gives rise to a further increase in the value of the screening parameter. The cumulative effect of this increase in counterions is to reduce the value of rtherm and the fraction of counterions that partially neutralize Zp. Hence, a smaller fraction of the counterions are associated with the kinetic unit. Thus, Zeff/Zp again increases but never attains the value that we associate with the titration charge Ztit. This interpretation is in line with Yamanaka et al.,14 who suggest that an increase in screening at the higher charge densities is responsible for the solid f liquid reentry phase transition as the surface charge density of their colloidal particles. Another parameter that is arbitrary in the calculations is the choice of the “boundary” radius. If we allow rcell to expand without limit, then we may determine the residual fraction of counterions that remain in the vicinity of the macroion. The corresponding function for fZ is obtained from eq 16 by setting exp(-rcell/rtherm) ) 0. The calculations, summarized in Figure 5, indicate a “residual” number of counterions remains within the domain of the colloidal particle. A “Limiting expression” for condensed counterions is well documented for infinitely long rods as in the Manning condensation model4 and for planar surfaces as shown by Enstro¨m and Wennerstro¨m5 and Zimm and Le Bret.15 One of the reviewers of this manuscript noted that in Figures 4 and 5 the ratio Zeff/Zp exceeds the value of unity as (|Zp|/ap)1/2 f 0. This is not due to eqs 13-17 but rather is an artifact of using an analytical expression of the screened Coulomb form for |Upc| in the calculation of rtherm. We show in Figure 6 the value of rtherm as a function of (|Zp|/ap)1/2. Clearly, the lower bound of rtherm ) ap is violated (14) Yamanaka, J.; Yoshida, H.; Koga, T.; Ise, N.; Hashimoto, T. Phys. Rev. Lett. 1998, 80, 5806. (15) Zimm, B. H.; Le Bret, M. J. Biomol. Struct. Dyn. 1983, 1, 461.
Langmuir, Vol. 16, No. 5, 2000 2121
Figure 7. Simulations of effective and actual surface charge densities: scheme 1. The surface charge densities were computed from the charges Zeff and Zp using the relationship s ) qe/4πa2pZ and expressed in units of microcoulombs per cubic centimeter. Mathematica was used for the linear and quadratic fits of the generated values of σeff in terms of the input values of σa, the “actual” value of the surface charge density. The volume fraction was φp ) 5 × 10-4. The solid line represents the generated values of the surface charge density. Linear fit: (- - -) ln(σeff) ) 0.844ln(σa) - 0.906.The quadratic fit lies below the generated values: (- - -) ln(σeff) ) 0.1490[ln(σa)]2 + .844ln(σa) - 0.906.
for (|Zp|/ap)1/2 < 2 regardless of the volume fraction. It is easily shown that if rtherm < 1, then eq 17 gives fZ > 1. We now compare our results with experiment. The data shown in Figure 4 are consistent with the light scattering data of Gisler et al.6 on PLS particles. For particles of an average radius ap ) 513 Å and a bare charge on the order of Zp ) 2084 (from their Table 2), they obtained from light scattering measurements the effective charge of Zeff ) 560. Thus, they experimentally determined charge fraction fZ = 0.27. Their PB calculations using the cell model as summarized in their Table 2 gives a predicted charge ratio Zeff/Zp ) 602/2084 = 0.29 for φp ) 10-3. Using the values Zp ) 2084, ap ) 513 Å, and φp ) 10-3, we calculate from eq 17 the value fZ = 0.37 for scheme 1 and fZ = 0.51 for scheme 2. Yamanaka et al.16 reported that the effective surface charge density obtained from conductivity measurements, σe, exhibited a very weak dependence on the volume fraction. Their empirical relationship between σe and the actual value of the surface charge density, σa, was viz.,16
ln(σe) ) 0.49ln(σa) - 1.0
(23)
We have determined the effective surface charge density σeff ) σafZ as a function of σa for ap ) 600 Å and σp ) 5 × 10-4, where -1 e ln(σa (µC/cm2)) e 1. A least-squares program was used to fit these data to both a linear and a quadratic equation in the logarithms, with the result for scheme 1
ln(σeff) )
{
0.844 ln(σa) - 0.906 0.149[ln(σa)]2 + 0.844 ln(σa) - 0.961
(24)
and for scheme 2
ln(σeff) )
{
0.917 ln(σa) - 0.637 0.081[ln(σa)]2 + 0.917 ln(σa) - 0.666
(25)
Within the accuracy of the numbers shown, the inclusion of the quadratic term does not significantly affect the intercept or the linear coefficient. The quadratic term does, however, affect the visual fit of the curve as shown in Figure 7 for scheme 1. Through personal correspondence (16) Yamanaka, J.; Hayashi, Y.; Ise, N.; Yamaguchi, T. Phys. Rev. E 1997, 55, 3028.
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with Professors Ise and Yamanaka, the errors in the experimental slope and intercept are deemed to be the least likely source of explanation of the differences between eq 23 and eqs 24 and 25. However, it was assumed that the equivalent conductivity of the ions was the same as that for the free ion. Such an assumption might introduce error for the high surface charge density measurements, and the prefactor, 0.49, might be larger (J. Yamanaka, personal communication). We believe, therefore, that the primary source of the discrepancy is the analytical expression for the screened Coulomb interaction. A comment should be made on representing these data on a double-logarithmic plot. Such a plot, of course, is necessary for a power law determination. Because the original data are scattered about the least-squares fit curve (cf. Figure 8 of Yamanaka et al.16), it is not useful to compare the above equations with any one particular set of data points. It is thus the trend in the data that is important in this analysis. The above calculations are based on the analytical expression for the screened Coulomb interaction between a counterion and a macroion and are therefore subject to the limitations of this linearized PB model. Within these limitations, the agreement between the expressions employed herein and the experimental data as discussed in the previous paragraph are quite reasonable. It is to be emphasized that we used the “bare” macroion charge in the above calculations rather than the macroion charge as an adjustable parameter as is usually done when interpreting data with a screened Coulomb potential. There are three important contributions to the structure of the ion cloud not considered in either the linear or nonlinear PB equation that should have a profound influence on the microion distribution, which have to do with the finite size of the microions. The first is the presence of a Stern layer. The Stern layer is defined as surface-associated counterions whose primary effect is to reduce the surface charge density. These counterions, by their very proximity to the surface, are tightly bound and not readily exchanged with the ions of the surrounding medium. In regard to Brownian dynamics simulations, or biased Monte Carlo calculations, the ions in the Stern layer may be identified as those ions that do not move far from the macroion surface. The structure of the Stern layer is determined by the microion-microion interactions as well as the macroion-microion interactions. A second contribution omitted in the PB equation is the correlations between the microions, either through their long-range electrostatic interaction or short-range finite size effects. These correlation contributions would be most significant in the vicinity of the macroion surface where the microion concentrations are considerably larger than the bulk concentration. Inclusion of these correlations should lead to a smaller microion concentration and thus a larger value of rtherm than that calculated from eq 12. The third omission is the cumulative electric potential of other macroions in the system. Lo¨wen et al.17 reported ab initio calculations that indicated that at high packing fractions of the macroion the concentration of counterions between two macroions was larger than anticipated from the DLVO theory. Such a rearrangement of counterions will affect the value of Zeff obtained from a macroion-macroion pair potential analysis. Our Monte Carlo simulations of a cluster of macroions indicates a relatively constant counterion concentration in the interior of the cluster.18 (17) Lo¨wen, H.; Madden, P. A.; Hansen, J.-P. Phys. Rev. Lett. 1992, 68, 1081. (18) Schmitz, K. S. Langmuir 1999, 15, 4093.
Schmitz
Figure 8. Brownian dynamics simulation of the counterion distribution function. The counterion distribution function gpc(r) was obtained by standard Brownian dynamics simulation methods.11 The charge on the macroion was Zp ) 30, and there was no added electrolyte. All distances were represented in these calculations in units of the macroion radius, ap ) 113 Å. The radius of the univalent counterion was ac ) 1 Å. The volume fraction was φp ) 0.01, or a radius of the spherical computation cell Rcell ) 524.5 Å (cf. eq 21). The distribution function gpc(r) was calculated after 1 × 107 moves for each of the counterions. The surface of the macroion is at r ) 1. The “+” in the distribution function is at gpc(r) ) exp(1), which defines the location of the thermal radius, which has the value Rtherm ) 186.7 Å. Clearly, the bound counterions are not limited to surface-bound counterions (Stern layer) but extend well over half of the macroion radius into the surrounding solution. Inclusion of these bound counterions results in the charge fraction fZ ) 0.76.
This suggests that the collective potential of the macroion cluster may not have a value for rtherm if the counterions are free to roam the interior of the cluster in a manner similar to conduction electrons in a metal. Since the submission of this manuscript, Professor L. B. Bhuiyan of the Laboratory of Theoretical Physics at the University of Puerto Rico has performed numerical calculations of rtherm from the nonlinear PB equation (personal communication). The value of rtherm was calculated from the distance at which gpc(r) ) exp(1), and the effective charge was obtained from integration of gpc(r) to rtherm. For the set of parameters φp ) 0.03, ap ) 113 Å, and Rcell ) 364 Å, he found that 0.453 > Zeff/Zp > 0.045 for 200 < Zp < 2500. Calculation of fZ from eq 17 is in reasonable agreement for the low values of the charge but fails for the higher charges. For example, for Zp ) 300, eq 17 gives fZ ) 0.346 for scheme 1 and fZ ) 0.527 for scheme 2, whereas Bhuiyan’s calculations give 0.327. The worst case is for Zp ) 2500, where eq 17 gives for scheme 1 fZ ) 0.520 and for scheme 2 fZ ) 0.645 as compared to the PB results of fZ ) 0.045. The characteristic parameters for the latter example are κcap ) 3.8 and (|Zp|/ap)1/2 ) 4.7. Hence, the latter PB calculations are beyond the validity of the screened Coulomb interaction in accordance with the calculations shown in Figure 2. However, the PB results are in good agreement with the data summarized by Roberts et al.19 in which Zeff/Zp is plotted as a function of (|Zp|/ap)1/2. We have also performed Brownian dynamics simulations using scheme 1 for calculating the counterion distribution function gpc(r), where r is a reduced distance in units of ap (Schmitz, K. S.; Sanghiran, V., unpublished results). The standard BD methods were employed.11 We report herein results for Zp ) 30, ap ) 113 Å, and a volume fraction φp ) 0.01. The distribution function gpc(r) obtained after 1 × 107 moves of each of the 30 counterions is shown in Figure 8. The effective charge was calculated by subtracting the average number of counterions within the distance rtherm and ap from the charge |Zp|. The fraction (19) Roberts, J. M.; O’Dea, J. J.; Osteryoung, J. G. Anal. Chem. 1998, 70, 3667.
Charge of Spherical Colloidal Particles
fZ thus obtained was 0.76. The value of fZ ) 0.82 was obtained for scheme 1 and fZ ) 0.85 for scheme 2. Both schemes 1 and 2 exhibit similar qualitative behavior in regard to the calculation of Zeff/Zp as a function of surface charge and volume fraction. Scheme 1 shows better quantitative agreement with the experimental data and computer simulations (PB and BD) than those of scheme 2. This is due to the fact that the analytical form of the screened Coulomb interaction used in this analysis is a far-field mean field result. Thus, the distribution of the counterions in the vicinity of the macroion is grossly underestimated by the Yukawa form. By having a small radius for the computation cell, scheme 1 thus “forces” the counterion distribution to be greater in the vicinity of the macroion than does scheme 2. In regard to comparison with experiment, the presence of other macroions results in a further increase in counterions between the macroions.17 By its very nature, the scheme 2 constriction does not permit inclusion of such direct multibodied microion effects. The usual procedure in applying charge renormalization and condensation theories is to reduce the actual surface charge with a prescribed number of counterions to an effective surface charge and then calculate the screening parameter using the remaining microions in the solution. In contrast, the proposed method is to use the actual surface charge, adjust its value by the number of microions within a distance Rtherm, and then calculate the screening parameter using all of the microions in the system. The implication of using all of the microions in the calculation of the screening parameter is that counterions may be exchanged between the two classes of counterions, bound and free. One reviewer noted that in a previous publication20 we stated that the DLVO potential, for macroion-macroion interactions, was not valid for κap < 1.3, whereas our conclusion drawn in the present study for macroionmicroion interactions was that the screened Coulomb potential was valid for κap < 3. This comment deserves special attention because the difference in the two classes of interparticle interactions as described by a screened Coulomb potential has not generally been appreciated in the past. We submit that these two conclusions are not inconsistent when examined within the context of the physical properties of the two systems. The inequality κap < 1.3 for the invalidation of the DLVO potential was a conclusion drawn by Matsuoka and co-workers21 (and (20) Schmitz. K. S. Phys. Chem., Chem. Phys. 1999, 1, 2109. (21) Harada, T.; Matsuoka, H.; Ikeda, T.; Yamaoka, H. Langmuir 1999, 15, 573.
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references stated therein) on the basis of ultra-small-angle X-ray scattering (USAXS) data. They reported that the intermacroion spacing decreased as the product κap likewise decreased for κap < 1.3. This behavior is in contradiction to the prediction of the DLVO theory that the intermacroion spacing should increase until the maximum separation is obtained. The USAXS data was thus interpreted in terms of extensive overlap of the ion clouds and an attraction between the macroions in the clusters due to sharing of the counterions.20 In this regard, the so-called “cell model” generally used in computer simulations is not appropriate because the intercell macroion-macroion and macroion-microion correlations are not taken into consideration. In the present study, the validity of the screened Coulomb macroion-microion potential for κap < 3 was determined in the absence of other macroions. Furthermore, the so-called “phase separation” observed in the USAXS studies generally is limited to colloids of high surface charge densities. Colloidal particles that do not exhibit such a separation may fall within the domain of the restrictions placed on the derivation of the DLVO potential. Under these conditions, the microions might likewise be assumed to be influenced by only one macroion, and thus the screened Coulomb macroion-microion interaction should be valid for κap < 3. 10. Conclusions It is concluded that an effective charge of a macroion can be theoretically constructed by including the net charge of counterions distributed over the range ap to Rtherm, in which the reduced energy is |βUpc| > 1. In this calculation, all of the microions are included in the calculation of the screening parameter. The mathematical form of the screened Coulomb pair potential (Yukawa form) has limited application due to the fact it is based on the linearized PB equation and is therefore subject to the approximations from which it was derived. In this regard, the linearized PB solution is valid for κap < 3. In addition, we conclude that the Yukawa form is limited to low surface charges ((|Zp|/ap)1/2 < 2) due to the inadequate representation of the counterion density as implied by the far-field screening parameter κ. Acknowledgment. I thank Professors Ise and Yamanaka for the discussions of their conductivity data. I am very grateful to Professor Bhuiyan for informing me of his nonlinear PB calculations prior to publication. I also acknowledge the constructive comments and suggestions of the reviewers of this manuscript. LA990522W
