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On the Grier-Crocker/Tata-Ise Controversy on the Macroion-Macroion Pair Potential in a Salt-Free Colloidal Suspension Kenneth S. Schmitz,† L. B. Bhuiyan,*,‡ and Arup K. Mukherjee‡ Department of Chemistry, University of Missouri - Kansas City, Kansas City, Missouri 64110, and Laboratory of Theoretical Physics, Department of Physics, University of Puerto Rico, Box 23343, San Juan, Puerto Rico 00931-3343 Received January 15, 2003. In Final Form: May 8, 2003 Several aspects of a recent controversy between the groups of Grier and Crocker [Crocker, J. C.; Grier, D. G. Phys. Rev. Lett. 1994, 73, 352; 1996, 77, 1897. Grier, D. G.; Crocker, J. C. Phys. Rev. E 2000, 61, 980.] and Tata and Ise [Tata, B. V. R.; Ise, N. Phys. Rev. E 1998, 58, 2237; 2000, 61, 983.] over experimental data vis-a`-vis the appropriateness of the classical Derjaguin-Landau-Verwey-Overbeek screened Coulomb pair potential versus the Sogami-Ise pair potential with an “attractive tail” in a charge-stabilized colloidal suspension are discussed. In particular, the relevant and sometimes contentious issue of the interrelationship between the effective colloid charge Zeff and the effective Debye-Hu¨ ckel screening parameter κeff is explored. Through a screened Coulomb analysis of Monte Carlo (MC) simulations of an isolated colloidal particle and its ion cloud, it is seen that the effective charge and the effective screening parameter remain essentially uncorrelated in practice. Examination of the underlying theoretical premises of the two model pair interactions indicates that they are based on mutually exclusive physical situations and thus experiments cannot be designed to favor one form over the other. The implications of the MC results on the applicability of the screened Coulomb forms of the pair interactions are also discussed.
Recently there has been a spirited debate between the groups of Grier and Crocker1 on one hand and Tata and Ise2,3 on the other regarding some experimental data that Grier and co-workers4-6 published as evidence in support of the well-known Derjaguin, Landau, Verwey, and Overbeek (DLVO) pair potential7 over an alternative pair potential of Sogami and Ise (SI).8 The DLVO pair interaction between two spherical macroions with center-tocenter distance r is
(
)
UDLVO(r) exp(κa) 2 exp(-κr) ) Z2 λ kBT 1 + κa B r
(1)
where λB ) e2/(kBT) is the Bjerrum length, |e| is the magnitude of the electron charge, is the permittivity of the solvent, kB is Boltzmann’s constant, T is the absolute temperature, Z|e| is the magnitude with sign of the macroion charge, and a is the radius of the macroion. The parameter κ is the inverse screening length and is defined by the number density of “free” microions in the solution, nf, which for monovalent ions is
κ2 )
4πe2nf ) 4πλBnf kBT
(2)
In contrast, the explicit form of the SI pair interaction is † ‡
University of Missouri - Kansas City. University of Puerto Rico.
(1) Grier, D. G.; Crocker, J. C. Phys. Rev. E 2000, 61, 980-982. (2) Tata, B. V. R.; Ise, N. Phys. Rev. E 1998, 58, 2237. (3) Tata, B. V. R.; Ise, N. Phys. Rev. E 2000, 61, 983-985. (4) Crocker, J. C.; Grier, D. G. Phys. Rev. Lett. 1994, 73, 352. (5) Crocker, J. C.; Grier, D. G. Phys. Rev. Lett. 1996, 77, 1897, 1900. (6) Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230. (7) Verwey, E. J.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: New York, 1948. (8) Sogami, I.; Ise, N. J. Chem. Phys. 1984, 81, 6320, 6330.
USI(r) ) kBT sinh(κa) Z2 (κa)
(
)( 2
1 + (κa) cosh(κa) -
)
κr exp(-κr) (3) λ 2 B r
Clearly the DLVO form is a purely repulsive potential at all separation distances between the two macroions whereas the SI potential is repulsive at close and intermediate separation distances and exhibits an attractive tail at larger separation distances. The DLVO screened Coulomb pair interaction approach has enjoyed classical status in colloid science for well over 50 years since its inception. Within the past two decades, the discovery of phenomena such as the like-ion interactions has prompted some questions on the range of applicability of the DLVO theory. The SI theory and the ensuing debate are arguably the best such examples. Although the DLVO theory was derived under restricted theoretical conditions, its current usage on real systems tends to ignore these constraints, and consequently, the concepts of effective charge, effective screening parameter, and even effective radius have been championed in order to maintain the intrinsic DLVO mathematical form. The interrelationship among these notions, however, does not seem to have been explored at great lengths in the literature. In this note, we will examine some aspects of this debate vis-a`-vis the effective quantities. The key features of the Grier and co-workers1,4-6 and Tata and Ise2,3 perspectives that are the focus of the present note are as follows: (a) the basic theoretical assumptions underlying the derivations of eqs 1 and 3 governing their applicability to practical experimental situations, (b) the interrelationship between Z and κ and their effective counterparts (Zeff and κeff) at both theoretical and experimental levels, and (c) the interpretation of experimental data given in refs 1 and 4-6 in the light of (b) above. For the purposes of this discussion, it is of value first to summarize the experimental setup and the data as
10.1021/la030010o CCC: $25.00 © 2003 American Chemical Society Published on Web 08/01/2003
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given in the Grier-Crocker paper1 and then to sketch briefly some of the more recent relevant works in the literature. In a suspension of mixed latex spheres of radii 0.325, 0.485, and 0.765 µm in distilled water, two spheres of interest, denoted j and i, were isolated by means of optical tweezers. All six pairs of the three latex sphere preparations were examined. The pair distribution function gji(r) was determined from the relative displacements of the two chosen spheres. The pair potential was thus determined as
Uji(r) ) -kBT ln(gji(r))
(4)
The potential Uji(r) was then analyzed in terms of eqs 1 and 3 where both charges Zj and Zi and also κ were allowed to be independently varied. The range of data for Uji(r) versus r reported in their Figure 1 extended to r ≈ 7 µm, or approximately 5 times the diameter of the largest latex sphere used in their study. It was concluded that the DLVO potential gave a more reliable fit to the data than did the SI potential. One of the important points in their assessment was that the values of κ obtained from the DLVO formulation were virtually the same, for example, κ-1 (nm) ) 272:268:289 with Z1 ) 6000/Z2 ) 13 800/Z3 ) 22 800 for the preparations of radius a1 ) 0.325/a2 ) 0.485/a3 ) 0.765 µm, respectively. Grier and Crocker1 surmise accurately that the DLVO and SI potentials were derived from the linearization of the Poisson-Boltzmann equation. However, they then make the sweeping conclusion that “...both theories are intended to describe the interactions between an isolated pair of charged colloidal spheres surrounded by pointlike simple ions.” In their counterstatement, Tata and Ise3 assert that the SI theory is not for isolated particles but that “...the Sogami theory is for multiparticle systems while the DLVO theory is for two-particle systems.” The fact that the DLVO potential is based on the absence of all colloidal spheres except the two of interest is quite transparent in the derivation given by Verwey-Overbeek.7 In contrast, an explicit separation of the potentials due to the colloidal particles and the counterions is given in the Sogami-Ise8 paper and is referred to as a shift in the potential. While many readers of their papers may consider this distinction between two isolated pairs and a multiparticle system to be one of semantics, studies by Larsen and Grier6 indicate that such a distinction is real. After inducing a “crystalline formation” of colloidal particles by an externally applied field, Larsen and Grier noted that a colloidal particle at the boundary between crystalline and free regions underwent several oscillations before being “released” to the solution phase. Such behavior is evidence for an effective pair potential with a minimum in the distance profile. If one may be justified in choosing the DLVO potential over the SI potential by visualization of the “quality of fit” to the data for two isolated spheres as did Grier and Crocker1 in reference to their Figure 1, then one must favor the SI potential as the potential of choice in the description of the stability of these “metastable” crystalline structures. This difference in behaviors of the isolated pair and cluster of colloidal particles may be explained in terms of the net forces on the microions for the two situations. Using the juxtaposition of potential fields (JPF) method, Schmitz9 concluded on the basis of vector gradient plots that there should only be repulsion between two isolated charged spheres, whereas attraction might occur for a cluster of spheres of like charge. (9) Schmitz, K. S. Phys. Chem. Chem. Phys. 1999, 1, 2109, 2117.
One of the criticisms of Tata and Ise3 was that κ should not be an independently varied parameter but must be related to the charge Z. They further state that independent variation of Z and κ is valid “...only if the counterion concentration npZ , ni.” where np is the macroion number density and ni is the number density of microion impurities. This inequality assumes both the validity of the theory and the accuracy of the data, and whether κ is considered to be a bulk quantity or a “local” quantity. In the context of the current experimental arrangement and theoretical expressions, the screening parameter as a bulk quantity is
κ2 ) 4πλB(n1 + n2 + n3 + ni)
(5)
where n1 ) |Z1|np1, n2 ) |Z2|np2, and n3 ) |Z3|np3 are the number densities of the released counterions for the colloidal particles of densities np1, np2, and np3. The number density ni accounts for the contribution of all other ions, including those that arise from the dissociation of the solvent. To check whether Z and κ can be varied independently would require independent knowledge of the densities np1, np2, np3, and ni. Unfortunately, this information is not available. However, information is given in Grier and Crocker1 that might allow one to estimate the validity of eq 5. Using the average value κ-1 ) 276 nm for their three data sets, we obtain a number density of ions of 1.49 × 1021 ions/m3. Crocker and Grier4 cite the value κ-1 ) 276 nm for “pure water”, or an ion number density of 1.23 × 1020 particles/m3. Assuming no added electrolyte, the resulting number of counterions released from the colloidal particles is nc = 1.37 × 1021. Since neither the absolute nor the relative concentrations of the three latex preparations are known, we employ that largest charge to estimate the maximum value of the average separation distance between the colloidal particles. The maximum value is (Z/nc)1/3 ) (22800/(1.37 × 1021))1/3 ) 2.55 × 10-6 m, or 2.55 µm. This maximum average separation distance is not consistent with the range of distances given in Figure 1 of Grier and Crocker,1 where the data indicate no spatial correlation between two spheres up to a distance of 7 µm. In other words, if there is to be a “free volume” of two isolated colloidal particles that spans up to 7 µm as shown by their data, then there must be some “compensating region” in the suspension in which the collection of particles has an average separation distance much less than that calculated from the bulk concentration. We would like to reiterate that the aim of this exercise has not been to favor one side over the other, viz., DLVO versus SI. We have merely tried to analyze the GrierCrocker experimental data from a different, perhaps a nontraditional, perspective, namely, in terms of the effective quantities κeff and Zeff (and the relationship between them), which are acknowledged to be the principal ingredients of the screened Coulomb formulation. Given these two forms as the only choices, the DLVO form appears to visually fit the data for the isolated pair of colloidal particles whereas the metastability of crystalline structures and the oscillatory behavior of an “interface particle” requires a potential form with a minimum in its distance profile. A natural question that now arises concerns the validity of the screened Coulomb form of the pair interaction. As noted by Fushiki,10 the “effective charge” Zeff and “effective screening parameter” κeff are related by an expression similar to that of eq 5, where we have included ionic impurity contributions. Failure to adhere to this relation(10) Fushiki, M. J. Chem. Phys. 1992, 97, 6700, 6713.
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Figure 1. (A) -ln(gc(r) versus r/a for floating values of Zeff and κeff. Monte Carlo simulations of gc(r) for Z ) 150, a ) 11.3 nm, and φp ) 0.01 are represented by the discrete points and the nonlinear least-squares fit curve by the solid line, where the parameters were allowed to “free float”. The best fit parameters in the above figure are A ) 0.777, Zeff ) 127, and κeff-1 ) 6.18 × 107 m-1, which gives a value Zκ ) 256. For emphasis, the least-squares line extends beyond the range of MC data. (B) -ln(gc(r) versus r/a for the constraint Zeff ) Zκ. Monte Carlo simulations of gc(r) for Z ) 150, a ) 11.3 nm, and φp ) 0.01 are represented by the discrete points and the nonlinear least-squares fit curve by the solid line, with the restriction and Zeff ) Zκ. The best fit parameters in the above figure are A ) 1.002, κeff ) 4.015 × 107 m-1, and Zeff ) Zκ ) 108. For emphasis, the least-squares line extends beyond the range of MC data.
ship would bring into question the validity of any theory or set of data, or both. Recall that the DLVO theory is based on a small perturbation of the ion cloud about one macroion when in the presence of a second macroion, where the screening parameter for the isolated macroions is the same that appears in the DLVO potential. It is therefore reasonable to test the DLVO theory indirectly by examining a system in which all parameters are known, viz., the relationship between Zeff and κeff for the distribution of counterions about an isolated macroion, gc(r). To this end, we have employed Monte Carlo (MC) simulations for a “salt-free” system using the Wigner-Seitz spherical cell model, where the precise charge and the number of counterions are known. The input parameters for monovalent counterions for the set of parameters {charge Z, radius a, volume fraction φp} are {150, 11.3 nm, 0.01} (set 1) and {1000, 11.3 nm, 0.03} (set 2). The counterion distribution function gc(r) was fitted to an exponential function of the form
(
gc(r) ) exp(A) exp -
)
Ueff(r) kBT
(6)
where the parameter A accounts for the baseline of gc(r) in the limit Ueff(rf∞) f 0, and the effective pairwise reduced interaction potential is
(
)
exp(κeffa) exp(-κeffr) Ueff(r) ) ZeffλB kBT 1 + κeffa r
(7)
where the charge of the counterion is taken to be unity. Associated with κeff is another charge Zeff defined by
κeff2 ) 4πλB|Zκ|np )
3λB|Zκ|φp a3
(8)
Shown in Figure 1 are plots of -ln(gc(r)) versus r/a with the least-squares fit to eqs 6 and 7 for parameter set 1.
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In Figure 1A, both Zeff and κeff were allowed to independently “float”, whereas in Figure 1B the constraint was that the values of the two charges be the same, that is, Zeff ) Zκ. The “effective charge” parameters for the leastsquares curve in Figure 1A are Zeff ) 127 and Zκ ) 256, whereas in Figure 1B these values are Zeff ) Zκ ) 108. Visually, the least-squares fit in Figure 1A is better than that in Figure 1B, but at the expense of any correlation between Zκ and Zeff. Similar visual characteristics in the result for the graphics of set 2 were also seen, where the best fit values are Zeff ) 317 and Zκ ) 625 when both parameters are allowed to float. Since the number and identity of the particles in the MC simulations are known, there cannot be any additional source of ions or charge to account for the failure of Zκ and Zeff to be in agreement in the “floating values” analysis. For both sets of parameters, Zeff/Z < 1. There is a “physical” interpretation for the inequality Zeff/Z < 1, namely, that the “real” charge of the colloidal particle is screened by the counterions. However, the direction of the inequality of the ratio Zκ/Z relative to unity differs for the two data sets. One might use the interpretation of a “screened charge” for Zκ/Z < 1 obtained for set 2, but there is no physical reason for Zκ/Z > 1 for set 1. That is, for Zκ/Z > 1 that might be obtained from real experimental data one might conclude there are “impurities” in the system as illustrated in eq 5. However, there are no such impurities in the simulations since the number of counterions is fixed by electroneutrality conditions. The results would indicate that within the framework of the screened Coulomb approach the effective charge and screening parameters may not be truly independent in a salt-free suspension with a finite volume fraction. Since both the charge and the screening parameter are equally important parameters in the theory, one must provide a justification of choosing one parameter over the other in the assignment of the “true” effective charge of the macroion. In summary, it would appear that the DLVO and SI potentials relate to two different physical situations and that there are subtle differences in the motivation behind the two approaches. The apparent motivation behind the DLVO potential was to provide an explanation for the stability of colloidal systems. The assumptions underlying the DLVO potential are that the interactions between only two colloidal particles are operative and there is an excess of added electrolyte. Indeed, this view is underlined in the work by Verwey and Overbeek,7 in which they insist that κ must be measured far from the macroions so that there is no contribution of the counterions as they are “localized” in the vicinity of the macroion, hence the concept of a “double layer”. In this regard, their definition of κ is “localized” and their model cannot allow extensive overlap of the double layers of neighboring colloidal particles. In contrast, the motivation behind the SI potential was to
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explain the instability of colloidal systems as evidenced by a “two-state” structure. In this regard, the SI theory is based on the Gibbs free energy of interaction between a collection of colloidal particles and the role of the counterions in establishing the dense-sparse regions. Since the concept of a phase separation has no meaning when only two particles are present, the form of the SI potential likewise has no meaning when only two particles are present. We therefore conclude that studies on isolated pairs of colloidal particles cannot invalidate effective pair potentials that may exhibit an attraction component whose origin is multibodied interactions. We would also argue that there is a fundamental flaw in making inferences about the colloidal charge based on the value of κ obtained from least-squares fits of experimental data to the screened Coulomb pair potential. In our MC simulations, Z and nc are fixed by electrical neutrality conditions. If both Zeff and κeff are allowed to “free float” in the least-squares characterization of gc(r), then one obtains the inequalities Zκ/Z > 1 and Zeff/Z < 1 for set 1, that is, low macroion charge. While the current paradigm of “charge renormalization” may give some reasonable physical significance of the latter inequality, there is no corresponding physical basis to account for the former inequality. If these inequalities likewise obtain for the DLVO potential analysis of real data, then the ionic strength of 1.2 µM estimated by Grier and Crocker1 from their value of κeff may be a gross overestimate of the “true” microion concentration. With this as a possibility, then our reanalysis of the average interparticle spacing of 2.55 µm may likewise be a gross underestimate, and hence there would be no need to postulate a coexisting region of higher colloidal particle density outside of their field of vision in the experiment. Within the context of the mathematical significance of κeff, and hence Zκ, we simply note that it is a measure of the rate of decay of the influence of the screened Coulomb interaction with separation distance between the two particles. The rate of decay is faster than anticipated on the basis of the value of Zeff because Zκ/Zeff > 1. The existing data, coupled with our MC simulations, are suggestive of the scenario that in a charge-stabilized colloidal system with a high macroion charge, these effective quantities remain essentially uncorrelated. Under such circumstances, we feel it is difficult to make any definitive statement about the validity of a theory based on the relative behavior of only one of the effective parameters. Acknowledgment. The partial support of National Science Foundation Grant 0137273 is acknowledged. L.B.B. also acknowledges an internal, institutional grant through FIPI, University of Puerto Rico. LA030010O
