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© Copyright 1996 American Chemical Society
MARCH 20, 1996 VOLUME 12, NUMBER 6
Letters Effect of Finite Ion Size on the “Overbeek Correction” to the Sogami-Ise Interaction Potential between Macroions of Like Charge Kenneth S. Schmitz Department of Chemistry, University of MissourisKansas City, Kansas City, Missouri 64110 Received September 16, 1995. In Final Form: January 19, 1996X The Sogami-Ise (SI) interaction potential has a “long range attraction” tail that is not present in the DLVO potential. Overbeek suggested that an “error” was present in the SI theory and that the inclusion of the solvent contribution eliminated the source of this long range attraction. The model used by Overbeek assumed that the macroions, by virtue of being fixed in their equilibrium positions, did not contribute to the electrical free energy of the solution. It is shown herein that the “Overbeek correction” is not rigorously zero for arbitrary polyion concentrations, as it must be in order to eliminate the Gibbs minimum for actual experimental conditions. It is shown herein on the basis of standard statistical thermodynamics expressions that the presence of a minimum in the pairwise Gibbs electrical free energy occurs for all screened Coulombic expressions for the Helmholtz electrical free energy, and therefore the correction term of Overbeek is in error. It is concluded that the concepts of screened Coulombic interactions and the equivalence of the pairwise Gibbs and Helmholtz electrical free energies are incompatible: one concept must be discarded if the other is to be retained.
A little over a decade ago Sogami and Ise1 proposed a model in which the potential of interaction between two macroions of like charge exhibited a minimum at a separation distance of a few thousand angstroms. The Sogami-Ise model (hereafter referred to as the SI model) was developed on the assumption that the distributions of counterions and macroions were mutually dependent, viz., the “counterion cloud” functioned for macroions in a manner similar to the “electron cloud” in molecules in the formation of chemical bonds. In this development Sogami and Ise (SI) obtained the following relationship between the electrical contribution to the Gibbs free energy (which we denote as Gelec SI ) and the electrical Helmholtz free energy (which we denote as Aelec SI )
Gelec SI
[
]
∂ ) 2+κ Aelec SI ∂κ2 2
Aelec SI )
2 (qeff m ) exp(-κr) r
(2)
where the effective charge macroion of radious am is qeff m ) Zjqe sinh(amκ)/amκ and the factor sinh(amκ)/amκ results from the assumption of a uniform distribution of surface charge on the macroion, and we have assumed a system of identical macroions separated by a distance of r. Substitution of eq 2 into eq 1 gives the SI result
[
Gelec SI ) 1 + amκ coth(amκ) -
]
κr elec A 2 SI
(3)
(1)
where κ is the Debye-Hu¨ckel screening parameter and X
the Helmholtz potential is of the form
Overbeek2 has suggested that an “error” in the SI theory was the omission of the “solvent contribution” and, once corrected, eliminates the long range minimum in Gelec SI .
Abstract published in Advance ACS Abstracts, March 1, 1996.
(1) Sogami, I. S.; Ise, N. J. Chem. Phys. 1984, 81, 6320.
0743-7463/96/2412-1407$12.00/0
(2) Overbeek, J. Th. G. J. Chem. Phys. 1987, 87, 440.
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The “Overbeek correction” is easily obtained if we write κ in terms of the solvent molar volume vj s as the simple ratio
κ2 ) 4πλB
∑j
Z2j
nj V
) 4πλB
∑j
nj
Z2j
nsvs
(4)
where λB ) q2e /kBT is the Bjerrum length, qe is the magnitude of the electron charge, Zj is the magnitude with sign of the jth electrolyte, nj is the number of moles of the jth species, kB is the Boltzmann constant, T is the absolute temperature, and V is the total volume of the solution or suspension. The Overbeek correction thus becomes
( ) ( )( )
( )
∂Aelec ∂Aelec ∂κ2 ∂Aelec ) ns ) -κ2 nsµs ) ns 2 ∂ns ∂ns ∂κ ∂κ2
(5)
Quite clearly inclusion of eq 5 in the operator of eq 1 offsets the second term in brackets and thereby the minimum in eq 3. For a system with one type of counterion ion, denoted by the subscript i, the “Overbeek correction” in terms of the integrated form of the Gibbs-Duhem expression gives
+ niµelec ) ns nsµelec s i
( ) ( ) ∂Aelec ∂Aelec + ni ∂ns ∂ni
( ) ( )
2
) -κ
∂Aelec ∂κ2
2
+κ
∂Aelec ∂κ2
(6)
the inclusion of eq 5 in the DLVO theory as given by Verwey and Overbeek7 also eliminates the electrical interaction of the Helmholtz potential, i.e., the standard DLVO potential. In order for the Overbeek theory to be correct, the minimum in the Gibbs electrical free energy must be a general result and not dependent upon theoretical constraints that cannot be met by experimental conditions. What is overlooked in both the Sogami-Ise1 and Overbeek2 papers is the effect of the finite size of the particles in the calculation of the ionic strength and, thus, κ2. In the case of finite ions the value of the ionic strength should be based on the generalized expression
κ2 )
∑j κ2j ) 4πλB∑j Z2j V )
where the superscript “elec” denotes that only the electrical part to the chemical potential is considered. To quote Overbeek,2 “This remarkable result shows that the solvent and the small ions together just give a zero contribution to Gel (and incidently also to Fel).” That inclusion of the “Overbeek correction” term in the Sogami-Ise theory eliminates the minimum in that theory is a position held by a portion of the scientific community.3-6 There is a problem, however, if the “Overbeek correction” is applied to the “standard DLVO” theory7 (DerjaguinLandau-Verwey-Overbeek). Verwey and Overbeek7 first point out that the chemical and electrical work on the surface of the macroion cancel each other, and thus obtain “The very simple result is that we find the total free energy of the double layer if only we calculate the electrical work necessary to discharge stepwise all ions of the solution.” with the further simplification is that,7 “...we need only to consider those ions which are present in excess in the solution and carry the liquid charge of the double layer, since for the mutual neutralization of all other ions, present in equivalent amounts in each point in space, the net work is zero.” Within the context of the DLVO model, the introduction of the “solvent correction term” eliminates all electrical interactions in the system since the chemical work on the surface of the macroion eliminates its contribution and the paired ions of opposite charge eliminate each others contributions. Smalley8 was the first to point out that (3) Weill, G. J. Phys. (Paris) 1988, 49, 1049. (4) Okubo, T. J. Chem. Phys. 1988, 88, 6581. (5) Badirkhan, Z.; Tosi, M. P. Phys. Chem. Liq. 1990, 21, 177. (6) Ferrari, M. E.; Bloomfield, V. A. Macromolecules 1992, 25, 5266. (7) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier Publishing Co., Inc.: New York, 1948. (8) Smalley, M. V. Mol. Phys. 1990, 71, 1251.
nsvj s + nmvj m +
∑i
(7) nivj i
where the subscript “m” refers to the macroion and the two summations are over all types of the electrolyte ions present in solution. Since both the Sogami-Ise and Overbeek papers give the charge neutralization condition in terms of the total volume, eq 7 must therefore apply to both models. Using eq 7 rather than eq 4 to define the screening parameter gives for the Overbeek model,
+ nsµelec s )0
∑j Z2j nj
4πλB
nj
( ) ( ) ∑ ( ) ( ) ∑
) ns ∑j njµelec j
∂Aelec
+
∑j nj
∂Aelec
∂ns ∂nj ∂Aelec ) [-κ2φs + (κ2 - κ2 2 ∂κ ∂Aelec ) κ2 [1 - φs - φj] j ∂κ2 ∂A elec ) κ2 φm ∂κ2
φj)]
j
( )
(8)
where κ2j ) 4πλBnj/V is the screening length contribution of the jth species and φk is the volume fraction of the kth species that make up the total volume of the solution or suspension. Quite clearly eq 8 is equal to zero only in the limit of zero concentration of the macroions, where of course no pairwise interaction occurs due to the absence of particles. Hence the “Overbeek correction” is rigorous only in the limit of no macroions. Therefore eq 6 does not constitute a proof that there is no minimum in the Gibbs free energy for finite corrections of real macromolecular systems. Let us now look at a system in which the macroionic species is included in the integrated form of the GibbsDuhem expression, viz.
+ nmµelec nsµelec s m +
( ) ( ) ( ) ∑ ( )
) ns ∑j njµelec j ) κ2 )0
∂Aelec
∂ns ∂Aelec ∂κ2
+ nm
∂Aelec
+
∂nm
∑j nj
[1 - φs - φm -
∂Aelec ∂nj
φj]
j
(9)
There are two points to be made from the results of eq 9. The apparent motivation of Overbeek2 was to introduce a term into the electrical free energy expression that could eliminate the source of the “long range attractive” part of the Sogami-Ise potential1 as outlined above. In doing so
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Langmuir, Vol. 12, No. 6, 1996 1409
Overbeek defined the macroions as being fixed in their equilibrium positions, thereby justifying their omission in the calculation of the electrical free energy and thus arriving at the “Overbeek correction” term. As indicated from eq 8 the desired effect of having the sum of free energies equal to zero only when no macroions are present, as eq 8 is not zero even if only two macroions are present in the system. In order to obtain the desired zero sum, however, the macroions must also be included in the free energy expression as shown by eq 9. Overbeek therefore erred in his attempt to obtain the desired result by omitting the macroions from the electrical free energy of the solution. Overbeek’s omission of the macroions in the Gibbs-Duhem expression was previously criticized in a footnote by Ise and co-workers.9 The second point to be made is that the Gibbs-Duhem expression was not correctly applied to this problem. Overbeek’s simplified method of obtaining his desired result is his eq 15, viz., in present notation
nsµelec + s
) ∫e)0(ns dµelec + ∑ni dµelec ∑i niµelec i s i )P,T i e
(10)
where the integral is stated to be the Debye-Hu¨ckel type of charging process. Overbeek then applied the GibbsDuhem relationship to the integrand to set the sum of electrical contributions to the free energy equal to zero. If this is the case, then the solvent itself is being charged from the integration of the charging parameter e ) 0 to e ) e. The correct interpretation of the Gibbs-Duhem relationship is that it gives restrictions on the change of the chemical potential on one component in terms of all of the other components in the system.10,11 If indeed Overbeek was correct in his application of the GibbsDuhem expression that terms are canceled, then the correction to the SI potential comes at the expense of having an isolated macroion phase with a net charge while the solution itself is electrically neutral! This absurd result does not obtain when the macroions are included in the free energy expression as indicated by eq 9 along with a correct interpretation of the Gibbs-Duhem expression. The Gibbs-Duhem expression is rearranged to read elec nm dµelec ∑ m + ∑ni dµi ) m i
) -( ns dµelec s
(11)
where more than one macroion species is now allowed. Thus, the change in the electrical component of the chemical potential of the solvent is related to the changes in the electrical component of the chemical potential of the solute ionic species by the relationship in eq 11. This does not mean, therefore, that the presence of the solvent cancels the electrical free energy of the solute species. For example, the solvent particles may become polarized due to the presence of the charged solute species and therefore alter the physical properties of the solvent. Let us examine the effect of eq 9 on the Sogami-Ise theory. Their starting expression relating Gelec and SI Aelec SI is (9) Ise, N.; Matsuoka, H.; Ito, K.; Yoshida, H.; Yamanaka, J. Langmuir 1990, 6, 296. (10) Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry; John Wiley & Sons: New York, 1980; p 697. (11) Atkins, P. W. Physical Chemistry; W. H. Freeman and Company: New York, 1990; p 158.
Gelec SI
)
( )
∑j nj
∂Aelec SI ∂nj
+
V,T,Z
( )
ZM ∑ M
∂Aelec SI ∂ZM
(12) V,T,Z
where the first summation, the “source term” that gives κ2(∂/∂κ2) in eq 1, is restricted to only the electrolyte ions, whereas the second summation involves only the macroions. We refer to this as the “molecular” approach since molecular species are identified in the expressions. The SI theory made the assumption κ2j ∝ nj/V whereas the Overbeek theory made the assumption κ2j ∝ nj/ns. Clearly if κ2 is defined by eq 7, then from eq 9 the “source term” in the SI theory is rigorously zero. However, it has been demonstrated that if eq 12 is applied in a symmetric manner in which both of the summations are over the electrolyte ions and macroions, then the κ2(∂/∂κ2) contribution comes from the second summation, i.e., over the charge derivatives. This derivation is presented in a more expanded paper with numerical results.12 However, the correct operator relating the two free energies can be obtained in a simplified manner using standard statistical thermodynamic relationships that do not depend upon specific molecular assumptions. Let us assume that these electrical free energies are a function of the screening parameter κ, viz., Gelec(κ) and Aelec(κ). The thermodynamic relationship between these two quantities is therefore
Gelec(κ) ) Aelec(κ) + PV
(
(
) Aelec(κ) + κ2
∂κ2 elec κ ∂A (κ) ) Aelec(κ) + 2 ∂κ
(
)
∂ ln(Q) ∂V ∂Aelec(κ)
) Aelec(κ) - VkBT
)
)
(13)
where the standard statistical thermodynamic relationship between the pressure P and the canonical ensemble partition function Q has been employed in going between the first two expressions on the right-hand side of eq 13 and the substitution ∂/∂V ) (∂κ2/∂V)(∂/∂κ2) with κ2 defined by eq 4 in going between the middle two expressions. We therefore draw the following conclusion from eq 13: If the electrical interactions between ionic species in solution or suspension are a function of κ, then the electrical parts of the pairwise Gibbs and Helmholtz free energies are not equal. In other words, the concept of electrical screening is incompatible with the concept of the equivalence of the Gibbs and Helmholtz electrical free energies. As a consequence any theory that has a screened Coulombic form for the Helmholtz free energy must exhibit a minimum in the Gibbs free energy, even the DLVO potential! The screening parameter therefore introduces an “internal pressure” in the macroionic system that is not considered in the Helmholtz free energy. One of the reviewers pointed out the papers by Crocker and Grier13 and by Vondermassen, Bongers, Mueller, and Vermold14 indicating an experimental “proof” that the DLVO theory accurately describes the interaction between pairs of macroions. These experiments were done under very low ionic strength conditions, less than 10-5 M, and the results were said to be in agreement with the DLVO potential. In order to distinguish between the various models for interparticle interactions, one must assure the (12) Schmitz, K. S. Submitted for publication. (13) Crocker, J. C.; Grier, D. G. Phys. Rev. Lett. 1994, 73, 352. (14) Vondermassen, K.; Bongers, J.; Versmold, H. Langmuir 1994, 10, 1351.
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following two conditions prevail: (1) the theoretical expression give predictions that are outside the experimental error range; and (2) that the average interparticle spacing is much greater than the location of the minimum in the Gibbs free energy for a particular model. Let us consider the study by Crocker and Grier.13 In their Figure 2 the “best fit” DLVO calculation gives κ-1 ) 161 nm for polystyrene sulfate spheres for the stated radius in the text of am ) 32 nm. Hence κam ) 0.1987 for this system. In regard to the “corrected” Sogami-Ise Gibbs free energy, one has from comparison of eqs 1 and 13 the expression
[
Gelec(κ) ) amκ coth(amκ) -
]
κr elec A (κ) 2
(14)
with a “cross-over” reduced distance of x0 ) 2 coth(κam) ) 2 coth(0.1987) ) 10.2. The data given in their Figure 2 are for values of x < 9. In other words, these data are in the range where the Sogami-Ise theory predicts a repulsion between the macroions, thus these authors have not unequivocally demonstrated that the interaction energy does not have a long range attractive tail. It is to be emphasized that the focus of this communication is not whether the SI theory is valid or not, but rather on the error in the “Overbeek correction” term with its implications to theories of macroion solutions and suspensions. As shown above, the “minimum” in the electrical component of the Gibbs free energy that the Overbeek correction was designed to eliminate is a natural result of basic statistical thermodynamic expressions for free energies that are a function of κ. One therefore has two options: (1) screened Coulombic interactions exist and therefore the electrical component of the Gibbs and
Helmholtz free energies are not equivalent; or (2) the electrical components of the Gibbs and Helmholtz free energies are equivalent but the concept of a screened interaction is in error. The concepts of Gelec(κ) ) Aelec(κ) and a screened Coulombic interaction are mutually exclusive! It is also concluded from eq 13 that Gelec(κ) ≈ Aelec(κ) for κ f 0. Since all theories in the literature give virtually the same expression for Aelec(κ) for κam , 1, one cannot distinguish between the various models by carrying out experiments for very dilute solutions and suspensions under salt-free conditions. This is because the “crossover” from repulsive to attractive interactions increases in interparticle separation distance as the ionic strength is lowered. Acknowledgment. I am grateful to reviewer 1 for pointing out an algebraic error in one of my equations which, when corrected, led to the more satisfying result given by eq 9. It was pointed out by reviewer 2 that the product nmµelec m must be considered in regard to whether it can be neglected in the infinite dilution limit. Even though nm, hence φm, may be quite small, the term µelec m may be quite large for highly charged macroions. The present paper does not address this problem, but it is certainly something that must be considered in view of the conclusions above that the macroion must be included when addressing the question of the electrical contributions to the free energy. Finally, I wish to give special acknowledgment to the spirited discussions with Julia P. Fisher in the preparation of this manuscript. LA950781V
