A Many-Bodied Interpretation of the Attraction between Macroions of

Langmuir 1997, 13, 5849-5863

5849

A Many-Bodied Interpretation of the Attraction between Macroions of Like Charge: Juxtaposition of Potential Fields Kenneth S. Schmitz Department of Chemistry, University of MissourisKansas City, Kansas City, Missouri 64110 Received May 8, 1997. In Final Form: August 6, 1997X There has existed since the 1930s a dichotomy among practitioners in the field regarding the electrostatic interactions in colloidal suspensions and polyelectrolyte solutions, viz., whether the electrical free energy is attractive or repulsive. The well-known Derjaguin-Landau-Verwey-Overbeek (DLVO) theory views the electrostatic interactions between a pair of charged macroparticles as a repulsive interaction, whereas the many-bodied “salt-crystal” structure of colloidal suspensions of Langmuir gives rise to a net attractive electrostatic interaction. Experimental data likewise are not definitive regarding these two apparently opposite points of view on the role of electrostatic interactions in these complex systems. For example, recent digital video studies by Grier and co-workers on polystyrene latex spheres (PLS) indicate that the pairwise interaction potential is repulsive and of the Yukawa form, whereas a collection of PLS particles has a long-range attractive component. A method is described herein based on a chemical model and the juxtaposition of potential fields (JPF) for multibody interaction systems. The JPF method is based on a topological procedure for identifying chemical bonding in molecules by vector gradient mapping. Vector gradient mapping methods applied to the JPF method indicate the importance of the distribution of counterions in determining the stability of a system of highly charged spheres. It is this distribution of the counterions in response to the potential fields of the macroions that provides a “crossover” from a repulsive interaction between two isolated spheres (unstable structure) to an attractive interaction for a group of spheres (stable cluster). Salient features of the JPF approach to charged colloidal systems are that it (1) does not rely on a particular form of the interaction potential, (2) identifies the shortcoming of the pair potential in the describing many-body effects, and (3) provides three mechanisms for the stability of clusters of macroions of like charge.

1. Introduction The assumption that the stability of a system of highly charged particles is the competition between a repulsive electrostatic interaction and an attractive van der Waals interaction has a long history. In 1938 Langmuir challenged this assumption when he addressed the role of attraction and repulsion in the formation of tactoids, thixotropic gels, protein crystals, and coacervates.1 In regard to the unipolar coacervation, Langmuir criticized the assumption of Kallmann and Willsta¨tter2 and the subsequent embracement by Freundlich3 and Hamaker4-6 that van der Waals attraction was of sufficient range to provide the necessary attraction for the observed stability. Also criticized was the use of energy diagrams in which the Debye-Hu¨ckel electrostatic repulsion and van der Waals attraction are superimposed and plotted as a function of the interparticle separation distance.6,7 Langmuir noted three objections to the energy diagram approach: (1) omission of thermal kinetic energy of the colloidal particles that would disperse the particles throughout the liquid; (2) the long-range electrostatic attraction between the charged micelle and the ion clouds of other micelles, which exceeded the repulsion between the micelles; and (3) the assumption of constant charge on the micelles, which should vary with the concentration of the micelles. As an alternative model, Langmuir assumed that the counterions lie between the micelles in a saltlike structure. In this configuration the electrostatic interaction of the collection of particles, both the micelles and simple elecX

Abstract published in Advance ACS Abstracts, October 1, 1997.

(1) Langmuir, I. J. Chem. Phys. 1938, 6, 873. (2) Killmann, H.; Willsta¨tter, M. Naturwissenschatten 1932, 20, 952. (3) Freundlich, H. Thixotropy; (Paris; 1935. (4) Hamaker, H. C. Physica 1937, 4, 1058. (5) Hamaker, H. C. Recl. Trav. Chim. Pay-Bas 1937, 56, 727. (6) Hamaker, H. C. Recl. Trav. Chim. Pay-Bas 1937, 56, 1.

S0743-7463(97)00479-4 CCC: $14.00

trolyte, was a net attraction. The repulsive interaction was due to the osmotic pressure buildup by the interspersed electrolyte ions. The model by Langmuir apparently was not attractive enough for colloidal scientists over the past half century. Instead, the interaction between highly charged macroions in solution and suspension was generally interpreted in terms of the DLVO (Derjaguin-Landau-Verwey-Overbeek) theory for pairwise interactions.8,9 Nonetheless there were observations that inferred long-range attractive interactions which could not be explained in terms of the DLVO model, which led Sogami and Ise10 to propose another form of the pairwise potential based on the Gibbs free energy (referred to as the SI model). The potential in the SI model exhibited a long-range attractive electrostatic tail and thus a stable minimum under certain ionic strength conditions. Over the past decade there has been copious amounts of work directed to the virtues and shortcomings of both the DLVO and SI models, of which the present author has also contributed. Some of these studies have an implicit challenge to some of the basic explicit precepts of the pairwise interaction approach. For example, Crocker and Grier11 reported that the pairwise interaction between two isolated spheres was repulsive, whereas Larsen and Grier12,13 reported the formation of metastable colloidal structures for a collection of 10 or more particles. The hidden implication of these studies was that the pairwise superposition principle cannot be used to determine the physical properties of a system. (7) Houwink, R.; Burgers, W. G. Elasticity, Plasticity and Structure of Matter, Cambridge University Press: Cambridge, England, 1937; 338-343. (8) Derjaguin, B. V.; Landau, L. Acta Physiochim. 1941, 14, 633. (9) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; (Elsevier Publishing Co., Inc.: New York, 1948. (10) Sogami, I.; Ise, N. J. Chem. Phys. 1984, 81, 6320. (11) Crocker, J. C.; Grier, D. G. Phys. Rev. Lett. 1996, 77, 1897. (12) Larsen, A. E.; Grier, D. G. Phys. Rev. Lett. 1996, 76, 3862. (13) Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230.

© 1997 American Chemical Society

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Schmitz

A vast majority of the efforts in the past 50 years focused on the determination of the pairwise potential, and the criticisms of Langmuir have been largely ignored. In the present communication a chemical model is proposed for multi-bodied system based on the juxtaposition of potential fields of the macroions (JPF model). It is shown that the distribution of electrolyte ions plays a major role in the determination of stability of the structure. In particular, a cluster of two particles (dimer) is unstable whereas a cluster of several particles (four in the present paper) leads to a stable structure. Another result is that the distribution of counterions in a multi-bodied cluster is the same as proposed by Langmuir, i.e., an accumulation of counterions within the interior of the cluster. 2. Structure of This Paper The purpose of this communication is not to solve this many-bodied problem but rather to explore the physics of a multibody cluster that may be stabilized by the shared electrolyte ions. The approach used herein is based on thermodynamics and the topology approach to quantum mechanical chemical bonding. The following section reviews the development of the DLVO and SI potentials, the experimental data of Grier and co-workers, and other experimental studies that indicate the inadequacy of the DLVO potential. The fourth section describes a multibodied interaction interpretation of data that are outside the domain of the pairwise interaction models. This is followed by a section that describes the system under consideration and the underlying thermodynamic approach based on a chemical model of adsorption of ions onto highly charges surfaces. Given in section 6 is a review of two experimental studies that distinguish between thermodynamically “bound” and “free” ions in colloidal systems, a distinction that is crucial to the present analysis. Because of the parallels of the present analysis to quantum mechanical chemical bonding, a gradient approach to the chemical bond problem in molecules is summarized in section 7. The juxtaposition of potential fields (JPF) method is described in section 8, applied to dimer and tetramer arrangement of charged spheres, and compared with computer simulations of a system of two interaction spheres, which is then followed by a general discussion in section 9. 3. Pairwise Interaction Potentials and Experimental Systems The DLVO Potential. The DLVO model is based on the solution to the linearized Poisson-Boltzmann (P-B) equation with the ad hoc addition of the van der Waals attraction term. The electrolyte ions are first charged in accordance with the Debye-Hu¨ckel process.14 The charge on the macroparticle is attained by adsorption of ions onto its surface. Under the assumption that the chemical work of adsorption identically cancels the electrical work of the adsorbed ions, the net free energy change in the system is due to the charging of the counterions. It is the presence of these counterions about the macroion that define the electric double layer. The form of the electrostatic interaction energy between the electric double layers surrounding two charged spheres, A and B, is a modified Yukawa potential, viz.,

UAB(r) ZAZBλB exp(-κ(aA + aB)) exp(-κr) ) kBT r (1 + κaA)(1 + κaB)

(1)

where kB is the Boltzmann constant, T is the absolute (14) Debye, P.; Hu¨ckel, E. Phys. Z. 1923, 24, 185.

temperature, ZA is the magnitude (with sign) of the charge for macroion A with radius aA,  is the bulk dielectric constant of the medium, λB ) qe2/kBT is the Bjerrum length, qe is the magnitude of the electron charge, and r is the distance of separation between the centers of the two spheres. The screening parameter κ in the DLVO model is calculated on the basis of the excess electrolyte concentration, i.e., added electrolyte, viz.

∑j zj2〈nj〉

κ2 ) 4πλB

(2)

where zj is the magnitude (with sign) of the charge of the j-species of electrolyte of bulk (average) number concentration 〈nj〉. The van der Waals term is

[

( )]

VA,A(x) AH 1 x2 - 1 1 + 2 ln + )kBT 12kBT x2 - 1 x2 x2

(3)

where AH is the Hamaker constant (which has a value in the range of 10-12 to 10-19 J) and the reduced distance x ) r/aA. Hence, the reduced DLVO interaction potential is

UDLVO(x) VA,A(x) UA,A (x) ) + kBT kBT kBT

(4)

In general the van der Waals term is significant only under high salt conditions because of the magnitude and long-range nature of the screened Coulombic term in eq 4. An example is the quasielastic light scattering (QELS) data of Corti and Degiorio15 on sodium dodecyl sulfate. As the salt is varied from 0.1 M to 0.5 M the slope of the concentration dependence of the mutual diffusion coefficient changes from positive to negative, which is consistent with the variation in the relative magnitudes of the two terms in eq 4. Polystyrene latex spheres (PLS) are model systems commonly used to study interparticle interactions and the various “states” of a fluid. The PLS particles can be prepared with uniform size, have a large scattering cross section for light scattering thus enabling one to go to very low concentrations, and are large enough to be seen in the microscope. The PLS particles spontaneously form “crystalline-like” structures under very low ionic strength conditions.16-25 In accordance with the DLVO theory, the long-range ordering is presumed to be due to the electrostatic repulsion between the PLS particles. For example, Rundquist et al.24 interpret that their observation of a decrease in interplanar spacing upon heating a dyed sulfonated PLS sample is due to the temperature dependence of κ in eq 1. The SI Potential. Over the past 25 years there have been reports that under certain conditions the interparticle spacing of the ordered structures was smaller than the average distance based on concentration.26-35 These observations were not within the domain of the DLVO (15) Corti, M.; Degiorgio, J. J. Chem. Phys. 1987, 86, 6616. (16) Clark, N. A.; Hurd, A. J.; Ackerson, B. J. Nature 1979, 281, 57. (17) Clark, N. A.; Ackerson, B. J.; Hurd, A. J. Phys. Rev. Lett. 1983, 50, 1459. (18) Ackerson, B. J.; Clark, N. A. Phys. Rev. A 1984, 30, 906. (19) Okubo, T. Acc. Chem. Res. 1988, 21, 281. (20) Okubo, T. J. Am. Chem. Soc. 1990, 112, 5420. (21) Okubo, T. J. Chem. Phys. 1991, 95, 3690. (22) Okubo, T. Naturwissenschaften 1992, 79, 317. (23) Wagner, N. J.; Krause, R.; Rennie, A. R.; D’Aguanno, B.; Goodwin, J. J. Chem. Phys. 1991, 95, 494. (24) Rundquist, P. A.; Jagannathan, S.; Kesavamoorthy, R.; Brnardic, C.; Xu, S.; Asher, S. A. J. Chem. Phys. 1991, 94, 711. (25) Reed, W. F. J. Chem. Phys. 1994, 100, 7825. (26) Bernal, J. D.; Fankuchen, I. J. Gen. Physiol. 1941, 25, 11.

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Langmuir, Vol. 13, No. 22, 1997 5851

theory. In an attempt to provide a quantitative explanation for these observations, Sogami and Ise10 developed a pairwise interaction potential based on the Gibbs free energy. In the SI model, the macroions were fixed in number and position, and the charging process was equated with the release of the counterions into the solution. The distributions of the macroions and the counterions were deemed mutually interdependent in the linearized form of the P-B equation, i.e.

(∇2 - κ2)Ψ(r) ) -

ν 4π N ( ZnqeFn(r) + 〈nio〉ziqe) (5)  n)1 i)1

∑

∑

where Fn(r) is the distribution of macroion charge, N is the number of macroions, and 〈nio〉 is the number concentration of counterions. After first shifting the potential, the P-B equation for the potential φ(r) for the small ions in the SI theory was

(∇2 - κ2)φ(r) ) -

4π

N

∑ ZnqeFn(r)  n)1

(6)

After a substitution of variables to the screening parameter was made, the resulting relationship between the Gibbs and Helmholtz free energies for identical spheres was given as

(

GSI(r) ) 2 + κ2

( ))

∂ ASI(r) ∂κ2

(

) 1 + y coth(y) -

κr A (r) 2 SI

)

(7)

where y ) κaA and ASI(r) is a screened Coulombic form similar to that in eq 1,

ASI(x) Zeff2λB exp(-x) ) kBT aA x

(8)

where the effective charge Zeff is

Zeff )

Z sinh(y) y

(9)

A feature of eq 7 not contained in the DLVO potential is the presence of a minimum at a location given by

RSI,min ) aA[(1 + y coth(y) + {[1 + y coth(y)][3 + y coth(y)]}1/2)/y] (10) The SI theory thus achieved what it was designed to do, namely provide some explanation for interparticle spacing that was less than the value based on a uniform distribution of macroions as inferred from the purely repulsive DLVO electrostatic interaction. (27) Kose, A.; Ozaki, M.; Takano, K.; Kobayashi, Y.; Hachisu, S. J. Colloid Interface Sci. 1973, 44, 330. (28) Plestil, J.; Mikes, J.; Dusek, K. Acta Polym. 1979, 30, 29. (29) Yoshiyama, T.; Sogami, I.; Ise, N. Phys. Rev. Lett. 1984, 53, 2153. (30) Ise, N.; Ito, K.; Okubo, T.; Dosho, S.; Sogami, I. J. Am. Chem. Soc. 1985, 107, 8074. (31) Ito, K.; Ise, N. J. Chem. Phys. 1987, 86, 6502. (32) Ito, K.; Okumura, H.; Yoshida, H.; Ueno, Y.; Ise, N. Phys. Rev. D 1988, 38, 10852. (33) Matsuoka, H.; Murai, H.; Ise, N. Phys. Rev. B 1988, 37, 1368. (34) Ise, N.; Matsuoka, H.; Ito, K.; Yoshida, H.; Yamanaka, J. Langmuir 1990, 6, 296. (35) Yoshida, H.; Ito, K.; Ise, N. J. Chem. Soc. Faraday Trans. 1991, 87, 371.

A Digital Video Microscopy Experimental Test of the DLVO and SI Potentials. There is much controversy in the field of colloidal science regarding the SI potential, and several papers have been written on the failure and successes of either the DLVO or SI forms of the pairwise interaction. Digital video microscopy (DVM) methods involve the least number of assumptions to interpret the data on the interactions between colloidal spheres. By videotaping the spheres through a microscope and calculating their probable distribution, viz., the radial distribution function g(r), the pairwise interparticle interaction energy U(r) is obtained through the relationship

U(r) ) -kBT ln(g(r))

(11)

Crocker and Grier11 recently reported digital video studies on the interaction between two isolated PLS particles. Optical tweezer techniques were used to bring the spheres into close proximity with each other, and then the relative motions were digitized from the video frames and analyzed. The spheres used in these experiments had diameters of 1.53, 0.97, and 0.65 µm. The adjustable parameters in the fit of the data to eqs 1 and 7 were an effective charge Z* and the screening parameter κ. A logarithmic plot of rU(r)/kBT vs r was reported to be linear. By using spheres of different size, the linear superposition approximation used in the derivation of eq 1 was tested. It was reported that the charges used in the identical pair study were recovered in the mixed pair study. These authors concluded that the DLVO form of the potential gave quantitatively consistent results. Experimental Observations Not Consistent with the Pairwise Potential Approach. Even though the careful analysis displayed in the Crocker and Grier11 paper indicates a preference of the DLVO form of the pairwise interaction potential over the SI form, one cannot conclude that the DLVO theory has been proven to be the final form of the pairwise interaction potential. One major problem with the DLVO expression is that the magnitude of the calculated charge of the macroion is consistently too low. Crocker and Grier, for example, mention incomplete dissociation of the surface groups and the “tightly” bound electrolyte ions about the macroion due to nonlinear effects not included in the DLVO approximation. Indeed, the consistently low charge values calculated from the DLVO theory have stimulated “charge renormalization” theories for highly charged systems.36,37 Curiously, the equation employed by Alexander et al.36 in their charge renormalization model, viz.

∇2φ(r) ) -

[

(

)]

qeφ(r) 4π Fm(r) + Fo exp  kBT

(12)

is precisely the nonlinear form of the P-B equation used by Sogami and Ise10 (cf. eq 6). Other observations that cannot be explained in terms of the DLVO theory are: the coexistence of highly ordered “crystalline” structures and random “gas” structures,32-35,38 at distances sufficiently far from the glass surface to make surface effects negligible, and the presence of void structures, large (36) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P.; Hone, D. J. Chem. Phys. 1984, 80, 5776. (37) Gisler, T.; Schulz, S. F.; Borkovec, M.; Sticher, H.; Schurtenberger, P.; Klein, R. J. Chem. Phys. 1994, 101, 9924. (38) Ito, K.; Nakamura, H.; Yoshida, H.; Ise, N. J. Am. Chem. Soc. 1988, 110, 6955.

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Schmitz

regions of the medium in which there are no macroions,12,13,39-43 which extend as far as 50 µm from the cell wall.42 The “partitioning” of macroions into two domains is also an observation in some polyelectrolyte solutions, where QELS methods were employed. Lin, Lee, and Schurr44 reported that the apparent diffusion coefficient (Dapp) of poly-L-lysine as a function of added NaBr. In going from high to low NaBr concentrations, Dapp first increased in value, as expected from theories of diffusion of macroions, but then underwent a catastrophic drop in value, by a factor of 20, at an ionic strength of 10-3 M NaBr for the 1 mg/mL solution. This phenomenon was labeled an “ordinary-extraordinary transition” (o-e transition). Drifford and Dalbiez45 reported that the location of the o-e transition for linear polyions followed the empirical relationship

Co-e m 〈b〉 FDD )

∑Zj2Cj

ZsλB

)1

(13)

where Co-e m is the molar concentration of monomeric units of the polyion at the transition location, 〈b〉 is the average charge spacing along the polyion contour length, and Zs is the salt valency. It was shown46 that the DriffordDalbiez ratio FDD could be expressed in terms of the relative volumes of the monomeric units (Vm) and the “ion cloud” cylindrical volume (VDH)

FDD )

4VDH )1 Vm

(14)

Thus the Drifford-Dalbiez ratio indicated that the o-e transition occurs at the onset of overlap of the ion atmospheres about the charged groups of the macroions. (There are several reasons for the presence of slow modes, such as entanglements of flexible linear polyions, that exhibit a broad transition. The signature of the o-e transition is (1) that it is extremely sharp, as given by eq 13, and (2) that there is a concomitant catastrophic decrease in the value of Dapp. The “anomalous slow mode” in the o-e transition is referred to herein as jeu des mole´ cules somnolentes, or the jms mode.) Further evidence that the o-e transition is related to the ionic atmosphere about the macroions is the result of using QELS methods with an external sinusoidal electric field (QELS-SEF). Schmitz and Ramsay47 reported that for poly-L-lysine in either 10 mM or 1 mM ionic strength solvent, the line widths broadened as either the frequency or the amplitude of the applied electric field increased. The values of Dapp calculated from these single ionic strength data spanned those obtained through the o-e transition. That is, in the narrow line width regime Dapp was comparable to that for the jms mode whereas the high-frequency/high-field-strength line width was comparable to that of the “ordinary” regime in the QELS experiment. (39) Ise, N.; Matsuoka, H.; Ito, K.; Yoshida, H. Faraday Discuss. Chem. Soc. 1990, 90, 153. (40) Ito, K.; Yoshida, H.; Ise, N. Chem. Lett. 1992, 1992, 2081. (41) Ito, K.; Yoshida, H.; Ise, N. Science 1994, 263, 66. (42) Yoshida, H.; Ise, N.; Hashimoto, T. J. Chem. Phys. 1995, 103, 10146. (43) Tata, B. V. R.; Yamahara, E.; Rajamani, P. V.; Ise, N. Phys. Rev. Lett. 1997, 78, 2660. (44) Lin, S.-C.; Lee, W. I.; Schurr, J. M. Bioplymers 1978, 17, 1041. (45) Drifford, M.; Dalbiez, J.-P.; Biopolymers 1985, 24, 1501. (46) Schmitz, K. S.; Ramsay, D. J. J. Colloid Interface Sci. 1985, 105, 388. (47) Schmitz, K. S.; Ramsay, D. J. Macromolecules 1985, 18, 933.

4. A Many-Bodied Interpretation of Cluster Stability Although experimental data for a two-state structure for a system of macroions of like charge has been around for many decades, an important aspect of the work of Grier and co-workers is that a “true” pairwise interaction potential is inadequate for the description of metastable “crystalline-like” structures. A “temporal aggregate” (TA) model was proposed47-50 to explain the jms mode. The TA is composed of a collection of macroions in which their ion clouds overlap, thus forming “spheres of influence” in which the shared electrolyte ions respond to the electric fields set up by the participating macroions. The TA is thus a dynamic structure in which the fluctuating electric fields serve, partially, to stabilize the structure as a counter action to the random dissipative forces of Brownian dynamics. The characteristics of the TA model were greatly influenced by the theoretical works of Kirkwood and Shumaker,51 who showed that net charge fluctuations on a polyion surface gave a large negative component to the chemical potential, and of Fulton,52 who showed that fluctuations in the dipole moments resulted in long-range correlations. Early support for the temporal nature of these “clusters” of PLS particles is in the paper by Clark, Ackerson, and Hurd.17 In this study PLS particles are confined to a “wedge”-shaped gap between quartz plates, and the scattered intensity is video monitored. A 6-fold pattern, similar to the crystalline state, appeared for a period of a few tens of milliseconds and then reappeared with a random orientation. Larsen and Grier12,13 created “metastable” colloidal structures with the application of a low frequency (60 Hz) electric field with a 10 V peak-to-peak amplitude. The effect of this field is to “compress” the PLS particles into a “superheated” crystalline state, and the subsequent time dependence of the cluster is followed by computer digitization of the video display. The dynamics of the system is described as follows. In the “gaslike” region the particles roam freely. At the interface the particles appear to be held by a potential barrier and eventually escape after several attempts. Within the crystalline structure the particles appear to rattle around in a potential well set up by the surrounding particles. Another result of this study is that a minimum “cluster size” of about 10 spheres may be necessary for the formation of metastable clusters. 5. Thermodynamics of the System: Theory The thermodynamics model for highly charged systems follows, close to but not identical with, the discussions of Padova,53,54 Hall,55-60 Pethica,60,61 and Ash et al.62 (col(48) Schmitz, K. S.; Parthasarathy, N. In Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems; Chen, S. H., Chu, B., Nossal, R., Eds.; Plenum Press: New York, 1981; p 83. (49) Schmitz, K. S.; Parthasarathy, N.; Vottler, E. Chem. Phys. 1982, 66, 187. (50) Schmitz, K. S.; Lu, M.; Gauntt, J. J. Chem. Phys. 1983, 78, 5059. (51) Kirkwood, J. G.; Shumaker, J. B. Proc. Natl. Acad. Sci. U.S.A. 1952, 38, 855. (52) Fulton, R. L. J. Chem. Phys. 1978, 68, 3089. (53) Padova, J. Electrochimica Acta 1967, 12, 1227. (54) Padova, J. J. Phys. Chem. 1968, 72, 796. (55) Hall, D. G. Trans. Faraday Soc. 1971, 67, 25167. (56) Hall, D. G. J. Chem. Soc., Faraday Trans. 2 1972, 68, 2169 (57) Hall, D. G. J. Chem. Soc., Faraday Trans. 2 1973, 69, 975. (58) Hall, D. G. J. Chem. Soc., Faraday Trans. 2 1973, 69, 1391. (59) Hall, D. G. J. Chem. Soc., Faraday Trans. 2 1978, 74, 405. (60) Hall, D. G.; Pethica, B. A. Proc. R. Soc. London A 1977, 354, 42. (61) Pethica, B. A. Colloids Surf. 1986, 20, 151. (62) Ash, S. G.; Everett, D. H.; Radke, C. J. Chem. Soc., Faraday Trans. 2 1973, 69, 1256.

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Langmuir, Vol. 13, No. 22, 1997 5853

lectively referred to as the model papers) regarding isolated electrolyte ions and macroions in solution and the change of state of the system when two charged surfaces are brought from infinity to within a distance τ. One major difference in the present approach is that the macroion substitutes for a charged plate used in the model papers. Hence the volume elements far from the macroions in the present study cannot be electrically neutral for point ion distributions as given by the solutions to the P-B equation. This is obvious for “salt-free” systems but is not so obvious in the case of added electrolytes. Likewise the charge on the macroion is generated concomitant with the charging process of the counterions initially placed in the solution medium. It is assumed that conclusions drawn from the studies on electrolyte ions also apply to macroions since, in the vicinity of the macroion, the individual charged groups of the macroion are distinguishable. Following the model papers the electrolyte ions, once charged, are allowed to adsorb onto the surface until equilibrium is achieved. This process is viewed in a manner analogous to the Gibbs adsorption isotherm for interfaces. As a review of the adsorption of gas onto a surface, the fundamental thermodynamic expression for the change in the internal energy U is given by

dU ) TdS - Psda + µdn

(15)

where Ps is the spreading pressure, da is the differential area, S is the entropy, µ is the chemical potential, and dn is the differential change in the number of adsorbed gas particles. One first integrates eq 15 and then differentiates to obtain the Gibbs-Duhem expression for the adsorption process

0 ) -SdT + adPs - ndµ

(16)

To illustrate the general approach, consider two identical volumes, V′. Let the subscript “c” represent a component of the solvent that may be adsorbed (or desorbed) by the solute particle and the subscript “s” denote the added solute particle. The two volumes are sufficiently far from each other that direct influences of the solute particles are absent in the former volume. It is assumed, however, that the two volumes are in osmotic equilibrium at constant T and µ. The Gibbs-Duhem expression for the volume containing the solute particle is given by

0 ) -S′′dT + V′dp - V′〈nc′′〉dµc - V′〈ns′′〉dµs

(17)

where the 〈nj′′〉 are unit volume concentrations, and for the solvent only volume

0 ) -S′dT + V′d(p - πos) - V′〈nc′〉dµc

(18)

where πos is the osmotic pressure. Let us now consider a volume V >> V′ but considerably smaller than the total volume of the system such that both situations given by eqs 17 and 18 are appropriate numbers to represent the solution. Let the number of solvent component particles in the “solution” situation be Nc′′ and in the “pure solvent” situation be Nc′, and the number of solute particles is given by Ns′′. The next step is to compare the average concentrations of the solvent components in the two volumes V. In the “solvent only” situation one has 〈nc′〉 ) Nc′/V. In the “solution” situation one has 〈nc′′〉 ) Nc′/V + ns(Nc′′ - Nc′)/V, where the second term accounts for the adsorption (desorption) of the solvent component due to

the presence of the solute particles. Let us express the difference 〈nc′′〉 - 〈nc′〉 as

〈nc′′〉 - 〈nc′〉 ) ns

Nc′′ - Nc′ ) ns〈N*es〉 V

(19)

Subtraction of eq 18 from eq 17 and rearrangement under conditions of constant T and ns gives the relationship Nc

N*js dµj ∑ j)0

dµs ) -

(20)

where we have taken the liberty to generalize the result to Nc components of the solution, including the solvent (j ) 0). The relevance of eq 19 to the system under consideration is found in its relationship with the pair distribution function,55 viz.

∫0∞[gcs(r) - 1]r2 dr

nsN*cs ) nsnc4π

(21)

It is easily shown that eq 21 represents the “excess” concentration in the vicinity of a surface above that of the concentration at which gcs(r) ) 1. A general description of the systems to be examined is that they are composed of (1) one or more uncharged macroparticles with sites to be charged, (2) solvent particles that are electrically neutral but may exhibit polarization or other electrical properties in the presence of a charged particle, and (3) additional solvent components in which the particles are uniformly dispersed in the aqueous solvent. These additional solvent components are initially uncharged, but have a future destiny to serve as counterions and added electrolyte ions. Isolated Surface. We first consider the case of zero added electrolyte for a system composed of a single macroparticle. For a volume element V′ located far from the macroparticle the number density for the neutral “counterion” particles is simply given by the uniform distribution

〈nc〉 )

Nc V′

(22)

Let us now simultaneously charge up the groups on the macroparticle and the counterions. Clearly this “instantaneous” state of the system is no longer in equilibrium. The counterions are attracted to the charged surface of the macroion and, if left unopposed, would completely neutralize the surface to minimize the energy. However, the energetics of the system is in competition with the dispersing entropic and kinetics contributions of both the solute and solvent particles. Eventually thermal equilibrium is established for the distribution of both the counterions and the aqueous solvent molecules. The distributions of both the uncharged and charged systems are illustrated in Figure 1. The number density now depends upon the distance the volume V′ is from the surface, i.e.

〈nc(r)〉 )

Nc(r) V′

(23)

For sufficiently large distances the number of particles may be taken, for convenience, to be constant with a value that is dependent upon the size of the container because

5854 Langmuir, Vol. 13, No. 22, 1997

Schmitz

Figure 1. Distribution of counterions in salt-free system. The macroparticle surface is represented by the rectangle on the left. In the top figure neither the particle surface nor the future counterions are charged; hence, the distribution of small particles is uniform. The bottom figure is the distribution of the counterions after charging (solid line) and is compared with the uniform distribution. The relationship between the uniform distribution and the counterion distribution for r f ∞ is given by eqs 22-24.

there is only one macroparticle present. Hence, one must have the relationship

〈nc(∞)〉 < 〈nc〉

Figure 2. Distribution of electrolyte ions in low and high surface charge density. Top: These curves represent the linearized Poisson-Boltzmann distribution of counterions (s) and co-ions (- - -) relative to the uniform distribution of ions (- -). At all points from the surface the total number of ions, regardless of identity, is the same as for the uniform distribution. Bottom: These curves represent the distribution of counterions and co-ions for highly charged surfaces, relative to the uniform distribution. The solution to the P-B equation represented here does not take into consideration interactions between the electrolyte ions. Hence, for r f ∞ the sum total of the ions may not equal that for the uniform distribution.

from the macroion must be a constant, which is denoted by µ, viz.

µ(r b) ) µ

(24)

for the identical volumes V′. The situation is somewhat different for the added electrolyte case, where it is assumed for convenience that the ions are symmetric in charge. Again the solvent components and surface of the macroparticle are uncharged. After the charging process, the counterions (defined as being either from the macroion to ensure charge neutrality or the identical ions from the added electrolyte) are drawn to the surface as before, and the co-ions are repelled from the surface. If the surface potential is low, then the number of counterions that are attracted is equal to the number of co-ions that are repelled. Hence, the number concentration of the ions is uniform, but the composition differs as one moves from the surface. On the other hand, if the surface charge is very high, then the co-ions are virtually excluded from regions in the vicinity of the macroion surface, and a much larger concentration of counterions accumulates in this region. If it is assumed that the distributions of both counterions and co-ions exhibit monotonic behavior as one goes from the surface of the macroion, then one has the result that, for either limiting case of the surface charge, there is a net charge in the volume V′ far removed from the macroion surface. Such a distribution is illustrated in Figure 2. The only way the neutral region far from the macroion surface can be achieved is if the distribution of ions is not monotonic. Such a “modified” distribution obtains only if one includes interactions between the ions. However, these modifications should not be sufficiently large as to give a gross distortion of the distribution as obtained from the P-B calculations If the system has achieved equilibrium, then the chemical potential of a region at an arbitrary distance b r

(25)

The components of µ(r b) are given by

µ(r b) ) µpot(r b) + µions(r b) + no′′(r b)µo(r b)

(26)

b) is the potential field due to the macroion where µpot(r b) represents the chemical potential of the surface, µions(r electrolyte ions, which is of the assumed form for all of the J ion types J

b) ) µions(r

nj′′(r b)[µj°(r b) + kBT ln(γj(r b)nj′′(r b))] ∑ j)1

(27)

b) is the local activity coefficient of the jth ion where γj(r b), µo°(r b) is the standard state type at local number nj′′(r b) is the local chemical potential chemical potential, and µo(r of the solvent, given in a form similar to eq 27

µo(r b) ) µo°(r b) + kBT ln(γo(r b)no′′(r b))

(28)

Since µ in eq 25 is constant, one has, from the GibbsDuhem relationship for eq 26, the general force-balance expression

dµpot(r b) dr

J

zjqeD(r b)] ) ∑ j)1

)[

[∑ j)0

]

b)nj′′(r b)) d ln(γj(r

J

-kBT

nj′′(r b)

dr

(29)

where the solvent, denoted by the subscript “o”, is included in the summation of the rightmost expression, zj is the magnitude (with sign) of the jth ion, qe is the magnitude of the electron charge, and D(r b) is the average electric displacement at the location b r.

Juxtaposition of Potential Fields

Langmuir, Vol. 13, No. 22, 1997 5855

Far Field. The volume element of interest is taken to be far from the macroion. To simplify the calculation without loss of generality, let us consider only one ionic b) ) 1 and assume that the species in eq 29 with γj(r contribution of the solvent is constant over the range of interest. The potential φ(r b) due to the presence of the macroion is

φ(r b) )

∫∞ D(rb) dr r

(30)

Integration of eq 29 yields the relationship

( ) nj′′(r b)

∫∞rD(rb) dr ) zjqeφ(rb) ) -kBT ln

zjqe

nj(∞)

(31)

Since the volume element is far from the macroion, one b)/kBT > 1 prevails. The work of moving inequality zjqeφ(r an ion from infinity to some region near the macroion surface must now take into consideration the effect of the field on the dielectric properties of the solvent. Padova53 assumed the free energy of a dielectric to be of the form

∆G )

∫∫E‚dD dV

1 4π

(39)

Because the overall system is electrically neutral, the first sum on the rhs of eq 34 is not in general equal to zero since, by definition, the macroion is far from this region. That is, there is no bounds for the region of electrically neutrality in an unconfined solution. As a consequence, the sign of this term is dependent upon the sign of the macroion charge because of the electroneutrality condition for the entire system, including the macroions. If the usual Debye-Hu¨ckel philosophy is evoked, the average potential is substituted for the actual potential, with the resulting more compact form of eq 34

where E is the electric field and D is the electric displacement. The differential dielectric constant was given as

∇2〈φ(r b)〉 ) - ϑD + K2〈φ(r b)〉

where n is the index of refraction, o is the static dielectric constant, and b is a constant. In a related study, Fenley, Manning, and Olson64 used a distance-dependent dielectric constant as a refinement in the counterion condensation model of Manning.65 The distant-dependent dielectric constant used in their study was that of Lavery66

(35)

where we have introduced a “Donnan” term ϑD, defined by

ϑD ) and K2 is defined as

4π 

J

zjqe〈nj′′(∞)〉 ∑ j)1

(36)

d )

dD dE

(40)

and the following form was obtained63

d )

o + n2 1 + bE2

(41)

(63) Padova, J. J. Chem. Phys. 1963, 39, 1552. (64) Fenely, M. O.; Manning, G. S.; Olson, W. K. Biopolymers 1990, 30, 1191. (65) Manning. G. S. Q. Rev. Biophys. 1978, 11, 179.

5856 Langmuir, Vol. 13, No. 22, 1997

(r) ) o -

Schmitz

o - 1 [(rS)2 + 2rS + 2] exp(-rS) 2

(42)

where r is the separation distance and S is the slope of a sigmoidal function described in Figure 5 of Lovery.66 Fenley et al. used values of S in the range 0.14-0.36. From eqs 41 and 42 the dielectric constant of water can vary from the value of 80 to d f 0 as E f ∞. The significance of a distance-dependent and fieldstrength-dependent dielectric constant is that the number of electrolyte ions in the vicinity of a surface is larger than anticipated on the basis of the bulk dielectric constant. Two Surfaces. For convenience we consider the interaction between two flat surfaces of finite extension and apply the above concepts under the condition of constant potential. The separation distance between the two surfaces is τ and the area of each surface is A. As in the previous section, the groups on the macroion surfaces and the pending counterions and added electrolyte particles are initially uncharged. The net charge on the macroion surfaces is determined after the charging process and the establishment of equilibrium. Although the volume between the two plates is rectangular, the particles between the two plates must also be in equilibrium with the surrounding media in accordance with an extension of eqs 25 and 26. Consider the events that occur when the two plates are brought from an infinite distance of separation to a distance of separation τ. The electrical components are charged while at infinite separation and the electrolyte ions are allowed to come to equilibrium with each surface. In this process an “outer boundary” concentration of electrolyte ions, i.e., ϑD as defined by eq 36, is established with the far field distribution given by eq 38. As the two surfaces approach each other, the influence of one potential field begins to influence the far field ions of the other macroion surface. Further movement of the two surfaces results in a readjustment of the ions to maintain a constant potential between the two surfaces. The constant potential is achieved by adsorption of ions onto the surface and/or by transport of ions from the “external” medium to the region between the two surfaces. This procedure continues to occur until the desired separation distance τ is attained. Under these circumstances eq 26 remains valid for each volume element of the solution, except that the potential b) now includes the contribution of both surfaces. The µpot(r work involved in moving the two surfaces from infinity to a distance τ is given by the integrated form of the GibbsDuhem expression, viz.,

∆G )

∫∞τ[∂µpot(rb)/∂h] dh + J

kBT[

∫∞τF (h) dh

ξOM )

U(r) kBT

(43)

where F (h) is the force exerted between the two surfaces b) now includes upon a movement of the distance dh, µpot(r the potential fields of both surfaces, and the sum from j ) 0 to J includes the solvent as well as the solute particles. b) on the distance Implicit in eq 43 is the dependence of nj′′(r h, as described in the previous paragraph. 6. Thermodynamics of the Electrolyte Ions in the System: Experiment Crucial to the chemical model for macroion systems is the concept that the electrolyte ions exist in at least two (66) Lavery, R. In Unusual DNA Structures; Wells, R. D., Harvey, S. C., Eds.; Springer-Verlag: New York, 1988; p 189.

(44)

The criterion for thermodynamically bound counterions was therefore defined on the basis of the interaction energy70

Ubound(r) g kBT

(45)

and it was also shown that, for screened Coulombic interaction of the DLVO form, the maximum distance for which the equality in eq 45 holds is at

amax ) aA + κ-1

τ ∑∫∞ nj′′(rb)[∂[ln(γj(rb)nj′′(rb))]/∂ h] dh] + j)0

thermodynamic regimes, the “thermodynamically free” ions and the “thermodynamically bound” ions. This is the basis, for example, of the charge renormalization theories36,37 and the anomalous small value of the screening parameter κ when used as an adjustable parameter to model fits to the data.67 There is experimental evidence that a surprisingly low fraction of counterions fall into the category of “thermodynamically free” ions. Ito, Ise, and Okubo68 reported transference and conductance measurements on linear polystyrene sulfonate (PSS) and polystyrene latices (PLS). The fraction of “free” counterions for the PSS was reported to be f ) 0.4, whereas f < 0.06 for the PLS particles. It was also noted that f seemed to decrease as the charge density of the particles increased. Roberts, Linse, and Osteryoung69 reported voltammetry studies on sulfonated PLS particles in which the hydrogen ion was the counterion. The fraction of free counterions was determined from the ratio of the diffusion coefficients of the ions in the presence and absence of the PLS particles. It was reported that f ) 0.024 for the PLS in deionized water. Criterion for Thermodynamically “Bound” and “Free” Ions. In the chemical model described above, the “excess” concentration of ions was calculated from eq 21. This “excess” concentration is not an accurate theoretical definition of the thermodynamically bound ions since the limit gcs(r) f 1 includes those ions distributed at random with respect to the macroion. The criterion used to determine the degree of bound ions is contained in the Oosawa-Manning condensation parameter for rodlike polyions,65 ξOM ) λB/b, where b is the average charge spacing along the line charge. If the pair interaction energy is strictly Coulombic, the OosawaManning parameter can be rewritten in terms of the ratio of the electrostatic energy for adjacent sites to the thermal energy

(46)

Thus the “thermodynamically bound” counterions are those located at a distance

r(bound) e amax

(47)

Roberts, Linse, and Osteryoung69 reported calculations that supported the criterion of eq 45. They reported nonlinear P-B calculations for ions distributed about a central macroion in a spherical cell. Using eq 45 as a criterion for “bound” ions, they obtained a value fcalculated ) 0.038, which is in reasonable agreement with their experimental value of f ) 0.024. (67) Crocker, J. C.; Grier, D. G. Phys. Rev. Lett. 1994, 73, 352. (68) Ito, K.; Ise, N.; Okubo, T. J. Chem. Phys. 1985, 82, 5732. (69) Roberts, J. M.; Linse, P.; Osteryoung, J. G. Langmuir, in press. (70) Schmitz, K. S. Langmuir 1996, 12, 3828.

Juxtaposition of Potential Fields

Langmuir, Vol. 13, No. 22, 1997 5857

electrons, the location of the macroions provides a potential field for the electrolyte ions. To a first approximation, it is assumed that the resulting potential field is a juxtaposition of the unscreened Coulombic potential fields of the participation macroions. The equilibrium distribution of electrolyte ions and solvent in the present of M macroions is obtained by combing eqs 25-28 M

µ)

∑

J

φm(r b) +

m)1

)

kBT φ(r b) + qe

nj′′(r b)[µj°(r b) + kBT ln(γj(r b)nj′′(r b))] ∑ j)0 J

nj′′(r b)[µj°(r b) + kBT ln(γj(r b)nj′′(r b))] ∑ j)0

(48)

where the normalized potential field is defined as Figure 3. Vector gradient plot of the probability function for the σ2sp bond. The hybrid bond for the two carbon atoms is a linear combinations of hydrogen-like orbitals, Ψ( ) (2s ( 2py)/ x2, where the bond is taken along the y-axis. The bonding orbital for the two atoms A and B is σ2sp ) (Ψ+(A) + Ψ-(B))/ x2, for the common coordinate axes. The gradient of the *, is shown above. probability density in the x-y plane, σ2spσ2sp The atoms are located at the terminal points, and the bond between the two atoms is indicated by a “corridor” outlined by the horizontal vectors leading to the terminal points. The heavy oval outlined by vectors parallel to its surface defines the “subregion” of the system in which the electrons are “confined” to be within the oval. The lengths of the vectors are on a logarithmic scale.

φ(r b) )

ZλB σ

M

∑b m)1|r

1

m,red|

(49)

As in the case of molecules where the location of the atoms serves to determine the potential field for the

bm|/σ is the reduced distance from the where |r bm,red| ) |r center of the mth macroion of diameter σ and charge Z and λB is the Bjerrum constant. The gradient in the potential field is related to the gradient in the ion and solvent distribution in accordance with eq 29. To establish that the JFP method can provide some insight as to the thermodynamics of macroion clusters, calculations for a dimer are compared with the three point extension (TPE) of the hypernetted chain calculations of Sa´nchez-Sa´nchez and Lozada-Cassou.74 In the TPE method, the macroions are represented as a dumbbell at a fixed interparticle separation, and two ions are permitted to roam about the volume surrounding the macroions. The interactions explicitly considered in the TPE calculations are the ion-macroion and ion-ion interactions to obtain the density map for the ions. Calculations of φ(r b) were carried out with the macroions in the z-y plane at a separation distance R/σ ) 8 and Z ) 50, and a contour map is shown in Figure 4. The figure at the top, viz., φ(r b), appears identical to the three-dimensional maps of the ion density of Sa´nchezSa´nchez and Lozada-Cassou,74 as it should by eq 48 for constant µ in each subregion. The contour map of φ(r b) is shown in the bottom of this figure. The gradient vector field of φ(r b) is shown in Figure 5. As in the case of Figure 3, the location of the two macroions is clearly indicated as the terminal points of the vectors. There is also a “corridor” set up by the gradient vectors connecting the two macroions, thus indicating an interaction between them. What is absent in the above figure is evidence of a subregion that satisfies the condition ∇F‚n b ) 0. The interpretation is that the ions are not “confined” to any region of space involving only two macroions. The nonlocalization of the ions is also illustrated through slices of φ(r b) taken along the macroion axis and perpendicular to the bonding corridor, as shown in Figure 6. The top profile in Figure 6 is a slice through the bonding corridor, where the location of the two macroions is clearly indicated. The bottom profile indicates a maximum in φ(r b) midway between the two macroins. These two slices together indicate a saddle point exists midway between the two macroions.

(71) Ray, J.; Manning, G. S. Langmuir 1994, 10, 2450. (72) Bader, R. F. W.; Slee, T. S.; Cremer, D.; Krata, E. J. Am. Chem. Soc. 1983, 105, 5061.

(73) Bader, R. F. W. Acc. Chem. Res. 1985, 18, 9. (74) Sa´nchez-Sa´nchez, J. E.; Lozada-Cassou, M. Chem. Phys. Lett. 1992, 190, 202.

7. Colloidal Systems: A Quantum Mechanical Analogy The attraction between colloidal particles of like sign is said to have similar characteristics of the chemical bond in quantum mechanics.10,71 If this comparison is valid, then methods used to identify chemical bonds in a molecular system can be applied to colloidal systems to determine those particles that are strongly “coupled” through their potential fields. The topological method of Bader and co-workers72,73 is explored in this section. As applied to molecules, this method determines the gradient in the electron density (probability function) for a fixed arrangement of atoms. In this gradient density profile, atoms are identified by the termination of gradient vectors at a single point, or a basin. A chemical bond is identified by a line that connects two nuclei. A closed surface in which no electrons cross the boundary, given by the mathematical criterion ∇F‚n b ) 0, where n b is a unit vector normal to the surface, is a subsystem of the molecule. To illustrate this graphics method the σ2sp hybrid bonding orbitals between two carbon atoms are shown in Figure 3. The two terminal points indicate the location of the two atoms, and the bond between them is clearly indicated by the horizontal “line” between the two atoms. One can also discern an “oval” shape outlined by the vectors parallel to its surface, i.e., ∇F‚n b ) 0, which defines the subsystem. Within the subsystem the gradient vector point in the direction of the chemical bond, which indicates the direction of maximum electron density. 8. Juxtaposition of Potential Fields

5858 Langmuir, Vol. 13, No. 22, 1997

Schmitz

Figure 5. Vector gradient plot of the potential for two charged spheres. The vector gradient of the potential in Figure 4 is shown. The macroions are located at the terminal points, and an interaction between the two spheres is indicated by the corridor, as in the hybrid bond case in Figure 3. In contrast to the hybrid bond calculation in Figure 3, there is no “subregion” for these two spheres. The lengths of the vectors are on a logarithmic scale.

Figure 4. Contour map of the juxtaposed potential fields for two charged spheres. The two spheres of charge Z ) 50 are separated by a reduced distance R/σ ) 8, where σ is the diameter of one sphere. The potential fields are in reduced form, where the amplitudes are defined as ZλB/σ, where λB is the Bjerrum length and σ is the sphere radius. In the above calculations, ZλB/σ ) 3.5, the number of points is 100, and the number of contours is 30.

Attention is now drawn to the configuration of four macroins, located at the positions (z, y) ) (4σ, 0), (0, 4σ), (-4σ, 0), and (0, -4σ). The gradient density profile is given in Figure 7. The four terminal points representing the macroions are again clearly indicated, and six bonding corridors representing all possible pairwise interactions are evident. An interesting feature emerges, however, that is not present in the pairwise system shown in Figure 5. The interior of the cluster exhibits a minimum that is bounded by a surface that almost qualifies as a subregion. A contour plot of this configuration is given in Figure 8, where 70 contour lines are shown to emphasize the interior region of the cluster. The JPF interior of the cluster appears to be constant. To further explore this phenomenon “slices” through the JPF profile, shown in the contour plot in Figure 8, are shown in Figure 9.

Figure 6. Juxtaposed potential field profiles. Top: The potential field profile is for a line drawn through the centers of the macroions. Bottom: The potential field profile is for a line drawn perpendicular to the line of centers.

The profile at the top of Figure 9 passes through two macroion, and exhibits a single minimum as in the case of two spheres, shown in Figure 6. The bottom profile in this figure, taken through the “bonding corridors”, exhibits a very shallow minimum. Hence the potential field interior to the macroion cluster is virtually con-

Juxtaposition of Potential Fields

Langmuir, Vol. 13, No. 22, 1997 5859

Figure 7. Vector gradient plot of the potential for four charged spheres. The vector gradient of the potential for four spheres located at (0, 4σ), (4σ, 0), (0, -4σ), and (-4σ, 0) is shown. The macroions are located at the terminal points, and interactions between the spheres are indicated by the interconnecting corridors, as in the hybrid bond case in Figure 3. Note that now, in contrast to the two-sphere case in Figure 5, there appears to be a “subregion” in the interior of the collection of spheres. The length of the vectors are on a logarithmic scale. Figure 9. Contour map of the juxtaposed potential fields for four charged spheres. The slices through the reduced potential field are given in Figure 8. Top: The potential field profile is for a line drawn through the centers of the macroions. Bottom: The potential field profile is for a line drawn perpendicular to the line of centers.

Figure 8. Contour map of the juxtaposed potential fields for four charged spheres. In the above calculations, ZλB/σ ) 3.5, the number of points is 100, and the number of contours is 70. The dashed lines indicate the directions taken for the profiles shown in Figure 9.

stant, which is a direct result of the symmetric arrangement of macroions. 9. Discussion The interpretation of data on polyelectrolyte and colloidal systems is not a simple matter in view of the long-range of Columbic interactions. Because of the complex nature of the interactions between particles and their surroundings, one can find in the literature papers that support either view regarding the nature of electrostatic interactions, viz., attractive or repulsive. This situation is exemplified by the fact that respected scien-

tists, in particular Langmuir and Overbeek, are associated with theories that have opposite predictions as to the character of electrostatic interactions in colloidal systems. It is our opinion that elements of truth are present in the majority, if not all, of the publications on highly charged systems. Hence there must be some “common thread” that connects these studies even if on the surface they appear to support opposite points of view. The task is to extract some form of a coherent picture of these complex systems through a critical comparison of these studies. The Langmuir and DLVO Views of Colloidal Systems. Both the Langmuir and DLVO models portray colloidal stability as a balance between attractive and repulsive forces in the system. By construction, the DLVO potential is limited to infinitely dilute systems in which only a pair of particles need be considered. Furthermore, the separation distance between this pair of particles is limited to large distances such that the linearized DebyeHu¨ckel theory is appropriate. As pointed out by Langmuir,1 such conditions apply only to properties of the dilute phase and are not applicable to situations in which phase separation occurs. It is in this Debye-Hu¨ckel region that both the Langmuir view and the DLVO view of colloidal systems may be perceived to be the same. That is, the electrolyte structure of the ions is uniform in the presence of added salt (cf. top of Figure 2). Langmuir also anticipated “charge renormalization” theories in his statement that the effective charge on the micelle in the Debye-Hu¨ckel region “...is roughly equal to the charge on a micelle reduced by the firmly bound negative ions in the sheath.” Whereas the Langmuir and DLVO views are the same in the dilute phase, these views diverge at the other concentration regime where phase separation occurs. In

5860 Langmuir, Vol. 13, No. 22, 1997

Schmitz

the DLVO model there is the ad hoc addition of a van der Waals attraction term. It is important to note that the concentration of electrolyte ions remains uniform if one ignores the identity of the ions. That is, in the linearized Debye-Hu¨ckel approximation the deficit of co-ions in the vicinity of the macroion is compensated for by the accumulation of counterions in the case of added electrolyte (cf. Figure 2). In the Langmuir view, the counterions accumulate in the region between the micelles giving rise to the “saltlike” crystalline structure of alternating positive and negative charges. It is this accumulation of particles in the “condensed phase” that provides the repulsive interaction that counteracts the attractive electrostatic force within the saltlike structure. This repulsive interaction between the electrolyte ions takes on the form of an “internal pressure” which, with the first-order Debye correction,75 is

P)

∑j 〈nj〉kBT -

λBκkBT

∑j zj2〈nj〉 )

6

(

κ2

1-

(4πnT)6

)

Pid

(50)

where

∑j 〈nj〉 ) 〈nT〉kBT

Pid ) kBT

(51)

∑j

∑j

nj

()

∂κ2 ∂

∂ ) κ2 ∂nj ∂κ2 ∂κ2

(52)

where it is assumed that for the individual ions κj2 ) nj/V and the volume V is constant. Overbeek argued that the solvent must be included in the derivative chain as introduced through the volume, viz., κj2 ) nj/ns where ns is the number of solvent molecules. Quite clearly this derivative cancels the term in eq 52 and therefore the attractive branch in the SI potential disappears. To be consistent in the mutual treatments of the DLVO and SI theories, Smalley first pointed out that inclusion of the solvent term in the original DLVO theory resulted in no electrostatic interactions at all!77 On the basis of thermodynamic arguments rather than molecular modeling, it was suggested that the relationship between the “Gibbsian” (Gelec) and Helmholtz (Aelec) free energies is70

(

)

∂ Gelec ) 1 + κ2 2 Aelec ∂κ

(53)

Rather than continue the arguments that emphasize the differences between these two models and which is (75) Debye, P. Phys. Z. 1924, 25, 97. (76) Overbeek, J. Th. G. J. Chem. Phys. 1987, 87, 440. (77) Smalley, M. V. Mol. Phys. 1990, 71, 1251.

∞

UDLVO )

is the ideal osmotic pressure and 〈nT〉 is the total number concentration of the electrolyte ions. As emphasized by Langmuir,1 introduction of electrostatic interactions between the electrolyte ions introduces an attraction term in the osmotic pressure. The DLVO and SI Theories of the Pairwise Interaction. There have been many comments on the differences between the DLVO and SI models for pairwise interactions between macroions. Perhaps the most extensively read criticism is that of Overbeek.76 The center of the criticism is the crucial step in the SI model10 for the introduction of the charged counterions

∂ nj ) ∂nj

“correct”, we wish to draw attention to the similarities and complementary interrelationships between these two potentials. Both the DLVO and SI potentials are based on the linearized Debye-Hu¨ckel approach, which means that the electrostatic interaction is extremely weak and/or the particles are greatly separated. Both of these theories resulted in a screened Coulomibic interaction. The DLVO theory further assumes that this pair of particles is isolated from all other particles in the system and that there is an excess of added electrolyte. In this regard the only contribution to the screening parameter is the added electrolyte. As first pointed out by Beresford-Smith and co-workers78 the DLVO potential is a “true” pairwise potential since it is insensitive to the concentration of macroions in the system. The SI model, however, takes into consideration the distribution of the other macroions as manifested in eqs 5 and 6. Hence the SI model is sensitive to changes in the ionic strength that results from changing the macroion concentration. Let us represent the sensitivity of the pairwise interaction in the DLVO theory in terms of a Taylor series expansion in κ2 about the point κ2 ) 0, where the DLVO and SI theories give identical forms of the pairwise interaction potential. Hence one can write

( )

n (o) 1 ∂ UDLVO

∑ n)0n!

(

2 n

∂(κ )

)

(κ2)n

κ2)0

∂ (o) = 1 + κ2 2 UDLVO ∂κ

(54)

(o) is the original form of the DLVO potential where UDLVO for two macroions in the absence of other macroions. Comparisons of eqs 50, 53, and 54 indicate that the SI potential is a theory for the first-order correction to the DLVO theory to account for the presence of other macroions. As in the case of the correction to the osmotic pressure for the presence of electrostatic interactions, the first-order correction to the repulsive interaction potential of the DLVO theory is an attraction due to the electrostatic interactions of the counterions of the additional macroions not considered in the DLVO theory. The Digital Video Microscopy Data of Grier and Co-workers. If the interpretation of the experimental studies on PLS particles put forth by Grier and coworkers11-13 is correct, then one corollary that results is that not all many-body phenomena are contained in the information provided by the pairwise interaction potential. In other words, the superposition principle of pairwise interactions is a questionable approach to providing theoretical models for the interpretation of concentrated macroion systems. The basis of the superposition principle has been questioned on theoretical grounds,79 but this is the first time, to our knowledge, that such a position is directly contained within experimental data. The motivation behind the development of the chemical model and the JPF method proposed in the present communication was not the experimental results of Grier and co-workers but rather the dichotomy that exists in (1) the theoretical predictions of the multi-bodied approach of Langmuir and the generally applied pairwise potential of the DLVO theory and (2) the controversy regarding the

(78) Beresford-Smith, B.; Chan, D. Y. C.; Mitchell, D. J. J. Colloid Interface Sci. 1985, 105, 216. (79) Lozada-Cassou, M.; Dı´as-Herrera, E. In Ordering and Organisation in Ionic Solutions; Ise, N., Sogami, I. Eds.; World Scientific Publishing Co.: Singapore, 1988, p 555.

Juxtaposition of Potential Fields

DLVO and SI theories that were specifically developed to explain experimental data. However, the JPF model proposed herein does not depend on the validity of the results of Grier and co-workers. These studies are included in this communication because the work of Crocker and Grier11 was specifically designed to determine the pairwise interaction potential between two isolated PLS particles. It is nonetheless pertinent to discuss these studies and those of Larsen and Grier12,13 because of the growing use of digital video microscopy methods to examine interparticle interactions. Three major concerns regarding DVM methods relevant to the present discussion are (1) the effect of the cover slip wall interactions on particles at the surface, (2) the propagation of cell wall effects to interparticle interactions as one moves away from the surface, and (3) the effect of confinement on the motions of the particles, even in the absence of interactions. Tanimoto, Matsuoka, and Yamaoka80 used evanescent wave light scattering methods to examine the interaction of an isolated sedimenting particle with the glass wall. The gravitational field presumably provided the “attractive” interaction and the electrostatic interaction between the charged particle and the cover glass provided the “repulsive” interaction. From the time dependence of the scattered light intensity, these authors concluded that the particles “vibrated” in the potential well set up by these two opposing forces and that the magnitude of the vibration increased with addition of salt. The interaction of the PLS particle with the glass wall is, therefore, a repulsive interaction at very close distances to the cell wall. Kepler and Fraden81 used DVM methods to study the interaction potential between PLS particles in a density matching D2O/H2O mixture, thus eliminating any possible sedimentation effects. The separation distance between the glass cell walls was sufficiently close that only a monolayer of PLS particles existed. They extracted a “true pairwise potential” by Brownian dynamics simulations to match the experimental determination of the pair distribution function, g(r), which exhibited an attractive interaction component. Both the DLVO and LennardJones pair potentials were used in these simulations. It was determined that the value of the Hamaker constant needed to fit the data were several orders of magnitude too large, and thus the DLVO potential was not adequate. A speculation was put forth that the origin of the attractive component of the interparticle interaction was due to the influence of the confining glass plates. A similar conclusion was drawn by Carbajal-Tinoco, Castro-Roma´n, and Arauz-Lara.82 Rather than confine the particles to monolayers, confocal laser scanning microscopy (CLSM) methods were used to study the distribution of particles as one moves away from the glass surface.83-85 Yoshida, Ito, and Ise83 followed the time-development of the crystallization structure of PLS particles in an D2O/H2O mixture at distances from the interface up to 40 µm. Subsequent layers formed sequentially from the hexagonal close-packed initial layer, reaching 22 layers in about 8.5 h. The subsequent layers were not fully developed during their formation, but (80) Tanimoto, S.; Matsuoka, H.; Yamaoka, H. Colloid Polym. Sci. 1995, 273, 1201. (81) Kepler, G. M.; Fraden, S. Phys. Rev. Lett. 1994, 73, 356. (82) Carbajal-Tinoco, M. D.; Castro-Roma´n, F.; Arauz-Lara, J. L. Phys. Rev. E. 1996, 53, 3745. (83) Yoshida, H.; Ito, K.; Ise, N. Phys. Rev. B 1991, 44, 435. (84) Ito, K.; Muramoto, T;, Kitano, H. J. Am. Chem. Soc. 1995, 117, 5005. (85) Muramoto, T.; Ito, K.; Kitano, H. J. Am. Chem. Soc. 1997, 119, 3592.

Langmuir, Vol. 13, No. 22, 1997 5861

proceeded through an “anisotropic stage” before attaining cubic symmetry. It was concluded that the interfacial attractive forces responsible for the formation of the first layer were somehow propagated throughout the crystalline system. Ito and co-workers84,85 examined the density of adsorbed particles as one moved from the glass surface. They found that under deionized conditions there was a net accumulation of particles near the surface, with a well-defined maximum in the distribution as one approached the surface. Identical profiles were obtained for H2O and a density matched D2O/H2O mixture, thus indicating that the accumulation of particles was not due to sedimentation. The difference in the interface and bulk concentrations of PLS particles decreased as the added salt concentration increased. Hence this source of attraction between the glass surface and the particles was electrical in nature. These observations were consistent with the single particle data of Tanimoto, Matsuoka, and Yamaoka80 described above. Hence it may be inferred that the potential in that study may also be purely electrical in origin without resorting to sedimentation effects. The picture that emerges from these studies is that there is an attraction between the glass surface and the PLS particles of like charge that is electrical in origin. This attraction appears for monolayered particles in a region of confinement on the order of a particle diameter, or develops in time in the presence of other particles that leads to a “crystalline growth” into the medium. Crocker and Grier11 reported experiments in which the spacing between the glass surfaces was varied by applying a negative pressure that bowed the cover slip. Under these conditions they found, in agreement with other DVM studies,81,82 that there is an attractive tail in the pairwise interaction for surface-surface separation distances 3.5 µm e d e 4 µm. In contrast, however, Crocker and Grier obtained a purely repulsive pairwise interaction for separation distances outside of this range. In their test of the DLVO and SI theories, the distance of either sphere to the glass surface was maintained by optical tweezers to be greater than 8 µm. It was implied that surface effects for the isolated pair of particles were negligible at these distances. Under these conditions the pairwise interaction potential was found to be purely repulsive, and the screened Coulomb form of the DLVO potential was verified by mixing the pairs of particles of different size and charge. Since these pairs of particles were isolated and not allowed to be in contact with the glass surface, a direct comparison cannot be made with the studies in which “crystalline growth” was observed for more concentrated suspensions in which the particles allowed to make wall contact.83-85 These observations were consistent with, but do not constitute proof of, the notion that surface interaction effects might be propagated through the crystalline structure.83 In the studies by Larsen and Grier,12,13 a low frequency electric field induced a “crystalline” structure. When the field was turned off, these metastable structures persisted for several seconds. This is a direct visualization of the QELS experiments performed in this laboratory47 and the “temporal aggregate” model47-50 proposed to explain the jms mode observed for some polyelectrolyte systems, 44,45 as described in sections 3 and 4. The DVM data appears to be consistent with the Langmuir view of colloidal systems. In the dilute phase, the Debye-Hu¨ckel approximation is valid, and if one ignores the charge, the symmetric electrolyte ion distribution is uniform, resulting in a single value of the screening parameter. Under these conditions the DLVO potential should be a valid approximation, and the

5862 Langmuir, Vol. 13, No. 22, 1997

pairwise interaction potential is purely repulsive. This situation obtains in the Crocker and Grier11 study for truly isolated pair of particles. Likewise, under these conditions the SI potential should not apply since it is based on the distribution of a system of macroions. As suggested from eq 54 and the discussion preceding this equation, the SI model represents a first-order correction model to the DLVO potential when the ions from the surrounding macroions make a significant contribution to the screening parameter. The DVM and CLSM results for finite PLS concentrations are also consistent with the Langmuir description of colloidal systems. Under these conditions the distribution of the electrolyte ions now takes on a structural form that influences the distribution of the macroions. Under these conditions the initial electrostatic repulsive interaction now becomes an attractive interaction as mediated by the electrolyte ion distribution. The SI model may now be viewed as a mean-field potential that attempts to incorporate long-range multibody effects into a pairwise potential. In view of this “switch over” in the role of electrostatic interactions, one may reconsider the effect of the glass surface interaction on the growth of a crystalline structure in terms of the Langmuir picture. Particles near a surface of like charge are, for some as yet unexplained reason, attracted to the surface by means of surface-particle electrostatic interactions. This accumulation of particles results in an increase in the concentration relative to the bulk concentration. As the number of layers grow, the influence of the glass surface diminishes. However, the accumulation of the macroion layers in turn results in a solution structure for the electrolyte ions in accordance with the Langmuir multibody model. The source of attraction between macroions now switches from surface effects to that which stabilizes the crystalline state in free suspension. Thus, while the surface-macroion interactions may nucleate a cluster of macroions, these effects may not be responsible for the growth of layers beyond a certain distance from the glass surface. This mechanism may be partially responsible for the “anisotropic” to “cubic” structural change reported by Yoshida, Ito, and Ise.83 The JPF Approach and the Salt Fractionation Model of Smalley, Scha1 rtl, and Hashimoto. The JPF chemical model is similar to the “salt fractionation” model proposed by Smalley, Scha¨rtl, and Hashimoto (SSH).86 The SSH model interprets the presence of clusters and voids in terms of a differential in the concentrations of salt interior to both the clusters and the voids. The excess concentration of counterions inside the cluster cavities is a result of the potential profiles shown in Figure 9. The excess concentration of ions in the void regions, i.e., coions and accompanying ions to achieve electroneutrality, results from the expulsion of the co-ions from the cluster cavity. Thus the osmotic pressure within the void tends to cause the voids to grow in size over a period of time. As these authors state, they “draw heavily on the SogamiIse interaction potential for spherical macroions” and the condition of constant potential for the charged spheres. From the SI potential they are able to make certain predictions regarding the salt range for stable void regions, viz., in the range 0.816 e κa e 3.05. The JPF chemical model likewise contains a salt fractionation mechanism for the stability of the clusters but makes no statement as to the form of the interaction potential between the participating macroions. The Chemical Model and the JPF Approach and Cluster Stability. Any approach designed to describe (86) Smalley, M. V.; Scha¨rtl, W.; Hashimoto, T. Langmuir 1996, 12, 2340.

Schmitz

the physics involved in these complex systems must be able to explain the instability of pairwise encounters while multibody interactions form stable structures. The JPF method provides such a picture. As indicated in Figure 5, the JPF for the two macroions does not result in a closed region in which the counterions reside. However, the JPF profile for the four body cluster shown in Figures 7-9 does indicate such a region. The basis of the JPF method is the constancy of the chemical potential throughout the system and the explicit relationship between the potential field set up by the macroions and the electrolyte and solvent particles. Just as one sets up the Coulombic field of atoms and allows the electrons to seek the lowest energy states for that particular atomic arrangement, the macroions are first set in place and the ions and solvent particles are allowed to equilibrate to satisfy the constancy of chemical potential criterion. In this regard the JPF approach as presented in this communication is used as a qualitative guide, indicating where the excesses and deficiencies of concentrations of electrolyte ions may be located. One may not anticipate, for example, that extensive computer simulations that incorporate the electrolyte specifies such as finite ion size and ion-ion interactions would invert the anticipated distributions. That is, one would not expect an excess of co-ions with the exclusion of counterions in the cavity of the cluster from extensive computer simulations. This speculation is supported by the comparison of the potential field of the dimer with the TPE calculations of the counterion density reported by Sa´nchez-Sa´nchez and Lozada-Cassou.74 The implication is that by using the gradient vector analysis of the JPF one might find interesting “stable” structures to be examined by the more exhaustive computer simulation methods. The following scenario illustrates the properties of the JFP chemical model. The groups on the macroions are initially uncharged, as are also the co-ions and counterions that are uniformly distributed throughout the medium. The macroions and electrolyte ions are now simultaneoulsy charged and allowed to come to their equilibrium distribution about the macroion cluster. The field in the interior of the cluster of macroions, i.e., the cavity, is higher than outside the cavity. Therefore, inside the cluster cavity the counterion concentration increases whereas the coion concentration decreases as equilibrium is established. Counterions also adsorb onto the macroion surface to establish equilibrium in accordance with the constant chemical potential criterion. However, one must consider the local concentration of counterions in the vicinity of macroion sites inside and exterior to the cluster cavity. The charge distribution on the surface of the macroions is no longer uniform as a result of this differential adsorption of counterions. There are now at least four mechanisms that contribute to an attractive interaction between the macroions comprising the cluster and thus to the stability of the cluster. First, the potential field inside the cluster is considerably higher than outside the cluster. This means that the concentration of counterions is higher than the bulk concentration and that the co-ions are largely excluded. The alternating “pockets” of positive and negative charge lead to an electrostatic attraction in accordance with the Langmuir model. The three remaining mechanisms involve fluctuating electric fields, which give rise to attractive long-range interactions.51,52 The potential inside of the cluster is constant. This means that the counterions are not “localized” to any one macroion but are free to roam within the interior of the cluster. This sharing of counterions between the participating macroions gives rise to a fluctuating electric field of the ion cloud. Second, any translational movement of any

Juxtaposition of Potential Fields

Figure 10. JPF chemical model for macroion clusters. The cluster of four macroions with surface sites (O) and the distribution of counterions (b) in the JPF chemical model is shown. The concentration of counterions interior to the cluster is greater than that of the exterior ones, and the adsorption of counterions on the interior surface sites is greater than that of the exterior surface sites. The three possible mechanisms for cluster stability are shown in the figure: (1) the rotational motion indicated by the curved arrow gives rise to an attractive interaction due to fluctuations in the interaction potential for an asymmetric surface charge distribution; (2) translational motion in a radial direction from the cluster, as indicated by the double arrow, results in net surface charge fluctuations due to the local counterion concentration that arise from volume fluctuations of the cavity; (3) the sharing of counterions within the cavity is as indicated by the dashed circle.

one macroion in a direction that causes a volume change of the cluster results in a readjustment of the net charge on the cavity side surface of the macroion. That is, the net interior surface charge fluctuates in response to the local concentration changes due volume fluctuations as the macroion moves in the radial direction. The third mechanism involves the rotation of the spheres. Since the net surface charge density is lower interior to the cluster than exterior, any rotation of spheres with an asymmetric charge distribution also gives rise to fluctuat-

Langmuir, Vol. 13, No. 22, 1997 5863

ing electric fields that stabilize the cluster. The structure of the cluster deduced from the JPF chemical model is illustrated in Figure 10. Although the JPF chemical model allows for chemical adsorption and desorption of ions on the macroion surface, the interfacial region is not identical to thermodynamic models of Hall and Pethica,60,61 and Ash et al.62 The “thermodynamically bound” counterions in the JPF model are defined by the statistical weighting factor at which point the thermal and Coulombic energies are equal (cf. eq 45). Hence, part of the “exponential tail” of the distribution function gcs(r) in eq 21 lies within the domain of the “thermodynamically free” counterions. If the pairwise interaction potential is purely repulsive as Grier and co-workers suggest, then what might be the mechanism for the “creation” of these clusters? One possible answer can be found in the superheated crystal experiment of Larsen and Grier.13 In the presence of a low-frequency electric field, the PLS particles are “compressed” into the ordered structure. A video of this process clearly shows that the ordered regions that result from the field remain long after the field is turned off. The video also shows that the superheated fluid melts at its edges and not within a cluster. Recall that the DriffordDalbiez ratio locates the onset of the o-e transition at a monomer/salt concentration marking the onset of the overlap of the neighboring ion atmospheres of the macroions. In other words, the solution and suspension is “crowded” if one takes as the criterion of overlap an “effective radius” given by eq 46. The “seed” of the crystalline-like ordered domain may therefore result from concentration fluctuations of the surrounding macroions, thus forcing the interior macroions into a superheated state. Once the seed has been established, long-range attractive interactions of the type envisioned by Langmuir1 may then further stabilize the crystalline domain. Acknowledgment. I am very grateful to Professor Grier for many discussions on the work carried out in his laboratory, and for the video tape showing the melting of the “superheated” suspension of PLS particles. I wish to acknowledge the spirited discussions with Julia P. Fisher that took place in the course of the preparation of this manuscript. LA970479H