Langmuir 1999, 15, 4093-4113
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JPF: A General Approach to Select Stable or Metastable Clusters of Macroions of Arbitrary Shape† Kenneth S. Schmitz Department of Chemistry, University of MissourisKansas City, Kansas City, Missouri 64110 Received August 28, 1998. In Final Form: March 9, 1999 The juxtaposition of potential fields method (JPF) is based on the Gibbs-Duhem expression relating potential gradients of the various components in the system, and the Bader vector gradient field interpretation of bonding in molecules. Stability of a cluster of macroions is thus inferred by the presence of a subregion in the vector gradient potential plots. The primary premise of the JPF interpretation is that the distribution of electrolyte ions set up by the field of the macroion cluster has a profound effect on the stability of the cluster. One can discern in such a plot at least four distinct regions related to the dynamics of the electrolyte ions. The sharing of counterions within overlapping nonlinear regions of neighboring macroions are a source of “attraction” between macroions in a cluster. The JPF method is applied herein to a cluster of spheres and of rods, with the suggestion that macroions of arbitrary shape may also be studied with this method.
1. Introduction There is an increased interest in the properties of polyelectrolyte, as attested to by the variety of papers presented at the Second International Symposium on Polyelectrolytes, to which the present issue of Langmuir is devoted. Because of the restriction of charged groups along the polymer backbone by chemical bonds and the long-range nature of the electrostatic interactions, the physical properties of both polyelectrolyte solutions and colloidal suspensions can exhibit totally unexpected behavior. An underlying feature of the complexity of these systems is that the fundamental nature of the electrostatic interaction has no universal following: whether the net electrostatic interaction of a cluster of macroions is attractive or repulsive. This is exemplified on the theoretical battlefield by two models proposed by respected researchers in the field. The DLVO potential (DerjaguinLandau-Verwey-Overbeek)1 is based on a pairwise interaction between two isolated charged spheres. The counterions were treated the level of a linearized approximation to the Poisson-Boltzmann (PB) equation and appear in the DLVO potential as a screening term for the electrostatic interaction between the two spheres. The result was a repulsive screened Coulombic interaction with an ad hoc addition of a van der Waals attractive interaction term. At the other extreme, Langmuir2 viewed micellar systems in terms of a crystalline structure. The charged micelles were located at specific lattice points. The counterions, collectively viewed as a macroion, were also located at lattice points. Under these conditions of alternating positive and negative deposits of charge the net electrostatic interaction is attractive. The repulsive part of the Langmuir potential, needed for the electrical stability of the structure, came from the osmotic pressure of the confined counterions. The experimental arena fares no better in deciding whether the net electrostatic interactions are attractive or repulsive. Consider, for example, the photothermal compression studies of Rundquist et al.3 Polystyrene sulfonated spheres of
diameter 83 nm were labeled with a light-absorbing dye. The structure of the colloidal crystals was monitored by the shifts in the Kossel ring diameter. Using a pump/ probe combination of light beams, these authors reported a contraction in the Kossel rings due to local heating. These observations were interpreted in terms of a “weakening” of the repulsive interaction between spheres within the heated region, followed by a reduction in the interparticle spacing due to the repulsive interactions of spheres outside the heated region. Conversely, these observations could also be interpreted in terms of a “strengthening” of attractive interactions between the spheres within the heated region.4 Thus observations alone on an experiment designed to monitor a subregion of a system are not sufficient to draw conclusions about the nature of electrostatic interactions as the interpretation of the results is dependent upon the implicit assumptions of the observer regarding the origin of forces that respond to the perturbation. A repulsive interaction outside the subregion cannot be distinguished from an attractive interaction within the subregion, and vice versa. Our entry into this area of interparticle interactions was to find an explanation for the so-called “ordinaryextraordinary” (o-e) transition of flexible polymers as determined from dynamic light scattering (DLS) studies. In 1978 Lin, Lee, and Schurr5 reported rather bizarre behavior of the dynamics of poly-L-lysine as a function of added electrolyte. For fixed polyion concentration, the ionic strength profile of the apparent diffusion coefficient, Dapp, first increased with a decrease in the added electrolyte. This behavior was expected on the basis of the theories for the salt dependence of the mutual diffusion coefficient. However, at a well-defined salt concentration, Dapp catastrophically dropped in value, by a factor of 20. Accompanying this change in the character of Dapp was a slight decrease in the intensity of scattered light, by about a factor of 4. Thus the lower salt regime was dubbed the “extraordinary” phase. This unexpected ionic strength profile was later reported for linear polystyrene sulfonate
† Presented at Polyelectrolytes ’98, Inuyama, Japan, May 31June 3, 1998.
(3) Rundquist, P. A.; Jagannathan, S.; Kesavamoorthy, R.; Brnardic, C.; Xu, S.; Asher, S. W. A. J. Chem. Phys. 1991, 94, 711. (4) Ise, N.; Smalley, M. V. Phys. Rev. B 1994, 50, 16722. (5) Lin, S.-C.; Lee, W. I.; Schurr, J. M. Bioplymers 1978, 17, 1041. (6) Drifford, M.; Dalbiez, J.-P. Biopolymers 1985, 24, 1501.
(1) Verwey, E. J.; Overbeek, J. Th. G., Theory of the Stability of Lyophobic Colloids; Elsevier Publishing Co., Inc.: New York, 1948. (2) Langmuir, I. J. Chem. Phys. 1938, 6, 873.
10.1021/la9811214 CCC: $18.00 © 1999 American Chemical Society Published on Web 06/08/1999
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by Drifford and Dalbiez,6 who characterized the location of the o-e transition by the empirical relationship
[m]〈b〉
RDD )
(1.1)
∑j [j]Zj
2
ZsλB
where λB ) qe2/kBT ) βqe2/ is the Bjerrum length, with qe the magnitude of the electron charge, the bulk dielectric constant of the medium, and kBT ) β-1 the thermal energy; [m] is the molar concentration of the monomer units of the polyelectrolyte with average charge spacing 〈b〉; [j] is the molar concentration of the electrolyte ions of type j and charge Zj; and Zs is the valency of the counterion of the polyion, which is also one component of the added electrolyte. The location of the o-e transition is at RDD ) 1, i.e., at the molar concentration of monomer
∑j [j]Zj2
ZsλB [m]o-e )
〈b〉
(1.2)
An unusual feature of the o-e transition is that no other measurement on the system through this ionic strength range exhibits such dramatic changes. These measurements on poly-L-lysine include low shear relative viscosity,7 conductivity,8 and tracer diffusion coefficients (Dt) using fluorescence recovery after photobleaching (FRAP) methods.9 A puzzling feature of the o-e transition is contained in the DLS and FRAP studies of Zero and Ware.9 These authors report an ionic strength profile of Dapp obtained by DLS methods that shows a dramatic decrease in value at the o-e transition location, in agreement with that of Lin, Lee, and Schurr.5 However, the corresponding profile for Dt only shows a gradual decrease throughout the entire salt range, eventually reaching at zero added salt a value of about half its high salt value. Zero and Ware report that this decrease in the value of Dt could easily be accounted for by an electrostatic expansion of the polyelectrolyte chain.9 This is direct evidence that the o-e transition is not a result of aggregation of the poly-L-lysine molecules. To emphasize the situation that existed in the early 1980s in regard to the o-e transition, we digress to discuss the standard relationship between Dapp, Dt, and the total intensity of scattered light as represented by the dynamic structure factor S(Q) ) 〈I(Q)〉 , where Q ) (4πn/λo) sin(θ/2) is the scattering vector, n is the index of refraction of the solvent, λo is the wavelength of incident light in a vacuum, and θ is the scattering angle. This relationship is10
Dapp (Q) )
Dt S(Q)
(1.3)
It is in the limit Q ) 0 that one obtains the value of the mutual diffusion coefficient, viz., Dt/S(Q ) 0) ) Dt(∂πos/ ∂cp) ) Dm, where the osmotic susceptibility is (∂πos/∂cp) ) 1/S(Q ) 0), πos is the osmotic pressure and cp is the macroion concentration. Since the value of 〈I(Q ) 0)〉 decreases through the o-e transition by a factor of 45 and (7) Martin, N. B.; Tripp, J. B.; Shibata, J. H.; Schurr, J. M. Biopolymers 1979, 18, 2127. (8) Shibata, J. B.; Schurr, J. M. Biopolymers 1979, 18, 1831. (9) Zero, K.; Ware, B. R. J. Chem. Phys. 1984, 80, 1610. (10) Schmitz, K. S., Macroions in Solution and Colloidal Suspension; VCH Publishers: New York, 1993.
Dt exhibits very little change in this region of the salt profile,9 Dapp for the slow mode cannot be identified with the mutual diffusion coefficient. In honor of the work of Schurr and Drifford, we have referred to the slow mode as jeux de mole´ cules somnolentes, with the notation Djms.10 To understand the lack of strong experimental evidence of the extraordinary regime other than DLS data, one must take into account what is being measured by each technique. In the case of DLS it is the relaxation of index of refraction fluctuations that is being monitored. The origin of the anomalous slow mode is thus the relaxation of a long-lived fluctuation in the index of refraction. On the basis of the FRAP results, the macroions involved in this slow decay process must necessarily diffuse as if they were “free” particles, yet somehow their movement relative to each other (mutual) must be retarded. In other words, the physical picture that emerges is that of a “swarm of bees” where the individual bees move in an unhindered manner yet the bees retain the cluster of the swarm. If one takes the classical view that like charges repel, then what must be the source of the “attraction” between the macroions that give rise to the correlated motion of the macroion cluster? The answer lies, we believe, in the role the electrolyte ions play in determining the dynamics of the macroions.11-14 This position implies a fluctuation mechanism may be responsible for the o-e transition.13,14 Three studies available at that time had a great influence in the formulation of a model for this fluctuation mechanism interpretation of the o-e transition. Kirkwood and Shumaker showed that fluctuations in the net surface charge could account for the measured dipole moments of some globular proteins15 and that these surface charge fluctuations gave rise to an attractive force between proteins of like charge.16 Oosawa and co-workers17,18 showed that correlations in the lateral fluctuations of counterions along a rod resulted in an attraction between the rods. Fulton19,20 showed that short-range correlations of the ions with a rotational relaxing dipole leads to the time development of long-range correlations between dipoles. On the basis of these studies we proposed that the slow mode represented a correlation of the dynamics between these polyelectrolytes that resulted from the adsorption or release of counterions that altered their net surface charge.11,12 Further consideration of these phenomena led to a “temporal aggregate” (TA) view of the slow mode and the o-e transition,13,14 which we quote as the mechanism for the metastable stability for these TA’s14 These “temporal aggregates” are stabilized by a delicate balance between the attractive forces arising from the fluctuating dipole filed generated from the sharing of small ions by several polyions and the repulsive forces due to the random Brownian motion and direct interactions between polyions of like charge. (11) Schmitz, K. S.; Kent, J. C.; Parthasarathy, N.; Radhakrishman, G.; Ramanathan, B. Biophys. J. 1980, 32, 246. (12) Schmitz, K. S.; Parthasarathy, N.; Gauntt, J.; Lu, M. In Biomedical Applications of Laser Light Scattering; Sattelle, D. B.; Lee, W. I.; Ware, B. R., Eds., Elsevier Biomedical Press: New York, 1982; p 61. (13) Schmitz, K. S.; Lu, M.; Gauntt, J. J. Chem. Phys. 1983, 78, 5059. (14) Schmitz, K. S.; Lu, M.; Singh, N.; Ramsay, D. J. Biopolymers 1984, 23, 1637. (15) Kirkwood, J. G.; Shumaker, J. B. Proc. Natl. Acad. Sci. U. S.A. 1952, 38, 683. (16) Kirkwood, J. G.; Shumaker, J. B. Proc. Natl. Acad. Sci. U. S.A. 1952, 38, 855. (17) Ohnishi, T.; Imai, N.; Oosawa, F. J. Phys. Soc. Jpn. 1960, 15, 896. (18) Oosawa, F. Biopolymers 1968, 6, 134. (19) Fulton, R. L. J. Chem. Phys. 1978, 68, 3089. (20) Fulton, R. L. J. Chem. Phys. 1978, 68, 3095.
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Langmuir, Vol. 15, No. 12, 1999 4095
Experimental support of the notion that fluctuations in the molecular electric fields may be responsible for the anomalous extraordinary regime is found in quasi-elastic light scattering studies in which an external sinusoidal electric field is applied across the sample (QELS-SEF).21 The added electrolyte concentrations of these poly-L-lysine solutions were 0.1 and 10 mM in KCl, chosen such as to lie below and above, respectively, the o-e transition location. The frequency of the driving field was 90 Hz, and the apparent diffusion coefficients (Dqels-sef) were obtained from the line width at half-height of the frequency-shifted spectral density. The electric field strength was varied over the range 2-30 V/cm. It was reported that, in both cases, the line widths broadened as the magnitude of the applied electric field increased. The low-field values of Dqels-sef were numerically equivalent to Djms and that the high field values of Dqels-sef were numerically equivalent to the high salt limit of Dapp. In other words, the dynamics associated with the o-e transition were reproduced by simply varying the strength of the applied electric field across a single sample thus establishing the electrical origin of Djms. The speculation that the counterions are shared between the macroions is based in the reinterpretation of the Drifford-Dalbiez ratio given by eq 1.1. We have rewritten the Drifford-Dalbiez ratio as a ratio of the volume of the ion cloud, V′DH to the volume of the monomer unit, V′m22
RDD )
n′m〈b〉
∑j
ZsλB
4πn′m〈b〉
)
Zsκ2
n′jZj2
V′DH )4 V′m
(1.4)
where the concentrations n′k are now given as the number density (particles/cm3) of the kth species at the o-e transition point
κ2 ) 4πλB
∑j
n′jZj2 ) κ2DH
(1.5)
(added)
defines the Debye-Hu¨ckel screening parameter κ in terms of the added electrolyte, V′m is the volume in cm3 of the monomeric units, and V′DH ) π〈b〉/Zs κ2 is the DebyeHu¨ckel volume per unit charge along the cylindrical unit of the polyion. Hence when RDD ) 1 one obtains for the ratio of the radii of equivalent spheres for the rods (V′m/ V′DH)1/3 ) 41/3 ) 1.59. Thus the location of the o-e transition is that salt concentration in the vicinity at which the ion clouds of the monomer units begin to overlap. The Drifford-Dalbiez ratio, therefore, gives further credibility to the TA model in which the polyion dynamics are correlated by the sharing of electrolyte ions in the overlapping ion clouds. An implied premise of the TA model is that the structure and dynamics of the ion cloud surrounding the macroions play an important role in the dynamics of the macroions. Because of the magnitude of the charge and the dimensions of the macroion relative to those of the supporting electrolytes, it may be accepted that the electric field of the macroions provides a blueprint for the structure of the surrounding ion cloud. This mutual interdependence of the two types of electric fields thus determines the stability or metastability of a cluster of macroions. We have recently proposed a juxtaposition of potential fields (JPF) method to provide some insight as to the disposition (21) Schmitz, K. S.; Ramsay, D. J. Macromolecules 1985, 18, 933. (22) Schmitz, K. S.; Ramsay, J. D. J. Colloid Interface Sci. 1985, 105, 388.
of the electrolyte ions in the vicinity of a cloister of macroions.23 The present communication expands further the application of the JPF approach to clusters of general shape as a guideline for more detailed computer simulations of the relative stability of these structures. Before embarking on this journey, we first shall review some relevant concepts regarding simple salt systems and theories of the macroion pair potential. This review is intended to provide insight as to the interpretation of the meaning of “overlap of ion clouds” in the context of contemporary theories and the relationship to experimental observations. In section 2 we review pertinent features and criticisms of the Debye-Hu¨ckel theory since the distribution of the electrolyte ions plays a major role in the interpretation of the observations on charged colloidal particles. A feature emphasized in the review of the original Debye-Hu¨ckel paper24 is the use of the screening parameter κ as a thermodynamic variable. In section 3 the pair potential expressions for the DLVO, Kirkwood-Shumaker, and Sogami-Ise are discussed in some detail, as well as the generalized thermodynamic approach of Michaeli, Overbeek, and Vroon on phase separation in colloidal suspensions. This is followed in section 4 by multibodied approaches to colloidal systems. The physical significance of κ in the pair potential forms is discussed in section 5 in terms of an “effective” radii at which the electrical interaction of the counterion with the macroion is set equal to the thermal energy. Because of the ubiquitous use of the screening parameter in theories of charged systems, the variety of theoretical expressions for κ is reviewed in section 6. Since the calculation of κ depends on the concentration of ions in solution, experimental evidence for “bound counterions” is given in section 7. It is argued in section 8 that a “screened Coulomb charge” composed of the macroion charge and the counterions at energies greater than the thermal energy is a better representation of the “effective charge” than the “renormalized charge” approach in which surface bound counterions reduce the net charge of the macroion. Since the basis of the present paper is on nonadditive effects in the nonlinear region of the multibodied interaction range, the limitations of the linearized forms of the pair potentials are discussed in section 9. The juxtaposition of potential fields (JPF) method is reviewed in section 10, which is followed by applications to a cluster of spheres and rods in sections 11 and 12, respectively, and concludes with a general discussion in section 13. 2. The Debye-Hu 1 ckel Theory Revisited In 1923 Debye and Hu¨ckel24 proposed a model to interpret data on the freezing point depression and related phenomena for strong electrolytes. Although the DebyeHu¨ckel (DH) theory is well-known and can be found in almost any textbook on physical chemistry, there are features of their work not generally emphasized that are relevant to the present development. A motivating issue of their study was that the observed behavior of the osmotic coefficient, as indicated through freezing point lowering and boiling point raising, did not follow a law-of-mass action expression. As shown in their Figure 6, the experimental data on the freezing point depression of KCl exhibited a maximum. They further pointed out that the electrical energy of the system, denoted in this text by Uelec, could not be calculated on the basis of the average separation distance between ions based on their concentration, which is 〈R〉 ) c-1/3 where (23) Schmitz, K. S. Langmuir 1997, 13, 5849. (24) Debye, P.; Hu¨ckel, E. Phys. Z. 1923, 24, 185.
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Schmitz
c is the total concentration of the ions. Rather, associated with each ion was a “cloud” of ions, and the relevant distance in these electrical systems was λDH ) 1/κDH, where κDH is defined by eq 1.5. The energy of the electrolyte system is written as
Usys ) Uclass + Uelec
potential 〈φj〉 for Wj
〈∇ h ‚∇ h Wj〉 f ∇2〈φj〉
(2.9)
and the second is the use of the average number distribution based on the mean potential
(2.1) s
where Uclass is the classical energy and Uelec is the desired electrical part. Their preferred thermodynamic function to describe the system is the function Φ
U + PV Φ)ST
(2.2)
It is readily recognized that Φ ) -G/T, where G is the Gibbs free energy. The rational of the selection of G rather than A, the Helmholtz free energy, as the free energy of choice is that experiments are usually done under constant pressure-temperature conditions rather than constant volume-temperature conditions. The electrical pressure is given, in the present notation, by
( )
Pelec ) -
∂Aelec ∂V
(2.3)
T
These authors then pointed out24 that it is incorrect to use the classical expression for the volume without taking into account the electrical effect of the ions on the volume. They estimated a relative volume change of 0.001 for water using an electrical pressure of 20 atm for a 1 M solution of KCl. If one neglects the electrical contribution to the pressure, one may write
Φsys ) Φclass + Φelec
(2.4)
Fc f 〈Fc〉 )
Φelec ) -
A ) T
∇2 〈φj〉 ) κ2〈φj〉
dT
(2.5)
with
Uelec )
1
∑i
2
NiqiWi
qj exp(κaj) exp(-κr) (r > aj) r (1 + κaj)
〈φj〉 )
∇ h ‚(E h) ) ∇ h ‚(∇W h ) ) -4πFc
qjn′j,o exp(-βqjWj) ∑ j)1
(2.8)
where qj ) Zjqe and n′j,o is the bulk concentration. There are two approximations employed by Debye and Hu¨ckel that received considerable scrutiny by their contemporaries. The first is the substitution of the mean
qjκ
(2.13)
(1 + κaj)
Hence the total electrical energy of the system, obtained by a charging process against the ion field,24,25 is given upon substitution of eq 2.13 into eq 2.6:
U
elec
)-
qi2κ
1
∑i Ni (1 + κa )
2
(2.14)
i
To perform the integration in eq 2.5 leading to the free energy, Debye and Hu¨ckel24 first converted to the new integration parameter, κ
dT
kB
)-
T2
2πq2p
κ dκ
∑i niZi
(2.15)
2
The DH expression for the electrical free energy is
Aelec DH T
s
Fc )
〈φj〉other ) -
(2.7)
where W denotes the potential of mean force of the entire collection of ions, E h is the electric field vector, ∇ h is the Laplacian operator (taken to be in spherical coordinates and a function only of the radial distance r), and Fc is the charge density expressed as a Boltzmann distribution
(2.12)
and the potential at the surface of the cavity due to all of the other ions present in the solution
(2.6)
where qi is the charge on the ith ion with number Ni and Wi is the potential of mean force felt by the ith ion due to all of the other ions present. The starting point of the DH theory is the Poisson equation
(2.11)
where the first term of the expansion was set equal to zero to satisfy the electroneutrality conditions. An expression for 〈φj〉 is given in any standard textbook account of the DH theory:25 (1) isolate the jth ion in a cavity of radius aj (which is the sum of the radii of the two charged particles or the diameter of the particle if they are of identical size); (2) set up expressions for the electrical potential and electric field at the cavity surface; then (3) solve the linearized PB equation using appropriate boundary conditions. The results of this analysis give for the particle of interest
elec
∫UT2
(2.10)
The average of the exponential is transferred to the average of its argument through the linearization approximation 〈exp(-B)〉 = 1 - 〈B〉, which leads to the DH equation
which, in accordance with eq 2.1, has as the electrical part elec
qjn′j,o〈exp(-βqjφj)〉 ∑ j)1
kB
)4π
∑i n′iZi
∑
2 j
κ2
∫0κa 1 + κa
NjZj2
j
dκ
(2.16)
j
where the emphasis on this limiting form is denoted by the subscript DH. Evaluation of the integral is carried (25) McQuarrie, D., Statistical Mechanics; Harper and Row Publishers: New York, 1976; Chapter 15.
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Langmuir, Vol. 15, No. 12, 1999 4097
out with the substitution yj ) κaj, with the result
Aelec DH
)-
∑j
Njqj2 κ
3
χj(yj)
(2.17)
where
χ(yj) )
3 3 1 + ln(1 + yj) - 2(1 + yj) + (1 + yj)2 3 2 2 yj (2.18)
[
]
The classical expression for Φclass is obtained from the canonical partition function for an ideal system of mobile N /N! and the relationship Aclass ) particles, Qideal ) qideal ideal -kBT ln(Q ), with the result for a system of s solute particles s
Φ)
s
qj2 κ
Nj[kB ln(Njqideal) - 1] + ∑Nj ∑ j)0 j)1
3T
Note that the first summation in eq 2.19 includes the solvent, denoted by the subscript “0”. A feature of the DH paper relevant to the present communication is the use of the derivatives with respect to κ, as given in eq 2.15. The plan is to change from the derivative with respect to the number of particles to the derivatives with κ as the variable. This substitution is effected through the definition of the screening parameter given by eq 1.5 and the total volume of the solution s
njvj j ∑ j)1
(2.20)
where vj k is the molar volume of component j, and “0” again denotes the solvent. The derivatives of the function F with respect to the particle number Nj thus become
( ) ( )( ) ∂F ∂F ∂κ ) ∂Nj ∂κ ∂Nj
)
d[κχ(yi)] dκ
[
1 3 1 + yj - 2 ln(1 + yj) yj 1 + yj
]
d(ΦA + ΦB) ) 0
(2.23)
with respect to the net particle change between phases. To illustrate, we outline the DH result for the freezing point depression. Since only the solvent passes between the pure and solution phases, j ) 0 and (∂κ/∂n0) ) -(κ/ 2)(vj 0/V) in accordance with eqs 1.5 and 2.20. The DH expression for the freezing point depression is thus
To
2
R
[
]
λBZj2κ ) Cj 1 ηj(yj) 6 j)1 s
∑
(2.25)
qj〈φij〉 * qi〈φij〉
(2.26)
The inequality of eq 2.26 is easily deduced when one considers a macroion and an electrolyte ion. In the case of the macroion there is a surrounding “cloud” of counterions whereas there is not a corresponding “cloud” of macroions about a counterion. 3. The Pair Potential and Macroion Systems
(2.22)
The equilibrium between two functions is determined by minimization of the function
h ofus To - Tf ∆H
4π 〈F 〉 ) -β[〈(∇W)2〉 - 〈∇W〉2] c
The terms on the rhs of eq 2.25 are collectively referred to as the fluctuation potential. Onsager28 noted that substitution of the average number of particles cannot be correct if only electrostatic interactions are present since the integrals do not converge for small values of r. A finite hard sphere interaction had to be added. Kirkwood29 redefined the average number of particles to include the fluctuation potential and then noted that by this definition the fluctuation potential did not have to be small. It was further concluded that the linearized form of the DH theory was consistent with the precepts of statistical mechanics, but the nonlinearized form was not so accommodating. An important shortcoming of the DH theory relevant to macroionic systems was noted by Onsager.28 From statistical mechanics, the work associated with the charging process should be invariant with respect to the interchange of particles, i.e., wij ) wji. However, the average potential about two dissimilar particles is not the same. Hence
(2.21)
Upon setting F ) κχ(κaj) ) κχ(yj), we define the function ηj(yj)
ηj(yi) )
∇2φij +
χj(yj) (2.19)
V ) n0vj 0 +
where Tf is the freezing point of the solution, To is the freezing point of the pure solvent, R is the molar ideal gas constant, ∆Hofus is the heat of fusion for the solvent, and Cj is the molar concentration of the jth solute. The function defined by eq 2.24 exhibits a maximum as a function of ionic concentration, in qualitative agreement with the data given in their Figure 6. Criticisms of the Debye-Hu 1 ckel Theory. There have since been many refinements in the DH theory. Most notably is that of Bjerrum,26 in which the experimental deviations from the DH predictions was interpreted by as the formation of “ion pairs” that reduced the number of charged particles in the system. The DH theory has also been examined on the basis of its foundations in statistical mechanics. Fowler27 objected to the substitution of the mean potential for the potential of mean force. He concluded that a fluctuation potential be added to the DH expression, viz.
Attention is now turned to systems of macroions in which the physical properties are described in terms of an “effective” pair potential. The classic and highly subscribed model for such a description is the DLVO (DerjaguinLandau-Verwey-Overbeek) mean field potential for the interaction between an isolated pair of spherical macroions. The derivation of this potential as described by Verwey and Overbeek1 is reviewed in this section. Also presented in this section are two other forms of the pair potential, which, in essence, represent modifications to the DLVO potential. Kirkwood and Shumaker15,16 considered the effect of surface charge fluctuations on the chemical potential, with the result that there is an (26) Bjerrum, N. K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 1926, 7, 1.
(2.24)
(27) Fowler, R. H. Trans. Faraday Soc. 1927, 23, 434. (28) Onsager, L. Chem. Rev. 1933, 13, 73. (29) Kirkwood, J. G. J. Chem. Phys. 1934, 2, 762.
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Schmitz
attractive component to the mean field potential for the isolated pair. In the theory of Sogami and Ise30 the presence of other macroions in an ordered array is taken into consideration in their model for an “effective” pair potential. 3.1. The DLVO Pair Potential. The motivation behind the development of the DLVO potential was to provide a theoretical basis for the stability and flocculation properties of lyophobic colloids. In this regard the colloid was considered to be in a different phase than the solvent. The charge on the colloid surface built up a corresponding layer of counterions, thus forming the electrical double layer. The work involved in the charging process involved three terms: the surface charge of the colloid; the charging of the counterion layer; and the charging of the added (excess) salt. The surface work was further partitioned into two parts: chemical work and electrical work. The work associated with the double layer was thus associated with the Helmholtz free energy, viz. chem elec elec Aelec ) Acolloid + Acolloid + Acounterion + Aelec salt (3.1)
It was argued that the chemical and electrical work on the surface of the colloid mutually canceled each other and that the presence of equal amounts of positive and negative ions of the added salt resulted in mutual cancellation of added salt terms. Hence, for the DLVO theory, the free energy of the double layer consisted only of the discharge all of the counterions in solution elec Aelec DLVO ) Acounterion
(3.2)
Verwey and Overbeek calculated the interaction potential for both the fix charge and the fixed potential cases and for large particles with thin ion clouds and spherical particles with thick ion clouds. For the present we can summarize the calculation in which the interaction potential between two approaching macroions of fixed charge. The relationships between the interaction potential and the free energy are then VR ) Aelec ) Qcol(φr - φ8), where Qcol is the charge of the colloid and φr is the potential at the distance r. To obtain a relationship between the potential and the surface charge, the linearized form of the PB equation was solved for the interaction between two spherical macroions in the limit of small values of κDHap, where ap is the radius of the macroion and κDH is limited to added electrolyte (cf. eq 1.5). Because of the symmetry about the interparticle axis, the potential was expressed in terms of Legendre polynomials and functions of r. The next step in the development is the ad hoc addition of a van der Waals attraction term, VA. Hence the DLVO pairwise interaction potential as a function of surface charge becomes
UDLVO ) VR + VA q2p exp(2κap) exp(-κr) ) r (1 + κa )2 p
[
( )]
AH 1 1 x2 - 1 + 2 + 2 ln 2 12 x - 1 x x2
(3.3)
where x ) r/2ap is the reduced distance of approach and AH is Hamaker’s constant, which lies in the range of values 10-12J < AH < 10-19 J. (30) Sogami, I.; Ise, N. J. Chem. Phys. 1984, 81, 6320.
Under highly deionized conditions, colloidal systems exhibit an iridescence that can be attributed to Bragg diffraction of the visible light.31 When the colloid concentration is increased, the cubic structures go from bcc to fcc.32 The size of these iridescent crystals can be quite large.33-35 These crystals are “soft” as indicated by their being easily deformed under low shear conditions.36-38 The structures undergo transitions from a three-dimensional crystal, to a two-dimensional crystal, and then to a one-dimensional string.37 Neutron scattering data indicate that the shear behavior of more highly concentrated suspensions (14%, w/w) is different from the more dilute suspensions (1%, w/w).39 Under high shear the crystalline structures break up to form an amorphous order, with an eventual return to a reordering in layers at higher shear rates. The large crystalline domains can be interpreted in terms of the repulsive part of the DLVO potential, in which the distance between the charged spheres is directly related to the concentration of the colloidal particles. It must be emphasized, however, that great care must be taken to obtain these highly ordered crystalline structures. The colloids must be suspended under demanding deionization conditions for several weeks.36,40 Yoshiyama, Sogami, and Ise41 reported Kossel line analysis on highly charged polystyrene latex spheres (PLS), in which the interparticle spacing was found to be significantly different from that expected on the basis of concentration. For example, the expected relationship between the interparticle distance to its radius, R/ap, was given as 1.75/φp1/3 for the bcc structure, where φp is the volume fraction. The observed ratio was reported to be 1.46/φp1/3. This difference was interpreted in terms of a long-range attraction between the colloidal particles. 3.2. The Kirkwood-Shumaker Pair Potential. Kirkwood and Shumaker showed that fluctuations in the net charge of some proteins could account for the experimentally determined values of permanent dipole moments.15 This result led these authors to examine the effect of a fluctuation in the net charge of the proteins on the electrical free energy of the system.16 The pairwise interaction potential between the ith and jth macroions was thus approximated in a manner similar to that of eq 2.10
〈exp(-βqjφij)〉 = 1 - β〈qjφij〉 +
β2 2 2 〈q φ 〉 + ... 2 j ij
(3.4)
The average term 〈qjφij〉 represents the Coulombic repulsion term as in the DLVO theory, where qp2 ) 〈qp〉2 in VR of eq 3.3. The second expansion term of eq 3.4 employs the fluctuation charge qp ) 〈qp〉 + ∆qp for the polyion, which is expressed in terms of the individual charges in the polyion (31) Vanderhoff, J. W.; van den Hul, H. J. J. Macromol. Sci. Chem. 1973, 7, 677. (32) Williams, R.; Crandall, R. S. Phys. Lett. 1974, 48A, 224. (33) Okubo, T. J. Am. Chem. Soc. 1990, 112, 5420. (34) Okubo, T. Naturwissenschaften 1992, 79, 317. (35) Okubo, T. Prog. Polym. Sci. 1993, 18, 481. (36) Clark, N. A.; Hurd, A. J.; Ackerson, B. J. Nature 1979, 281, 57. (37) Ackerson, B. J.; Clark, N. A. Phys. Rev. Lett. 1981, 46, 123. (38) Okubo, T. J. Chem. Soc., Faraday Trans. 1 1984, 84, 1171. (39) Ackerson, B. J.; Hayter, J. B.; Clark, N. A.; Cotter, L. J. Chem. Phys. 1986, 84, 2344. (40) Okubo, T. J. Chem. Phys. 1991, 95, 3690. (41) Yoshiyama, T.; Sogami, I.; Ise, N. Phys. Rev. Lett. 1984, 53, 2153.
Clusters of Macroions of Arbitrary Shape np
qp )
∑
Langmuir, Vol. 15, No. 12, 1999 4099
np
〈qp,m〉 +
m)1
∑
∆qp,m )
m)1 np
qe[
∑
np
〈zp,m〉 +
m)1
∑ ∆zp,m]
(3.5)
m)1
where np is the number of charges of the polyion p. This charge is evaluated on the basis of occupation numbers at each site on the polyion and the orientation of each macroion. We ignore the orientation terms by setting them to unity. Kirkwood and Shumaker proceed to evaluate the term 〈φj2φij2〉, with the general result
β 2 2 〈q φ 〉 ) 〈( 2 j ij
∑
∑〈zj,m〉 + ∑∆zi,m)2(∑〈zi,m〉 + λB2kBT exp(2κdj,i) exp(-2κr)
∆zi,m)2〉
2(1 + κdj,i)2
r2
(3.6)
where dj,i ) aj + ai is the distance between centers of the jth and ith polyion at contact. Although Kirkwood and Shumaker were primarily interested in the role of net charge fluctuations in relationship to measured dipole moments, their formalism is readily applied to the present focus on charge fluctuations and attraction. If one assumes that the fluctuations in either the distribution of charges on the macroion surfaces or the net charge are correlated, then the cross correlation terms are not rigorously zero. This has profound consequences on both the leading and the first correction term in the expansion given by eq 3.4. In the case of the leading term, we have
the basis of concentration (2Do). This observation led to the postulate of a long-range attractive tail in the electrostatic component of the pair interaction potential. A “two-state” model was suggested30 to explain this discrepancy, where “ordered” and “disordered” regions coexisted in the medium. Visualization of the two-state structure and the relative dynamics of both phases was achieved by photographing monodisperse PLS particles.46 It is quite clear that the classical DLVO potential could not be used to explain the stability of the two-state structure. In an attempt to explain these observations, Sogami and Ise30 (SI) proposed a theory for an effective pair potential based on the Gibbs free energy. At the onset of this review, we point out that the SI theory has met its fair share of opponents. Some of the more controversial points are brought out at the end of the present presentation. The SI model initially assumes that the macroions are in fixed lattice positions, and that the volume of the crystalline region, V, is contained within the volume of the system, V*. The starting point of the SI theory was the linearized PB equation
∑
qnFn(r) - 4π
n)1
qin′io ∑ i)1
(3.9)
Comparison of eqs 2.7, 2.11, and 2.25 with eq 3.9 indicates that the SI model is a macroion counterpart to the fluctuation potential. After the introduction of a shift in potential
ψ(r) ) φ(r) +
〈qjφij〉 ) exp(2κap) exp(-κr) (3.7) (〈qj〉〈qi〉 + 〈∆qj(r)∆qi(r)〉) r (1 + κa)2
ν
N
(∇2 - κ2)ψ(r) ) -4π
4π
ν
∑n′ioqi i)1
(3.10)
and subsequent substitution into eq 3.9, these authors obtained for the shifted field N
∑ qnFn(r) n)1
where
(∇2 - κ2)φ(r) ) -4π 〈∆qj(r)∆qi(r)〉 )
∑ ∑〈∆qj,m(r)∆qi,p(r)〉 m p
(3.8)
Although the separation distance r is not explicitly taken into consideration, 〈∆qj(r)∆qi(r)〉 must be a function of the separation distance. This is because at large distances one may expect 〈∆qj(r)∆qi(r)〉 ) 〈∆qj(r)〉〈∆qi(r)〉 ) 0 since there is no correlation between the two macroions. If there is a negative correlation between the charge groups on the macroions, then the double sum in eq 3.8 gives rise to an attraction between the two macroions. However, this term is expected to be most important near the isoelectric point. For highly charged macroions the presence of the double sum term acts to reduce the value of the effective charge when the DLVO potential is used to analyze the data. On the other hand, a positive correlation gives rise to an increase in the repulsion relative to the expectations of the DLVO potential. 3.3. The Sogami-Ise Pair Potential. For several years Ise and co-workers42-45 have pointed out that that average spacing between particles in the crystalline state (2Dexp) was significantly smaller than that expected on (42) Ise, N.; Okubo, T. Acc. Chem. Res. 1980, 13, 303. (43) Ise, N.; Okubo, T.; Yamamoto, K.; Kawai, H.; Hashimoto, T.; Fujimura, M.; Hiragi, Y. J. Am. Chem. Soc. 1980, 102, 7901. (44) Ise, N.; Okubo, T.; Sugimura, M.; Ito, K., Nolte, H. J. J. Chem. Phys. 1983, 78, 536. (45) Ise, N.; Okubo, T.; Kunugi, S.; Matsuoka, H.; Yamamoto, K. J. Chem. Phys. 1984, 81, 3294.
(3.11)
This latter expression was solved by Fourier transformation, with the result
φ(r) )
1
N
∑ qn∫ 2 n)1
2π
1
F˜ (k) exp(ik‚r) dk k + κ2 2
(3.12)
Hence the charge density was now expressed as
Fc(r) ) -
N κ2 qnFn(r) φ(r) + 4π n)1
∑
(3.13)
The potential φ(r) was thus identified with the distribution of small ions. The development thus far is standard. For example, eq 3.9 is precisely the same form as used by Beresford-Smith et al.47 in their study of colloidal systems in low added electrolyte conditions. Also, Fc(r) defined by eq 3.13 results in the Poisson equation given by eq II.2 of Alexander et al.48 in their theory of charge renormalization. The controversy therefore does not lie in the basic equations (46) Ise, N.; Okubo, T.; Ito, K.; Dosho, S.; Sogami, I. J. Colloid Interface Sci. 1985, 103, 292. (47) Beresford-Smith, B.; Chan, D. Y. C.; Mitchell, D. J. J. Colloid Interface Sci. 1985, 105, 216. (48) Alexander, D.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P. J. Chem. Phys. 1984, 80, 5776.
4100 Langmuir, Vol. 15, No. 12, 1999
Schmitz
of the SI theory, in which the interdependence of small ion and macroion distributions is defined. The controversy is in the application of the Gibbs free energy rather than the Helmholtz free energy as the function of choice. Recall in the review of the DH theory, the preferred function of choice was the Gibbs over the Helmholtz free energy for simple salt systems (cf. eq 2.2). It was postulated by Sogami and Ise that the relationship between the Helmholtz electrical free energy (denoted in this text by Aelec) and the Gibbs electrical free energy (denoted by Gelec) is
∑i
Gelec )
( ) ∂Aelec
n′i
electrolyte
∂n′i
∑n
+
( )
Zn
macroion
∂Aelec ∂Zn
(3.14)
where the variation in the valency of the macroion (second term) led to the introduction of ions into the solution (first term). The reason that they did not include the derivative with respect to the number of macroions was that their number was assumed to be fixed. Upon taking the volume of the solution to be fixed, viz., κj2 ∝ nj/V, and using the Debye-Hu¨ckel24 transformation relationship given by eq 2.21, the first term in eq 3.14 was found to be
∑i
( )
n′i
∂Aelec ∂n′i
)
∑i
n′i
( )( ) ( ) ∂Aelec ∂κ2
∂n′i
∂κ2
) κ2
∂Aelec ∂κ2
(3.15)
The second term in eq 3.12 was evaluated on the basis that the charge in the pairwise interaction enters as the square, with the result
( )
∑n
Zn
∂Aelec ∂Zn
) 2(A - Ao)
(3.16)
where the difference A - Ao refers only to electrical interactions, where Ao is the nonelectrical part to the Helmholtz free energy. The SI operator is thus obtained:
G
SI,elec
(
)
∂ ) 2 + κ 2 (A - Ao) ∂κ 2
(3.17)
The SI result for the pairwise interaction between the mth and nth macroion in the Gibbs formulation, GSI,elec mn was thus reported to be
GSI,elec mn
q/mq/n exp(-κRmn) (2 + 2C - κRmn) ) 2 Rmn
(3.18)
where the effective charge is
sinh(κam) q/m ) qm κam
(3.19)
and
C)
κam coth(κam) + κan coth(κan) 2
A minimum occurs in
Rmin )
GSI mn
∑p |Zp|n′pZ2j ) κ2DH + κ2c
κ2 ) κ2DH + 4πλB
(3.22)
where Zj is the charge on the counterion, n′p is the number density of macroions of charge magnitude |Zp|, and κDH2 is defined by eq 1.5, i.e., is based solely on the added electrolyte, and κc2 is the contribution from the released counterions. In contrast, the SI potential is an “effective pair potential” that explicitly attempts to take into consideration all of the other macroions in the derivation of the pair potential as manifested through the “fluctuation potential”. This bridge between the small ion and macroion domains is made through the use of the Gibbs free energy as defined by eq 3.14. All of the major criticisms of the SI potential stem from the preference of the Gibbs free energy over the Helmholtz free energy, even though this was also the preference in the DH paper. The Overbeek “Solvent Correction Term”. The first major paper to question the SI potential was due to Overbeek.55 Rather than express the screening parameter as κj2 ∝ nj/V, Overbeek chose to express the screening parameter as κj2 ∝ nj/no, where no is the number of solvent particles. Hence by placing the “correction term” no(∂A/
(3.20)
at
C + 1 + [(C+1)(C+3)]1/2 κ
a stable minimum for crystalline structures coexisting with liquid structures. Since that time there have been several studies that confirm the experimental observations of a “two-phase” system. The most reliable methods to study the two-state structure are video methods, since there is no need for a theory to interpret observations made by eye. Using video imaging techniques, Ito et al. 49 examined the dynamics of the two states and their interface. It was reported that the crystalline region displayed some distortion and that the particles at the interface sometimes “evaporated” into the solution or “condensed” into the order region. Larsen and Grier 50,51 also observed the kinetics at the crystalline-liquid interface, and concluded that an attraction occurred between these interfacial particles and the lattice based on the numerous vibrations prior to “evaporation” into the less dense phase. Ise and co-workers52,53 observed “voids” within the suspension, i.e., regions with no colloidal particles present. The voids were observed 60 µm into the medium,52 thus indicating that their existence was not a surface effect. Yoshida, Ise, and Hashimoto53 reported that the size of the void regions grew with time, thus supporting the notion that long-range attractive interactions were operative. Monte Carlo simulations by Tata and Ise54 using the SI potential indicated a phase separation and the development of void structures with disordered and ordered domains. Controversies Over the Sogami-Ise Pair Potential. As first pointed out by Beresford-Smith and coworkers,47 the DLVO potential may be considered to be a “true pair potential” since it is independent of the density of colloidal particles. One can make the DLVO potential sensitive to the concentration of other colloidal particles by the ad hoc addition of counterions to the screening parameter, viz.
(3.21)
The SI potential, in the “Gibbs” form, thus achieved what it set out to do, namely to describe the presence of
(49) Ito, K.; Nakamura, H.; Yoshida, H.; Ise, N. J. Am. Chem. Soc. 1988, 110, 6955. (50) Larsen, A. E.; Grier, D. G. Phys. Rev. Lett. 1996, 76, 3862. (51) Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230. (52) Dosho, S.; Ise, N.; Ito, K.; Iwai, S.; Kitano, H.; Matsuoka, H.; Nakamura, H.; Okumura, H.; Ono, T.; Sogami, I.; Ueno, Y.; Yoshida, H.; Yoshiyama, T. Langmuir 1993, 9, 394. (53) Yoshida, H.; Ise, I.; Hashimoto, T. J. Chem. Phys. 1995, 103, 10146. (54) Tata, B. V. R.; Ise, N. Phys. Rev. B 1996, 54, 1. (55) Overbeek, J. Th. G. J. Chem. Phys. 1987, 87, 440.
Clusters of Macroions of Arbitrary Shape
Langmuir, Vol. 15, No. 12, 1999 4101
∂no) ) -κ2(∂A/∂κ2) in eq 3.14 the source of the attraction in the SI theory thus vanishes. However, correction terms that are applied to one theory should also be applicable to all theories. Recall that in the DLVO theory the work (Helmholtz free energy) was due solely to the discharge of ions in solution, where the italics appeared in the work of Verwey and Overbeek1 (p 58). Smalley56 pointed out that if the “correction” term were also applied to the DLVO potential, then there would be no electrical interactions in highly charged systems. We have proposed an alternative approach that circumvents the problem of the “solvent correction” term altogether.57 We criticized the asymmetric use of eq 3.14 in which the introduction of particles was limited to the counterions and the charging operation to the macroions. The approach we used was to include all particles of the system, including the solvent, in both derivative sums. If all of the components are held at fixed ratios, then the number derivatives in eq 3.14 rigorously add to zero. It was then assumed that the charging process involved pairwise interactions of the general Helmholtz free energy elec ) ZRZβHRβ(κ), where HRβ(κ) need not be a form, ARβ known function of κ. The charging process is thus described by the integration25
1
(
) (
∂ZRZβHRβ(κ)
∑R ∑β 2ZR
∂ZR
)
∂ λ dλ ) 1 + κ2 ZRZβHRβ(κ) ∂κ2 (3.23)
This differs from the SI operator in eq 3.17 because of the factor 1/2 in eq 3.23 to account for the double counting of the charging process. It was also noted that eq 3.23 obtains from the standard thermodynamic relationship between Gibbs and Helmholtz electrostatic free energies57
(
)
∂ Gelec ) Aelec + PelecV ) 1 + κ2 2 Aelec (3.24) ∂κ where the internal electrostatic pressure Pelec is defined as
( ) ( )( ) ( )
Pelec ) -
κ2 ∂Aelec ∂Aelec ∂κ2 ∂Aelec )) 2 ∂V ∂V ∂κ V ∂κ2
(3.25)
where we have used the DH change in variables as given in eq 2.15. We have identified the additional term in the SI potential with the internal electrostatic pressure of the system, in agreement with the DH theory. Since this term was obtained by two different methods, via DH charging process for the macroions and the electrolyte ions and also via basic thermodynamic relationships, the introduction of the solvent correction term of Overbeek becomes a moot point. The indefensible asymmetric application of the “solvent correction term” -κ2∂/∂κ2 to the SI potential and not to the DLVO potential is thus avoided. Woodward and the External Osmotic Pressure. A second paper of note is that of Woodward,58 who investigated in great detail the use of the Gibbs free energy in the SI theory. As mentioned above, Sogami and Ise differentiated between the volume of the system (V*) and the volume to which the crystalline state was confined (V). Woodward argued that there was no mechanism to confine the ions to any region of the solution. He then proceeded to perform an exact calculation on a system (56) Smalley, M. V. Mol. Phys. 1990, 71, 1251. (57) Schmitz, K. S. Langmuir 1996, 12, 3828. (58) Woodward, C. E. J. Chem. Phys. 1988, 89, 5140.
with two thin walls of negative surface charge that are enclosed by neutral surfaces. Although an attraction was obtained, he argued that it would not be sufficient to overcome the repulsive part under any conditions. However, Woodward’s approach was from the classical point of view of the relationship between the Gibbs and Helmholtz free energies, where his calculations were tailored to the effect of some external osmotic pressure. It is our present position that the colloidal crystals are subject to an internal electrostatic pressure. Let us first examine what is meant in the SI paper regarding the two different volumes, V* and V, in relation to the charge renormalization theory of Alexander et al.48 As mentioned, both of these theories begin with the same PB expression and therefore should have similar, although not always obvious, interpretations of the physical nature of the colloidal system. The relevant quote from the Alexander et al. paper is48 More physically, one may argue that if the electrostatic potential energy gained by a counterion in reassociating on the surface of a macroion exceeds thermal energies, it will bind and therefore renormalize downward the effective energy for subsequent counterions. This process will continue until the chemical equilibrium described by eq I.7 obtains. The relevant quote from the SI paper is4 Henceforth the behavior of the system is described in terms of the variable V whose value is determined by the equilibrium conditions which depends on the configurations and conformations of the macroions [see eq 30]. If one extends the criterion of Alexander et al. that “bound ions” are identified as equivalence of electrostatic and thermal energies to include ions in the diffuse cloud, then clearly the range of this definition varies with volume as the ionic strength and the charge of the macroion varies. This is shown in the present discourse in Section 8 on the “screened” charge of the macroion. It is this range is what Sogami and Ise associate with the volume V. This interpretation of V is philosophically aligned with the extension of the counterion condensation theory of Manning59 as extended to include two parallel rods,60,61 To quote from Ray and Manning:60 The assembly of two lines at a separation distance F is associated with a spatial region within which the condensed counterions randomly translate. We shall see that the volume (and shape) of this region depends on F. Hence the philosophical basis of the Woodward paper is mutually exclusive of that of the Alexander et al.48 and Manning59-61 papers. One cannot embrace the Woodward paper as a viable criticism of the SI theory without simultaneously rejecting the concepts of charge renormalization and counterion condensation. It is noted that Debye and Hu¨ckel discussed in detail the electrostatic pressure (Pe in their notation) and make the following statement:24 Strictly speaking, it is incorrect to use the classical expression for V (as a function of T and P) without regard to the electrical effect of the ions, since the pressure Pe causes also a change in volume. (59) Manning, G. S. Q. Rev. Biophys. 1978, 11, 179. (60) Ray, J.; Manning, G. S. Langmuir 1994, 10, 2450. (61) Manning, G. S. Second International Symposium on Polyelectrolytes, Inuyama, Japan, 1998.
4102 Langmuir, Vol. 15, No. 12, 1999
Schmitz
They then proceeded to calculate the electrostatic pressure of a 1 M solution of KCl, with the result that the electrostatic pressure of 20 atm causes a relative volume change of 0.001 for water. Due to the incompressibility of water, the preferred Gibbs free energy gave way to the Helmholtz free energy in their study. Hence the electrostatic pressure should affect not the volume V* in the S-I paper, and does not contradict the conclusion of Woodward based on the classical distinction between Gibbs and Helmholtz free energies. However, the colloidal crystals are quite compressible and the associated volumes of the crystalline regions are readily responsive to the internal electrostatic pressure. As we see in the next paragraph, this internal pressure may arise from an inhomogeneous distribution of electrolyte ions. van Roij and Hansen and Spinodal Instabilities. Wishing to preserve cornerstone form of a repulsive pair potential as in the DLVO theory, van Roij and Hansen62 (RH) proposed another model to explain the two-state structure observed in PLS systems. The full Hamiltonian of the system used in the RH paper, which explicitly included the idea gas part contained in the Ao part of the SI theory, was
Heff m ) Km +
1
Nm
∑
2 i*j
(
vhs(rij) +
)
qp2 exp(-κrij)
rij Aid -
+
qp2 κNm 2 1 + κap
(3.26)
where Km is the kinetic energy, vhs(rij) is the hard sphere pair potential, ap is the radius of the macroion, and Aid is the ideal gas contribution to the free energy, which is analogous to Φclass in eq 2.4. The authors found it necessary to include the counterions and the ionized water in the calculation of the screening parameter
κ2 ) 4πλB(2n′s + Zpn′p + 2n′H+) ) κ2DH + κ2c + κ2H+ (3.27) where [H+] ) [OH-] ) 10-7 M. Van Roij and Hansen pointed out that the last term in eq 3.26 is the driving force behind the spinodal instability and that the pair potential remains as a purely repulsive potential. Graphs were presented as a function of the volume fraction of the macroions, using parameters similar to those in experiments. The authors reported that in the region of coexistence, the dense region of macroions had less salt than in the “void” regions. The RH interpretation of the two-state system as being a result of a spinodal instability necessarily results in an inhomogenious distribution of salt. Therefore, inherent in their model but not explicitly considered, is the electrostatic pressure that arises from such a distribution. In contrast to the solvent, the colloidal crystals are easily compressed by “local” heating,3 low shear rates,35,36,38,39 and gravitational fields.33 Therefore the “effective” pairwise interaction between the macroions must include the electrostatic pressure as defined by eq 3.25. This means that the following substitution must be made in eq 3.26:
(
)
2 q2p exp(-κrij) ∂ qp exp(-κrij) f 1 + κ2 2 (3.28) rij rij ∂κ
(62) van Roij, R.; Hansen, J.-P. Phys. Rev. Lett. 1997, 79, 3082.
Therefore the form of the pairwise interaction does have an attraction component even for phase separations driven by spinodal instabilities! We note that a phase separation into high and low salt regions (salt fractionation) was also predicted by Smalley, Scha¨rtl, and Hashimoto63 (SSH) on the basis of the SI potential alone. These authors deduced the following phase behavior: (1) 0 e κa e 0.816, where the particles congregate according to the SI potential but there is no two-state structure; (2) 0.816 e κa e 3.05, where the particles still congregate in SI structures but now the maximum fractionation of salt occurs which is accompanied by an increase in the osmotic pressure in the “void” regions of the macroion distribution; and (3) 3.05 e κa, where there is no macroion-ordered structure in the suspension. Since the DLVO and SI pairwise potentials are equivalent in the limit κ f 0, the first region of the SSH theory is identified with a crystalline structure whose average spacing is based on the concentration. Of course the two theories are in concert in the high salt limit. It is the intermediate region where the two potentials can be distinguished. We compare the RH and SSH theories in regard to the location of the transition to the two-state structure. Using a particle radius of 366 nm and charge Z ) 6881, Van Roij and Hansen state an instability occurs for ns < 2 × 10-6 M. In the SSH model the screening parameter is calculated to be κ = 0.816/a ) 2.23 × 104 cm-1, or a molar concentration of monovalent ions (such as NaCl) of [Na+] ) [Cl-1] ) 9.2 × 10-7 M. The two models are surprisingly in good qualitative agreement! It must be emphasized that there is no experimental evidence that the salt does indeed fractionate in a manner described by the RH or SSH theories. Hence agreement between the two models in regard to the predictions of the transition point does not endorse the spinodal instability mechanism nor does it verify the SSH mechanism. However, the above exercise does indicate that the two approaches are in concert with each other and thus cannot be distinguished by experiment. The experimental observation is an apparent attraction between macroions, which is contained in both theories. Both the RH and SSH theories are focused on the effective pair potential: the RH theory centers on the small ion spinodal properties with no explicit connection to the distribution of macroions with the intent to preserve the DLVO form of the pair potential, and the SSH theory is a study of the properties of the SI pair potential. However, the premise of the present paper is on the dominance of multibodied effects on the phase separation. More relevant to the present view is the paper on phase separation by Michaeli, Overbeek, and Voorn (MOV).64 In the present terminology, the MOV model assumed the existence of “macroion regions” and “void regions” whose separation stability was governed by the Gibbs free energy. The components of the Gibbs free energy were the electrostatic interactions between the macroions and counterions in the “macroion phase”, Ge, and the entropy of mixing of the solvent and counterions between the two phases, Gs. Using volume-normalized Gibbs free energies, viz., G h ) G/V, these authors obtained the following two inequalities (63) Smalley, M. V.; Scha¨rtl, W.; Hashimoto, T. Langmuir 1996, 12, 2340. (64) Michaeli, I.; Overbeek, J. Th. G.; Voorn, M. J. J. Polym. Sci. 1957, 23, 443.
Clusters of Macroions of Arbitrary Shape
∂2G hs 2 ∂φpolymer
Langmuir, Vol. 15, No. 12, 1999 4103
>0
(3.29)
ap, the concept of an “effective hard sphere radius” for a macroion does not have parallel properties of “true” hard spheres. For example, phase transitions in the true hard sphere system results from an increase in the concentration of the spheres, whereas the corresponding transitions in the Okubo model occur upon decreasing the macroion concentration. 6. What Is the Theoretical Expression for K As reviewed in the previous paragraph, the screening parameter κ appears as a parameter in the analytical expression for an “effective” radius Rtherm that differentiates “thermodynamically bound” counterions from the “thermodynamically free” counterions. But how does one calculate a value of κ? In the Debye-Hu¨ckel theory, κ is simply defined by all of the simple ions present, as given in eq 1.5. For colloidal systems, however, there is no general consensus as to its definition. The Beresford-Smith et al.47 definition of κ includes the counterions as well as the added electrolyte, as defined by eq 3.20. Kirkwood and Shumaker16 include the fluctuations in the counterion concentration that arise from the fluctuations in surface charge. Counterion condensation theories48,59,60 which partition the solution into two-states only include the “free” counterions in the calculation of κ. Van Roij and Hansen62 found it necessary to include the ions of the solvent in their definition of κ (cf. eq 3.25). In the coupled mode theory of Lin, Lee, and Schurr,5 the evaluation of κ includes the term Z2j nj for all species present: the added electrolyte, counterions, and the macroions. There is also the question as to whether there is only one “global” value of κ, or if there are “local” values of the screening parameter. Certainly if there is a phase separation leading to “local” concentrations of salt, as in the RH and SSH theories, then one might anticipate different screening factors in the different regions. In fact, Alexandrowicz and Katchalsky70 solved the PB equation by subdivision of the system into two regions. The “inner region” was the “salt free” solution, where it was assumed that the high charge of the cylinder excluded the co-ions. The “outer region” was assumed to have sufficiently high salt to screen the cylinder from other such cylinders in the solution. Hence κinner * κouter. Similarly, Podgornik and Parsegian67 distinguish between screening lengths inside a bundle of rods and between the bundles of rods. We are taking the position that “local” screening and nonlinear effects are intimately associated with each other in the interpretation of data on these complex macroion systems. We partially justify this view in the following section in which the definition of Rtherm as defined by eq 5.4 gives good results in the analysis of conductivity and voltammetry data on “free” counterion concentrations. 7. Experimental Evidence in Favor of “Thermodynamically Bound” Counterions According to the Gibbs-Duhem relationship, there is a mutual interrelationship between the activities of all components in a system. The interpretation of these changes are, to a certain extent, dependent upon the disposition of the interpreter. As in the case of the DebyeHu¨ckel model of simple salts, deviations from the limiting (70) Alexandrowicz, Z.; Katchalsky, A. J. Polym. Sci., Part A 1963, 1, 3231.
Clusters of Macroions of Arbitrary Shape
expressions may be interpreted in terms of either “ion pair association” or “fluctuation potentials” as discussed in section 2. In a similar manner this situation arises in macroionic systems, where the terms “thermodynamically free” and “thermodynamically bound” are devised to interpret the difference in measurements of ions in the presence and absence of the macroions. Operationally, the fraction of “free” counterions, denoted by “f”, in a macroion system is the ratio of measured concentrations to expected concentrations, the latter based on the theoretical (titration) charge of the macroion. Whether or not this partitioning of ions represents a situation in which case the “bound” counterions are removed from the solution, as required by condensation theories, or a deviation from a mean field approximation, in which case the counterions are allowed to roam the entire solution, as in the case of fluctuation potentials, has not yet been established. The “condensation” approach has been used to interpret conductivity data.71,72 Ito, Ise, and Okubo71 studied the conductance of linear polystyrene sulfonate (PSS) and PLS, and calculated the transference numbers for the counterions. They reported that the fraction of free counterions for the latices was in the range 0.03 < f < 0.16, whereas for the linear polyion they obtained the value f ) 0.37. These differences in f values were attributed to differences in the respective charge densities, where the higher bound fraction obtained for the higher charge density. For example, the highest charged latex (7.2 × 105 sulfates/ particle) had the value f ) 0.04 whereas the lesser charge latex (1.7 × 104 sulfates/particle) had the value f ) 0.1. Penafiel and Litovitz72 also used low frequency (500 Hz - 1 kHz) conductivity measurements in their studies of sodium polyacrylate solutions. The degree of ionization was varied by pH adjustments, and the conductivity increment ∆σ ) σT - σsolv(1 - v) was determined, where σT is the measured value corrected for the solvent contribution σsolv(1 - v) of volume fraction v. A discontinuity in the ∆σ versus ξOM profile occurred at ξOM ) 1.7, where ξOM is defined by eq 5.1. This discontinuity was interpreted as “counterion condensation” even though the value of 1.7 exceeded the predicted value ξOM ) 1. The measured conductivity was interpreted in terms of three contributions: the diffuse ion cloud about the polyion (σD), the bulk ions (σB), and the polyion (σP). The condensed counterions were said to be territorially bound to the polyion and thus did not contribute to the conductivity. Voltammetric studies were performed on linear polystyrenesulfonic acid73 (PSSA) and PLS particles.74,75 For a disk microelectrode of radius ro, the steady-state current was interpreted in terms of the diffusion-limited expression for the limiting current, il ) 4nFDc(H+)ro, where n is the stoichiometric number of transferred electrons, F is the Faraday, and c(H+) is the bulk concentration of the hydrogen ions. The results of the voltammetric studies exhibited similar behavior as the conductivity studies on PSS and PLS, 71 viz., the measured current for the linear polyelectrolyte was about 35% of that for the polyelectrolyte-free solution whereas for the PLS system it was about 4%. Rather than retain a constant value of D with a reduction in the value of c(H+) to that of “free” counterions, these data were interpreted in terms of an (71) Ito, K.; Ise, N.; Okubo, T. J. Chem. Phys. 1985, 82, 5732. (72) Penafiel, L. M.; Litovitz, T. A. J. Chem. Phys. 1992, 96, 3033. (73) Ciszkowska, M.; Osteryoung, J. G. J. Phys. Chem. 1994, 98, 3194 (74) Roberts, J. M.; Linse, P., Osteryoung, J. G. Langmuir 1998, 14, 204. (75) Roberts, J. M.; O’Deas, J. J., Osteryoung, J. G. Anal. Chem. 1998, 70, 3667.
Langmuir, Vol. 15, No. 12, 1999 4105
adjustment in the value of D due to the presence of the macroion. To evaluate D, the cell model of Jo¨nsson et al.76 was employed, in which a macroion was centered at the origin and the distribution of the “labeled” compound (the counterions) was expressed as an expansion in spherical harmonics. The “effective” diffusion coefficient, Deff, for this model was represented as
Deff ) Doc(b)U(b)/〈c〉
(7.1)
where Do is the self-diffusion coefficient at zero electrostatic potential, c(b) is the counterion concentration at the cell boundary, 〈c〉 is the bulk concentration, and U(R) satisfies a differential equation that results from the spherical harmonic expansion of the mole fraction of labeled particles and evaluated at the cell boundary. Using a cell model with the macroion at its center, Roberts et al.74 solved the nonlinear PB equation for the local potential was solved numerically, the concentration of counterions determined at the boundaries, and the selfdiffusion coefficient of monovalent counterions was then calculated from eq 7.1. Excellent results were obtained between their calculations and experiment. These authors also noted that if the number of “bound” counterions was identified with those with energies greater than kBT, then the fraction of free counterions was found to be f ) 0.038. 8. Surface Charge, Renormalized Charge, and the Screened Coulomb Charge It has been known for some time that the effective charged measured by some experiments is considerably smaller than the surface charge based on ionization of the surface groups. One interpretation of this observation led to the concept of “charge renormalization” where the unscreened Coulomb interaction of the macroion with the counterion effectively neutralizes a certain fraction of the surface charge.48 Using the expression for renormalized charge of Alexander et al.,48 Ito, Ise, and Okubo71 calculated for their sample #1 a value of f ) 0.001, which is considerably smaller than the experimental value of 0.04. We now show that “bound counterions” as defined by distances r < Rtherm may provide a more accurate account for the experimental observations. Let us define an “effective charge” Zeff in terms of the counterion distribution function nc(r)
Zeff ) Zp
∫aR
nc(r)r2 dr
therm
p
(8.1)
where nc(r) is a normalized distribution function for the counterions
∫a∞nc(r)r2 dr ) 1
(8.2)
p
Since Rtherm is identified as the 1/e point for the Boltzmann weighting factor, we assume nc(r) decays as exp(-r/Rtherm). Hence we obtain from eqs 8.1 and 8.2
exp nc(r) )
( ) ap Rtherm
2 Rtherm(a2p + 2apRtherm + 2Rtherm )
(
×
exp -
)
r (8.3) Rtherm
(76) Jo¨nsson. B.; Wennerstro¨m, H.; Nilsson, P. G.; Linse, P. Colloid Polym. Sci. 1986, 77-78, L961.
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Schmitz
We now substitute for Rtherm the equivalence given by eq 5.4, with the resulting expression for the fraction of released counterions f
Zeff 5 exp(Y - 1) )f )1- 2 Zp Y + 2Y + 2
(8.4)
where Y ) ap/Rtherm. We now relate the screening parameter κ to the volume fraction of the macroion, φp ) nc4πap3/3, and the assumption of zero added salt. Hence the product κcap, where defined by eq 3.22, is now expressed as
x
κcap ) x3λBφpZc2
|Zp| )s ap
x
|Zp| ap
(8.5)
and we have for the ratio f ) Zeff/Zp
Zeff )1Zp
(
5 exp -
1
)
1 + sxZp/ap
(1 + sxZp/ap)2
2 + 5s2(Zp/ap) + 6sxZp/ap
)f (8.6)
In the limit Y f 1 (Zp f ∞ or ap f 0) we have Zeff/Zp f 0, and in the limit Y f 0 (ap f ∞ or Zp f 0), we have Zeff/Zp f 0.0803. This reduced ratio is consistent with the experimental data summarized in Table 3 of Roberts et al.75 for the highly charged latex particles. The physical significance of Zeff in the present model is that it represents a “screened” charge pertaining to the unit of radius Rtherm rather than a “renormalized” charge with counterions condensed on the surface of the macroion of radius ap.48 To obtain the cell model conditions of the model of Roberts et al.,75 one takes the limit φp f 0, from which we have the limit Zeff/Zp f 0.0803. Hence the results of the two models are essentially identical with respect to the degree of charge reduction for the isolated macroions. Let us now apply eq 8.4 to sample #1 of Ito et al.71 The charge equivalent/L ) c for this study was reported to be 0.0028 and f ) 0.04, which results in a concentration of “free ions” as being cf ) 1.12 × 10-4 M. From this concentration the screening parameter is κc ≈ x4πλBcfNA/1000 ) 2.5 × 105 cm-1, or a screening distance of 1/κc ) 400 Å. The boundary for the linearized region for the macroion-electrolyte system is thus calculated from eq 5.4 to be at Rtherm ) 775 Å. Hence Y ) 375/775 ) 0.484, from which the fraction of free ions calculated from eq 8.4 is f ) 0.068. Thus the concept of a “screened Coulomb charge” that associates counterions with a screened interaction greater than the thermal energy as part of the dynamic macroion unit is in better agreement with experimental results than the charge renormalization model. Having now established that the extent of the ion cloud is related to the effective charge of the macroion, one must take into consideration possible effects that may result if the macroion concentration is sufficiently high that ion clouds overlap. 9. Limits of Applicability of Linearized Expressions The linearized form of the DH theory results in a uniform distribution of particles. That is, the deficit of co-ions is compensated for by the excess counterions such that a uniform density of particles obtains. There is no information to be gained, therefore, about the structure of the electrolyte ion distribution and κ may therefore be regarded as a global parameter. The situation changes,
however, for a system of highly charged macroions in which the deficit of co-ions is overcompensated for by the large excess of counterions. Thus one must consider the effect of this asymmetry in local concentrations of electrolyte ions. Such a discrepancy was taken into consideration by Alexandrowicz and Katchalsky70 in their solution for charged cylinders. Standard linearized solutions to the PB equations, as employed for example in theories such as the DLVO potential, do not contain any information about the structure of the ion cloud. The structure of the ion cloud must also be influenced by the proximity of two or more macroions. As indicated in section 4, nonadditivity effects associated with the distribution of electrolyte ions may lead to an attraction between colloidal particles and phase separation. It is therefore of value to determine the limits of validity for these linearized models for the pair potential between ions and the macroion as well as between macroions. Linearized solutions to the PB equation are rigorously valid only if the pairwise interaction is much less than the thermal energy kBT. We therefore use as a guideline in this exercise the “thermal separation distance” Rtherm as defined by eq 5.4 for the macroionelectrolyte (me) pair. To illustrate the range of validity of linearized theories we use the concepts developed in section 6 on the data of Ito, Ise, and Okubo.71 From the volume fraction φp ) 0.0011 for sample #1 we calculate an average distance between macroions to be (1/np)1/3 ) ap(4π/3φp)1/3 ) 5856 Å, which compares with the separation distance at which the nonlinear regions of two macroions touch, i.e., 2Rtherm ) 1550 Å. This means that approximately three-fourths of the average separation distance may be considered to be in the linear PB regime. The nonlinear region, or overlap region, accounts for a significant fraction of the allowed phase space and must therefore be taken into consideration in data interpretation. What is needed is a description of the macroion system that may provide some insight as to the nature of the electrical properties of the macroion system as a whole without relying on models limited to effective pair potentials. Such an approach, we believe, is the juxtaposition of potential fields23 method, which is outlined in the following section. 10. The JPF Method The JPF method22 is based on the equivalence of the chemical potential in all regions of the system that are at arbitrary distances from the macroion and the treatment of the macroion potential field in a manner similar to that of atoms play in molecules.77,78 We consider a small volume in the solution at location r that does not include any macroion but may contain electrolyte ions as well as the solvent. For simplification we write the contribution of the chemical potential at r in terms of the charged colloidal particles (the reduced potential, φcol(r), electrolyte ions (µions(r), and the solvent (µsolv(r)))
βµ(r) ) βµ ) φcol(r) + βµions(r) + βµsolv(r)
(10.1)
The next step is to treat the colloidal system in a manner that parallels the adiabatic approximation in the description of molecules. Due to the very large size of the colloidal particles relative to the electrolyte ions, the macroions are assumed to be “frozen in time” as the electrolyte ions establish an equilibrium distribution in accordance with (77) Bader, R. F. W.; Slee, T. S.; Cremer, D.; Krata, E. J. Am. Chem. Soc. 1983, 105, 5061. (78) Bader, R. F. W. Acc. Chem. Res. 1985, 18, 9.
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Langmuir, Vol. 15, No. 12, 1999 4107
eq 10.1. Since there are no macroions in the “test volume” one may write for the electrical potential due to all of the colloidal particles of fixed surface charge acting on a test particle of unit charge, in reduced form
Zm
M
φcol(r) ) λB
∑ m)1d
m|rm,red|
(10.2)
where |rm,red| ) |rm|/dm is the reduced distance from the center of the mth macroion of diameter dm and charge Zm, and λB is the Bjerrum constant. Note that the definition of the potential φcol(r) is independent of the electrolyte ions. The next step is to “introduce” the electrolyte ions into the system having the potential field set up in eq 10.2. As the ions are introduced into the system, the concentration is adjusted in accordance to the colloidal potential φcol(r) and also the interactions with the other ions. Hence the parameter βµions(r) contains not only the concentrations but also the ion-ion interactions, the latter being manifested as an activity coefficient. We therefore write in symbolic form Ns
b) ) µions(r
∑ j)1
Ns
[µoj + kBT ln(γj(r)Cj(r))] ∑ j)1
µj(r) )
(10.3)
where Ns is the number of ion types and Cj(r) is the molar concentration of the jth ion. It is assumed that the nonideality arises from purely electrostatic interactions between the ions. Hence we may write for the activity coefficient
γj(r) ) exp(zjφj(r))
(10.4)
where the reduced potential experienced by a test charge, Fj(r), is Ns
zn
∑ n*jd |r
φj(r) ) λB
n
(10.5)
jn,red
where the sum over n includes all of the ions except the jth ion, and |rjn,red| is the reduced distance between the jth and nth ions. within the macroion cluster. From the Gibbs-Duhem expression, we have the following relationship between the gradient of the electric potential set up by the macroions and the gradient in the chemical potential of the electrolyte ions
0 ) ∇φcol(r) + ∇βµions(r) ) ∇φcol(r) + ∇βµideal(r) + ∇βµnonideal(r)
(10.6)
where we have set ∇µsolv(r) ) 0 to simplify the development without loss of generality and have made the usual separation of contributions of ideal and nonideal parts Ns
[µoj + kBT ln(Cj(r))] ∑ j)1
µideal(r) )
(10.7)
Ns
µnonideal(r) )
kBT ln(γj(r)) ∑ j)1 Ns
φφ(r) ) kBTφions(r) ∑ j)1
) kBT
(10.8)
Let us now examine the gradient of the electrical potential in a volume at r in the vicinity of a macroion cluster in which the nonlinear expressions for the pair potential are invalid. The potential gradients -∇φcol(r) and -∇φions(r) are the electrical forces exerted on the test charge. Because of the high charge asymmetry and localization of the macroions in the cluster and the fact that the electrolyte ions are free to roam the entire volume of the container, one has the “conditional” inequality ∇φcol(r) > ∇φions(r). (The reason for the qualifier “conditional” will be made clear.) Let us therefore assume that the contribution -∇φions(r) can be treated as a perturbation on the force -∇φcol(r). Hence from eqs 10.6 and 10.7 we have
β∇µions(r) = ∇ln(Cj)(r) = -∇φcol(r)
(10.9)
We now employ some of the concepts associated with the stability of molecules as presented by Bader and coworkers.77,78 The probability density of the electron cloud is given by the square of the wave function ψψ* ) Fe. In this description the gradient vector field of the probability density, ∇Fe, is determined for a particular set of coordinates of the atoms. There are three regions of interest in this description. That region of space in which the vector gradients terminate at a single point is defined as the location of the atoms. Vectors aligned in such a way as to “connect” two atoms represents an interaction between the atoms, and represents a chemical bond. A “closed surface” defined by the equation (∇Fe)‚n ) 0, where n is a vector normal to the surface, is referred to as a subsystem of the molecule. A subsystem of the molecule is therefore a region in which the electrons are confined. It is this characteristic of the description of molecules that is of interest in the description of macroion clusters. In the JPF method the corresponding vector density gradient expression for the subregion is (-∇φcol)‚n ) 0, with the relationship to the chemical potential of the ions given by eq 10.9. The physical interpretation for colloidal clusters is that the force exerted on the electrolyte ions by the macroions of the cluster is solely along directions tangent to the cluster geometry. If such a surface exists within the interior of the cluster then the electrolyte ions are, with respect to the dominant interaction with the macroions, confined to be within this subregion. Thus the stability (or metastability) of a macroion cluster may be inferred from the presence or absence of a subregion (or partial subregion) interior to the cluster. In the following two sections we apply the JPF method to a cluster of spheres and rods. All calculations were performed with Mathematica, a means of doing mathematics on a computer. 11. The JPF Method and a Cluster of Spheres We have previously applied the JPF to a cluster of two and four spheres, respectively, in the X-Y plane.23 It was concluded from the vector gradient profiles that the interaction between the pair was purely repulsive, due to the observation that no subregion was present. However, the cluster of four macroions may be attractive due to the presence of a subregion in the vector density gradient profile. In the present case we apply the JPF method to a threedimensional cluster of six spheres of diameter dm ) 500 Å and charge Zp ) 100, symmetrically arranged at each location of a distance of 3dm a long the six directions of the axes. The radius of the computation cell is given as Rc ) 10dm, and all spatial distances are represented in terms of reduced coordinates, i.e., X ) x/dm. Constant
4108 Langmuir, Vol. 15, No. 12, 1999
Figure 1. Constant potential contours of cluster of spheres. Six macroions of diameters dm ) 500 Å and charge Zp ) 100 are located a reduced distance R/dm ) 3 along each axis within a cell of relative radius Rc/dm ) 10. The constant potential contours are shown in the X-Y plane.
Figure 2. Profiles of the potential for a cluster of six spheres. The potentials shown above are for a cluster of six spheres as described in Figure 1. Note the constant potential region over the range -1 to +1. Key: (solid line) along the X-axis that passes through the two spheres; (dashed line) along the line passing between the spheres.
potential contours for the array of six macroions projected in the X-Y plane are shown in Figure 1. As in the case of the planar geometry of four macroions,23 there is a region of constant potential within the cluster at the level of resolution of this profile. Ions within this region are not localized about any one macroion. As one moves toward any one macroion, the contours encompass only one macroion. As one moves outside and away from this cluster, the contours become spherical in shape about the cluster. This means that the cluster, when viewed at large distances, appears as a single point source of charge. The potential profiles in the X-Y plane and passing through the macroions (solid line) and between the macroions (dashed line) are superimposed in Figure 2. The location of the macroions is defined at zero potential, by choice in these calculations. The extent of the constant potential region interior to the cluster is now obvious. Ions within this region do not experience any force exerted by the colloidal particles of the cluster. It is also noted that the potential at the surface of a macroion facing toward the interior of the cluster is greater than the diametrically opposite location on the macroion but facing exterior from the cluster. The implication is that the local concentrations of counterions adsorbed onto the macroion surface is greater for the interior surface than the exterior surface of the macroion. The gradient vectors of the potential field are shown in Figure 3. The macroion locations are indicated by the vectors directed to a single point, which is also referred to as the basin.77,78 We move along a line originating at the origin and bisects two basin points in this plot. The
Schmitz
Figure 3. Vector gradient potential in the X-Y plane. The vector gradient potential for the arrangement of spheres described in Figure 1 is shown above.
first encounter of the vector field is that with lines perpendicular to the direction of our observation line of sight. This set of vectors defines a subregion in which the interior is that of the constant potential. Ions within this boundary do not experience any force from the macroions in the cluster. We define this as region I. Upon further movement along this observation line we encounter a family of vectors perpendicular to the observation line. This second family of vectors is identified by their “connecting” of two adjacent basin points, and collectively represent region II of the gradient vector profile. While on the line connecting two adjacent basins, we now turn to one of the basins and move along this line. An important feature along this new line of observation is that the direction of the arrows changes in the vicinity of these basins. We refer to this “inner region” about a basin point as region III. The region exterior to the cluster is referred to as region IV. 12. The JPF Method and a Cluster of Rods There is current interest on the stability of a cluster of rods, with emphasis on rod dimensions that mimic DNA molecules.65-67,79 We therefore examine the possible stability of rod clusters using the JPF approach. In the following the rods are aligned parallel along the Z-axis. The diameter and length of the rods are dm ) 20 Å and L ) 10dm ) 200 Å, respectively. The total charge on each rod is the magnitude Zm ) -120. In these calculations each rod is composed of 10 subunits, each of diameter dm ) 20 Å and centrally located subunit charge of zm ) -12 units. These rods, therefore, mimic a 60 base pair length of DNA. The reduced coordinates are now given in terms of the length of the rod, i.e., X ) x/L, rather than the diameter dm. Two-Rod System. Constant potential contours of the potential field set up by two rods, whose central beads are located at the reduced coordinates ((0.25, 0, 0) and extended symmetrically along the Z-axis, are shown in Figure 4 (top) for the X-Z plane passing through the origin and in Figure 4 (bottom) for the X-Y plane passing through the origin. These contours indicate that the discrete nature of the surface distribution is lost within one or two diameters of the subunit. For the finite length rods, the potential profiles clearly indicate that the concentration (79) Grønbech-Jensen, N.; Mashl, R. J.; Bruinsma, R. F.; Gelbert, W. M. Phys. Rev. Lett. 1997, 78, 2477.
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Langmuir, Vol. 15, No. 12, 1999 4109
Figure 6. Cutaway of the potential surfaces for two rods. Shown above is a cutaway three-dimensional contour plot of the constant potential surfaces φcol(r) ) 0.99φcol((0,0,0)) (inner surface) and φcol(r) ) 0.90φcol((0,0,0)) (outer surface).
Figure 4. Constant potential contours of two rods. Rods of length L ) 200 Å and diameter dm ) 20 Å and total charge magnitude Zp ) -120 are separated by the relative distance r/L ) 1.0. Key: (top) contours in the X-Z plane at Y ) 0; (bottom) contours in the X-Y plane at Z ) 0.
Figure 5. Potential profiles for two rods. The potential profiles for the two-rod system are shown along the X-axis at Y ) 0 and Z ) 0 (dashed line) and along the Y-axis at X ) 0 and Z ) 0 (solid line).
of counterions is greatest at the centers of each rod and that they accumulate in the region between the rods. The potential profiles along the X-axis (passing through the rods) and the Y-axis (passing between the rods) are shown in Figure 5. Shown in Figure 6 is a threedimensional contour plot of the constant potential surfaces φcol(r) ) 0.99 φcol(0,0,0) and 0.90 φcol(0,0,0) as cut through the two rods. The shape of the profiles in these two figures is important. In Figure 5 the maximum along the Y-axis coincides with the minimum along the X-axis. Note the narrow taper of the 0.99 φcol(0,0,0) surface between the two rods in Figure 6. This indicates that the tightly bound ions are highly localized about each rod, with little sharing between the two rods. Vector gradient potential profiles (not shown) do not contain any hint of a subregion, and are similar to those for two spheres in Figure 5 of ref. 23.
Figure 7. Potential contours for six rods in a hexagonal array. The separation distance between rods diagonal in the array is r/L ) 0.5. Key: (top) contours of the potential in the X-Y plane at Z ) 0; (bottom) contours of the potential in the X-Y plane at Z ) 0.4.
Hexagonal Packed Six-Rod System. A hexagonal array of rods was generated, with the center of each rod located a reduced distance of 0.5 from the origin and aligned along the Z-axis. The shape of the constant potential contour in the X-Z plane along the Y-axis for the case of 6 parallel rods in a hexagonal cluster is similar to that shown in Figure 4 (top) for the two-rod case. However, there is an increase in the region of overlap in the central region of the cluster. A major difference between the pair and hexagonal arrays emerges, however, for the constant potential profile in the X-Y plane. Shown in Figure 7 (top) are the contours in the X-Y plane at Z ) 0 and in Figure 7 (bottom) for the X-Y plane at Z ) 0.4.
4110 Langmuir, Vol. 15, No. 12, 1999
Schmitz
Figure 8. Potential profiles for six rods in a hexagonal array. The potential profiles for the six-rod system are shown along the X-axis at Y ) 0 and Z ) 0 (dashed line) and along the line passing through the cluster between two rods (solid line).
Figure 10. Cutaway of the potential surfaces for six rods in hexagonal array. Shown above is a cutaway three-dimensional contour plot of the constant potential surfaces for φcol(r) ) 0.99φcol((0,0,0)) (inner surface) and φcol(r) ) 0.99φcol((0,0,0)) (outer surface).
Figure 9. Vector Gradient Potential for Six Rods in the X-Y Plane. The negative of the vector gradient field of the macroion potential, -∇φcol(r), is shown above for the X-Y plane at Z ) 0.
The “circle” interior to the cluster in Figure 7 (top) but absent in Figure 7 (bottom) indicates a “bowl shaped” structure of the potential profile in the interior of the cluster. This is verified by the potential profiles passing through the X-axis (through the rods - dashed curve) and the Y-axis (between the rods and solid curve) at Z ) 0, as shown in Figure 8. The profile of ∇φcol(r) in the X-Y plane at Z ) 0 is shown in Figure 9. The presence of a subregion is indicated by the 180° directional change of the vectors along the line between the rods. Shown in Figure 10 is a threedimensional contour plot of the constant potential surfaces φcol(r) ) 0.99 φcol(0,0,0) and 0.90 φcol(0,0,0). Comparison of the surfaces in Figure 10 with those in Figure 6 clearly shows a greater degree of overlap between the rods in the hexagonal cluster for the φcol(r) ) 0.99 φcol(0,0,0) ions. 13. Discussion Linearized theories of the pair potential have limited application when applied to systems of highly charged polyelectrolytes and charged colloidal particles. To partially compensate for this, the concept of “charge renormalization” was introduced48 with the idea that the “condensed” counterions, which provide the neutralization, are surface bound. This definition and placement of the condensed counterions was used by Robbins, Kremer, and Grest80 as a justification of using the DLVO potential in their computer simulations of charged spheres without (80) Robbins, M. O.; Kremer, K.; Grest, G. S. J. Chem. Phys. 1988, 88, 3286.
regard to the range of the interactions. However, if one defines the “bound” counterions by the condition of equivalence of electrostatic and thermal energies as given in section 5 with the experimental verification discussed in detail in section 7 and the constraints of the screened Coulomb charge (section 8) as discussed in section 9, then there is a wide range of configurations in computer simulations for which the linearized form of the pair potential is inappropriate. It is our supposition that nonlinear regions of the Coulombic interactions between macroions result in redistribution of the counterions with significant impact on the physical properties of the system under examinations. Parallels are drawn from the Debye-Hu¨ckel paper24 which lead to the conclusion that modifications to mean field theories may be made through the variation in the screening parameter to account for observations not within the domain of the mean field potentials (cf. sections 2 and 3). However, these modifications to an effective pair potential are deemed to be valid only under dilute concentrations of the macroions as defined by the minimal occurrences of overlap of units of effective radius Rtherm (cf. sections 5-9). Systems with extensive overlap thus fall well outside the domain of the linearized theories, and more qualitative approaches may have to be employed to provide the insight needed to interpret unusual observations in these complex systems. One such approach is the JPF method, as adapted from the methods used to study the stability of molecules. It was stated in section 11 that, according to the JPF approach, there are at least four distinct regions for the counterions associated with a cluster of more than two macroions, and therefore at least four dynamic regions for the electrolyte ion. For pedagogical reasons one may envision the potential in the X-Y plane as a mountain range, where the mountains are represented as the magnitude of the potential in the third dimension. This metaphor is readily visualized with the slice of the constant potential contours of Figure 10 in conjunction with the vector gradient potential fields in Figure 3. Region I is defined as the constant potential in the center of the cluster and may be likened to a meadow in the center of the surrounding mountain range. Region I appears as a “hole” in the center of the profile of Figure 10. Since there is no force originating from the macroions, counterions in region
Clusters of Macroions of Arbitrary Shape
I may be considered to behave as “free” counterions. The second region of interest corresponds to the slope of the mountain range in which one may travel between mountains. Region II begins at the base of the mountains, and is confined to constant contour lines that traverse the interior of the cluster. Region II in Figure 10 is that region bounded by region I and the minimum in the direct line along the surface between adjacent basin locations. There is no force originating from the macroions on the counterions along these constant potential lines in region II. In contrast, the constant potential contour lines in region III encircle only one of the macroions in the cluster. This region is identified in Figure 10 by the constant contour line forming an ellipsoid about each basin point. Region IV lies outside the cluster, and at large distances becomes spherical in the three real dimensions. This region is not present in Figure 10 but appears in Figure 1. The linearized mean field DLVO theory is thus limited to region IV and therefore may not accurately describe the equilibrium and dynamic properties in the vicinity of a cluster of macroions. We now address two phenomena that have recently gained attention in regard to the stability of macroion clusters: nonadditivity of the interactions and specific ion effects. Ha and Liu65,66 and Podgornik and Parsegian67 have shown that nonadditivity effects may account for the apparent attraction between clusters of rods. These effects are due to the redistribution of counterions in the presence of the macroions (Professor Andrea Liu, personal communication). To demonstrate that similar effects occur for a cluster of spherical particles, we have performed Monte Carlo (MC) simulations on a cluster of six spheres arranged as described in section 11. In the MC simulations the input parameters are the macroion diameter dm ) 1000 Å and charge Zp ) 50 and a counterion diameter of dc ) 5 Å and charge -1. The movement of the electrolyte ions is determined by standard MC methods.82 In these simulations the distribution of the counterions is first achieved by a random number generation, making sure that the ions are within the spherical cell of radius Rc ) 10dm and do not overlap with any of the other particles in the system. The location of each ion is advanced through small but finite changes in the coordinates, qj(n + 1) ) qj(n) + ∆max*RAN(n + 1), where qj(n + 1) is the new value of the generalized coordinate qj for the jth electrolyte ion for the n + 1 iteration, qj(n) is the old value of that coordinate, ∆max is the maximum value for the displacement for any one iteration and RAN(n + 1) is the value of the n + 1 iteration of a random number from a uniform probability distribution between -1 and +1. For the present calculation ∆max ) 4(dc/dm), i.e., four times the diameter of the ions, in reduced coordinates. After each successful move, the reduced energy of the system is then determined by summing up the pairwise Coulombic interactions. All energies that were greater than 5% of the previously accepted energy were rejected. The calculations were performed on the DEC Alpha AXP2100/ M500 at the University of Missouri Computing Facilities. In these calculations a “preaveraging” series of 106 iterations were performed before any data was retained for averaging purposes. In the preaveraging cycle only those moves that resulted in a lower energy for the system were retained. The number of iterations in calculations used for averaging was also 106. Since the purpose of this exercise was to determine the distribution of counterions (81) Hribar, B.; Vlachy, V. J. Phys. Chem. 1997, 101, 3457. (82) Allen, M. P.; Tildesley, D. J. Computer Simulations of Liquids; (Clarendon Press, Oxford Science Publications: Oxford, England, 1993.
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Figure 11. Comparison of excess distribution function G[r] 1 of the Counterions. The excess distribution function G[r] 1 of the counterions is measured from the origin of the coordinate system. The location of the macroions is indicated by the horizontal line at G[r] - 1 ) 2. The solid curve is G[r] - 1 for the counterions in the macroion cluster of six charged particles. For comparison, G[r] - 1 for a similar cluster of six macroparticles, with the one located at (3,0,0) having a charge of 50 and the others uncharged, is also shown (points).
in the cluster, we did not calculate the usual pair distribution function represented by gM,j[r] for the electrolyte ions about a macroion. Instead, the distribution function was determined in the normal way, but in reference to the origin of the coordinate system. Hence the notation G[r], for the counterions located a distance of r from the origin. The number of bins, or discrete “onion layers” of thickness ∆r, used in these calculations was 500, or a thickness ∆r ) 10/500 ) 0.02. The reduced diameter of the electrolyte ion was 5/1000 ) 0.005. The calculated value of the “excess” distribution of counterions, G[r] - 1, is given in Figure 11. The line at G[r] - 1 ) 2 indicates the location of the macroions; hence, the distribution function G[r] is reduced in the shell due to the volume occupied by the macroions. The solid line is for the case when all six macroions are charged, and the dashed line is for the case of five neutral spheres and one charged sphere. If the superposition of ion distribution is valid, then the solid and dashed curves should coincide as these are normalized curves. These calculations clearly indicate a large concentration of the counterions in the interior of the cluster that cannot be accounted for by the additivity assumption. Specific ion effects include the finite size of the ion and the net charge of the ion. The JPF approach is limited to the potential field set up by the macroions as “experienced” by a test charge of unit magnitude. As such the ion specific effect of charge cannot be considered. Recall, however, that in the original Langmuir model the counterions were collectively treated as a “macroion” and their lattice placement led to the net electrostatic attraction of the system. If the counterions were treated in a similar manner for a pair of macroions then one would likewise expect a net attraction between this pair of macroions. However, this is opposite to direct experimental observation for univalent ions in which the interaction between isolated pairs of macroions is a repulsive interaction.50,51 Hence one might expect that an “attraction” might result for multivalent ions somewhere in the list of charges from monovalent to macroion charge. To illustrate this hypothesis, we next consider the Monte Carlo (MC) calculations of Hribar and Vlachy81 for a solution of macroions with a macroion/ion charge ratios of 20:1 and 20:2 and a size ratio of 15:2. The ions in the Hribar and Vlachy81 simulations interacted through a hard sphere potential and an unscreened Coulombic interactions. They reported calculations using 64 macroions and the corresponding
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number of counterions for charge neutrality, at macroion concentrations of c ) 0.01 mol/dm3 and 0.02 mol/dm3. The pair distribution functions gpp(r), gpc(r), and gcc(r) were determined, with MC statistics covering 70 million attempted configurations after first equilibrating the system with 5 million configuration attempts. Both the MC and hypernetted chain (HNC) methods were used. For the monovalent ion calculations, the pair distribution functions behaved as anticipated on the basis of a repulsion only potential. As the concentration decreased, the first peak in gpp(r) extended to a larger distance for both the monovalent and divalent ions. In contrast, however, the first peak for the divalent counterion did not vary with c. Furthermore, there was very little long-range structure in the solution, as evidenced by the lack of higher order peaks. This difference from the expectations based on the DLVO potential was interpreted as an attraction of electrostatic origin between the macroions. Their calculations further showed that divalent ions accumulated around the macroion, and that the probability of finding a second divalent ion near the first is on the opposite side of the macroion for the more concentrated solutions. They concluded that the attraction observed in the divalent ion case was due to a sharing of the layer of counterions. Attraction in the Hribar and Vlachy81 study involve fluctuations in the distribution of ions in the vicinity of the surface of the macroions, whether by sharing of the ions or correlations between the surface charge distributions. This result is also contained in the KirkwoodShumaker theory, as we now demonstrate using eqs 3.5 and 3.6 We make a simple “minimum effort” (ME) calculation of the cross correlation as defined by the following postulates: (1) the charge sites on the two macroion surfaces can be ordered in a parallel manner, i.e., j1, j2, j3, ..., jn for one macroion and i1, i2, i3, ..., in for the conjugate sites of the second macroion; (2) the cross correlation pertains only with its conjugate site, viz., j1 T i1, j2 T i2, etc; (3) there is no correlation between sites on the same macroion. The significance of the second postulate is that neighbor site interactions are absence. The significance of the third postulate is that the charge on the macroion is not conserved. That is, the absence of an adsorbed counterion on one site necessarily requires it to be found on another site within the same macroion. With these postulates we need only to calculate the average 〈zj,mzi,m〉 for one pair of sites. Let us now write the charge at one site in terms of an “occupation number” nc for the bound counterion, zj,m(nc) ) zj,m + zcnc. Hence if nc ) 0 then zj,m(0) ) zj,m and if nc ) 1 then zj,m(1) ) zj,m + zc. Thus if one assumes an equal a priori chance for all possible adsorbed states then for this set of postulates we have
〈zj,mzi,m〉 ) zj,m(0)zi,m(0) + zj,m(1)zi,m(0) + zj,m(0)zi,m(1) + zj,m(1)zi,m(1) 4 (13.2) If the site charge is zj,m ) -1, then for univalent counterions 〈zj,mzi,m〉 ) 1/4 since zj,m(1) ) 0. Consider now a strong negative correlation between bound ions on conjugate sites, then the last term in the numerator of eq 13.2 is omitted. Thus 〈zj,mzi,m〉 ) 1/3 for the strongly negative correlation case. It is clear that for either case the pair potential remains repulsive for univalent counterions. For strong negative cross correlations the effective charge is larger than that obtained from equal a priori adsorption of counterions. Consider now the adsorption of divalent ions that are allowed to bind at each site. In this case
Schmitz
zj,m(1) ) +1 and the sum given by eq 13.1 gives the equal a priori average 〈zj,mzi,m〉 ) 0. However, if the negative cross correlation is strong then 〈zj,mzi,m〉 ) -1/3, which gives rise to an attraction between the two macroions. If one employs ions of finite size, the monopole charge at the macroion site is replaced by an electrical dipole. Hence the adsorption of a counterion of charge magnitude equal to that of the site does not eliminate the electrical interaction for that site, as employed in the above ME calculations. Likewise the inclusion of neighbor interactions with the finite ion effects has an effect on the quantitative aspects of the charge fluctuation calculation. These additional effects do not alter, however, the qualitative nature of the conclusions drawn in the ME calculations. We are presently studying the interaction of macroions in the presence of a charged planar surface. There are two theoretical studies of note in the literature, however, that involve interactions with a planar surface that have direct relevance with the JPF method. Rouzina and Bloomfield83 examined the interaction between two planar surfaces in which there was a correlation between ions on the bound sites. Their results indicate that attraction occurs between the surfaces if ions bound on one surface interact with unbound sites on the other surface. These calculations, therefore, fall into the same category as the ME calculations above. Bowen and Sharif84 solved the PB equation for the force between two charged spheres in the presence of a wall of like charge. They conclude that the ion cloud about the two spheres is distorted in the presence of the charged wall and that this distortion results in an attraction between the two spheres. This result is therefore in concert with the premise of the JPF method, viz., that the distribution of the counterions has a profound effect on the electrical interactions between charged particles. We have not included the effect of ion-ion interactions in the cluster. In the evaluation of the force -∇φions(r), one must consider the effect of the macroion field. Because of the high charge density of the macroions, co-ions are excluded from the interior of the cluster whereas counterions are drawn into the interior of the cluster. The latter is shown in Figure 11 for our MC calculations. The co-ion and counterion distributions are supported by our MC and Brownian dynamics (BD) simulations, the details of which will be published elsewhere. This distribution is also consistent with the model of the ion distribution about a highly charged rod by Alexandrowicz and Katchalsky.70 Thus within the interior of the cluster the potential φions(r) is due primarily, if not exclusively, to the counterions. As in the Alexandrowicz and Katchalsky model, it necessarily follows that local rather than global screening parameters should be employed in the description of the system. Furthermore the dominance of the macroion field dictates that the distribution of counterions about a “probe” counterion is not symmetric as in the Debye-Hu¨ckel theory summarized in section 2. That is, there is a greater concentration of counterions about the probe ion on the side facing the macroions for regions II and III. In contrast, φions(r) in region I may be more symmetric about the probe counterion due to the constant potential of the macroions in the cluster. For this reason it is not a simple matter to obtain an analytical expression for the distribution function of ions in the vicinity of a macroion cluster. One must resort, in our opinion, to computer simulation techniques to adequately describe such structures in solutions and suspensions. (83) Rouzina, I.; Bloomfield, V. A. J. Phys. Chem. 1996, 100, 9977. (84) Bowen, W. R.; Sharif, A. O. Nature 1998, 393, 663.
Clusters of Macroions of Arbitrary Shape
The theories of Ha and Liu,65,66 Podgornik and Parsegian,67 and Rouzina and Bloomfield83 involve short range effects that are related to fluctuations in the surface charge density. Such effects are contained within the formalism of Kirkwood and Shumaker as reflected in the ME calculations above. The JPF method focuses on long-range effects as dictated by the potential field set up by the cluster of macroions. We have shown herein that the potential field set up by an array of spheres and rods greatly distorts the distribution of the ions relative to the simple superposition principle of pairwise interactions. Similar distortion of the potential for spheres in the presence of a wall indicates that this is the source of attraction between spheres.84 Since spheres, rods, and planes are the fundamental geometric objects used to describe macromolecules of general shape, we therefore suggest that the JPF method may be applied to macroions of general shape. An interesting feature of the colloidal cloister is that the discrete distribution of the macroion charges becomes “lost” within a few diameters of the subunits from the cluster, as shown in Figures 1, 4, and 7. That is, the cluster itself appears as a single unit of spherical symmetry. Thus there is the occurrence of a second double layer that may be associated with the cluster itself! There is another interesting feature of the current paradigm regarding the “screened” Coulomb charge and the definition of Rtherm: counterion condensation as employed in the charge renormalization theories involving surface adsorption may
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actually act in the direction of decreasing the range of validity of linearized theories. This conclusion is readily obtained from eq 5.4 in the limit of zero added electrolyte. According to condensation models (cf. section 6) the “condensed” counterions are not included in the calculation of the screening parameter κ. Hence a reduction in the value of κ necessarily results in an increase in the value of Rtherm, or the range of nonlinear electrostatic effects. The JPF view of a macroion cluster appears to be consistent with the model of Michaeli, Overbeek, and Voorn64 on the phase separation of colloidal particles, where the attractive electrical interaction is identified with the collective interactions of the macroions and associated counterions. These interactions give rise, in the JPF approach, to the dominance of nonlinear effects as being the source of attraction, where the counterions are subject to nonlinear regions of the collective group of macroions within the defined cluster. Or in other words, nonlinear effects are the origin of “ion mediated” attraction between the macroions. Acknowledgment. I extend my gratitude to Professor Andrea Liu for discussions on her work, in particular, and nonadditive effects, in general. I wish to acknowledge the spirited discussions with Julia P. Fisher in the preparation of this manuscript. LA9811214
