Liquid State Theory of Polyelectrolyte Solutions - The Journal of

 Copyright 2009 by the American Chemical Society

VOLUME 113, NUMBER 6, FEBRUARY 12, 2009

CENTENNIAL FEATURE ARTICLE Liquid State Theory of Polyelectrolyte Solutions† Arun Yethiraj Theoretical Chemistry Institute, Department of Chemistry, UniVersity of Wisconsin, 1101 UniVersity AVenue, Madison, Wisconsin 53706-1396 ReceiVed: August 5, 2008; ReVised Manuscript ReceiVed: October 24, 2008

The molecular modeling of polyelectrolyte solutions is considered one of the grand challenges of condensed phase physical chemistry. There have been many advances in our understanding of these systems, and the insight obtained from liquid state approaches is reviewed in this article. Integral equation theories have been successful in describing the structural properties of polyelectrolytes in good solvents. The theory provides an accurate description (when compared to computer simulations and experiment) of the static structure, conformational properties, surface forces, and osmotic pressure of polyelectrolyte solutions. Challenges remain for the description of strongly coupled systems and poor solvents, and some possible future directions are discussed. 1. Introduction and Background Polyelectrolyte solutions are composed of ionizable polymer molecules, which become charged when dissolved in water. The physical properties are the result of a balance between the hydrophobic interactions of the backbone and electrostatic interactions of the polar side groups, and their fascinating behavior has been of fundamental interest for decades. For example, single polyelectrolyte chains are predicted to display a wide range of conformations including rods, globules, pearlnecklace structures, torii, and helices depending on the balance between electrostatic and solvent-induced interactions and the local chemistry of the molecules. Their practical importance arises from the large number of applications: as processing aids † 2008 marked the Centennial of the American Chemical Society’s Division of Physical Chemistry. To celebrate and to highlight the field of physical chemistry from both historical and future perspectives, The Journal of Physical Chemistry is publishing a special series of Centennial Feature Articles. These articles are invited contributions from current and former officers and members of the Physical Chemistry Division Executive Committee and from J. Phys. Chem. Senior Editors.

such as flocculants, dewatering agents, demulsifiers, and drag reduction agents; as additives in detergents and cosmetics; and in the manufacture of membranes, ion-exchange resins, gels, and modified plastics. The worldwide production of superabsorbent polymers alone was 850 000 t in 1999. They are also important as biological molecules such as proteins, sugars, and nucleic acids. The study of polyelectrolytes therefore spans diverse areas and has seen active research that has been extensively reviewed.1-5 Examples of materials commonly studied in basic research are solutions of polystyrene sulfonate (PSS) and poly(acrylic acid) (PAA). PSS is widely studied and characterized but also has practical applications: The sodium salt of PSS is used in ion-exchange resins, and to treat hyperkelemia (high potassium levels). The backbone of PSS is hydrophobic, and the ionizable moieties are side groups. In principle, each monomer can carry a unit charge, but often, not all of the monomers in PSS are sulfonated. PSS is a good example of the complexity of intermolecular interactions in polyelectrolyte solutions. In addition to the electrostatic interactions, one expects important

10.1021/jp8069964 CCC: $40.75  2009 American Chemical Society Published on Web 11/26/2008

1540 J. Phys. Chem. B, Vol. 113, No. 6, 2009 Arun Yethiraj is a Professor of Chemistry at the University of Wisconsin, Madison. He received a B. Tech. (Indian Institute of Technology, Bombay, 1985), an M.S. (Louisiana State University, 1987), and Ph.D. (North Carolina State University, 1991), all in Chemical Engineering. He was a postdoctoral research associate at the University of Illinois and has been at UW-Madison since 1993. Yethiraj is a Senior Editor of the Journal of Physical Chemistry. Professor Yethiraj’s research focuses on theoretical and computational studies of soft condensed matter, with an emphasis on coarse-grained models of complex fluids. He has used liquid state methods such as computer simulation, density functional theory, and integral equations to study several systems including confined liquids, polymer melts and blends, liquid crystals, polyelectrolyte solutions, and the dynamics of fluids in complex media.

contributions from dispersion interactions and hydrophobic (solvent-induced) interactions. Intramolecular interactions and correlations are also nontrivial because bond-angle and torsional potentials will influence the presentation of charged moieties in solution. Finally, it is not clear a priori if all of the ionizable groups are dissociated in solution. The goal of this article is to review work on simple models of polyelectrolyte solutions. In these models, the polymer molecules are modeled as chains of charged spheres and the other components are either treated implicitly or explicitly as structureless spheres. Many of the complexities of real systems (such as degree of ionization or dispersion interactions) are not considered. The past decade has seen considerable advances in our understanding of the behavior of these primitive models. This article describes research using computer simulations and liquid state theory, e.g., integral equations and density functional theory, on the properties of polyelectrolyte solutions. Polyelectrolyte solutions are still considered one of the least understood materials in soft matter, especially when compared to solutions of neutral polymers. For neutral polymers onecomponent models (where only the polymer is treated explicitly) provide a qualitatively correct description of the conformational and thermodynamic properties, and these solutions are generally considered well understood. In good solvents, the characteristic chain size, R, scales with the degree of polymerization, N, as R ∼ N3/5. The scaling exponent changes as the solvent quality is decreased, with R ∼ N1/2 in theta solvents and R ∼ N1/3 in poor solvents. The situation is far more complicated in polyelectrolyte solutions where at least three components, e.g., polymer, counterion, and solvent, play an important role. It has been shown experimentally that the conformational properties depend not only on the charge on the polymer and on the concentration of added salt but also on the chemical nature of the salt ions.1 In a one-component model of the solution, the intermolecular interactions between the polymer sites can be a nontrivial function of the solution parameters. Not only does the added salt affect the screening between charges on the chains, but the degree of dissociation of the ionizable groups can also change, thus altering the charge on the polymer molecules. Elucidating the physical chemistry of electrolyte solutions that include polymeric species is one of the challenges of condensed phase theoretical chemistry. Scaling theories can be used to understand the properties of semidilute solutions of neutral polymers.6 In dilute solutions of neutral polymers in a good solvent, R ∼ N3/5and the correlation length, ξ ∼ R. The behavior of ξ in semidilute solutions is obtained from the scaling ansatz ξ ∼ R(c/c*)γ, where c* is the overlap threshold concentration. c* is defined as the concentration at which the internal concentration of the chain, i.e., N/R3, is equal to the average concentration. Since R ∼ N3/5, this gives c* ∼ N-4/5. In semidilute solutions, the polymer molecules

Yethiraj interpenetrate and form a mesh-like structure. In this regime, one expects ξ ∼ N0 and R ∼ N1/2 (according to Flory’s ideality hypothesis). The above conditions give ξ ∼ c-3/4 and R ∼ c-1/8. Furthermore, the osmotic pressure, π, is predicted to scale as, π ∼ 1/ξ3 ∼ c9/4. All of these predictions have been verified experimentally. Even for primitive models and good solvents, scaling theories have mixed results for polyelectrolyte solutions. In dilute solution, the chains are expanded and the chain size, R, and correlation length, ξ, scale with the degree of polymerization, N, as R ∼ N and ξ ∼ N, and thus, c* ∼ 1/N2. Using the scaling ansatz and the result ξ ∼ N0 in semidilute solutions, one obtains ξ ∼ c-1/2.7 The theory predicts a peak in the static structure factor with a position q* that scales with concentration as q* ∼ c1/2, which is consistent with simulation8,9 and experiment.10-16 This prediction was one of the important successes of scaling theory in this field. The scaling theory predictions for the osmotic pressure, on the other hand, are not in agreement with simulations.8,17 In particular, the theory predicts π ∼ 1/ξ3 ∼ c3/2 in semidilute solutions, where π is the osmotic pressure, while simulations show that π ∼ c9/4. This is, of course, the polymer contribution, and it has been suggested that the osmotic pressure of polyelectrolyte solutions is dominated by the counterions,18,19 but this is not supported by computer simulations.17 It has also been argued that scaling theory is not appropriate for “marginal” solvents20 or rigid rods.21 Scaling theory, for example, is in qualitative error for the scaling of the osmotic pressure with concentration in solutions of uncharged rigid rods.21 That scaling theories are not an unqualified success is not entirely surprising given the multiple important length scales in the problem. Furthermore, liquid state correlations and excluded volume interactions are known to be important in polyelectrolyte solutions.9,22-34 There have been a number of computer simulation studies, mostly of generic models for the polyions. Stevens and Kremer8,35-37 reported molecular dynamics simulations for the static properties of polyelectrolyte solutions at nonzero concentrations and in good solvents. This bench-mark calculation established the behavior of a primitive model of polyelectrolyte solutions. There have been several other studies of single chains and solutions that have demonstrated the importance of correlation effects.38-40 For example, increasing the charge of a single flexible chain can result in an expansion or collapse of the chain, depending on the charge density, and added salt can cause a collapse and re-expansion of chains even in good solvents.41 Divalent counterions cause a decrease in size of single chains and the formation of bundles in semidilute solutions.42 A condensation into compact structures, however, requires trivalent or tetravalent counterions.43 There have been some recent simulations of atomistic models of polyelectrolytes including DNA,44,45 PAA,46-49and PSS.50 In this article, we review computer simulation and liquid state theory approaches that have been applied to polyelectrolyte solutions. We discuss the static structure, conformational properties, thermodynamics, surface behavior, and dynamic behavior of primitive models of polyelectrolyte solutions, as predicted by integral equation theory and computer simulations. A review of the entire literature is not the scope of this article, and instead, topics are chosen from the standpoint of personal interest. We address the ramifications of treating counterions, salt ions, and solvent molecules in an explicit fashion, and compare to experiments where possible. Much of our understanding of the static behavior of synthetic polyelectrolyte

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solutions comes from scattering (light, neutron, and X-ray) experiments.51-59 We show that theories are at the stage where they can provide a good qualitative description of these experiments. The rest of this paper is organized as follows. Integral equation theory is described in section 2, the static structure of rods, conformational properties of flexible chains, surface forces, osmotic pressure, and solvent effects are discussed in sections 3-7, and section 8 presents some conclusions and future outlook. 2. Integral Equation Theory A quantity of central importance in the theory of simple liquids is the pair correlation function, g(r), which is the probability (relative to an ideal gas) of finding a particle at position r given that there is a particle at the origin. All other structural and thermodynamic properties can be obtained from g(r).60 In simple liquids, g(r) can be calculated by solving the Ornstein-Zernike (OZ) integral equation with a second closure relation. The OZ equation is60

h(r) ) c(r) + F

∫ c(r') h(|r - r'|) dr'

Figure 1. Scheme of the self-consistent integral equation procedure: Starting with an initial guess for the single chain structure factor, the PRISM equation with closure is solved for the medium-induced “solvation” potential. The structure of a single chain interacting via the “bare” plus medium-induced potential is obtained from a single chain theory or simulation. This gives the next guess for the single chain structure factor. The process is continued until convergence.

a long-ranged electrostatic interaction, the choice of closure approximation is not obvious. Schweizer and Yethiraj66-68 suggested a class of molecular closure approximations that are fashioned on a two-molecule level rather than on a site-site level.AnexampleofamolecularclosureistheLaria-Wu-Chandler (LWC) closure:69

(1)

ω / c / ω(r) ) ω / (-βu) / ω(r) + h(r) - ln g(r) (6)

where F is the number density, h(r) ) g(r) - 1, and c(r) is the direct correlation function, for which the OZ equation may be considered a definition. In Fourier space, the OZ equation takes the form

where β ) 1/kBT, where T is the temperature and kB is Boltzmann’s constant, u(r) is the site-site interaction potential, and the asterisks denote convolution integrals, i.e.,

hˆ(k) ) cˆ(k) + Fcˆ(k) hˆ(k)

(2)

where k is the Fourier transform variable, and

4π hˆ(k) ) k

∫0∞ rh(r) sin(kr) dr

(3)

Given another closure relation between h(r) and c(r), one can solve for g(r). For hard spheres, the popular Percus-Yevick (PY) closure60 sets c(r) ) 0 for r > σ, where σ is the hard sphere diameter, with g(r) ) 0, for r < σ. The generalization of the OZ equation to polymers is the polymer reference interaction site model (PRISM) equation of Curro and Schweizer,61-64 which for one-component polymers is given by

hˆ(k) ) ω ˆ (k) cˆ(k) ω ˆ (k) + Fω ˆ (k) cˆ(k) hˆ(k)

(4)

where the correlation functions h(r) and c(r) are averaged over all the sites on the polymer molecules, F is the number density of polymer sites (or monomers), and ω(k) is the single chain structure factor (Fourier transform of the average intramolecular correlation function). If ω(k) is known, as is the case for rigid molecules, the PRISM equation in conjunction with a closure approximation, can be solved for the pair correlation function. For hard sphere chains, the PY closure is known to be accurate.65 The static structure factor can then be obtained from

Sˆ(k) ) ω ˆ (k) + Fhˆ(k)

(5)

When the interaction potential contains a slowly varying component, such as a short-ranged van der Waals attraction or

f / g(r) )

∫ f(r') g(|r - r'|) dr'

(7)

A closure of the form of eq 6 cannot be used, of course, when the interaction potential contains a hard sphere part. A standard method for treating this case is to use the “reference” form of the closure,60 i.e.,

ω / c / ω(r) ) ω / c0 / ω(r) + ω / (-βu) / ω(r) + g(r) (8) h(r) - h0(r) - ln g0(r) where the interaction potential consists of a hard core plus tail potential u(r) and c0(r) and h0(r) are the direct and total correlation functions, respectively, of a fluid at the same density but interacting via only a hard core (or reference) potential. We refer to this closure as the reference LWC or R-LWC closure. The PRISM equation is solved using a Picard iteration procedure. Space is discretized, usually with 212 or 213 grid points. The correlation functions are separated into a long-ranged part (which is known) and a short-ranged part, which is solved for in an iterative fashion. The PRISM equation is enforced in reciprocal space, and the closure is enforced in real space. For flexible chains, the intramolecular correlation functions are not known and must be determined in a self-consistent fashion. Within the PRISM framework, this procedure consists of two parts.61,70,71 For a given set of intramolecular correlation functions, the PRISM equations can be solved for the intermolecular correlation functions, as described above. An effective “solvation” potential is determined from the intermolecular correlation functions. The next guess for the intramolecular correlation functions is obtained by solving a single chain problem where the sites interact via the sum of the original “bare” interaction and a solvation potential. The procedure is depicted in Figure 1. The solvation potential, W(r), has a simple

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form, and for a one-component polymer system, it is given by72

ˆ (k) ) -Fcˆ(k) Sˆ(k) cˆ(k) W

(9)

The most straightforward method, conceptually, for solving the single chain problem is via Monte Carlo simulations, where simulations are performed for a single chain interacting via the bare plus solvation potential and the single chain structure is monitored. Other approximate methods have also been implemented, for example, using a thread model for the polymer chain. For solutions of flexible polyelectrolytes, the PRISM predictions are relatively insensitive to the choice of method used for solving the single chain problem. The extension of the one-component PRISM theory to multicomponent systems is straightforward.12,13,15 One has to solve a set of PRISM equations, which can be written in matrix form:

ˆ )Ω ˆC ˆΩ ˆ +Ω ˆC ˆH ˆ H

(10)

ˆ ij(k) ) where the elements of the matrices are given by H FiFjhîj(k), Cîj(k) ) cîj(k), and Ωij(r) ) Fiδijωi(r). The molecular closures take the form

Ω / C / Ω ) Ω / C0 / Ω + Ω / ∆C / Ω

(11)

where asterisks denote matrix multiplications and convolution integrals in real space, and the elements of the matrix C0 are the direct correlation functions of the reference hard chain (or sphere) mixture fluid. The matrix ∆C involves closure approximations. In the R-LWC closure, for example,

[Ω / ∆C / Ω]ij(r) ) [Ω / C0 / Ω]ij(r) + Hij(r) FiFj ln gij(r) (12) It is not necessary to use the same closure for all correlation functions. For example, it is convenient to use the meanspherical approximation for the counterions and the R-LWC closure for the polyions. The mixture PRISM equations are solved via a Picard iteration procedure. 3. Static Structure of Solutions of Rod-like Polyelectrolytes Solutions of rod-like polymers are a good model system because complications arising from chain conformational changes are not present. Experimentally, solutions of tobacco mosaic virus (TMV) are an excellent system because these viruses are rigid, rod-like, and highly charged in solution. In most solutions, the static structure becomes uninteresting (similar to an ideal gas) as one goes to very low concentrations. In polyelectrolyte solutions, however, several interesting features are observed in the static structure factor in dilute solutions.10,12,15 At low concentrations, a primary and sometimes secondary peak is observed in the structure factor, suggesting significant liquidlike layering on a length scale of the order of the size of the molecules. The primary peak in the structure factor broadens and moves to higher wave vectors as the concentration is increased, suggesting that the liquid-like order diminishes as the concentration is increased. As the concentration is increased further, the peak reappears (at higher wave vectors) and grows in intensity with increasing concentration. The scaling of the

Figure 2. Site-site pair correlation function between charged rods for L ) 20, Ψ ) 1, and various volume fractions (as marked).

position of the peak, kmax, with concentration shows two distinct regimes, with kmax ∼ c1/3 in dilute solutions and kmax ∼ c1/2 in semidilute solutions. Shew and Yethiraj25,31,73,74 implemented the PRISM theory for solutions of rod-like charged particles. The shape of the TMV molecules, which are rigid cylindrical objects, were approximated using a “squashed shish kabob” model where each molecule was a line of overlapping spheres and used sufficiently many spheres to approximate a cylinder. Each sphere carried a charge, and the interaction between any two sites was taken to be a hard sphere plus screened Coulomb interaction, i.e., u(r) ) ∞ for r < σ and u(r) ) Ψ exp(-κr) for r > σ, where Ψ is the strength of the interaction and k is the inverse Debye screening length. In terms of the charge of the site, Ψ ) q2lB/(1 + κσ), where lB ) e2/4πεkBT is the Bjerrum length, e is the charge of an electron, and ε is the dielectric constant. In this simple model, the solvent is a dielectric continuum and counterions and salt are treated in a mean-field fashion. The PRISM theory with the R-LWC closure was in quantitative agreement with simulations for charged rods at infinite dilution.73 This was a surprise because for one-component polymers the PRISM theory is least accurate at low densities and becomes more accurate for high density melts.65 In dilute solutions, the theory predicted peaks in the pair correlation function g(r). Figure 2 depicts g(r) for rods for various volume fractions φ (defined as φ ) πcσ3/6, where c is the concentration of sites and σ is the hard sphere diameter or thickness of rods). For very low concentrations, e.g., φ ) 10-5, g(r) displays a peak at distances of the order of the length of the molecules. As φ is increased from 10-5 to 10-3, the peak becomes more pronounced and some liquid-like layering is evident from the oscillations in g(r). At a higher concentration (φ ) 0.01), the peak disappears. Since the potential of mean force is given by -kBT ln g(r), the theory predicts an attractive well in the potential of mean force at very low but nonzero concentrations. This attractive well disappears at higher concentrations. A physical picture emerges from these results. At low concentrations, the highly charged particles prefer to remain widely separated. One could imagine each rod being confined to a sphere with a diameter of the order of the length of the rods. These spheres “pack” among each other in a manner similar to simple liquids of hard spheres. In the hard sphere analogy, the effective volume fraction of these spheres is larger than that of the rods by a factor of L2 (where L is the length of the rods) and high enough to show liquid-like order. This explains the peak in g(r) at low concentrations. At higher concentrations, the rods begin to interpenetrate and this causes the layering to disappear. Interpenetration of the rods becomes possible because the mean separation of the rods and the range


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Figure 3. Scaling of position of peak in structure factor with concentration. The symbols are experimental data,12,14 and the dashed line is the theoretical prediction31 with no adjustable parameters. The solid lines are a linear least-squares fit to the theory in the low and high concentration regimes with slopes as marked.

of the Coulombic repulsion become smaller as the concentration is increased. This interpenetration is possible only because the actual volume fraction of molecules inside the “spheres” referred to above is very small. In suspensions of spherical colloidal particles, for example, the magnitude of liquid-like layering will monotonically increase with increasing concentration. At much higher concentrations, steric interactions can become important and one might observe strong liquid-like ordering on a length scale of σ or (for long enough rods) nematic and other liquid crystalline phases. A consequence of this liquid-like order is a peak in the static structure factor. Figure 3 compares theoretical predictions for the dependence of the position, kmax, of the peak in the static structure factor, on concentration to experimental data for TMV12 and fd-virus10,14 solutions. The theory is in very good agreement with the experiments with no adjustable parameters. The limiting slope, i.e., scaling exponent, is 1/3 and 1/2 at low and high concentrations, which is consistent with experiments and predictions of scaling theory. If the charge on the molecules is treated as an adjustable parameter, the theory can fit experimental data. Figure 4 compares theoretical predictions to experimental data for TMV solutions. The dimensions of TMV particles are well-known, and the only adjustable parameter is the charge on the molecules, which is not known. When the charge (denoted QTMV) is treated as an adjustable parameter, the theory is in excellent agreement with experimental data. Also shown in the figure are predictions of the PRISM theory where counterions are incorporated explicitly.74 There is almost no difference between the predictions of the theory where counterions are incorporated explicitly or implicitly via the Debye-Huckel (DH) approximation! Even the charge required to fit the data is the same, except for the two highest concentrations. Interestingly, the charge required to fit the TMV data is the same whether the counterions are including using a DH approximation or whether they are incorporated explicitly using a two-component model for the TMV and counterions in saltfree solution. Shew and Yethiraj74 argued that the behavior of the fit charge could not be attributed to a Manning-type counterion condensation75 because the PRISM theory with explicit counterions included nonuniform counterion distributions at an approximation better than the nonlinear PoissonBoltzmann equation. They suggested that TMV particles behave like a weak polyacid in dilute solution, with a chemical equilibrium between ions and polyions. The integral equation theory has also been used to investigate the partial structure factors in DNA solutions.76,77 In order to

Figure 4. Static structure factor of TMV solutions: Comparison of theoretic predictions to experimental data12,14 for various concentrations (as marked). The solid lines are predictions using the Debye-Huckel approximation,31 and the dashed lines are predictions where counterions are treated explicitly.74 The charge of the molecules (in the theory) is adjusted to reproduce the height of the structure factor.

Figure 5. Partial structure factor of DNA solutions: Comparison of theoretical predictions76 (lines) to neutron scattering data77 for the polyion-polyion (S11) and polyion-counterion (S12) partial structure factors. The fraction of charged and counterion bound sites on the DNA are treated as adjustable parameters.

fit the data, the binding of counterions to the polyions had to be explicitly incorporated and three types of sites on the polyions, namely, bare charged sites, counterion occupied sites, and protonated sites were used. With two of these fractions treated as adjustable parameters, the theory was able to fit experimental data for the polyion-polyion (S11) and the polyion-counterion (S12) partial structure factors. This is shown in Figure 5, which compares theoretical predictions to neutron scattering data.77 The above comparison had some interesting conclusions. First of all, it was necessary to invoke a chemical binding of salt to DNA, or incomplete dissociation of the DNA, in order to fit the data. Second, the polyion-polyion partial structure factor depended primarily on the charge of the polymer and was insensitive to the fraction of sites that had counterions bound to them. The polyion-counterion partial structure factor, on the


Yethiraj

other hand, is dominated by the scattering from bound counterions, which is more than an order of magnitude greater than thatfromthefreecounterions.Thisresultsinthepolyion-counterion partial structure factor being very similar to the polyion-polyion partial structure factor. The integral equation theory work on the structure of rods demonstrated that liquid state theory could be effectively used to investigate the structure of polyelectrolyte solutions. Very few parameters were required to compare to experiment, and some of the conclusions were interesting. This work made possible the application of this theoretical approach to other problems in polyelectrolyte solutions, including the surface behavior and thermodynamic properties. 4. Conformational Properties and Static Structure of Solutions of Flexible Polyelectrolytes In flexible polyelectrolytes, there is an interesting balance between intramolecular and intermolecular correlations. The conformational changes can be quite large, ranging from rodlike to globular conformations depending on the charge of the molecules and solution conditions such as the concentration of added salt. The conformational properties and the distribution of ions in solution are therefore strongly coupled. Until the 1990s, theoretical research on the properties of flexible polyelectrolytes focused on single chains, and experiments focused on semidilute solutions, resulting in very little synergy between theory and experiment. In 1993, Stevens and Kremer8 reported results of molecular dynamics simulations of flexible polyelectrolytes in salt-free solutions. This was a landmark paper because of the quantitative results it provided for the testing of existing theories. There were several striking observations. The rod-like scaling R ∼ N has always been assumed to be true for polyelectrolyte solutions, but the simulations showed that this regime might be physically unrealizable because chains contracted at concentrations much below the overlap threshold concentration, and experiments were possible only for semidilute solutions. In addition, no theory reproduced simulation results for the static structure factor. The self-consistent PRISM theory has been used to investigate the structure and conformational properties of solutions of flexible polyelectrolytes.9,26,29,30 For polyelectrolyte solutions, the approximations used to calculate the single chain structure do not seem to have a significant impact on the conformations or structural properties. Many different single chain approaches have been investigated: Monte Carlo simulations where the single chain structure is calculated exactly,9 a field theoretic thread model for the polymer molecules where the single chain problem is solved using perturbation theory or a variational approach,29,30 and an even simpler Flory-type approach with an additional term to account for the other chains.26 All of these approaches give similar results for the chain conformational properties, provided the effect of other chains is incorporated through a solvation potential obtained from the PRISM theory. If the conformational properties are known, the PRISM theory is very accurate for the static structure. Figure 6 compares theoretical predictions for the root-meansquare end-to-end distance from a self-consistent PRISM theory to molecular dynamics simulations of flexible polyelectrolytes. In the implementation of the theory shown, a field theoretic thread model is used for the polymer molecules, and the single chain conformational properties are obtained via a perturbation on the mean-square end-to-end distance, 〈R2〉, about a reference semiflexible chain.29 The theory predicts that the chains are extended but not rod-like at low concentrations, and the collapse

Figure 6. Mean-square end-to-end distance of flexible polyelectrolytes: Comparison of theoretical predictions29 (lines) to molecular dynamics simulations8 for various chain lengths (as marked). The dashed line marks the overlap threshold concentration from the theory.

Figure 7. Static structure factor of flexible polyelectrolytes: Comparison of theoretical predictions29 (lines) to Monte Carlo simulations9 for N ) 32 and various concentrations.

of the chains from this extended state to a coil-like state occurs at concentrations much smaller than the overlap threshold (dashed line in Figure 6). The agreement between theory and simulation is quite remarkable considering the simplicity of the molecular model. The theory overestimates the size of the chains at low concentrations but becomes more accurate as the concentration is increased. The predictions of the theory do not deteriorate as the chain length is increased. The theory is also accurate for the static structure factor. Figure 7 compares theoretical predictions for the static structure factor to Monte Carlo simulations9 for N ) 32. The behavior of the static structure factor is similar to that of rods. At low concentrations, there is a strong peak at low wave vectors, and as the concentration is increased, this peak broadens and moves to higher wave vectors. The theory is in good agreement with the simulation results without any adjustable parameters. A physical picture for the behavior may be obtained by considering the three relevant length scales: R, κ-1, and rc ∼ c-1/3, where κ is the inverse Debye screening length. At low concentrations, κ-1 . rc . R and the Coulomb repulsion dominates the structure. As the concentration is increased, κ-1 ∼ rc and liquid-like order begins to set in. At higher concentrations, R > κ-1, the chains begin to interpenetrate and the liquidlike order dissipates. The onset of structure is not a polymeric effect, but the disappearance of order is. The drop in R occurs at concentrations much below the overlap threshold concentration because the chains are already in repulsive contact at these low concentrations. This set of results shows that the integral equation theory can provide an accurate description of the conformational and structural properties of primitive models of polyelectrolyte


Figure 8. Solvation forces for surfaces immersed in polyelectrolyte solutions: Predictions of the wall-PRISM theory.87 The overlap threshold concentration is cl3 ) 10-4.

solutions, i.e., in good solvents. The theory is also in qualitative agreement with small-angle X-ray scattering experiments for the dependence of the position of the peak in the structure factor with concentration or charge fraction. 5. Surface Forces The surface behavior of polymers is of considerable fundamental and practical interest, with many applications. Polymeric additives to colloidal suspensions, for example, can either enhance or diminish the stability of the dispersion, depending on the nature of the polymer-induced surface forces. An understanding of the influence of polymers on the effective interaction between colloidal particles is therefore of considerable interest. There are several types of experiments in which surface forces can be measured. One approach is to measure the disjoining pressure in thin liquid films of polyelectrolyte solutions as a function of film thickness.78,79 Other methods are atomic force microscopy,80-82 total internal reflection microscopy,83,84 or via the surface forces apparatus.85 Measurements have shown that the force between surfaces immersed in dilute polyelectrolyte solutions is an oscillatory function of separation.79,82 The PRISM theory has been extended to investigate forces between surfaces immersed in a polymer solution.86 The theory is implemented for a mixture of spherical particles and a polymer solution. The spherical particle consists of an internal hard core (of radius R1) and an annular hard shell of inner radius R1 + H and outer radius R2. The interaction between a monomer, at a distance r from the spherical particle, is such that the interaction potential is zero if R1 < r < R1 + H and infinite if r < R1 or R1 + H < r < R2. In the limit as R1 and R2 tend to ∞, and the concentration of spheres is zero, this model is identical to a solution between two (hard) surfaces at a separation H. The theory requires as input the structure factor of the bulk solution, and this can be obtained from the approaches described in the previous section. Since the surfaces are hard walls, the pressure is simply related to the density of polymer sites at the surface. The predictions of the PRISM theory87 for the concentration profiles for polyelectrolytes between surfaces are in good agreement with simulation.88 The solvation force between two surfaces is an oscillatory function of their separation with a period of oscillation that corresponds to the length scale on which liquid-like order is present in the bulk solution.87 Figure 8 depicts the solvation force as a function of surface separation for charged hard sphere chains for N ) 200 and for various reduced concentrations. The

J. Phys. Chem. B, Vol. 113, No. 6, 2009 1545 reduced concentration, c, is the number of chain sites per unit volume multiplied by l3, where l is the bond length. The overlap threshold concentration for this system is approximately c*l3 ) 10-4. The period of the oscillations in the solvation decreases as the concentration is increased consistent with the peak in S(k) moving to higher wave vectors. The amplitude of the oscillations in the solvation force decreases as the concentration is increased consistent with the peak in S(k) becoming broader; i.e., the liquid-like packing becomes less efficient. For long enough chains, the period of the oscillation in the solvation force scales as c-1/3 and c-1/2 in dilute and semidilute solutions, respectively, consistent with the scaling of the position of the peak in the static structure factor with concentration. The predictions for these scaling exponents are in accord with experiments of the disjoining pressure in polyelectrolyte solutions at charged surfaces.79,82 The oscillatory forces can be explained in terms of liquidlike ordering in the solution because liquid-like ordering in bulk solution results in liquid-like layering when the solution is confined between surfaces. This effect is well-known in simple liquids where oscillatory density profiles are observed for hard spheres and Lennard-Jones liquids near surfaces. In polyelectrolyte solutions, the length scale of this liquid-like order is quite large and depends on polymer concentration and charge. With the addition of salt, the period of the oscillations increases, and the amplitude of the oscillations is also reduced. An important prediction is that, for intermediate salt concentrations, the solvation force is predominantly attractive, but for a large amount of excess salt, the solvation force becomes shortranged and purely repulsive. Sober and Walz83,84 reported attractive forces between a polystyrene latex particle and a glass surface in the presence of a nonadsorbing NaPSS solution with added salt. The PRISM predictions are in qualitative accord with their experiments. 6. Osmotic Pressure The conformational changes of polyions, coupled with electrostatic and excluded volume effects, can have significant and interesting effects on the volumetric properties. The osmotic pressure is therefore an important quantity for understanding electrostatic and excluded volume effects in polyelectrolyte solutions and for testing theories. One of the interesting assumptions often used in polyelectrolyte solutions is that the osmotic pressure is dominated by the free, i.e., unbound to polyion, counterions, which behave like an ideal gas. The effective charge on the polyions is often estimated by assuming that the concentration of free counterions can be obtained from the osmotic pressure via the ideal gas equation of state.89 The remaining counterions are presumably bound to the polymers, thus decreasing their charge. The origin of this hypothesis might be traced to the fact that in ideal gases the osmotic pressure is proportional to the number density of molecules and there are many more counterions than polymer molecules. The hypothesis is somewhat baffling because even in simple electrolytes nonideality effects are important at very low ionic strengths. The situation is expected to be more complicated in polyelectrolyte solutions. The osmotic pressure of solutions of salt-free polyelectrolytes has been investigated via computer simulation8,17,19 and liquid state theory.8,17 Molecular dynamics simulations8 were consistent with the scaling exponents of π ∼ c in dilute solutions (although they did not try to not distinguish between c1 and c9/8) and π ∼ c9/4 in semidilute solutions. These exponents are predicted by the integral equation theory for rods, whether counterions are


Yethiraj

Figure 10. Simulation results17 for the three (polymer, counterion, and cross) electrostatic contributions to the total excess osmotic coefficient. The terms are grouped into parts that are electrically neutral.

Figure 9. Simulation results17 for the volumetric properties of saltfree polyelectrolyte solutions. (a) Osmotic coefficient, φ ) βπ/c, as a function of reduced monomer concentration for monovalent counterions and lB ) 1 (squares) and lB ) 3 (triangles), and divalent counterions and lB ) 1 (circles). (b) Hard sphere and electrostatic contributions to the Γ ) φ - 1.

incorporated explicitly, or implicitly via a DH approximation. As mentioned in the Introduction, scaling theory fails dramatically for the osmotic pressure of polyelectrolyte solutions. Chang and Yethiraj17 performed simulations of a primitive model (charged hard sphere flexible chains and charged hard sphere counterions) of polyelectrolyte solutions with the goal of delineating different contributions as well as testing liquid state theories. The osmotic compressibility factor (or osmotic coefficient), defined as φ ) βπ/c, is a nonmonotonic function of concentration. It is convenient to consider φ rather than π because φ ) 1 for an ideal gas, and φ is in the range 1-10 for the cases considered. Figure 9a depicts φ as a function of reduced concentration cσ3 for three cases, lB ) 1 and 3 with monovalent counterions and lB ) 1 with divalent counterions (all for N ) 16). As the degree of electrostatic correlations increases, i.e., with increasing lB or going from monovalent to divalent counterions, the nonideality in φ increases; i.e., it becomes less than 1. This is because the electrostatic contribution is negative and increasing electrostatic effects decreases the osmotic pressure. Also shown in the figure is a line with slope 5/4, which is the scaling behavior of φ in neutral semidilute polymer solutions; the simulation results are consistent with this scaling. In the simulations, the pressure is calculated from the pressure equation, which involves integrals over correlation functions. The expression for the pressure naturally separates into several contributions, which may be attributed to the hard sphere interactions or electrostatic interactions. Of course, the correlation functions themselves depend on all of the interactions, so this separation into different contributions is not completely clean. Figure 9b shows that the hard sphere contribution to the excess osmotic coefficient (defined as Γ ) φ - 1) is a monotonically increasing function of concentration, and is very

similar to that of a neutral system, and the electrostatic contribution is a monotonically decreasing function of concentration, as in solutions of simple electrolytes. The excess part of the osmotic coefficient is dominated by the electrostatic contribution at dilute concentrations and by the hard sphere contribution at higher concentrations. As a consequence, polyelectrolyte solutions in concentrated solutions show the same scaling behavior as that of neutral polymer solutions. However, there is a transition regime where the electrostatic and hard sphere contributions are comparable. The concentration in this regime is generally higher than c*, and therefore, electrostatic effects can be important even in semidilute solutions. In other words, the neutral chain scaling π ∼ c9/4 is not expected over the entire range of semidilute concentrations but only when the concentration is high enough for excluded volume (hard sphere correlations) to become dominant. Consequently, this scaling is not clear for the more correlated systems in Figure 9a. It is not possible to divide the osmotic pressure into solely “polymeric” and “counterion” contributions. For the hard sphere contribution, an examination of the virial shows that there are three terms: these involve correlations between polymer molecules (Γpp), between counterions (Γcc), and between polymer molecules and counterions (Γpc + Γcp ) 2Γpc). Although the electrostatic contributions can also be written as the sum of these three parts, they cannot be calculated separately. In this case, terms in the virial have to be collected so that the integrals over the correlation functions are convergent, which is possible if we consider contributions for pairs that are electrically neutral. For the electrostatic contribution, we therefore consider two terms, Γcc + Γcp and Γpp + Γpc. The former may be considered the counterion contribution because it contains all of the effects due to the counterion-counterion correlations. The dominant contribution to the excess osmotic pressure comes from the correlations between the polymers and counterions, and not from correlations between the counterions. Figure 10 depicts the electrostatic part of Γcc + Γcp and Γpp + Γpc for N ) 16 and lB ) 1. The figure shows that the term containing Γcc makes a negligible contribution to the total. Note that Γcc and Γpp are both positive quantities, and the total electrostatic Γ is negative. Therefore, the important contribution to the osmotic pressure comes from the polymer-counterion contribution, Γpc. Interestingly, Liao et al.19 arrived at the opposite conclusion, namely, that the counterions make the dominant contribution to the osmotic pressure, from their molecular dynamics simulations of polyelectrolyte solutions. They did not calculate different contributions to the osmotic pressure from their simulations but


Figure 11. Comparison of the PRISM integral equation theory (lines) to simulation results for the osmotic pressure for lB ) 1 and 3. The circles are simulation results for lB ) 1 from Chang and Yethiraj17 (filled circles) and Stevens and Kremer17 (open circles), and the squares are simulation results for lB ) 3. The straight line has a slope of 9/4. The PRISM predictions for the two cases are almost indistinguishable.

instead estimated the polyion contribution, πp, to the osmotic pressure from βπp ∼ 1/ξ3, with ξ calculated from the simulations, and obtained the counterion contribution by subtracting this from the total. Note that they ignored the cross term, which we find is dominant. Furthermore, their scaling ansatz does not give a quantitative estimate of the polymeric contribution to the osmotic pressure. The conclusion that the counterion contribution is not dominant comes from simulation studies on only one model system, namely, the primitive model of chains of charged spheres, and for a limited range of degrees of polymerization, namely, N ) 16, 32, and 64. One might, however, expect these conclusions to be generally valid for the following reasons. First of all, there is not any discernible chain length dependence in the results, which suggests that the relatively short chains studied are not an issue. Second, because the Coulomb interaction is long-ranged, local chemical details are not expected to be as important in dilute solutions. It would be interesting, of course, to support these findings with simulations of other systems. The PRISM theory is in good agreement with the simulation results except under conditions where electrostatic correlations are dominant. Figure 11 depicts the osmotic pressure for N ) 16 and lB ) 1 and 3. The predictions of the theory are in excellent agreement with simulations for lB ) 1 but not for lB ) 3. In fact, the theory predicts a negligible difference in the osmotic pressure for the two cases, which is not borne out in the simulations. It is possible that this deficiency comes from the use of the mean-spherical approximation (MSA) closure60 for the counterions, and this is known to underestimate the ion correlations. It would be more appropriate to employ the hypernetted chain (HNC) closure,60 but this closure has convergence problems in dilute solutions. 7. Polyelectrolytes in Poor Solvents: Importance of Explicit Solvent Computer simulations have been very useful in establishing the behavior of polyelectrolytes in good solvents. In most experiments, e.g., PSS in water, however, the solvent quality is poor for the hydrophobic backbone. The standard way to decrease solvent quality in simulations is to include an attractive

J. Phys. Chem. B, Vol. 113, No. 6, 2009 1547 interaction between the polymer sites.90,91 In polyelectrolyte solutions, this leads to a strange situation where the total interaction between two charged sites becomes attractive! Simulations of these models are plagued by difficulties in equilibration. At high concentrations, these simulations show glassy agglomeration and gel-like metastable structures, which have not been seen experimentally.92 It has also increasingly become clear that an explicit incorporation of the solvent molecules could be important.93-95 An explicit incorporation of solvent molecules, as an additional component, is particularly important in the study of polymers in poor solvents. Chang and Yethiraj95 demonstrated that the dynamics of polymer collapse were qualitatively different if the solvent molecules were included explicitly rather than implicitly, even for neutral chains. The main reason is that the solvent-induced effects are patently not pairwise decomposable. When the solvent molecules are included implicitly, via a pair interaction, two polymer sites on the interior of a collapsed structure (e.g., globule or cylinder) feel a very strong attractiVe interaction. In reality, however, the interaction should be just the bare interaction between monomers, which is strongly repulsiVe in polyelectrolytes. It is well-known that “averaging” over a component in a multicomponent system results in nonpairwise additive interactions between the molecules of the remaining components, and the collapse of polymers or folding of proteins are specific cases where the pairwise approximation can have dramatic consequences because there is a vast difference between the interaction between beads depending on their local environment (i.e., inside a globule or outside). A first step in incorporating a realistic solvent model was taken by Chang and Yethiraj,93,94 who developed a simple explicit solvent model for polyelectrolyte solutions. For saltfree solutions, their model consisted of polyions, counterions, and solvent molecules. The nonelectrostatic interaction between any two monomers and two solvent spheres included an attractive component, the strength of which controlled the quality of the solvent. The solvent quality is characterized by a single parameter λ, which is the well depth of the attractive interaction; for λ ) 0, the solvent quality is good, and as λ increases, the solvent quality becomes poorer. The solvent does not alter the electrostatic interactions between the charged groups. A single neutral chain collapses for λ ≈ 0.1 and a single charged chain collapse for λ ≈ 0.4. Snapshots of single polyelectrolyte chains in explicit solvent did not show the pearl-necklace structures predicted by theory and seen in implicit solvent simulations. Figure 12 depicts snapshots from simulations with N ) 128 and for various values of the solvent quality parameter (for clarity, the solvent molecules are not shown). There are several interesting features. The counterions are in the immediate vicinity of the polymer under conditions where the chain is far from collapsed. The collapse of the chain is accompanied by the counterions being gathered by the chain. This is expected because the counterions can bridge two charged beads on the polymer. Even when the chain is completely collapsed, the globule is not electrically neutral; i.e., there are some counterions “free” in solution. The collapse of the polymer chain is accompanied by a dramatic decrease in the counterion self-diffusion coefficient, suggesting that measuring the diffusion of counterions could be a useful marker for the collapse of the polymer. An analysis of the conformations using cluster algorithms gives results consistent with the theoretical predictions; i.e., the chains appear to have pearl-necklace structures on average. In explicit solvent simulations, the globules are held together quite loosely and are


Yethiraj For single chains, explicit solvent simulations predict that the polymer adsorbs flat onto the surface, whereas implicit solvent simulations predict that the chain collapses in the region away from the surface. In many-chain systems, thick adsorbed layers are formed in simulations with explicit solvent, which does not happen in implicit solvent. A very simple effective solvent model, called the solvent-accessible-surface-area (SASA) model,98 is in qualitative agreement with explicit solvent simulations, at a fraction of the computational cost. 8. Conclusions and Outlook

Figure 12. Snapshots of single chains from simulations94 with explicit solvent, for various values of the solvent quality parameter λ (a larger λ corresponds to a poorer solvent and a neutral chain collapse for λ ≈ 0.1). The solvent molecules are not shown.

Figure 13. Simulation results93 for the variation of the mean-square radius of gyration of polylectrolyte chains in explicit solvent with concentration for different values of λ.

therefore much more mobile and dynamic objects than might be inferred from simulations with implicit solvent. Results for the average chain size were similar to what is seen in simulations with implicit solvent. The impact of incorporating the solvent explicitly on the solution properties is dramatic. Under similar solvent conditions, previous simulations observed polymer collapse, gelation, and glassy dynamics,91 whereas explicit solvent simulations showed phase separation.93 The reason for this is that the many-body nature of solvent-induced interactions plays a crucial role in the properties of polymers in poor solvents. The effective pair interaction is an oversimplification that results in qualitatively different results for the solution properties. In explicit solvent, the chains do not collapse. Figure 13 depicts the mean-square radius of gyration as a function of concentration for various values of the solvent quality. In good solvents, the chain size decreases as the concentration is increased. In poor solvents, however, the chain size increases as the concentration is increased. The simulations are consistent with the chains adopting conformations similar to that in a polymer melt. The many-body nature of the solvent-induced interactions is also important for the surface behavior of polyelectrolytes.96,97

Polyelectrolyte solutions are one of the most fascinating systems in soft matter because the static and dynamic properties are very interesting and reflect a delicate balance between many interactions. In the past decade, there has been considerable progress in our understanding of these solutions, and some of these have been reviewed in this paper. Despite significant advances, there are still many open questions in polyelectrolyte solutions with debates of fairly elementary questions such as chain conformations, osmotic pressure, and domain formation. Many of the theories use the concept of counterion condensation, a molecular understanding of which is elusive. Some challenges and avenues are discussed below. The behavior of primitive models of polyelectrolytes in good solvents can be considered well understood. Extensive computer simulations have elucidated the properties of these systems, and liquid state theories have been very successful. For example, the PRISM integral equation theory is in good agreement with simulation results for the static structure, conformational properties, surface behavior, and thermodynamics of polyelectrolyte solutions in good solvents. With a minimal number of adjustable parameters, the theory can fit experimental data. This has to be considered a significant success of polymeric liquid state theory. The behavior of polyelectrolytes in poor solvents, however, is not very well understood. For good solvents, the study of minimal models such as bead-spring polyions and counterions played an important role in developing an understanding of the physical chemistry. There is, however, no widely accepted minimal model for polyelectrolytes in poor solvents. For a variety of problems, including the conformational properties of single chains, the surface behavior of polyelectrolytes, and the thermodynamics and phase behavior of polyelectrolyte solutions, it has been shown that the simplest implicit solvent models are in qualitative error (when compared to simulations with explicit solvent). Explicit solvent simulations are computationally intensive, however, and more sophisticated implicit solvents might play a role. Many-body implicit solvent models where the solvation free energy is a function of polymer conformation appear to be promising. The simplest such model, namely, the solvent-accessible-surface-area (SASA) model, is in qualitative agreement with explicit solvent simulations.96 These solvent models, however, only treat the nonelectrostatic solvent effects, and do not effectively incorporate, e.g., dielectric solvation effects. The most straightforward way of extending the integral equation approach is to incorporate the solvent as an additional component. The only challenge is that appropriate closure relations have to be investigated and tested. It is important to note that most experimental systems do involve poor solvents because polyelectrolytes generally contain both hydrophilic and hydrophobic groups. Biological polymers are a classic case where this is true. Therefore, the solvent quality is likely to be poor for at least some of the segments and the solvent effects discussed above could therefore be important in most practical situations.

Centennial Feature Article Strongly coupled systems remain a challenge for theory. The behavior of systems with divalent and trivalent counterions is not well established, although they are quite common in experiments. The behavior of the PRISM theory for this situation is not known because the theory runs into convergence problems (with the accurate closure approximations). For the simpler case of electrolytes in the vicinity of an infinite charged cylinder, density functional theories have been shown to be accurate.99,100 Investigating density functional theories for the structure and thermodynamics of polyelectrolyte solutions is a promising avenue for research. A wide-open avenue for research is the dynamics of polyelectrolyte solutions. Simulations101,102 have demonstrated that the structure and dynamics are closely coupled in these systems, and there is interesting static structure on many length scales. One would expect a successful theory for the dynamics to incorporate the static structure and the coupling between various components in the system. The simplest liquid state theories for the dynamics, such as mode-coupling theory,103 have not been tested experimentally, and more experiments on simple systems would greatly add to our understanding of the viscosity and diffusion coefficient of polyelectrolyte solutions. The advances in theory have not impacted our understanding of experiments as much as one might hope. Part of the reason is that the majority of experimental systems contain both the above-mentioned complications of poor solvents and strong coupling. It is becoming clear that we do not completely understand what the important interactions are in polyelectrolyte solutions. It would appear that we have learned as much as possible from “spherical cow” models of bead-spring chains and threads and further progress must come from understanding the impact of “details” on polyelectrolyte behavior. For example, do the constraints on the placement of charged groups in PSS (caused by the local chemistry of the polymers) play an important role on their physical properties? Advances in experimental methods, for example, single molecule spectroscopy, promise to provide more detailed information about the static and dynamic properties of polyelectrolyte solutions. Fluorescent correlation spectroscopy (FCS) measurements104 for the diffusion of polycarboxylic acids in aqueous solutions and polarized Raman spectroscopy measurements for PAA conformations105,106 are examples of the type of detail that one could get from spectroscopic measurements. An attractive avenue therefore is the investigation of realistic models of polyelectrolytes. Such simulations are routine in biological chemistry, and should be interesting in polymers as well. The one caveat is that in polyelectrolytes we are interested in phenomena on much larger length scales and time scales and this will push the envelope of feasibility. Chemically realistic simulations should become even more relevant in the near future with advances in experimental techniques promising to provide detailed information regarding polyelectrolyte structure and dynamics. The development of chemically realistic simulation models and methods, perhaps borrowed from biophysics but tuned to the needs of polymer science, might be an important avenue for future research. Acknowledgment. This material is based on work supported by the National Science Foundation through grants CHE9502320, CHE-9732604, CHE-0315219, and CHE-0717569, and S. C. Johnson, through a S. C. Johnson distinguished fellowship. The important contributions of Professor ChwenYang Shew, Professor Rakwoo Chang, and Dr. Govardhan Reddy are gratefully acknowledged.

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