chemistry of polyelectrolytes

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iii

Preface

Polyelectrolytes are polymers bearing dissociated ionic groups. Their uniqueproperties, dominated by strong long-range electrostatic interactions, havebeen studied over the past few decades. Substantial theoretical and experi-mental efforts have been made, for example, to understand the origin ofslow domains or loose clusters in semidilute solutions of highlycharged polyelectrolytes. This kind of attractive interaction between macro-ions is not consistent with the standard theory based on the overlap of theelectrical double layers between charged flat surfaces. Charge-fluctuationforces between several polyions due to sharing of their counterions or at-traction by expansion of the condensed layers between charged rods havebeen suggested to explain the appearance of these formations. Particularfocus has also been placed on polyion interactions with counterions, sincetheir condensation on the polyion surface is one of the most characteristicproperties of the polyelectrolytes. The interaction of polyions with othercharged or neutral species and, in particular, the adsorption of ionizablepolymers at interfaces, is the second aspect of the physical chemistry ofpolyelectrolytes that has been extensively studied due both to the funda-mental importance of this phenomenon and to its central role in numerousindustrial processes.

The interest in polyelectrolyte investigations has increased in the last fewyears as evidenced by the first two International Symposiums on Polyelec-trolytes, held in 1995 and 1998. The number of papers dealing with poly-electrolytes has also increased substantially. This is not surprising consid-ering the wide application of natural and synthetic polymers in medicine,

iv Preface

paper making, mineral separation, paint and food industries, cosmetics andpharmacy, water treatment processes, and soil remediation. The fabricationof layer-by-layer assembled multicomposite films, which fall in a categoryof novel nanomaterials, presently hold a central place in this area.

The purpose of this volume is to collect results that show the currentunderstanding of the fundamental nature of polyelectrolytes. I hope that itsappearance will stimulate the research efforts toward solving many problemsin this interdisciplinary field. Practical utilization of these results is beyonddoubt. The book is addressed to scientists working in the fields of biochem-istry, molecular biology, physical chemistry of colloids and ionizable poly-mers, and their applications in related technical processes.

The volume consists of three parts. The first deals with static and dynamicproperties of salt-free polyelectrolyte solutions and of solutions with addedsalts. An extension is presented of the counterion condensation theory to thecalculation of counterionpolyion, coionpolyion, and polyionpolyionpair potentials and the appearance is predicted of inverted forces leading tothe formation of loose clusters in solutions of polyelectrolytes. The originof counterion-mediated attraction between like-charged chains is also dis-cussed within a charge fluctuation approach that reconciles the thermal fluc-tuation approach with the ionic crystal one. A new criterion for counterioncondensation is introduced through molecular dynamics simulations of acell-like model for stiff polyelectrolytes; the effects considered include poly-ions overcharging, charge oscillations, and attractive interactions. MetropolisMonte Carlo simulation is also applied to calculate counterion distributions,electric potentials, and fluctuation of counterion polarization for model DNAfragments. Theoretical approaches developed for the description of coilglobule transition of polyelectrolyte molecules are treated in two limitingsituationsfor a single macromolecule at infinite dilution and for a poly-electrolyte gel. Although emphasis is placed on the recent developments inthe theory of polyelectrolytes, this first part provides a partial review of thenew experimental results that try to explain different aspects of the physicalchemistry of polyelectrolytes.

The second part is devoted to adsorption of polyelectrolytes at interfacesand to flocculation and stabilization of particles in adsorbing polymer so-lutions. A recent theory of the electrostatic adsorption barrier, some typicalexperimental results, and new approaches for studying the kinetics of poly-electrolyte adsorption are presented in the first chapter of this part. In thefollowing chapters, results are collected on the electrical and hydrodynamicproperties of colloidpolyelectrolyte surface layers, giving information onthe structure of adsorbed layers and their influence on the interactions be-tween colloidal particles; examples and mechanisms are analyzed of poly-electrolyte-induced stabilization and fragmentation of colloidal aggregates;

Preface v

self-assembled monolayers from synthetic polyelectrolytes on water or solidsurfaces and the role of amphiphilic polyelectrolytes for the emulsion sta-bility are considered. Special attention is given to surface force measure-ments that show how association between polyelectrolytes and surfactantsat solidliquid interfaces influences surface interactions and structure of ad-sorbed layers.

The third part discusses polyelectrolyte complex formation and complex-ation of polyelectrolytes with surfactants and proteins. Mobility of shortchains and dynamic properties of polyelectrolyte gels are also considered.Phase transitions in ionic gels are explained with simple models in whichpolymerpolymer interactions are taken into account at a molecular level.In the second chapter of this part, recent experimental and theoretical ad-vances are summarized for gel electrophoresis, which is invaluable in pre-dicting conformation and structural changes of biologically significantmacromolecules. In the following chapters, results are grouped for the stoi-chiometry, structure, and stability of highly aggregated polyelectrolyte com-plexes; for the role of hydrophobicity and electric charge of the partners inthe protein binding to amphiphilic polyelectrolytes; and for the micellar-likeaggregation of surfactants bound to oppositely charged polyelectrolytes.

I wish to thank first Professor Arthur Hubbard, who invited me to edit avolume on this rapidly expanding field. Acknowledgments are due to allauthors for their valuable contributions and willing cooperation. I acknowl-edge with gratitude the assistance of Ani Pesheva in the correspondence, aswell as the efforts of our production editor, Paige Force, and of all my friendswho contributed to the production of this volume.

Tsetska Radeva

vii

Contents

Preface iiiContributors xi

Part I Structure and Properties of Polyelectrolyte Solutions

1. Structure and Dynamics of Polyelectrolyte Solutions byLight Scattering 1Marian Sedlak

2. Molecular Dynamics Simulations of the CylindricalCell Model 59Markus Deserno, Christian Holm, and Kurt Kremer

3. Inverted Forces in Counterion Condensation Theory 111Jolly Ray and Gerald S. Manning

4. Polyelectrolyte Solutions with Multivalent Added Salts:Stability, Structure, and Dynamics 135Maurice Drifford and Michel Delsanti

5. Physical Questions Posed by DNA Condensation 163Bae-Yeun Ha and Andrea J. Liu

6. Conformational Transition in Polyelectrolyte Molecules:Influence of Osmotic Pressure of Counterions 181Valentina V. Vasilevskaya

viii Contents

7. Conductance of Polyelectrolyte Solutions, Anisotropy andOther Anomalies 203Hans Vink

8. Electrical Polarizability of Polyelectrolytes by MetropolisMonte Carlo Simulation 223Kazuo Kikuchi

9. Polyelectrolytes in Nonaqueous Solutions 245Masanori Hara

Part II Polyelectrolytes at Interfaces

10. Kinetics of Polyelectrolyte Adsorption 281Martien A. Cohen Stuart and J. Mieke Kleijn

11. Electric Light Scattering of Colloid Particles inPolyelectrolyte Solutions 305Tsetska Radeva

12. Monolayer Assemblies of Poly(L-Glutamic Acid)s atTwo-Dimensional Interfaces 347Nobuyuki Higashi and Masazo Niwa

13. Emulsions Stabilized by Polyelectrolytes 363Patrick Perrin, Frederic Millet, and Bernadette Charleux

14. PolyelectrolyteSurfactant Interactions at SolidLiquidInterfaces Studied with Surface Force Techniques 447Per M. Claesson, Andra Dedinaite, and Evgeni Poptoshev

15. Fragmentation of Colloidal Aggregates byPolyelectrolyte Adsorption 509Emile Pefferkorn

16. Interactions Between Polyelectrolytes and Kaolin 567Joachim Kotz and Sabine Kosmella

Part III Polyelectrolyte Complexes and Gels

17. Phase Transitions in Polyelectrolyte Gels 591Etsuo Kokufuta

18. Anomalous Migration of DNA in Gels and thePolyelectrolyte Nature of DNA 665Udayan Mohanty and Larry W. McLaughlin

Contents ix

19. Complexation Between Amphiphilic Polyelectrolytes andProteins: From Necklaces to Gels 687Christophe Tribet

20. Polyelectrolyte Complex Formation in Highly AggregatingSystems: Methodical Aspects and General Tendencies 743Herbert Dautzenberg

21. Surfactant Binding to Polyelectrolytes 793Ksenija Kogej and Joze Skerjanc

22. Metal Complexation in Polyelectrolyte Solutions 829Tohru Miyajima

Index 875

xi

Contributors

Bernadette Charleux, Ph.D. Laboratoire de Chimie Macromoleculaire,Universite Pierre et Marie Curie, Paris, FrancePer M. Claesson, Ph.D. Department of Chemistry, Royal Institute ofTechnology, and Institute for Surface Chemistry, Stockholm, SwedenMartien A. Cohen Stuart, Ph.D. Laboratory of Physical Chemistry andColloid Science, Wageningen University, Wageningen, The NetherlandsHerbert Dautzenberg, Dr.habil.nat Department of Colloid Chemistry,Max Planck Institute of Colloids and Interfaces, Golm, GermanyAndra Dedinaite, Ph.D. Department of Chemistry, Royal Institute ofTechnology, and Institute for Surface Chemistry, Stockholm, SwedenMichel Delsanti, Ph.D. Service de Chimie Moleculaire, Commissariat alE nergie Atomique Saclay, Gif sur Yvette, FranceMarkus Deserno, Ph.D. Max-Planck-Institute for Polymer Research,Mainz, GermanyMaurice Drifford, Ph.D. Service de Chimie Moleculaire, Commissariata lE nergie Atomique Saclay, Gif sur Yvette, FranceBae-Yeun Ha, Ph.D.* Department of Chemistry and Biochemistry,University of California, Los Angeles, California

*Current affiliation: Department of Physics, Simon Fraser University, Burnaby, British Co-lumbia, Canada

xii Contributors

Masanori Hara, Ph.D. Department of Chemical and BiochemicalEngineering, Rutgers University, Piscataway, New Jersey

Nobuyuki Higashi, Ph.D. Department of Molecular Science andTechnology, Doshisha University, Kyoto, Japan

Christian Holm, Ph.D. Max-Planck-Institute for Polymer Research,Mainz, Germany

Kazuo Kikuchi, Sc.D. Associate Professor, Department of Life Sciences(Chemistry), Graduate School of Arts and Sciences, University of Tokyo,Tokyo, Japan

J. Mieke Kleijn, Ph.D. Laboratory of Physical Chemistry and ColloidScience, Wageningen University, Wageningen, The Netherlands

Ksenija Kogej, Ph.D. Department of Physical Chemistry, Faculty ofChemistry and Chemical Technology, University of Ljubljana, Ljubljana,Slovenia

Etsuo Kokufuta, Ph.D. Institute of Applied Biochemistry, University ofTsukuba, Tsukuba, Ibaraki, Japan

Sabine Kosmella, Ph.D. Institute of Physical and Theoretical Chemistry,University of Potsdam, Potsdam, Germany

Joachim Kotz, Ph.D., Dr.rer.nat.habil Department of ColloidChemistry, Institute of Physical and Theoretical Chemistry, University ofPotsdam, Potsdam, Germany

Kurt Kremer Professor, Max-Planck-Institute for Polymer Research,Mainz, Germany

Andrea J. Liu, Ph.D. Department of Chemistry and Biochemistry,University of California, Los Angeles, California

Gerald S. Manning, B.A., Ph.D. Department of Chemistry, RutgersUniversity, Piscataway, New Jersey

Larry W. McLaughlin, Ph.D. Department of Chemistry, BostonCollege, Chestnut Hill, Massachusetts

Frederic Millet, Ph.D. Service de Physique LE tat Condense,Commissariat a` LE nergie Atomique (Saclay), Gif-sur-Yvette, FranceTohru Miyajima, Ph.D. Department of Chemistry, Faculty of Scienceand Engineering, Saga University, Saga, Japan

Udayan Mohanty, Ph.D. Department of Chemistry, Boston College,Chestnut Hill, Massachusetts

Contributors xiii

Masazo Niwa, Ph.D. Department of Molecular Science and Technology,Doshisha University, Kyoto, Japan

Emile Pefferkorn, Ph.D. Department of Physics of Dispersed Mediaand Interfaces, Institut Charles Sadron, Strasbourg, France

Patrick Perrin, Ph.D. Laboratoire de Physico-chimie Macromoleculaire,Ecole Superieure de Physique et de Chimie Industrielles, Paris, France

Evgeni Poptoshev, M.Sc. Institute for Surface Chemistry, Stockholm,Sweden

Tsetska Radeva, Ph.D. Institute of Physical Chemistry, BulgarianAcademy of Sciences, Sofia, Bulgaria

Jolly Ray Department of Chemistry, Rutgers University, Piscataway,New Jersey

Marian Sedlak, D.Sc. Institute of Experimental Physics of the SlovakAcademy of Sciences, Kosice, Slovak Republic

Joze Skerjanc, Ph.D. Department of Physical Chemistry, Faculty ofChemistry and Chemical Technology, University of Ljubljana, Ljubljana,Slovenia

Christophe Tribet, Ph.D. Laboratory of Physical Chemistry ofMacromolecules, Ecole Superieure de Physique et de Chimie Industrielles,and Centre National de la Recherche Scientifique, and University of Paris6, Paris, France

Valentina V. Vasilevskaya, Ph.D. Nesmeyanov Institute ofOrganoelement Compounds, Russian Academy of Sciences, Moscow,Russia

Hans Vink, Ph.D. Department of Physical Chemistry, University ofUppsala, Uppsala, Sweden

11Structure and Dynamics ofPolyelectrolyte Solutionsby Light ScatteringMARIA N SEDLA K Institute of Experimental Physics of the SlovakAcademy of Sciences, Kosice, Slovak Republic

Static light scattering and dynamic light scattering are useful experimentaltools for the investigation of the structure and dynamics of polyelectrolytesolutions as well as for the characterization of polymer systems in general.The structure can be probed by static light scattering (SLS) on a certainlength scale (typically from 20 nm to several microns) limited mainly bythe light wavelength. Therefore the accessible structural information is usu-ally on the level of a whole polymer chain, interchain correlations, and thesolution structure on a large length scale exceeding single chain dimensionsand interchain separation distances. Light scattering does not yield directinformation on the local structure inside the chain, referred to also as theprimary and secondary structure. However, smaller length scales can beprobed by very similar techniques in which x-rays or neutrons are usedinstead of visible light (small-angle x-rays and small-angle neutron scatter-ing). The dynamics probed by dynamic light scattering (DLS) is limited bythe technical possibilities of currently available instrumentation (from ca 0.5s to seconds). Naturally, the dynamics is related also to the structure.Therefore some structural information can be calculated from the dynamicdata, too. In this case, the length scale probed is not limited by light wave-length and extends below 1 nm. Charge interactions in polyelectrolyte so-lutions dominantly influence the structure and dynamics on the above-men-tioned length and time scales, and therefore light scattering is a source ofinformation on the character of these interactions. Absolute values of scat-tering intensities are thermodynamic quantities that enable us under certaincircumstances to calculate such important parameters as polymer molecularweight or the second virial coefficient in the virial expansion of osmotic

2 Sedlak

pressure. As will be shown throughout this chapter, light scattering frompolyelectrolyte solutions is a rather complex and variable subject, and theinterpretation of light scattering data is often a nontrivial task. The aim ofthis chapter is to present the issue of the structure and dynamics of poly-electrolyte solutions by light scattering in a compact form. It is based onthe authors own work, which is complemented by selected literature results.It is not intended to be a complete review, and therefore the selection ofcited literature is to some extent personal. We acknowledge at this pointmany valuable works in the field that are not included. Our presentation isfocused mainly on linear flexible polyelectrolytes, but many properties areuniversal and apply also to other polyelectrolyte classes unless explicitlyspecified in another way. Excluded is the case of polyelectrolyte solutionswith multivalent counterions, which will be discussed in a different chapter,and some other special cases specified in the text.

I. BRIEF THEORETICAL BACKGROUNDIn general, the source of light scattering from a polymer (or polyelectrolyte)solution is the existence of spatial and temporal fluctuations in refractiveindex in this scattering medium. The overall scattering intensity measuredin an SLS experiment is proportional to the average of the square of refrac-tive index fluctuations in the scattering volume,

2I(q ) n (q) (1)ifwhere the tensor n(q ) is a Fourier transform of the refractive index fluc-tuation n(r) at a place r of the scattering volume, and q is the scatteringvector, with an absolute value of

4n q = q = k k = sin (2)f i 20

with n the refractive index of the medium, 0 the light wavelength in vac-uum, the scattering angle, and k

i and k

f the wavevectors of the initial andfinal (scattered) beam, respectively. q1 defines the length scale at which thestructure is probed. The scattered electric field time autocorrelation functionmeasured in a DLS experiment is proportional to the time autocorrelationfunction of the fluctuations in refractive index

(1)g (t) n (q, 0)n (q, t) (3)if ifwhere the tensor n(q, t) is a Fourier transform of the refractive indexfluctuation n(r, t) at a place r at a time t. The correlation function as givenby Eq. 3 reflects temporal correlations of refractive index fluctuations and

Structure and Dynamics by Light Scattering 3

can be expressed as a Laplace transform of the spectrum of relaxation timesA()

(1) t/g (t) = A()e d (4)0

The spectrum of relaxation times obtained from the correlation function byinverse Laplace transform is frequently a multimodal distribution where sep-arate peaks can be ascribed to different modes. Each mode represents acertain type of motion giving rise to a fluctuation with given frequency(relaxation time) and strength. The position of the peak on the time axiscorresponds to the relaxation time i of the particular mode. If the mode isdiffusive, a diffusion coefficient Di = (1/i)q2 can be ascribed to this relax-ation time. The peak area corresponds to the strength of the particular fluc-tuation (the portion of scattering intensity due to this mode). The total scat-tering intensity as measured in an SLS (often referred to as integral lightscattering) experiment is given as

I A() d (5)0

Fluctuations in refractive index in two-component systems (particles andsolvent) arise due to density and concentration fluctuations. Density fluctu-ations are determined by the isothermal compressibility of the scatteringmedium RT(/p), where R is the gas constant, T is absolute temperature,

is density, and p is pressure. For moderate concentrations (up to 20%),the density fluctuations in solution can be assumed to be equal to densityfluctuations in the pure solvent. These are relatively fast compared to thetime window probed by DLS and therefore not covered in the spectrum ofrelaxation times. Hence if the solvent scattering is subtracted, only concen-tration fluctuations are taken into account. The excess light scattering byparticles at a zero scattering angle I(0) = Itotal(0) Isolvent(0) is given byconcentration fluctuations and can be related to osmotic compressibilityRT(c/), where

Kc 1 = (6)

I(0) RT c

where is the osmotic pressure against the solvent, c is the solute concen-tration in g/L, and the constant K is defined for vertically polarized incidentlight as

4 Sedlak

2 2 24 n (dn/dc)K = (7)4 N0 A

where NA is Avogardros number, n is the solution refractive index, and dn/dc is referred to as the refractive index increment, which represents thescattering contrast of particles in a particular solvent.

The intensity I() scattered at a nonzero angle can be expressed viaosmotic compressibility when the size of particles and the range of inter-particle forces is reasonably small compared to 2/q:

Kc 1 1 = (8)

I() RT p() cwhere p() governs both intra- and interparticle interference. The former isdue to the interference of light rays scattered from different segments of thesame particle, the latter to the interference of light rays scattered from seg-ments belonging to different particles. In the case of homogeneous spheres,p() can be factored as p() = P()s(), where P() is the so-called singleparticle form factor, reflecting only the intraparticle interference, and s() isreferred to as the solution structure factor, reflecting only the interparticleinterference. This factorization does not hold in general for other particlearchitectures and is certainly not valid for rigid rods [1]. Nevertheless, it canbe considered as a good approximation in most cases. Usually the normal-ization is such that P(0) = s(0) = 1.

Scattering from a collection of identical particles can be also expressedin terms of the so-called particle approach to light scattering as

2I() = kNm P()S() (9)where k is experimental constant covering the square of the scattering con-trast, N is the number of particles in the scattering volume, and m is theparticle mass. In this notation, S() = 1 in the absence of interparticle cor-relations. S() 1 covers interparticle interactions, which are implicitly in-cluded in c/ in Eq. 8. Thus S(0) does not have to be equal to 1 as distinctfrom s(0). S() can be calculated as [2]

sin(qr) 2S() = S(q) = 1 4 [g(r) 1] r dr (10)n qr0where n is the number density of particles and g(r) is the radial distributionfunction of particles reflecting the probability of finding a particle at distancer from another particle. g(r) = 1 evidently implies S() = 1.

From the point of view of the dynamics, the total excess scattering in-tensity in the simplest case of a dilute two-component system (solvent andweakly interacting particles with dimensions small compared to q1) corre-


FIG. 1 Schematic diagram of the apparatus for static and dynamic light scattering.NF, neutral optical filters; L1 , focusing lens; SC, scattering chamber; C, sample cell;TC, temperature controller; , scattering angle; IA, iris aperture; L2 , objective; V,viewer; P/VS, pinhole/vertical slit; PMT, photomultiplier tube; HV, LV, high voltage,low voltage power supply; AD, amplifier and discriminator; DC, digital counter;C/C, correlator/computer.

sponds to the scattering contribution from one modetranslational diffusionof particles. This does not hold in general, where more modes can be ob-served in the spectrum of relaxation times. This is the case for (1) multi-component systems, (2) systems with large particles where internal dynamics(dynamics of segments inside the particle) can be seen or rotational diffusioncan be seen when particles are asymmetric, (3) concentrated solutions ofparticles where short-time and long-time diffusion can be seen, etc. In fact,polyelectrolyte solutions are systems where multimodal spectra of relaxationtimes are obtained.

II. LIGHT SCATTERING APPARATUSA schematic diagram of the apparatus for static and dynamic light scatteringis shown in Figure 1 (top view). The initial laser beam passes horizontallythrough the sample and defines the wavevector k

i. Scattered light is detectedby a detection unit, which rotates in a horizontal plane. The position of the

6 Sedlak

detection unit (scattering angle ) defines the wavevector of the scatteredbeam k

f and hence the scattering vector q (Eq. 2). The scattering angle can be varied in typical setups from 15 to 150, which corresponds to valuesof q = 0.00420.031 nm1 (for water solutions and the laser wavelength 0= 514.5 nm). The scattered light is detected by photon counting. Integratedintensities (number of photons per defined, sufficiently long time interval)are measured in an SLS experiment. Fluctuations of the scattering intensitiesare analyzed by photon correlation in a DLS experiment. The measuredquantity is in this case a homodyne autocorrelation function of the scatteringintensity, which can be converted into the scattered electric field autocor-relation function g(1)(t) (see Eq. 3). More technical details on the light scat-tering experiment can be found elsewhere [3].

III. LIGHT SCATTERING FROM POLYELECTROLYTESOLUTIONSMULTIMODAL SPECTRA OFRELAXATION TIMES

Multimodal spectra of relaxation times arise as a consequence of the mul-ticomponent nature of polyelectrolyte solutions consisting of solvent (usuallywater), polyions, counterions originating from the dissociation of ionizablegroups on polyions, and low-molecular-weight salt composed of small mo-lecular or atomic ions. Small ions of the salt, which are likely charged,compared to polyions, are referred to as coions. We will also include in ourdiscussion the case of polyelectrolyte mixtures (polyions of the same chem-ical composition but different molecular weights). Polyelectrolyte solutionsin practice have a nonzero width of the polyion molecular weight distribu-tion and can be in principle considered as mixtures, too. Polyions are notalways distributed homogeneously in solutions and mixtures, but insteadform larger structures referred to as domains or clusters. These structuresalso contribute to the overall scattering intensity and can be therefore con-sidered as an additional component of the system.

In order to discuss scattering contributions from particular components,we can estimate several parameters coming into play: size, scattering con-trast, number concentration, and interparticle interference effects (see Eq.9). Upon assumption of negligible interparticle interactions [S() = 1], thescattering at zero angle [where P(0) 1] can be written as I(0) = kNm2

where p is the particle density, Vp is the particle volume,2 2 2 6kN V N R ,p p s p

s is the overall scattering contrast including density, and Rp is the particleradius assuming its spherical shape. With regard to sizes of particular com-ponents in polyelectrolyte solutions, these are largely differing. Radii ofsmall ions are on the order of angstroms (12 A in the case of atomic ions


as Na or Cl), the apparent radius of water molecules is approximately 1A , radii of gyration of polyions range typically from 5 to 80 nm dependingmainly on molecular weight, and radii of polyelectrolyte domains rangeapproximately from 30 to 300 nm. If we realize that the scattering is in ourrough approximation proportional to the sixth power of the particle radius,we see that the contributions may largely differ due to particle sizes. Alsonumber concentrations (number of particles per unit volume) can differlargely upon going from the so-called salt-free case (molar concentration oflow-molecular-weight salt cs 5 106 M) to the high-salt case (cs 1M) or by varying the polyion molar concentration in a range appropriate forlight scattering measurement (cp 1010 to 101 M). The scattering contrastof polyions given by the refractive index increment dn/dc is usually higherthan for typical pairs of neutral polymers and their solvents (for instance,the value for sodium polystyrene sulfonate (NaPSS) in water is dn/dc = 0.23mL/g). The contrast of small ions in water is comparable with typical valuesfor neutral polymers in their solvents (dn/dc 0.15 mL/g). The contrastof polyion domains and their number concentration are quantitatively notknown.

It can be deduced from the parameters outlined that the scattering con-tributions of particular components (giving rise to particular dynamic modes)can be very variable upon changing experimental conditions. In addition,there are also strong interparticle interference effects due to strong interpar-ticle interactions, which dramatically influence scattering intensities and in-crease the variability. The practical consequence of this discussion is a con-clusion that for correct understanding and interpretation of light scatteringdata, it is useful (if not necessary) to evaluate scattering contributions fromparticular modes in absolute units and then to discuss them separately. Thisevaluation is based on the combination of data from static and dynamic lightscattering. The total scattering intensity measured in an SLS experiment canbe normalized to the scattering from a standard (e.g., benzene). This nor-malized intensity I()/IB() can be then written as a sum of contributionsfrom particular modes I()/IB() = A1() A2() A3() . Since relativeamplitudes of the modes Ai()/Aj() are known from DLS data, absolutevalues of amplitudes (in units of benzene scattering), can be calculated, too.For bimodal spectra, where I()/IB() = A1() A2():

I()/I () I()/I ()B BA () = A () = (11)2 11 A ()/A () 1 A ()/A ()1 2 2 1

In order to be able to evaluate contributions from particular modes, thereare two requirements: corresponding peaks in the spectrum of relaxationtimes should not overlap completely, and the amplitudes should not decrease

8 Sedlak

FIG. 2 Relaxation times of dynamic modes observed in polyelectrolyte solutionsand mixtures over a broad range of experimental conditions: diffusion of lowmolecular weight salt; diffusion of polyions or polyion segments in semidilutesolutions; interaction mode in polyelectrolyte mixtures; and diffusion of poly-electrolyte domains (clusters). The data are based mostly on the work on linearflexible polyelectrolytes. Relaxation times correspond to scattering at 90. See textfor more details.

below approximately 2% of the total scattering intensity. Figure 2 showsrelaxation times of dynamic modes observed in polyelectrolyte solutions andmixtures in a whole range of accessible experimental conditions. The dataare based mostly on work on model polyelectrolyte systems [well-definedmolecular weight standards of NaPSS and weak polyelectrolytes with vari-able chargespoly(methacrylic) acid and poly(acrylic acid)] but also holdfor solutions of linear flexible polyelectrolytes in general. Most features arecommon also for solutions of globular, rigid, or semirigid (wormlike) poly-electrolytes. Excluded from the general scheme in Figure 2 are extremecases, such as extra large polyions, where in principle internal modes canbe observed, or large highly asymmetrical rigid polyions, where rotationaldiffusion and bending modes can be observed [4]. The modes shown inFigure 2 (from left to right on the time axis) correspond to (1) diffusion oflow-molecular-weight salt, (2) diffusion of polyions or polyion segments insemidilute solutions, (3) interaction mode in polyelectrolyte mixtures, and(4) diffusion of polyelectrolyte domains. It can be seen from Figure 2 thatthe four dynamic modes are well separated. There is only a more or lessapparent overlap of the polyion diffusion and interaction mode in this sche-matic diagram (apparent means that at low ionic strengths, where bothmodes are present, there is no actual overlap). The outline of the rest of thechapter is such that we discuss first aspects of the dynamics (particulardynamic modes) and afterwards aspects of static light scattering, keeping inmind the composite multimodal nature of the integral intensity. The problemof polyelectrolyte domains (clusters), which is equally reflected in both SLSand DLS, will be discussed at the end.


IV. DIFFUSION OF LOW-MOLECULAR-WEIGHT SALTSalt ions in polyelectrolyte solutions are atomic ions or low-molecular-weight molecular ions. These are referred commonly to as small ions. Com-pared to macromolecules or supramolecular structures, the dynamics ofsmall ions is fast and the scattering contribution is very low. Both effectsare due to their small dimensions. The measurement of the small-ion dy-namics by dynamic light scattering is therefore difficult. Only recently, wesucceeded in measuring the diffusion of small ions in our laboratory. Wepresent here the first brief results. Technically, the small-ion diffusion rateis relatively well within the possibilities of current dynamic light scatteringinstrumentation. For instance, the diffusion coefficient for NaCl in watercorresponds approximately to the relaxation time of 4 s at scattering angle = 45. The more important problem is the extremely weak scattering sig-nal. While the scattering contrast of most small ions in water is comparablewith the scattering contrast of neutral polymers in their solvents (dn/dc 0.15 mL/g), the small size of such ions compared to polymers or colloidsis the main factor. For illustration, the excess scattering from 3M NaClsolution in water is equal to 0.034 in units of benzene scattering. Scatteringfrom water is equal to 0.11 in the same units.

Figure 3 shows the correlation function and the corresponding spectrumof relaxation times for a solution of sodium poly(styrenesulfonate) (NaPSS)in 3.7 M NaCl. Two modes can be clearly recognized. The slower modecorresponds to the diffusion of polyions, which will be discussed in the nextsection. The faster mode corresponds to the diffusion of salt (NaCl). Asexpected for a diffusive process, the inverse relaxation time of this mode

vf (the subscript vf refers to very fast) is q2 dependent (Figure 4). Thediffusion coefficient of the salt small ions was calculated from the slope ofthe dependence vf = Dvfq2 in Figure 4 as Dvf = (1.7 0.1) 105 cm2s1.The scattering amplitude of the very fast mode varies proportionally withthe salt concentration and is q independent as expected. Figure 5 shows thecorrelation function and the corresponding spectrum of relaxation times fora pure solution of NaCl in water (no polymer added). Only one diffusivemode is present with the diffusion coefficient matching relatively closely thevalue of Dvf obtained in polyelectrolyte solution.

In the following we apply a rigorous theoretical treatment of the dynamicscattering from a system of oppositely charged point Brownian particles [5]to the case of the NaCl salt. It is assumed that particles a and b canbe characterized by diffusion coefficients Da , Db , charges Za , Zb , and numberdensities Na , Nb . The total charge neutrality implies that NaZa NbZb = 0.In a general case (Da Db), in the small-q limit, the calculated dynamicstructure factor S(q, t) = S(q)g(1)(t) is a double exponential S(q, t) = A1exp(1(q)t) A2 exp(2(q)t). It holds for 1(q) and 2(q):

10 Sedlak

FIG. 3 Correlation function and corresponding spectrum of relaxation times forsodium poly(styrenesulfonate) (NaPSS), Mw = 5,400, in 3.7 M NaCl. Polymer con-centration c = 1.9 g/L. Scattering angle = 30.

k TB2 2 3 (q) = D D (12)1 a a b b 62 2 2 D D (D D )(k T/6)a b a b b a B2 (q) = q (13)2 2 2 3D D (k T/6)a a b b B

where is the viscosity of the scattering medium, kB is Boltzmanns con-stant, and T is temperature. 1 is the total DebyeHuckel screening lengthdue to particles of both types in volume V:

4 N Na b2 2 2 2 2 = = Z Z (14)a b a b k T V V0 BThe first mode has a q-independent frequency 1(q) and is referred to as theplasmon mode by analogy with plasmas, in which the plasma frequency is


FIG. 4 Angular dependence of the inverse relaxation time vf = 1/vf of the veryfast mode corresponding to the diffusion of salt (NaCl) in solution of sodiumpoly(styrenesulfonate) (NaPSS), Mw = 5,400, in 3.7 M NaCl. Polymer concentrationc = 1.9 g/L.

also constant. However, with q 0, the amplitude A1(q) vanishes. Thereforethis mode is difficult to observe, and the dynamic structure factor reducesto a single exponential with a q2 dependent frequency 2(q). Upon neglectof hydrodynamic interactions, which are included in Eq. 13, the formula forthe frequency 2(q) simplifies, and hence the corresponding diffusion coef-ficient can be expressed as

(q) (1 Z /Z )D D2 b a a bD = = (15)2q D Z /Z Da b a b

This result, obtained upon several approximations, is identical to the clas-sical phenomenological NernstHartley formula for the coupled diffusion ofoppositely charged ions [6]. In order to apply this formula to the case of anNaCl solution, we use for the uncoupled diffusion coefficients Da = D Naand Db = where and are values of diffusion coefficient ofD , D D Cl Na Clparticular ions obtained from conductivity data extrapolated to infinite di-lution = 1.33 105 cm2s1 and = 2.03 105 cm2s1) [7].(D D Na ClUpon substitution into Eq. 15, we obtain D = =2D D /(D D ) Na Cl Na Cl1.61 105 cm2s1. The agreement with the value of the experimentallyobtained diffusion coefficient is satisfactory.

12 Sedlak

FIG. 5 Correlation function and corresponding spectrum of relaxation times for3.4 M NaCl in water. Scattering angle = 30.

V. POLYION DIFFUSIONDiffusion of polyions in polyelectrolyte solutions is in the majority of casesinfluenced by mutual long-range electrostatic interactions. Consequently dif-fusion coefficients measured by dynamic light scattering are mostly collec-tive (mutual) diffusion coefficients. The translational diffusion coefficient ofa single polyion is measurable only in dilute solutions. The necessary di-lution is inversely proportional to the concentration of added salt, whichscreens interactions. Most of the light scattering data on polyion diffusionare available at conditions where the size of polyions and the mean inter-polyion distances are smaller than q1 (q is the scattering vector length).The discussion in this section will be based on this assumption, althoughsome exemptions will also be discussed.

The diffusion of polyions is strongly influenced by several factors: theeffective charge of the polyion, the concentration of added low-molecular-weight salt (ionic strength), and the polyion concentration (mean inter-polyion distance). Because there are usually more diffusive modes observed


FIG. 6 Dependence of polyion fast diffusion coefficient Df on degree of neutrali-zation for poly(methacrylic acid) salt-free aqueous solutions. Mw = 30,000, poly-mer concentration c = 36.6 g/L. Polymer chains were ionized by neutralization withsodium hydroxide. (Adapted from Ref. 8.)

in polyelectrolyte solutions, the diffusion of polyions is frequently referredto as fast diffusion, and the corresponding diffusion coefficient as the fastdiffusion coefficient Df . Figure 6 shows the dependence of Df on polyioncharge. The system investigated is partially ionized solutions of poly-(methacrylic acid) (PMA) [8]. PMA is a weak polyacid dissociating in wateronly slightly (ca. 2%). A stronger dissociation (and hence also charge) isreached by neutralization with sodium hydroxide. The degree of neutrali-zation is defined as the ratio of the molar concentration of added NaOHto the monomer molar concentration of the polyacid (i.e., the number ofNaOH molecules per monomer). The diffusion coefficient increases uponincreasing and levels off at higher values of . This transition correlateswith the results of the OosawaManning theory [9,10], according to whichthe onset of counterion condensation corresponds to the situation where themean intercharge spacing along the chain Ac equals the Bjerrum length lBdefined as lB = e2/kBT (e is the electron charge, is the dielectric permit-tivity, kB is Boltzmanns constant, and T is temperature). For the PMA sam-ple investigated, the condition lB = Ac corresponds to c = 0.36. For > c,the charge density on polyions is stabilized by counterion condensation andso is the diffusion coefficient. In conclusion, the diffusion coefficient in-creases with the polyion charge. The small natural ionization at = 0 canbe suppressed by decreasing pH (e.g., by addition of HCl). This decrease of

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FIG. 7 Dependence of polyion fast diffusion coefficient Df on molar concentrationof added salt (NaCl). Aqueous solutions of sodium poly(styrenesulfonate) (NaPSS),c = 5 g/L, Mw = 5,000 (), 47,000 (), and 1,200,000 (). (Adapted from Ref. 12.)

charge is also reflected in lowering the value of the diffusion coefficient (notshown here). The value of c can be changed by changing the solvent die-lectric permittivity (and hence the Bjerrum length). The onset of conden-sation thus shifts to lower charge densities in organic solvents with lowerdielectric permittivity [11].

Figure 7 shows the dependence of the polyion diffusion coefficient onadded salt concentration [12]. Results for three different molecular weightstandards of sodium poly(styrenesulfonate) (NaPSS) with the addition ofNaCl are presented. NaPSS is a strong polyelectrolyte, i.e., strongly disso-ciating in aqueous solutions. The charge is therefore controlled only bycounterion condensation (corresponds to the case = 1 for PMA). Thegeneral feature of the results in Figure 7 is that the polyion diffusion accel-erates upon decreasing the added salt concentration cs. The symbol cs refersto molar salt concentration. At high concentrations of added salt cs, chargeeffects are screened, and values of diffusion coefficients resemble those ofequivalent neutral polymers. At low cs, very high values of diffusion coef-ficients are measured, which are not typical for neutral polymers at all.Polyions diffuse much faster than comparable neutral macromolecules atconditions of high charge density and weak screening. The polyion diffusioncoefficient increases continuously upon decrease of cs. The leveling off atvery low cs is due to the fact that the added salt concentration becomes inthis case low compared to the concentration of uncondensed counterions.


FIG. 8 Dependence of polyion fast diffusion coefficient Df on polyion molecularweight. Salt-free aqueous solutions of sodium poly(styrenesulfonate) (NaPSS), c =45.6 g/L. (Adapted from Ref. 13.)

The diffusion coefficient at high cs decreases with increasing molecularweight. This is similar to neutral polymers where the diffusion coefficientis inversely dependent on the friction factor, which is proportional by thepower law to polymer molecular weight, i.e., D f 1 The diffusionvM .wcoefficient at low cs is, on the other hand, independent of polymer molecularweight. This can be also documented by a more detailed data set on molec-ular weight standards of NaPSS [13] (Figure 8).

Figure 9 shows the dependence of the polyion diffusion coefficient onpolyion concentration c. This data was obtained on NaPSS standard withMw = 100,000, but similar dependencies were obtained also on standardswith different Mw [14]. The diffusion coefficient is within experimental errorindependent of c at c > 0.5 g/L and sharply decreases upon decreasing cbelow 0.5 g/L.

Without doing detailed quantitative analysis of the data, it can be statedthat the polyion diffusion can be qualitatively described by two theoreticalconcepts. The first concept capable of qualitative explanation of the polyiondiffusion is the concept based on considering polyions as interacting Brown-ian particles with direct interactions between polyions and hydrodynamicinteractions. The short-time collective diffusion coefficient for a system ofinteracting Brownian particles treated by statistical mechanics is calculatedfrom the first cumulant 1 of the dynamic structure factor S(q, t) as[1517]

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FIG. 9 Dependence of polyion fast diffusion coefficient Df on polyion concentra-tion. Salt-free aqueous solutions of sodium poly(styrenesulfonate) (NaPSS), Mw =100,000. (Adapted from Ref. 14.)

1 1 1 S(q, t) H(q)1D = = ln S(q, t) = = D (16a)t=0 02 2 2q q t q S(q) t S(q)t=0where S(q, t) = S(q)g(1)(t), D0 stands for free particle diffusion coefficient(infinite dilution, no interactions), S(q) is the static structure factor reflectingdirect interparticle interactions involved, and the term H(q) is known as thehydrodynamic function reflecting hydrodynamic interactions. Equation 16aholds exactly for spherical particles. In the case of coils or rods, the formfactor P(q) comes into play such that

H(q)D = D (16b)0 P(q)S(q)

The influence of P(q) may be physically viewed as the effect of internaldynamics [4]. The resulting diffusion coefficient D can be either lower orhigher than D0, according to the interplay between the influence of theabove-mentioned effects. The dominating factor in the case of polyions inpolyelectrolyte solutions is usually the static structure factor S(q). Low val-ues of S(q) yield high values of D. Low S(q) is found at low q, (q


are due to low osmotic compressibility (see Eqs. 8 and 9). The dependenceof osmotic compressibility or S(q) on fundamental parameters as polyionconcentration, added salt concentration, polyion charge and molecularweight is therefore reflected in the diffusive behavior via the diffusion co-efficient D. In accord with the above-outlined treatment of diffusion, thethermodynamic definition of the collective diffusion coefficient is D = (1/fcoll)(/c),T, where fcoll is collective friction and is chemical potential.

The second concept explaining high diffusion rates of polyions in lowsalt solutions is referred to as the coupled mode theory [4,6,18,19]. The fastpolyion diffusion is explained as a consequence of the coupling of the poly-ion motion with the dynamics of small and much faster counterions andcoions. Both polyions and small ions are charged species creating a commonelectrostatic field, which fluctuates due to their motion and reversally influ-ences their dynamics. The polyion diffusion coefficient is calculated by solv-ing a three-component diffusion equation in a matrix formalism (three partialdiffusion equations for polyions, counterions, and coions, which are coupledthrough the PoissonBoltzmann equation for the common electrical poten-tial). Solution of the matrix equation yields three frequency eigenvalues forthree eigenmodes, which are (1) the plasmon mode with a q-independentfrequency, named by analogy with plasmas, in which the plasma frequencyis also constant, (2) the polyion diffusion, and (3) the small-ion diffusion.The analytical solution for the polyion diffusion is given by a complexformula that is not reproduced here. A simpler expression for the zero anglelimit of the polyion diffusion coefficient is given upon several assumptions:the added salt is a symmetric 11 electrolyte, it has a common counterionwith the polyion, and there is the same charge and the same diffusion co-efficient of the counterion and coion, respectively

1D(q = 0) = [D (1 ) D (1 )] (17)p s2

with

D Z D [1 (2c /Z c )]p p s s p p = (18)

D Z D [1 (2c /Z c )]p p s s p pwhere D is the diffusion coefficient, Z is the charge, and the indexes p, srefer to polyion (p) and salt (s), respectively. The total molar concentrationof the 11 electrolyte ions is 2cs = Zpcp, where is the molaradd add2c cs sconcentration of the added 11 electrolyte, cp is the polyion molar concen-tration, and Zpcp is the molar concentration of counterions that dissociatefrom polyions. The diffusion coefficients Dp and Ds are infinite dilutionvalues for particular components in the absence of charges (if they were

18 Sedlak

neutral particles). The asymptotic cases are the excess added salt case andthe no added salt case, respectively:

2csD(q = 0) = D (19)pcp

D D (2 Z ) 2cp s p sD(q = 0) = Z (20)pD Z 2D cp p s pDependencies of polyion diffusion coefficient on added salt concentration(Fig. 7) can be fitted by using Eq. 17 with Zp as an adjustable parameter.Values of Zp giving optimum fits are, however, much smaller than stoichio-metric values. The same conclusion can be made upon fitting data on variouspolyelectrolytes [20].

Most of the data shown in Figures 69 are obtained at moderate polyionconcentrations, where the structure factor S(q) does not show maxima in theq range accessible by light scattering. The mean interpolyion spacing issmall compared to q1. In this case, the polyion diffusion coefficient is qindependent. At relatively low polymer concentrations (0.01 g/L) and lowadded salt concentrations, the polyion diffusion coefficient D(q) exhibitsangular minima corresponding to angular maxima in S(q) in accord with Eq.16 [21,22]. The diffusion coefficient at the minimum D(qm) describes thestructural relaxation of the dominant structure giving angular extremes ratherthan collective diffusion observed at q 0.5 g/L (see also Figures 8 and 9). Thisindicates that the collective diffusion of chain segments in semidilute solu-tion is roughly the same as the diffusion of whole chains. The concentrationand molecular weight independence of the fast diffusion coefficient in thesemidilute regime D was theoretically derived by explicitly consid-0 0c Mwering triple screening for coupled dynamics of ionic species associated withelectrostatic, excluded volume, and hydrodynamic interactions [23]. On theother hand, a dilutesemidilute transition can be clearly recognized in scat-tering data obtained at high cs [24]. The overall picture is similar to neutralpolymers. In the dilute regime, the diffusion coefficient is weakly dependenton concentration and strongly dependent on molecular weight. At the tran-


sition, a much stronger concentration dependence develops, and the diffusioncoefficient becomes molecular weight independent. Diffusion in the semi-dilute regime corresponds to blob diffusion, i.e., D 1, where is theblob size or the correlation length of the network. At high cs, scales as c3/4 [25,26], which agrees with experimental data [24].

Because of strong interactions in polyelectrolyte solutions without addedsalt, the use of the well-known StokesEinstein relation for free particlediffusion D = kBT/6Rh, where is viscosity and Rh is hydrodynamicradius, is rather limited. Even at very low concentrations, where intermo-lecular interactions can be neglected due to large intermolecular separations,the friction factor contains in addition to the StokesEinstein contributionfSE = 6Rh also a contribution from electrolyte dissipation, so that the totalfriction factor f = fSE feldis, where the electrolyte dissipation term reflectsthe retardation of the polyion motion due to the instantaneous distortion ofthe surrounding ion atmosphere as the polyion moves through the solvent[27,28].

The StokesEinstein relation for free particle diffusion can be applied atinfinite dilution upon partial screening of electrostatic effects by added salt.At higher added salt concentrations cs the polyion diffusion obeys therelation

D = D (1 k c) (21)0 Dwhere kD is referred to as the diffusion second virial coefficient. kD wasfound proportional to 1/cs (e.g., for NaPSS solutions between cs = 10 and500 mM [29]) and proportional to polyion molecular weight Mw at these saltconcentrations [29]. D0 increases with cs in this interval, which means thatthe hydrodynamic radius decreases with increasing cs. In this regime, it ispossible to compare data on hydrodynamic radii and radii of gyration fromstatic light scattering. The ratio Rg/Rh reflects macromolecular shape andinternal structure. Different values of this ratio correspond to spheres, rods,coils in theta or good solvent, etc. The hydrodynamic radius is related tothe ability of solvent to penetrate and flow through the macromolecule. Mac-romolecules with more open structures are drained deeply, whereas thosewith more closed structures are drained only at the periphery. This leads tohigher values of Rg/Rh in the former case and smaller in the latter. Limitingcases are referred to as the free-draining and nondraining case, respectively.In the free-draining case, the hydrodynamic radius should scale as Rh Mw,because each segment of the polyion (each increment of mass) contributesindependently to the friction factor f. In the nondraining case, Rh Rg

where < 1. The NaPSS behavior was found more close to nondrainingM ,wbehavior (least-draining). It was reported for NaPSS that 1/D0 for cs0.5Mw= 5 mM to 4 M [30] and 1/D0 in 100 mM NaCl [29]. The Rg/Rh0.6Mw

20 Sedlak

ratio for NaPSS decreases from 3.0 to 1.8 by increasing cs from 2 mM to 4M NaCl [30,31]. For some other polyelectrolytes, D0 was found independentof cs in spite of some cs dependence of Rg, which can be ascribed to lessbulky sidegroups on main chains in these polyelectrolytes leading to morefree-draining behavior, unless the direct relation between D0 and Rh wasviolated (problems due to viscosity, dissipation, etc.) [30].

Finally, we make a comment concerning the long-time self-diffusion insolutions with moderate and high polyion concentrations, i.e., diffusion ofa single polyion in a collection of other polyions on time scale t >> I, whereI is the relaxation time of the polyions configuration (I is equal to the timeneeded for the polyion to diffuse a distance equal to its radius). This dif-fusion is not measurable by dynamic light scattering. It is necessary to labelpolyions and consequently measure their diffusion over large distances (longtimes) by other techniques. This is done either by chemical labeling (FRAP,fluorescence recovery after photobleaching; FRS, forced Rayleigh scattering)or by spin labeling (PFG NMR, pulsed field gradient nuclear magnetic res-onance spectroscopy). The long-time self-diffusion of polyions is apprecia-bly slower than the collective diffusion as measured by light scattering. Dataon self-diffusion of NaPSS solutions can be found in Refs. 32 and 33.

VI. DYNAMICS IN POLYELECTROLYTE MIXTURESPolyelectrolyte binary mixtures (two polyions of the same chemical com-position but different molecular weights) exhibit at low ionic strength quitespecific dynamic behavior [34,35]. Figures 10 and 11 summarize DLS resultsobtained on binary mixtures of NaPSS without added salt. Figure 10 showsspectra of relaxation times obtained on binary mixtures of NaPSS with Mw= 5,000 and Mw = 1,200,000. The total polymer concentration was main-tained at c = c(P1) c(P2) = 25 g/L while the mixture composition x =c(P2)/[c(P1) c(P2)] was varied. c(P1) and c(P2) are concentrations ofcomponent 1 (NaPSS, Mw = 5,000) and component 2 (NaPSS, Mw =1,200,000), respectively. There are two modes in the spectrum that can beidentified with the modes occurring in binary solutions, referred to as thefast and slow mode (although the amplitude of the fast mode is relativelyweak). Corresponding diffusion coefficients are marked as Df and Ds andare discussed in detail in separate sections. In addition, a third mode clearlydevelops upon mixing of two polyelectrolyte samples. The relative ampli-tude of this mode reaches a maximum around x = 0.5 and decreases to zeroin the limits x 0 and x 1. This mode is diffusive in nature (relaxationtime scales with q2). With respect to the classification of modes as fast andslow, the new mode appearing in the mixture can be classified as the me-dium mode and therefore is marked as Dm in Figure 10. Relative amplitude


FIG. 10 Spectra of relaxation times for binary mixtures of two polyelectrolytesamples (P1, P2) in water: P1 = NaPSS, Mw = 5,000, P2 = NaPSS, Mw = 1,200,000.The total polymer concentration was c = 25 g/L. The mixture composition x = c(P2)/(c(P1) c(P2)) was from top to bottom x = 0, 0.046, 0.188, 0.488, 0.824, and 1.0.All measurements were performed at scattering angle = 90. Three diffusive modesobserved for 0 < x < 1 are marked as Df , Dm , and Ds , respectively. (Adapted fromRef. 34.)

of the medium mode increases upon decreasing the total polymer concen-tration (Figure 11). At lower polymer concentrations, it dominates the spec-trum of relaxation times. The amplitude of the medium mode is not angularlydependent, indicating that it is not directly related to the formation of anytype of larger structures in solution.

22 Sedlak

FIG. 11 Spectra of relaxation times for binary mixtures of two polyelectrolytesamples (P1, P2) in water: P1 = NaPSS, Mw = 5,000, P2 = NaPSS, Mw = 1,200,000.The mixture composition was x = c(P2)/(c(P1) c(P2)) = 0.49. The total polymerconcentration was from top to bottom c = 25.0, 15.0, 3.8, 1.7, and 0.58 g/L. Allmeasurements were performed at scattering angle = 90. (Adapted from Ref. 34.)

Analogical mixtures of neutral polymers (analogical means here the samepolymer chain lengths, same mixture composition, same total monomer con-centration, and very similar solvent quality) do not show this behavior. Fig-ure 12 summarizes results on binary mixtures of polystyrene samples withMw = 4,000 and Mw = 670,000 in benzene. The total monomer concentrationwas c = 25 g/L, and the mixture composition was varied from x = 0 to x =1. The faster mode in Figure 12 is associated with the diffusion of the low-molecular-weight component (Mw = 4,000) and the slower mode with thediffusion of the high-molecular-weight component (Mw = 670,000). When xincreases, the fraction of the low-molecular-weight component in the mixturedecreases and consequently the amplitude of the faster mode decreases. Onthe other hand, the slower mode amplitude increases as the fraction of the


FIG. 12 Spectra of relaxation times for binary mixtures of two polystyrene samples(P1, P2) in benzene: P1 = PS, Mw = 4,000, P2 = PS, Mw = 670,000. The totalpolymer concentration was c = 25 g/L. The mixture composition x = c(P2)/(c(P1) c(P2)) was from top to bottom x = 0, 0.0036, 0.0069, 0.022, 0.057, 0.63, and 1.0.All measurements were performed at scattering angle = 90. (Adapted from Ref.35.)

24 Sedlak

high-molecular-weight component in the mixture increases. The comparisonof the dynamic behavior of mixtures of neutral and charged polymers showsthat the medium mode in polyelectrolyte mixtures is clearly due to electro-static interactions and therefore we also refer to it as the interaction mode.

The number of eigenmodes (uncoupled modes that are detected if theiramplitudes are sufficiently large) in multicomponent mixtures equals thenumber of components in the mixture [5]. Therefore the appearance of anadditional mode in the mixture is expected. Regarding the interpretation ofthis mode, it should be noted that there is currently no exact theory ofpolyelectrolyte mixtures that could be directly confronted with our experi-ments carried out on given conditions. Anyway, eigenmodes in general canbe identified with simple types of motions (that can be easily visualized)only in very special cases. Therefore even in the presence of an exact theory,it could be that the motion giving rise to the interaction mode in thepolyelectrolyte mixture cannot be easily visualized. What can be clearlystated is that the appearance of the interactive mode in dynamic light scat-tering spectrum means that there is a fluctuation in refractive index withgiven relaxation time.

Theoretical treatment of the dynamics of multicomponent mixtures ingeneral [5,3638] predicts the existence of two modes in binary mixtures:the cooperative diffusion [fluctuations in the total concentration c = c(P2) c(P1)] and the so called interdiffusion [fluctuations in the local relativeconcentration of the two components c(P2)/c(P1)]. The latter mode is alsoreferred to as the composition fluctuation mode or heterogeneity mode. Theamplitude of the interdiffusion mode exhibits a maximum as a function ofthe mixture composition. Rigorously speaking, the exact identification ofeigenmodes as cooperative diffusion and interdiffusion can be done only inone special case: mixtures of components that differ in optical labeling butare otherwise identical. In other cases, these two modes are only linearcombinations of eigenmodes and vice versa. With respect to these theoreticalconsiderations and to experimentally observed characteristics, it can be con-cluded that composition fluctuations may significantly contribute to the fluc-tuations we detect as the interactive mode in a polyelectrolyte mixture. How-ever, a major role for other possible mechanisms cannot be excluded. Inevery case, the dynamics of the polymeric components in a polyelectrolytemixture at low ionic strength is not uncoupled as in analogical mixtures ofneutral polymers. Polyion species in polyelectrolyte mixtures with low ionicstrength do not diffuse independently. The interpretation of the fast diffusivemode in a polyelectrolyte mixture as the cooperative diffusion (fluctuationsin the total concentration) and its independence of the mixture compositionand the total concentration is supported by the molecular weight and con-


TABLE 1 Values of Medium Diffusion Coefficient Dm and Ratio of the MediumMode Scattering Amplitude vs. the Fast Mode Scattering Amplitude Am /Af forBinary Mixtures of Six Different NaPSS Samples in Water. The mixtureComposition Was x = 0.5 and the Total Polymer Concentration Was c = 5 g/L

Mw of mixture componentsDm

109 cm2/s Am /Af

100 000 1 200 00047 000 1 200 0005 000 1 200 0005 000 100 0005 000 47 000

90138237395634

0.620.517.954.845.22

centration independence of the cooperative polyion diffusion in comparablebinary solutions.

Polyelectrolyte solutions with broad polyion molecular weight distribu-tions may be considered also as multicomponent mixtures, and similar ef-fects can, in principle, be expected. Therefore NaPSS mixtures with variouscomponents (which can be considered as solutions with various bimodalmolecular weight distributions) were investigated [34,35]. The results arepresented in Table 1. It can be seen that the amplitude of the interactionmode (normalized with respect to the amplitude of the fast mode) is muchlarger in the case of smaller molecular weights of the components. The lowerthe molecular weights of the components, the faster the interaction mode.In all cases, however, the interaction mode is still well separated from themolecular-weight-independent fast diffusive mode. While a more detaileddiscussion of these results is beyond the current understanding of the mech-anism of dynamic processes in polyelectrolyte mixtures, the practical con-sequence is that the possibility of the occurrence of this mode in dynamiclight scattering from solutions of polyelectrolyte samples with broad or bi-modal molecular weight distributions should be kept in mind in order cor-rectly to interpret experimental data.

VII. POLYION MOLECULAR WEIGHTThe ability of light scattering to yield polymer molecular weight and otherimportant parameters such as the second virial coefficient in the expansionof osmotic pressure or the single chain structure is one of the most practicalapplications of this technique. However, because of the complex nature oflight scattering from charged polymers leading to multimodal spectra ofrelaxation times, some caution has to be used. Experimental conditions

26 Sedlak

should be optimally set so that the total scattering is due to one dynamicmodediffusion of weakly interacting polyions. In the worse case, the scat-tering contribution of this mode must be calculated from the total scatteringby a combination of static and dynamic light scattering (see Sec. III). Inpractice, the optimum condition is fulfilled at low polyion concentration andhigh added salt concentration. In this case the excess scattering (after sub-traction of solvent scattering) obeys a formula [39] similar to the generallight scattering equation for a two-component system based on fluctuationtheory (Eq. 6):

K*c 1 = (22) I(0) RT c s

The solvent in this case is a salt solution (solvent with small ions) whichis in Donnan equilibrium with the polyelectrolyte solution, the index smeans that the chemical potentials of all the solutes except that used indifferentiation are constant, and K* is given as [39]

2 2 24 n (dn/dc)sK* = (23)4 N0 Awhich is again formally equivalent to the formula for neutral two-componentsystems (Eq. 7). The only difference is that the refractive index incrementis at constant chemical potential of salt rather than at the constant concen-tration of salt. Constant chemical potential of salt is a consequence of theDonnan equilibrium.

The Donnan osmotic pressure in Eq. 22 can be expanded to the virialform

c 2 3 = RT (A ) c (A ) c (24)2 3 s sMwwhere Mw is the polyelectrolyte molecular weight and are virial co-(A )i sefficients at the Donnan pressure. Equation 22 can be then written in theform

K*c 1= 2(A ) c (25)2 sI(0) Mw

which is again formally equivalent to the classical Zimm formula commonlyused for neutral two-component systems except that experiments must bedone at a constant chemical potential of salt (Donnan equilibrium pressure).The polyion molecular weight is obtained via Eq. 25 by double extrapolation


of scattering intensity to zero scattering angle and zero concentration. Inpractice, a complete angular dependence is measured for several concentra-tions to perform the double extrapolation. In order to meet fully the conditionof constant chemical potential of salt, the experimental procedure should besuch that each polyelectrolyte solution is first dialyzed against a given sol-vent. Then the dialyzed solute and solvent, which are at the Donnan equi-librium after dialysis, should be used for light scattering measurement (ex-cess solution scattering over solvent scattering) and refractive indexmeasurement (difference in refractive index between solution and solvent).The difference between the excess scattering with and without previous di-alysis can in most cases be neglected. What cannot be neglected are differ-ences between refractive index increments measured at constant salt con-centration and constant chemical potential of the salt (differences betweenthe constants K and K*). If samples are not dialyzed and refractive indexincrements are measured only at constant salt concentration, then apparentrather than true molecular weights are measured. According to Eqs. 7, 23,and 25, the apparent molecular weight is related to the true molecularMwweight Mw as

2K (n/C )s cM = M = M 1 (C /c) (26)w w w s sK* (n/c)Cs

where Cs is the salt concentration in g/L. The term corresponds(C /c)s sto the adsorption of salt on the polyion, and the most obvious contributionis from the formation of the electrical double layer (excess of counterionsand deficit of coions in the vicinity of the polyion). can be esti-(C /c)s smated from the theory of Donnan membrane equilibrium or from solvingthe PoissonBoltzmann equation to calculate the concentration of counter-ions and coions in the vicinity of the polyion. It can be expressed as [39]

C M c Ms s s s= = Z a (27)p

c M c Mw p w s s

where cs and cp are molar concentrations, Ms and Mw are molecular weightsof the low-molecular-weight salt and the polyion, respectively, Zp is thepolyion valence, and the factor a = 0.5 for low Zp and a < 0.5 for high Zp .The quantity is the adsorption of salt on the polyion in molar(c /c )s p sconcentrations, i.e., the number of salt molecules adsorbed at one polyion.It is to be emphasized that it is a negative adsorption because the salt isexpelled from the polyion surrounding. Therefore there is a minus sign inEq. 27. After entering Eq. 27 into Eq. 26, the following relation between

and Mw is obtained:Mw

28 Sedlak

FIG. 13 Dependence of apparent molecular weight on molar refractive indexM wincrement of salt. Poly(methacrylic acid), half neutralized by NaOH, in solutions ofvarious salts. (Adapted from Ref. 39.)

(n/C ) Ms c sM = M 1 Z a w w p (n/c) MC wsn

= M 1 const M (28) w s Cs c

The apparent molecular weight is smaller than the true molecular weight.The difference is greater the heavier the salt is. The validity of Eq. 28 wasverified experimentally by its authors [39] (Figure 13).

In typical experiments, the solution and the solvent are brought into equi-librium by dialysis against each other using semipermeable membranes en-abling the salt ions to migrate across but preventing polyions from the mi-gration. The Donnan equilibrium is established by a redistribution of saltions in this way. In cases where polyelectrolyte is soluble in nonionizing (orweakly ionizing) organic solvent, it is possible to perform molecular weightdetermination without needing to perform equilibrium dialysis. As an ex-ample we can mention poly(methacrylic acid) in ethyleneglycolmonoeth-ylether [40]. This does not apply, however, when a mixed solvent is usedto enable solubilization by adjusting the solvent dielectric permittivity (e.g.,organic solvent plus water). Because many polyelectrolyte samples arestrongly hygroscopic (up to 30% water content), it is necessary for the pur-pose of accurate molecular weight measurement to determine true concen-trations. The same care must be taken regarding potential changes in con-


centration during dialysis. We note that the molecular weight measured bylight scattering in polydisperse samples is a weight average. By combinationwith the number average (obtained for instance by osmometry), the samplepolydispersity Mw/Mn can be estimated. The last comment concerns the factthat the measured value of Mw includes polyion plus a proportional amountof counterions, i.e., one counterion per each monomer unit. Mw = c/cp, wherec is the concentration of the polymer material in g/L (polyions plus coun-terions) and cp is the molar concentration of polyions.

VIII. SECOND VIRIAL COEFFICIENTUnder conditions of partly screened interactions in dilute solutions (highadded salt concentration cs and low polymer concentration c), the solutionosmotic pressure can be expressed via a virial expansion (Eq. 24). Thenlight scattering becomes a useful tool to obtain values of second virial co-efficients characterizing interactions in solution. The second virial coefficientcan be calculated from the slope of the dependence given by Eq. 25. Therelation between the true and the apparent second virial coefficient is similarto the relation between the true and the apparent molecular weight (see theprevious section for more details and the meaning of the symbols):

2K* (n/C )s cA = A = A 1 (C /c) (29)2 2 2 s sK (n/c)cs

The dependence of the second virial coefficient on basic quantities suchas the solution ionic strength or the polymer molecular weight is usually ofgreater scientific interest than the knowledge of true vs. apparent values.Therefore the time-consuming dialysis step leading to true rather than ap-parent values is usually omitted. Hence measured second virial coefficientsare usually only apparent values.

As expected, the second virial coefficient decreases upon increase of cs .The exact form of this dependence is shown in Figure 14 as a loglog plotof the data on sodium poly(styrenesulfonate) in aqueous NaCl solutionsobtained by different authors [30,41]. Values obtained by osmometry arealso shown for comparison [41]. The general trend is that the dependenceis stronger at lower cs and weaker at higher cs. It was proposed that A2

in the former case and A2 in the latter [41]. The dashed line in1 1/2c cs sFigure 14 represents calculated values of A2 according to Yamakawas theory[30,42]. Qualitatively similar dependencies of A2 on cs were found also insolutions of poly(methacrylic acid) [43] and some biological polyelectrolytes[22,44]. At very high cs, a deviation from such dependence leading to thetaconditions (A2 = 0) and even to macrophase separation (salting-out effect,A2 < 0) may be eventually reached in some systems. Temperature becomes

30 Sedlak

FIG. 14 Dependence of the second virial coefficient on added-salt concentration.NaPSS in NaCl solutions. Results by light scattering from Ref. 41 (), Ref. 30 ();and by osmometry from Ref. 41 (). The dashed line shows calculation of the secondvirial coefficient by incorporating electrostatic excluded volume effects after Ref.30.

an important factor in this situation [45]. The molecular weight dependenceof A2 is much weaker than in solutions of neutral polymers. A2 is almostindependent of Mw at lower cs and only weakly dependent on Mw at highercs [46].

The second virial coefficient of polyelectrolytes is treated theoreticallyeither by applying the theory for charged spherical colloids with a correctionfor the chain character of the polyion or as an extension of the theory ofthe second virial coefficient for nonionic linear polymers. An example of anextension of the theory of the second virial coefficient for nonionic linearpolymers to polyelectrolytes is the above-mentioned Yamakawa approach ofusing perturbation theory of excluded volume to calculate A2 [42].

IX. SINGLE CHAIN STRUCTUREIn order to extract information on the single chain structure from SLS data,it is necessary to fulfill the condition of a single mode scattering, similarlyto the extraction of molecular weight and second virial coefficient from SLSdata (see Sec. VII). A further condition is that the scattering from polyionsshould not be influenced by intermolecular interference due to intermolecularinteractions, i.e., the solution structure factor S() = 1. In this case the total


FIG. 15 Schematic representation of a Kratky plot q2P(q) vs. q for a Gaussianchain. P(q) is the chain form factor, q is the scattering vector. The inset shows aGuinier plot ln P(q) vs. q2 (see text for details).

static scattering intensity is proportional to the single particle form factorP() (in our case the polyion form factor) (see Eq. 9). The P() (or P(q))function contains information about the structure of the polymer chain. Inprinciple, three regions can be distinguished in the form factor of a polymerchain assuming a Gaussian coil conformation: (i) P(q) for2 2exp(q R /3)gq1 >> Rg , (ii) P(q) q2 for Lp < q1 < Rg , and (iii) P(q) q1 for q1 103 M) where interpolyion interactions are partlyscreened. In this case higher polymer concentrations yielding higher and

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FIG. 17 Dependence of the radius of gyration of NaPSS chains in dilute solutionon added salt concentration. Results by static light scattering for samples: () Mw =1,000,000 from Ref. 57; () Mw = 780,000 from Ref. 30; and () Mw = 400,000from Ref. 31.

more reliable intensities can be used. Results can be extrapolated from thesepolymer concentrations to infinite dilution, to assure that a pure form factoris obtained. In some cases, however, Rg for cs > 103 M was found inde-pendent of concentration in the range c = 0.02 to 0.2 g/L [31]. The mostfrequently treated problem regarding the polyion conformation is the depen-dence of Rg on cs . Figure 17 shows data on NaPSS collected from severalworks [30,31,57]. The general feature is that Rg decreases with cs approxi-mately by a power law Rg with exponent 0.15. Results on othercspolyelectrolytes such as quaternized poly(vinylpyridine) [31], ionizedpoly(acrylic acid) [58], and bacterial hyaluronate [44] show similar behaviorwith ranging from 0.15 to 0.25. The chains are appreciably coiled at highercs . For comparison with results in Figure 17, we can calculate Rg of chainsupon assumption of a fully extended rodlike conformation. For the sampleswith the smallest and highest Mw, Rg = = 142 and 534 nm, re-2L /12 cspectively (Lc is the contour length).

The chain conformation is determined by the interaction between neigh-boring segments and the interaction between distant segments along a poly-mer which, via chain flexibility, are located in each others vicinity. Theformer effect determines the local chain stiffness. The latter is referred to asthe excluded volume effect and influences the overall conformation. Bothtypes of interaction can be of electrostatic and nonelectrostatic origin. In theabsence of excluded volume effects (flexible polyions in a theta state or


semirigid polyions), the radius of gyration is referred to as unperturbed(Rg0). The local stiffening of the chain can be expressed in terms of a per-sistence length (distance along the chain over which the polymer maintainsa stiff rodlike structure). The total persistence length Lp consists of an in-trinsic and an electrostatic part, Lp = L0 Le . The intrinsic part L0 is dueto the flexibility of the backbone without charges and is given by fixed bondangles, rotameric states, helical structures, etc. The electrostatic part Le isgiven by the contribution of charge interactions to increased stiffness of thechain. The issue of Le was a subject of intense theoretical work [5963].Rg0 can be expressed in three different ways: (1) in the rod2 2R = L /12g0 climit, where Lp >> Lc ; (2) = LcLp /3 in the random coil limit, where Lp2R g0

36 Sedlak

Another approach to the interpretation of experimental data was based onthe assumption of a uniform expansion of the chain, where the interactionenergy between the segments was given by two terms: the usual short-rangeexcluded volume and the electrostatic interaction via DebyeHuckel poten-tial [31]. The resulting formula for the expansion factor r was used to fitexperimental dependencies of Rg with two fitting parameters: the degree ofionization and the intrinsic excluded volume pseudopotential proportional tothe FloryHuggins interaction parameter. This method was applied to a largeset of experimental data on Rg of quaternized poly(vinylpyridine) as a func-tion of added salt concentration, backbone hydrophobicity, solvent dielectricpermittivity, and the chemical nature of coions and counterions [31]. Al-though the physical significance of the obtained effective degree of ioniza-tion f and the intrinsic excluded volume w0 as fit parameters was criticallydiscussed [31], reasonable conclusions were drawn from such analysis ofthe data. The main results can be summarized as follows: Rg decreases withthe chain hydrophobicity and increases with the solvent dielectric permittiv-ity, and the exponent in the relation Rg varies from 0.15 to 0.25c sand does not significantly depend on the hydrophobicity below cs 102M. Above cs 101 M, a downward curvature in the dependence of Rg vs.cs (departure from Rg occurs and significantly depends on hydro-c )sphobicity. This curvature determines the solubility limit (salting out ef-fect) at high cs . In terms of the fitting parameters f and w0 , f dominatesat low cs , while w0 controls the curvature and the precipitation at high cs.A substantial effect on f and w0 was found upon changing the chemicalnature of the counterions, especially at high cs (cs > 102 M). Recent mo-lecular dynamics simulations on the effect of added salt on polyion confor-mation represent a new challenge. They are currently restricted to shorterchains, but they enable us to use full Coulomb interactions without approx-imations [66].

In the case of intrinsically rigid polyelectrolytes, such as DNA, experi-mental results [67] show that electrostatic persistence length calculated fromthe data shows no unique power law dependence on cs . Compared to theOSF theory [60,61], a much better agreement with these data was achievedlater by the calculation of Le via numerical solution of the PoissonBoltz-mann equation for a toroidal polyion geometry [59,62]. These calculationsshowed that the exponent in the scaling Rg varies from 1 toc s1/4 upon increase of cs . A breakdown of the OSF theory for flexible chains(unless Le >> Lp) was indicated by taking into account fluctuations in thechain configuration [63].

The last comment concerns polyions which are intrinsically rigid (notflexible or semiflexible). In this case, light scattering can provide informationon the size and partly also on the shape of the polyions by comparing


experimental form factors with those calculated for different shapes (spheres,ellipsoids, rods). The estimation of shapes requires monodisperse samples[68]. This requirement is met usually for biological polymers.

X. INTERPOLYION CORRELATIONSA common feature of scattering from polyelectrolyte solutions is the exis-tence of peak(s) in angular dependencies of scattering intensity. Figure 18shows an example of such a result obtained by static light scattering [50].The sample is the same as the one as in Fig. 16 (HPSS, Mw = 354,000); theconcentration is higher: c = 5.5 103 g/L to 2.8 102 g/L. The positionof the peak maximum qm changes as qm c 0.36. The same concentrationscaling was measured for another sample in this work (Mw = 1,060,000)[50]. Similar angular maxima were reported by SLS for NaPSS by differentauthors [21], and also for a variety of other polyelectrolytes with differentconformations (architectures), e.g., semiflexible proteoglycan monomers (qm c 1/3) [69], semirigid succinoglycan polysaccharide (qm c 0.45) [22], andrigid TMV particles (qm c 1/3) [70]. In accord with the fact that qm increaseswith c, angular maxima were also obtained at high q by small-angle x-rayscattering (SAXS) and small-angle neutron scattering (SANS) at correspond-ingly high concentrations (10 to 100 g/L) on a large variety of polyelec-trolytes [53,54,7174]. Apparently, qualitatively the same phenomenon isseen by various scattering techniques at different length scales according tothe polyelectrolyte concentration. The existence of the peak is clearly dueto electrostatic interactions. The addition of salt decreases the height of thepeak and eventually, at high cs , a complete disappearance occurs. The po-sition of the peak either stays constant within experimental accuracy or aslight decrease of qm is observed upon increasing cs [70,72]. In the case ofweak polyelectrolytes with variable charge densities, the peak gradually de-velops upon increase of charge density, and qm moves towards higher q [74].No change is observed upon stabilization of charge above the ManningOosawa counterion condensation limit. A shift of qm towards higher q uponincreasing charge density on variably sulfonated NaPSS was observed, too[75].

The presence of peaks in angular dependencies of scattering intensity canhave, in general, several physical origins. Multiple sharp peaks appear inscattering from highly ordered crystalline lattices. Peaks appear also in scat-tering from simple liquids; the first and largest peak is followed by severalothers with gradually decreasing peak height. In all cases peaks arise due tointerparticle correlations. These correlations are reflected in the solutionstructure factor S(q), which was defined by Eq. 10, where the radial distri-bution function of particles g(r) was introduced. We remind the reader that

38 Sedlak

FIG. 18 Normalized excess scattering from salt-free aqueous solutions ofpoly(styrenesulfonic acid) (HPSS), Mw = 354,000, at concentrations (g/L): (1)0.00553, (2) 0.00829, (3) 0.01106, (4) 0.01383, (5) 0.01660, (6) 0.01936, (7)0.02213, (8) 0.02490, (9) 0.02766. (Adapted from Ref. 50.)

g(r) reflects the probability of finding a particle at a distance r from anotherparticle. Thus for the simplest case of hard sphere particles g(r) = 0 for r Rp , the shape of g(r) depends onthe volume fraction . At low , g(r) = 1 for r > Rp (step function). At high ( exceeding 0.10.3), g(r) exhibits a specific shape reflecting the so-called liquidlike structure (Figure 19). At r equal to the mean interparticledistance, g(r) exhibits a peak [g(r) > 1] reflecting the nearest neighbor cor-


FIG. 19 Schematic representation of the radial pair distribution function g(r) andcorresponding solution structure factor S(q) for two cases discussed in the text:liquidlike structure (left) and correlation hole effect (right). See text for more details.

relation. The first peak is followed by several smaller peaks reflecting thepositional correlation between the central particle and the outer shells ofmore and more distant neighbors. These are gradually less and less correlatedleading to gradually smaller peaks with g(r) 1 for r . The corre-sponding S(q) function is shown in Figure 19, too. Figure 19 shows alsog(r) corresponding to the so-called correlation hole effect. It is assumed thatthere is a shell (hole) around each particle, from which other particles areexcluded. This assumption leads to a step function g(r) with the discontinuityat r equal the effective particle size or, more realistically, to g(r) of asigmoidal shape with inflection point at r given by the effective particlesize. This shape of g(r) does not lead to the presence of peaks in S(q). S(q)is a continuously increasing function of q approaching S(q) = 1 at q .For completeness, we note that g(r) has multiple delta peaks in the case ofideal crystal lattices leading to sharp multiple peaks in S(q), that for lessperfect crystals (lattice defects, paracrystals) a broadening of peaks occurs,especially for higher order peaks, which decrease in height, and that forliquidlike order of polydisperse spheres, the resulting S(q) is strongly mod-ified (the main peak shifts to lower q and decreases in height while theoscillations after the main peak are washed out) [76].

In the correlation hole concept, the S(q) function does not exhibit a max-imum. However, upon combination with a monotonically decreasing P(q)

40 Sedlak

function, it leads to a maximum in the scattering intensity I(q) P(q)S(q).The idea of an effective volume of polyions, very similar to the correlationhole concept, was originally created by Doty and Steiner in early studies onproteins [77]. De Gennes et al. [78] defined the scattering peak in a semi-dilute polyelectrolyte solution as a demonstration of the short range orderarising from the hard-sphere-like repulsion between correlation blobs. Thechain is considered to be a random walk of correlation blobs of size , whereon length scales smaller than the dilute solution scaling applies and onlength scales larger than all interactions are screened. The correlation blobs(correlation volumes) are space filling. The correlation volume can be con-sidered as a space around each segment, from which other segments areexcluded. The maximum in I(q) is expected for qm 2/ c 1/2 (1is the screening length). It was shown based on scaling arguments that I(q)> qm[78]. The correlation hole concept was used to calculate q dependencies ofscattering intensities for spheres and rods [79] and for coils [80]. In the lattercase the calculation is focused on semidilute solutions and therefore involvesthe radial distribution function of the coil segments instead of that of thecenters of gravity of the coils.

In order to interpret experimental data, the knowledge of S(q) is required.A direct evaluation of S(q) from the scattering intensity as S(q) I(q)/P(q)is, however,

chemistry of polyelectrolytes

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