Current Opinion in Colloid & Interface Science 17 (2012) 64–73

Contents lists available at SciVerse ScienceDirect

Current Opinion in Colloid & Interface Science

j ourna l homepage: www.e lsev ie r .com/ locate /coc is

New experiments for the quantification of counterion condensation

Klaus Huber a, Ulrich Scheler b,⁎a Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germanyb Leibniz-Institut für Polymerforschung Dresden e.V., Hohe Str. 6, 01069 Dresden, Germany

⁎ Corresponding author. Tel.: +49 351 4658 275; faxE-mail address: scheler@ipfdd.de (U. Scheler).

1359-0294/$ – see front matter © 2012 Elsevier Ltd. Alldoi:10.1016/j.cocis.2012.01.005

a b s t r a c t

a r t i c l e i n f o

Article history:Received 15 November 2011Received in revised form 20 January 2012Accepted 25 January 2012Available online 4 February 2012

Keywords:Counterion condensationCounterion distributionEffective chargePolyelectrolytesASAXSElectrophoresis NMRPFG NMR

The condensation of counterions is an important aspect of charged macromolecules. Therefore an experimen-tal characterization of the condensation of counterions is desirable. In this contribution two experimentaltechniques for the characterization of counterion condensation are introduced and compared: AnomalousSmall Angle X-Ray Scattering (ASAXS) is able to probe the spatial distribution of counterions and electropho-resis nuclear magnetic resonance (NMR) measures counterion condensation via the effective chargeobtained from the dynamic behaviour of molecules and complexes in an electric field.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Electrostatic effects play an important role in nature as well as inmany artificial materials. Often a large number of charges is locatedon a single object like a colloid or a macromolecule (polyelectrolytebehaviour) resulting in an electrostatic potential so strong that thecounterions do not have sufficient thermal energy to escape it [1,2].The individual groups remain charged (dissociated or protonated),the counterions however are located in the vicinity of the macromol-ecule. The charges remaining on the polyelectrolyte affect the mor-phology of the polyelectrolyte and its response to environmentalchanges. A better understanding of the processes of counterioncapturing by polyelectrolytes therefore is an important issue infundamental science and at the same time is highly relevant tothe development of new functional materials.

Only fewmethods are known, which permit to measure and quan-tify the condensed counterions. An indirect measure of condensationis the activity of counterions in the solution containing the polyelec-trolyte. This can be probed either by potentiometry [3–6] or conduc-tometry [7], where the difference of the measured effect between thetotal number of ions and the active number is attributed to condensedcounterions. The activity of chloride anions for instance in the pres-ence of poly(dimethyldiallylammoniumchloride) has been measuredby chloride ion selective electrodes [3]. More effort has been devotedto the investigation of metal cations in the presence of polyanyions[4–7]. By means of the latter contributions, specific binding of alkaline

: +49 351 4658 231.

rights reserved.

earth cations could well be distinguished from classical counter ioncondensation observed with alkali metal cations. An approach similarto the electrochemical methods is the measurement of the osmoticpressure [8] because only the non-condensed counterions are osmot-ically active and contribute to the osmotic pressure. In neutronscattering experiments the distribution of counterions in the vicinityof dendrimers has been evaluated and attributed to the condensationof counterions [9]. The interaction of polyelectrolytes with para-magnetic counterions has been observed both in the mobility asmanifested in the EPR lineshape [10] and the spin exchangebroadening of the spectra from distance-dependent interaction[11].

There are also numerous theoretical works and simulations on thesubject of counterion condensation. Holm and coworkers discusseddifferent estimators for the effective charge of macromoleculesderived from simulations [12]. These nicely correspond to criteriafound in different experiments

The subject of the present paper is to present two recent experi-mental developments giving a direct access to the condensation ofcounterions and probing the local distribution of the counterionswithin the domain of the hosting polyelectrolyte. Anomalous X-RayScattering reveals scattering curves with element selectivity andthus offers information on the spatial distributions of the “selectedelement”. Herewith, clustering of counterions on a macromoleculecan be observed and quantified. The combination of diffusion andelectrophoresis NMR inherently provides chemical selectivity fromthe NMR-spectroscopic information. Via the electrophoretic mobilityand the self diffusion coefficient the effective charge of macromole-cules and counterions is determined.

Fig. 1. Anomalous dispersion corrections obtained by Cromer-Liberman calculations[26,27]. The values at 16.105 keV result from the convolution with the energy resolutionof the JUSIFA beamline [25•]. Reproduced with kind permission from Springer Science &Business Media: The European Physical Journal E-Soft Matter.

65K. Huber, U. Scheler / Current Opinion in Colloid & Interface Science 17 (2012) 64–73

2. Anomalous Small Angle X-Ray Scattering

2.1. Evolution of the Field

Despite the fact that the work which applies ASAXS on the inves-tigation of counterion distribution in polyelectrolyte solution is stillsparse, some significant progress has been achieved in the fieldwhich is only partially covered by a more recent article on the benefitASAXS offers to an investigation of the behaviour of dilute polyelec-trolyte solutions [13]. Progress in the field includes the successfulisolation of the scattering pattern of the counterions and the quanti-fication of the amount of counterions bound to the domain of thepolyelectrolyte chains. Along this line, the following notes extendformer reports [14••,13] and at the same time intend to convince abroader audience on the potential ASAXS bears for the investigationof counterion binding by polyelectrolyte chains in solution.

The scattering contrast of a SAXS experiment in general is deter-mined by the excess electron density Δρ of the dissolved polyelec-trolyte salt with respect to the electron density of the solvent.Accordingly, the scattering amplitude from the polyelectrolyteA(q) is represented as a function of the momentum transfer q asfollows

A qð Þ ¼ ∫V

Δρ r→

� �exp −i q

→r→

� �d3r 1

with the momentum transfer q corresponding to the value of thescattering vector written as a function of the wavelength λ ofthe x-ray beam, and of the observation angle θ.

q ¼ 4πλ

sinθ2

Once the scattering sample includes an absorbing species, theamplitude A(q) in Eq. (1) can be subdivided as follows

A qð Þ ¼ ∫V

Δρnr r→

� �exp −i q

→r→

� �d3r þ ∫

V

Δρr r→

� �exp −i q

→r→

� �d3r 2a

A qð Þ ¼ Anr qð Þ þ Ar qð Þ 2b

In Eq. (2b), the first term comprises the entire non-resonant scat-tering Anr(q) from all species of the polyelectrolyte including thepolymer chains and the counterions and the second term capturesthe additional resonant scattering contribution Ar(q), which exclu-sively stems from the counterions as the absorbing species.

The excess electron densities appearing in Eq. (2a) are deter-mined by the density distributions u(r) of monomeric units formingthe polymers and by the density distribution of the counterions v(r)via

Δρnr rð Þ ¼ f 0;C−ρSVC

� �h i⋅v rð Þ þ f M−ρSVM½ �⋅u rð Þ 3a

Δρr r; Eð Þ ¼ f 0C Eð Þ þ if 0 0C Eð Þ� �⋅v rð Þ 3b

In Eq. (3a, 3b), VM and VC are the volumes of a monomer withinthe polymer and of a counterion respectively. The electron densityof the solvent is indicated as ρS. Further on, fM and f0,C represent thenumber of electrons of a monomer and a counterion respectively. Ithas to be emphasized that the excess number of electrons of thecounterion includes two extra terms, which depend on energy(wavelength) and which form the resonant term of Eq. (3b). Thesetwo energy dependent terms f’C(E) and f”C(E) and hence Δρr(r,E)only contribute significantly if experiments are performed closeenough to the absorption edge of the absorbing species and decreaseto zero a few percent beyond the value of the energy at the absorption

edge. An illustrative example for the evolution of f’C(E) and f”C(E) isgiven in Fig. 1 for Sr2+ which act as counterions in an example indi-cated below. Due to this dependence of f’C(E) and f”C(E) on the energythe scattering contrast of the polyelectrolyte-counterion system canbe varied by changing the wavelength of the primary x-ray beamclose to an absorption edge. This variation in turn provides additionalinformation on the internal structure of complex scattering spe-cies. To this end, the scattering pattern described as the differen-tial scattering cross section is written as follows[14••],.

dΣdΩ

q; Eð Þ ¼ Anr qð Þ⋅Anr� qð Þh i þ 2f 0C Eð ÞFnrC qð Þ þ f 0C Eð Þ2 þ f 0 0C Eð Þ2h i

SCC ð4Þ

where the first term represents the non resonant part also availablewith a conventional SAXS experiment, FnrC corresponds to the crossterm and SCC is the normalised scattering contribution from the coun-terions addressed by ASAXS.

Publications where this concept has been applied for the first timeto the investigation of polyelectrolyte systems refer to biologicalmolecules and date back to the early seventies [15,16]. Here, theloose shell of Cs+ counter ions condensed around the stiff backboneof anionic DNA chains had been addressed. Wavelength dependentSAXS measurements close to the L3 absorption edge of Cs+ indicateda decrease of the (apparent) cross section radius of the DNA strandupon approaching the absorption edge. The decrease was attributedto the dispersion of f’C(E) , which, as an increasingly negative value,lowered the scattering contrast of the condensed counterions. Theresulting decrease of the apparent cross section could only be madeconsistent with an adsorbed Cs+ shell larger than the bare radiusof the DNA strand.

It was only in 1999, when Sabbagh et al [17] presented anotherstudy on the condensation of Co2+ ions on sodium polyacrylate(Na–PA) chains. They performed an analysis of three scattering curvesclose to the K absorption edge of Co2+ revealing that the closer thewavelength gets to the edge, the lower the scattering curve lies. Asuitable ASAXS reference experiment with weakly charged poly-acrylic acid in the presence of Co2+ ions revealed the same scatter-ing behaviour as a pure CoCl2 solution and hence indicated thecomplete lack of correlations of Co2+ ions around the neutral poly-acrylic acid coils. Although, this ASAXS experiment was a purelyqualitative analysis, it showed that Co2+ condensation occurredonto anionic PA coils and that ASAXS is sensitive to ion specificscattering.

Further analysis of counterion condensation was propelled by theprogress in preparing new model polyelectrolytes with well defined

66 K. Huber, U. Scheler / Current Opinion in Colloid & Interface Science 17 (2012) 64–73

shapes. Whereas Rehahn et al. [18,19] succeeded to synthesize welldefined rod-like macroions based on a rigid poly(p-phenylene)backbone, Guo et al. [20] prepared sphere-like colloids with ananionic polyacrylate shell. It was expected that those regularlyshaped polyelectrolytes would enable to directly scrutinize thePoisson-Boltzmann cell model for counterion condensation. For therod-like poly(p-phenylene) backbone iodine anions had been ap-plied [18,19] and for the polyanionic colloidal spheres use had beenmade of rubidium cations [20]. Both types of counterions have ahigh number of electrons and a suitable absorption edge, whichmade them good candidates for ASAXS experiments. In two prelimi-nary reports [18,19] on energy dependent SAXS measurements atthe JUSIFA at DESY, it could be demonstrated, that the scatteringcurves expressed as the differential scattering cross section de-creased upon approaching the absorption edge of the respectivecondensing counterions. The decrease could nicely be explained bythe negative contribution of the cross term in Eq. (4), caused by thenegative value of f’C(E). The effect is shown in Fig. 2, which alsodemonstrates the successful prediction of the counter ion distribu-tion around the rod-like macroions [18] by means of the Poisson-Boltzmann cell model for rod-like polyelectrolytes [21].

A succeeding paper published by Das et al. [22] focused again onthe counterion distribution around DNA. The authors performed acomparative study with one piece of a 25-base-pair DNA beingsurrounded either by Na+, Rb+, Mg2+ or Sr2+ respectively. Analysisof dilute solutions of Mg-DNA and Sr-DNA at two energies close to theK absorption edge of Sr2+ ions revealed only an energy dependentvariation of the scattering curves from Sr-DNA. As already observedby Ballauff et al. [18–20], the curve recorded closer to the absorptionedge has slightly lower differential scattering cross sections than thecurve further apart from it. Successively, an interesting comparisonwas performed by the authors anticipating that a variation of thescattering contrast as observed upon variation of the wavelengthclose to an absorption edge of a cation can be “mimicked” by ex-changing counterions by corresponding homologues, with varyingorder number Z. Indeed, the difference curve of dΣ/dΩ(q,E) fromSr2+ recorded at two wavelengths overlay with the difference ofconventional SAXS curves from Sr-DNA and Mg-DNA once the dif-ference curves were normalized by the appropriate scattering con-trasts. The same effect was observed with Rb+ and the pair Rb-Narespectively (Fig. 3). The results also revealed that roughly twice

Fig. 2. Experimental ASAXS data on rod-like polycations based on a poly(p-phenylene)backbone with iodine counterions in comparison to theoretical calculations. The sym-bols indicate evaluated experimental data from measurements at 32 keV (●), at33.169 keV (□) and the difference of Eq. (4) recorded at the two energies (Δ). Thelines represent theoretical approaches which reproduce the data with Poisson-Botzmann cell model based on the following model parameters: charge parameterξ=6.65, contact distance a=0.8 nm, cell radius R=8.43 nm Δρrod=25 e/nm3 andΔfC=39 e/ion as the iodine contrast far away from the absorption edge [18]. Repro-duced with kind permission from Springer Science & Business Media: Colloid andPolymer Science.

as many counterions were condensed if monovalent counterionswere used instead of bivalent ions independent of the type ofcounterions.

In these early investigations, the focus lay on the expected lower-ing of scattering intensities inferred by the impact of a negative f’C(E)which increases its absolute value upon approaching the edge andusually led to a decrease of dΣ/dΩ(q,E) in Eq. (4), due to a dominationof the second term. As will be shown below, this shift is equal to theseparated scattering curves. In a significant step forward, accesscould be provided to the q-dependent scattering intensity of the res-onant (i.e. selected) element SCC(q). To this end a series of at leastthree suitable ASAXS experiments performed at different wave-lengths of the primary X-ray beam below and close to the absorptionedge of the selected element has to be performed. The breakthroughcame almost at the same time by two alternative strategies [23,24••,25•] They shall both be outlined in the following paragraphs. Ithas to be emphasized at this point that extraction only becamepossible due to further progress in the brilliance of synchtotronradiation and detector sensitivity and in an enhanced precision ofthe transmission measurements and in energy resolution. Theseinstrumental developments were a prerequisite since the SCC(q)-term in Eq. (4) is particularly small compared to the other twoterms and requires an extremely large accuracy in data treatment.The small size of this term stems from its prefactor, which is thesum of the two squares of very small numbers.

2.2. Access to the Resonant Scattering Contribution

One strategy to reach at the pure scattering contribution of thecounterions was introduced by Ballauff et al [23,24••] who used thefollowing procedure. Once dΣ/dΩ(q,E) is recorded at various wave-length, a plot of dΣ/dΩ(q,E) versus f’C(E) at distinct q-values enablesa quadratic fit with an f’C(E) and an [f’C(E)2+ f”C(E)

2] term resulting inSCC(q) as the coefficient of the [f’C(E)2+ f”C(E)

2] term. The only pa-rameters required to calculate SCC from such fits are the two energydependent contrast factors f’C(E) and f”C(E). The trends of theseanomalous dispersion corrections can be calculated according to aprocedure introduced by Cromer and Libermann [26,27].

Two polyelectrolyte model systems were analysed with this strat-egy by Ballauff et a. [23,24••]. One system is based on a rod-like

Fig. 3. Shapes of the difference of SAXS curves from Rb-DNA and Na-DNA and from Sr-DNA and Mg-DNA respectively in comparison to the corresponding anomalous differ-ence curves from Sr-DNA at two energies and from Rb-DNA at two energies. The sym-bols indicate the following differences: Difference between profiles taken off and onanomalous edge for Rb-DNA (□) and Sr-DNA (○); difference of SAXS curves of Rb-DNA and Na-DNA (■) and Sr-DNA and Mg-DNA (●). Experiments are compared tothe prediction based on a non-linear Poisson-Boltzmann approach for monovalentand divalent ion atmospheres (lines) [22]. Copyright (2003) by the American PhysicalSociety.

Fig. 4. A: ASAXS intensities measured at 6 different energies (4 below and two above the edge) of the incident beam, referring to a given q-value as indicated in the graph. Theintensities are plotted against the effective real part f’(E)eff of the scattering factor. The dashed line shows the fit according to Eq. (4) if f”(E)eff is disregarded. The index “eff” indicatesthat the finite width of the primary beam was taken into account appropriately for the calculation of the values for f’C(E) and f”C(E) [23]. B: The partial intensities represented byEq. (4): The upper data is the first term in Eq. (4) and refers to the intensity obtained far below the edge (○); the second curve represents FnrC(q) (□); the lowermost term is the selfterm SCC(q) that solely refers to the scattering contribution of the counterions (∇). ASAXS data are compared to the prediction of the Poisson-Boltzmann cell model [23]. Repro-duced by permission of the PCCP Owner Societies Royal Society of Chemistry.

67K. Huber, U. Scheler / Current Opinion in Colloid & Interface Science 17 (2012) 64–73

poly(p-phenylene) backbone with two positively charged tertiaryammonium side chains. Bromine ions with an absorption edgeclose to 13 keV served as counterions [23]. The high brilliance ofthe beam at the ID02 in the ESRF enabled to perform a series of 6SAXS experiments at variable energy close to the K-edge of Br−.Thereby, a plot of dΣ/dΩ(q) versus f’C(E) became possible at variableq-values. Since the term f”C(E)

2 was negligible, each of these plotscould be fitted by an equation quadratic in f’C(E) , yielding corre-sponding values for FnrC(q) and for SCC(q) as fit parameters. Herethe term SCC(q) describes solely the scattering behaviour of thecondensed Br− counterions. Successful extraction of the scatteringbehaviour from the condensing counterions SCC(q) clearly demon-strated the capability of ASAXS. In addition, the use of a regularlyshaped rod-like polyelectrolyte enabled a direct and quantitativetest of the Poisson-Botzmann cell model for counterion condensa-tion since this cell model was derived under the simplifying as-sumption of rod-like macroions. The cell model allowed derivation

Fig. 5. (Left) ASAXS measurements of the shrinking NaPA chains ([NaPA]=3.61 mM; [Sr2+

following data: Total scattering from the polymer and the Sr2+ ions (black squares); differeof the Sr2+ ions denoted as SCC(q) (red circles). The straight line indicates the power law ochains ([NaPA]=3.25 mM; [Sr2+]/[NaPA]=0.46) at the Sr-K absorption edge at 16.105 keV.ions (black squares); difference of Eq. (4). recorded at two energies (blue triangles); form fis a fitted model function. The model, denoted as pearl necklace, gives the scattering functbetween the spheres. For the size distribution of the spheres, a log-normal size distributiEDP Sciences.

of a radial distribution of counterions n(r) around the cylindricalmacroion [21] according to

n rð Þ ¼ n R0ð Þ 2βκ r cos β ln r=RMð Þ½ �

� �

with a brbR0 the regime between the outer cylinder wall at theradius a and the cell radius R0, which is accessible to counterions,κ the Debye screening constant and β and RM two integration con-stants. The integration constant RM corresponds to a radius in theregime of a bRMbR0 which defines a cylindrical subvolume of thecell comprising most of the condensed counterions. Since all param-eters are predetermined by the geometry of the rod-like polymersand the dielectric constant of the medium, the radial distributioncould directly be applied to predict all three scattering contributionsfrom Eq. (4) (Fig. 4).

]/[NaPA]=0.41) at the Sr-K absorption edge at 16.105 keV. The symbols indicate thence of Eq. (4). recorded at two energies (blue triangles); form factor of the distributionf q−2 expected for polymer coils. (Right) ASAXS measurements of the shrinking NaPAThe symbols indicate the following data: Total scattering from the polymer and the Sr2+

actor of the distribution of the Sr2+ ions denoted as SCC(q) (red circles). The solid lineion of N spheres with radius R being aligned on an invisible rod with fixed distances don was assumed with a predefined number of pearls N=3 [25•]. With permission of

0.2 0.40.000

0.001

0.002

0.003

4q2 S C

C /

cm-1nm

-2

q / nm-1

Fig. 6. Evaluation of the invariant QC based on Eq. (5) for a solution of sodium polyacry-late with a molar mass of 950 kDa in 0.01 M NaCl in the presence of Sr2+ cations. TheSr2+ concentration was 1.5 mM and the molar ratio of [Sr2+]/[COO−] was 0.458 where[COO−] refers to the molar concentration of sodium polyacrylate expressed as moles ofmonomers per volume. The red line indicates a model fit based on a dumbbell withpolydisperse spheres having an averaged outer sphere radius of 13 nm and being sep-arated by a distance of 71 nm within the dumbbell. The spheres represent a simplifieddescription of collapsed domains within the anionic polyacrylate coils, induced by thebinding of Sr2+ cations. The model calculation also provided a good estimation of con-tribution of the missing scattering to the resonant invariant beyond the resolution ofthe ASAXS experiment (gray area on the left). The determination of the invariant asthe area under the data indicated that that a fraction of 17% of the Sr2+ cations werebound to COO− residues within the polyacrylate coil [29•]. Reprinted with permissionfrom Journal of Chemical Physics. Copyright 2007, American Institute of Physics.

68 K. Huber, U. Scheler / Current Opinion in Colloid & Interface Science 17 (2012) 64–73

In the second system investigated [24••] to isolate the scatteringcontribution of condensing counterions, a regular spherical polyelec-trolyte brush had been applied. The colloidal particles are based on acrosslinked polystyrene core with a spherical shape. Short polyacry-late chains where successively grafted onto these colloids by meansof photoinitiated radical polymerisation. The carboxylate functionswere neutralized with monovalent Rb+ ions, exhibiting an absorp-tion edge at 15.1997 keV. This time, a series of 13 wavelength de-pendent SAXS experiments close to the K-edge of Rb+ have beencarried out at the ID02. The same procedure as outlined in thesucceeding paragraph again yielded FnrC(q) and for SCC(q). Herethe term SCC(q) describes solely the scattering behaviour of thecondensed Rb+ counterions. First and foremost, all three partialscattering contributions turned out to run almost parallel, whichindicated that the counterion distribution followed the sphericalsymmetry inferred by the colloidal particles. In more detail, thescattering behaviour could satisfactorily be interpreted by a coreshell model. The shells were formed by decaying density of coun-terions which went entirely parallel to the trend of the monomericCOO− residues.

An alternative strategy [25•] is based on three ASAXS measure-ments at three different energies (wavelengths) E1, E2 and E3. Sub-traction of the scattering curves of any two energies led to so calledseparated scattering curves, which have lost the first term ofEq. (4). A successive subtraction of two separated scattering curvesfrom two different combinations of Ei and Ej led to a term whichonly include Scc(q).

Subtraction of scattering curves and separated scattering curvesin all cases have to be performed at identical q-values for each con-tributing component. This usually requires interpolation of scatter-ing intensities as the 2-dimensional q-grid slightly shifts if thewavelength is shifted. As in the above mentioned strategy, the onlyparameters required to calculate SCC are the two energy dependentcorrection factors f’C(E) and f”C(E), which form the anomalousdispersion corrections to the excess electron number of the counter-ions ΔfC(E) as they modulate it close to the absorption edge of therespective species. As shall be discussed below, this does not onlylead to the scattering contributions of the counterions but in par-ticular cases also to the amount of counterions being capturedwithin the domain of the polyelectrolyte chains.

The strategy had been introduced with energy dependent SAXS ex-periments on long chain sodium polyacrylates in dilute solution. In thissystem an additional salinity of 0.01 M NaCl was used in order to sup-press interparticular correlations causing an undesired structure factorof the polyelectrolyte chains. A small fraction of the sodium ions hadsuccessively been replaced by Sr2+ ions, which were known to interactspecificallywith the COO− residues of the polyacrylate chains. The Sr2+

cations exhibit an absorption edge at 16.105 keV, below which threeSAXS curves had been established at three differentwavelengths. Appli-cation of the above described separation procedure now made accessi-ble the pure scattering contribution of the Sr2+ cations. As we will see,investigation of this system provided clear evidence for two strikingbenefits from quantitative ASAXS experiments introduced by Ballauffet al. [23,24••] and Goerigk et al. [25•,28,29•]: (i) the evolution of thescattering pattern of one species – here the Sr2+ cations – and its spatialdistribution; (ii) the quantitative measure of the amount of thisspecies being locally confined in the polyelectrolyte domain. Like inthe preceding ASAXS experiments on polyelectrolytes, the purelyresonant scattering followed a similar q-dependency as the one ob-served for the other contributions, thereby demonstrating that thecondensed counterions reproduce the shape of the “templating”polyelectrolytes. The interesting feature of the present system wasthe occurrence of specific interactions between Sr2+ and COO−,which cannot be reconciled anymore with the classical counterioncondensation. Such a specific interaction caused a preferentialadsorption of the Sr2+ cations, which led to partially collapsed

polyacrylate segments and which could be modelled as pearl-likesubstructures along the polymer backbone. Noteworthy, such pearl-like substructures only occurred if a certain ratio of [Sr2+]/[COO−]was exceeded [25•,28]. An overview on these experiments is providedby Fig. 5.

A further step toward a quantitative analysis of ASAXS data wasdone by Goerigk et al. [29•] in making use of the Porod-invariant[30] of the purely resonant scattering contribution from the ab-sorbing species. This invariant gets accessible by ASAXS and canbe established via the following integral

bv >¼ VC

2π2r20∫qSCC qð Þq2dq ð5Þ

where r0 corresponds to the absolute scattering length of an elec-tron and integration is limited to the experimentally accessible q-space. The only constant required in Eq. (5) is the radius of thecounterion, which enables calculation of the volume VC of thecounterion. With SCC(q) inferred from Eq. (4), the invariant of thepure resonant scattering leads to the averaged number density bv> ofthe (absorbing) counterions.

If the absorbing counterions occur in two different states, i.e. (i) asions bound to (or correlated with) the polyelectrolyte and (ii) asindependently moving (non-correlated) counterions, care has to betaken for the physical meaning of bv>. In such a case, a value forthe latter parameter can be unambiguously attributed to the counter-ions in a bound state if proper back ground subtraction has beenperformed. This background would include just as many counterionsdissolved in the solvent background as remain uncorrelated in the

69K. Huber, U. Scheler / Current Opinion in Colloid & Interface Science 17 (2012) 64–73

polyelectrolyte solution. Unfortunately, this is usually not known apriori. However, it could be nicely demonstrated, that in dilute solu-tion the scattering contribution to the experimentally accessiblePorod-invariant, stemming from freely moving counterions can beneglected in a typical small angle experiment [29•].

Along this line of thought, the amount of Sr2+ cations bound tosodium polyacrylate chains could be determined unambiguouslyand for the first time simply by determining the Porod invariant ofthe resonant scattering SCC(q) of these counterions. To this end, theexperiments presented in Ref. [28] for the system of a dilute solutionof sodium polyacrylate in 0.01 M NaCl in the presence of Sr2+ cationshad been reevaluated in Ref [29•]. An example thereof is presentedin Fig. 6. All four intermediates are partially collapsed polyacrylateanions. The invariants of SCC(q) revealed that between 9 and 23%of the Sr2+ cations were bound by COO− residues indicating thatsome 10–20% of the residues of these partially collapsed interme-diates were neutralized and predominantly located in pearl-likesubstructures [29•].

Despite this progress, only a small number of further ASAXSstudies on counterion condensation onto polyelectrolytes havebeen published since. Geissler et al. [31,32] presented an investiga-tion of mixed alkali and alkaline earth cations on hyaluronic acidin semidilute solution. Selection of a suitable q-window at fairlylarge values put the focus on the radial distribution around thepolyelectrolyte backbone, which was considered to be rigid and cy-lindrical in the respective spatial resolution. This enabled the authorsto express the partial scattering factors in terms of the product ofthe radial scattering amplitudes stemming from the backbone andfrom the surrounding counterions. With this simplification, in turn,the scattering amplitude and thus the scattering factor of the radi-ally distributed counterions became accessible. However, the actualvalue of their experiments lies in the applied combinations ofcounterions. In a first series they investigated the radial distribu-tion of (absorbing) 0.1 M Rb+ ions at variable amounts of (non-absorbing) Ca2+ ions covering a concentration regime in between0 M and 0.1 M. If no Ca2+ ions were present at all, the Rb+ densitydistribution profile could be interpreted with the simple Poisson-Boltzmann cell model for rod-like polyelectrolytes [21], in agreementwith the findings of Ballauff et al.[18,19,23]. Increasing amounts ofCa2+ partially displaced the monovalent counterions, though notentirely even in the presence of equal amounts of Ca2+. In a secondseries, the contrast situation was vice versa. Hyaluronic acid togetherat 0.1 M non-absorbing Na+ ions were investigated in the presence ofan increasing amount of absorbing Sr2+ ions. Two effects could beclearly outlined. At high concentrations of Sr2+ counterions, the ampli-tude of the Sr2+-shell approached a plateau value. The density profilecould be described by a cylindrical shell which was significantlythinner than the diffuse shell of the cloud from purely monovalentions. The latter feature is in agreement with an observation madealready by Das et al. [22].

In an interesting study on Nafion films, Sugiyama et al. [33] ap-plied the concept of counterion condensation onto polyelectrolytesto unravel the morphology of Nafion. Nafion consists of polytetra-fluorocarbon main chains and side chains with the grafted side chainsbeing capped with sulfonic acid groups. The material serves as mem-branes in electrolyte fuel devices where they separate the electrodesand at the same time provide paths for charge transport by means oftheir ionic groups. In this context ASAXS with Cu2+ cations had beenused in order to recover the hydrophilic domains formed by thesulfonic acid residues. The goal had been achieved by extractingthe pure resonant Cu2+ scattering factor from a Nafion samplesoaked in aqueous CuCl2 solutions according to the procedure sug-gested first by Goerigk et al. [25•]. The results are expected to shedlight on the problem of gradual loss of electrical output upon longtime operation, which is attributed in part to the absorption ofmetal cations from the environment. Expectations are justified as

ASAXS does not only yield the mode of distribution of cations withinNafion but its amount can be quantified and the results can succes-sively be related to conductivity measurements.

The latest paper on the application of ASAXS to counterion con-densation was published by Pollack et al. [34] who once moreaddressed the condensation of counterions around a short DNA seg-ment with 25 base pairs. Thereby, the authors succeeded to extractthe partial scattering factor of either the monovalent Rb+ counterionsor Sr2+ counterions by applying the procedure suggested byBallauff et al. [23,24••]. Pollack et al. also presented a method ofextracting the number of counterions condensed on a DNA chain.The method however had to be based on two assumptions, notbeing required with the method first suggested by Goerigk et al.[29•] based on the Porod invariant [30] of the purely resonant scatter-ing contribution. The two additional assumptions were (i) that thepartial scattering factors can be represented as a product of therespective amplitudes, which is only possible in correlations withcylindrical or spherical symmetry and (ii) that the initial values atq=0 have to be established for the partial scattering factors. Thelatter became possible only via a model fit to the scattering curves,where use had been made of the Poisson-Boltzmann cell model fora rigid cylinder. [34]. The latter certainly is adequate for the shortchain cylinder-like DNA segments but would fail for flexible longchain polyelectrolytes [8,35] and for cations exhibiting specific inter-actions with the ionic groups on the backbone [29•].

Despite the low number of studies on counter ion condensationonto polyelectrolytes, significant progress could be achieved. Progressincludes developments in the field of data quality, which usually hasto be extremely high for ASAXS studies, and which was accomplisheddue to new generations in synchrotron radiation sources and its pre-cise detection as well as procedures to analyze quantitatively thescattering results. Taken together, it became possible to determinethe structure of the condensing counterion cloud and to determinethe amount of condensed counterions, which represents a strikingdepth of information available on highly charged systems in solutionincluding polymers and colloids. Since numerous elementary anionsand cations are suitable for such studies, a large variety of systemsboth in synthetic materials as well as in biology will become acces-sible to such studies. In such systems counterions may affect thestructure of the hosting colloid or polymer and vice versa by eithernon-specific electrostatic interactions or specific interactions. Theresulting interplay may be part of a template-effect in mineraliza-tion or of a catalytic mechanism or of a hierarchical structure forma-tion in biological systems to give but a few exiting examples. It isthis progress and prospect which renders ASAXS a highly promisingtechnique in complex charged systems.

3. Electrophoresis NMR

3.1. General considerations

Nuclear magnetic resonance (NMR) is a powerful tool for struc-ture characterization because of the sensitivity of the NMR signal tothe chemical environment of the nucleus under study [36,37]. Thereare numerous applications of NMR to the study of polyelectrolytes[38]. Spatial information is encoded in NMR spectra by the applicationof magnetic field gradients [39]. The same can be utilized to encodedisplacements, which has been widely used to investigate diffusion[40], pulsed-field-gradient NMR (PFG NMR). For the distribution ofmolecular weights like in polymer the numerical inversion of theLaplace transform leads to satisfactory results, when appropriateboundary conditions are applied [46]. PFG NMR measures a two-time correlation function. No information is obtained on the posi-tion of the spins for any time between the two gradient pulses. Inpractical applications the diffusion time is in the order of tens ofmilliseconds.

1

10

100

1000

char

ge [e

]

N1 10 100 1000

Fig. 8. Effective charge of PSS as a function of the degree of polymerization in [55]. Withpermission of John Wiley and Sons.

-5

-4

-3

-2

-1

0

1

2

3

4

5

x 10-4

-5

-4

-3

-2

-1

0

1

2

3

4

5

x 10-4

b

-123-122-121-120-119-118-117

-5

-4

-3

-2

-1

0

1

2

3

4

5

x 10-4

012345

-5

-4

-3

-2

-1

0

1

2

3

4

5

x 10-4

mob

ility

[cm

²/V

s]

mob

ility

[cm

²/V

s]

mob

ility

[cm

²/V

s]

mob

ility

[cm

²/V

s]

F [ppm] H [ppm]

a

-123-122-121-120-119-118-117 012345F [ppm] H [ppm]

Fig. 7. Two-dimensional electrophoresis NMR spectra, left 19 F signal of perfluorinated succinic acid and right 1H signal of PDADMAC (a) both components separately and (b) suc-cinic acid as the counterion to PDADMAC: [52•]. With permission of John Wiley and Sons.

70 K. Huber, U. Scheler / Current Opinion in Colloid & Interface Science 17 (2012) 64–73

Coherent and incoherent flow can either be separated by dataprocessing [39] or by the design of the experiment [41]. Incoherentmotion like diffusion results in an attenuation of the NMR signalin the PFG NMR experiment, while coherent motion as that ofcharged moieties in an electric field results in a phase modulation.The application of PFG NMR to electrophoresis is termed electro-phoretic NMR or electrophoresis NMR [42–45]. It can be performedon a standard NMR spectrometer, however it requires a dedicatedprobe head enabling the application of an electric field in situduring the PFG NMR experiment. Details on instrumentation andpossible applications can be found in [44].

The phase shift resulting from electrophoretic motion of a singlespecies may be evaluated for each value of the electric field applied.There is a linear relation for all practical electric fields [47]. Thisis particular important for the comparison with simulations [48–51],which are usually performed on much shorter time scales, thus re-quiring stronger electric fields. In a more general case data evaluationis performed by a modified two-dimensional Fourier transform. As aresult a two-dimensional spectrum is obtained correlating the chem-ical shift for the identification of the chemical species under studywith the electrophoretic mobility as seen in Fig. 7. This spectrumhas pure absorption lineshape and resolves the sign of the electro-phoretic mobility, which is important if macromolecule and counter-ions are detected simultaneously.

The diffusion coefficient and the electrophoretic mobility are mea-sured independently but on the same time and length scales. Thusthey can be jointly evaluated to determine the effective charge ofthe species moving, the macromolecule including a fraction of thesolvation shell and the condensed counterions. From the force

balance between the force in the electric field and the hydrodynamicfriction the effective charge is determined from Eq. (6) just with theassumption of a steady-state velocity.

z ¼ μ⋅kB⋅Te⋅D ð6Þ

If the nominal number of charges per molecule is known, the dif-ference is attributed to condensed counterions.

0.0

0.1

0.2

0.3

0.4fr

actio

n of

cha

rge

molar fraction methanol [%]

exp. chargecharge according to Manning

0 20 40 60 80 100

80 70 60 50 40 30permittivity

Fig. 9. Fraction of the charge that is effective, not compensated by condensed counter-ions as a function of the relative dielectric permittivity adjusted in mixtures of metha-nol and water [53••]. Reprinted from Advances in Colloid and Interface Science, 158(2010). Copyright 2010, with permission from Elsevier.

71K. Huber, U. Scheler / Current Opinion in Colloid & Interface Science 17 (2012) 64–73

3.2. Applications

Using electrophoresis NMR the condensation of counterions hasbeen directly observed in a system of poly(diallyldimethyl ammoni-um chloride) (PDADMAC) and perfluorinated succinic acid (PFSA)as counterions. Here the proton NMR signal originates from thepolyelectrolyte only, while the fluorine signal originates from thecounterion. In the electrophoresis NMR spectra in Fig. 7a both com-ponents are investigated separately exhibiting an electrophoreticmobility of opposite sign. However, as seen in Fig. 7b PFSA is movingwith the same sign of the electrophoretic mobility as PDADMAConce they are brought together. The absolute value of the electro-phoretic mobility is smaller for PFSA than for PDADMAC. This isindicating that there is exchange between free and condensed coun-terions, which happens to be rapid on the time scale of the PFG NMRexperiment. This becomes even more evident, when the concentra-tion of PFSA is varied in the solution. With increasing concentrationof PFSA the averaged electrophoretic mobility, which is observed isshifted towards the value for the free PFSA, because the populationof the free counterions is increasing [52•].

Polystyrene(sulfonate) (PSS) is a flexible strong polyelectrolyte,which is available with a narrow distribution of molecular weights.As a strong polyelectrolyte PSS contains acid groups which are disso-ciated nearly independent of the conditions of the surroundingmedium. It is therefore well suited for an investigation of the effectsof the medium on the condensation of counterions [53••,54].

pH=7.7

0

10

20

30

40

50

60

char

ge

generation0 1 2 3

Fig. 10. Fraction of the charge that is effective as a function of the generation for PMAMA dtonated. Copyright 2011 by MDPI AG.

The effective charge of PSS has been investigated as a function ofthe degree of polymerization [55]. As seen in Fig. 8 the effectivecharge is equal to the nominal charge for the monomer as expectedand for an octamer. Further increase of the molecular weight leadsto an effective charge, that is smaller than the nominal charge.There is an intermediate size of molecules, where the effectivecharge coincides with that, which would be predicted by the simpleManning model [1] for a linear infinitely long polyelectrolyte. Atlarger molecular weights the difference becomes even larger. Thiscould be explained by folding of the polymer, so that no longer alinear arrangement of charges as in a fully extended chain could beassumed. This is supported by the fact that the fractal dimension ofPSS in salt-free solution is 1.6 as opposed to 1 for a rod-like polymer.The fractal dimension has been determined from the dependence ofthe hydrodynamic radius as a characteristic length on the molecularweight of PSS [55].

In addition the ionic strength [56] and the dielectric constant ofthe solution [57] have been varied. With increasing ionic strengthenhanced condensation of the counterions is observed. While in asalt-free solution of PSS with a degree of polymerization of 350, 75charges are observed, this value goes down to 18 at an ionic strengthof 7.5mM of NaCl. At the same time the fractal dimension of the poly-electrolyte goes to about 1.8 indicating increased folding of the poly-mer as a result of the smaller repelling forces between charged repeatunits.

The dielectric constant of the solution has been lowered mixingwater and methanol. If the content of methanol goes up, the effectivecharge of PSS goes down as expected. At a molar fraction of methanoljust below 80% there is a sudden drop of the effective charge, whichaccompanied by a similar drop in the hydrodynamic size of themolecule. It is understood, that the decreased effective charge leadsto decreased repelling electrostatic forces along the polymer. Theresulting more compact conformation implies a greater charge densi-ty per volume, which in turn requires more counterions to condense.As shown in Fig. 9 this is self accelerating [57].

Interesting effects have been observed by the variation of thekind of counterions [58]. For the first step monovalent counterionsin interaction with PSS have been investigated: The proton in theacid form, lithium and sodium. There is more counterion condensa-tion going from the proton to lithium to sodium. While this changeis rather moderate, there is a drastic drop of the hydrodynamic sizeof the polyelectrolytes at the same ionic strength.

Organic counterions are monitored simultaneously; they are iden-tified by the NMR chemical shift and can thus be evaluated separately.On the time scale of the PFG NMR experiment there is rapid ex-change of free and condensed counterions, and therefore a population-weighted average is observed. From the comparison with the diffusioncoefficient or the electrophoretic mobility in a low-molecular weight

pH=3.0

0

10

20

30

40

50

60

char

ge

0 1 2 3

generation

endrimers at pH=7.7 (left) and pH=3 (right), where the amino groups are fully pro-

• Of special interest.•• Of outstanding interest.

72 K. Huber, U. Scheler / Current Opinion in Colloid & Interface Science 17 (2012) 64–73

salt with the interaction with a macromolecule the fraction of con-densed counterions is determined. In a particular case the combina-tion of effective charge of the macromolecule with the fraction ofcondensed counterions the molecular weight of the polymer hasbeen inferred [59].

As a model for globular polyelectrolytes dendrimers with aminogroups have been used. The degree of protonation as a function of thepH is directly inferred from the chemical shift of the protons from theneighbouring CH2 groups [60]. As expected the primary amino groupsare protonated first, then the tertiary amino groups. At high pH, whenonly a minor fraction of the amino groups is charged, the effectivecharge is equivalent to the nominal charge. The chemical shift in theNMR signal of protons next to the amino groups is an indicator of thedegree of protonation and thus the nominal charge. Lowering the pHthe difference between the nominal and the effective charge increasesas depicted in Fig. 10. In the fully charged situation the nominal chargeincreases exponentially with the generation while there is an apparentlinear increase of the effective charge implicating that the fraction ofcondensed counterions keeps increasing with the increasing size ofthese three-dimensional molecules.

4. Conclusions

The effect of counterion condensation plays an important role inmany applications of charged macromolecules and complexes likefor instance for the triggering of processes in living systems, for thedesign of new responsive materials or for the development of fuelcells to name but a few examples. It is thus of fundamental interestand has received broad attention in theoretical work and simulations.Whereas most of the experiments available so far focus on the activityof the non-condensed counterions in the solution, two relativelyrecent experimental techniques are described in the present contri-bution giving access to counterion condensation and permitting itsquantification.

Anomalous SAXS allows the selective determination of the scat-tering pattern of individual elements. It may thus be applied toreflect the distribution of the counterions giving access to the structureof the trapped (i.e. condensed or bound) counterions coexisting withthe freely moving ones and may measure the fraction of these trappedions. In practical implementations the recording of several scatteringcurves at different energies is required with each curve being averagedover an extended period of time to achieve high quality scatteringdata precise enough for the data processing. It has to be emphasizedthat such selective investigation can even be performed in the pres-ence of additional mixed salt ions of variable species and composi-tion. The structural information in addition to the quantificationof the condensed counterions of a selective ion provides comple-mentary and unprecedented information.

In electrophoresis NMR the effective charge is inferred from a com-parison between the self diffusion and the electrophoretic motion.This in turn is compared to the nominal charge. The effective chargeof counterions in contact with the macromolecule maybe comparedwith that of the same ion in a low-molecular-weight salt. The frac-tion of counterions moving with the macromolecule is consideredas condensed. This in turn permits the simultaneous detection ofboth the effective charge of the macromolecule and the fractionof condensed counterions. The PFG NMR experiment measures atwo-time correlation function and thus inherently includes a timeaverage for a time in the order of milliseconds.

Two experimental approaches to the description of counterioncondensation have been described and compared. Though practicallyboth experiments require signal averaging, there is a fundamentaldifference on the time scales. While the scattering experimentyields a spatial distribution of the counterions, electrophoresisNMR measures displacements and thus averages over a flow time,which is important because free and condensed counterions

exchange on a faster time scale.While the electrophoresis NMRmea-sures the effective charge via its effect on the drift velocity in aknown electric field the ASAXS experiment measures a spatial distri-bution and correlation. ASAXS can selectively investigate differentinorganic counterions even in competition, while electrophoresisNMR can study organic counterions together with the polyelectro-lyte. Both experiments inherently provide an ensemble average.

References•,••

[1] Manning GS. In: Sélégny E, Mandel M, Strauss UP, editors. Polyelectrolytes.Dordrecht-Holland: D. Reidel Publishing Company; 1974. p. 9–34.

[2] Manning GJ. Chem Phys 1969;51:924.[3] Wandrey C, Hunkeler D, Wendler U, Jaeger W. Counterion Activity of Highly

Charged Polyelectrolytes. Macromolecules 2000;33:7136–43.[4] Miyajima T, Mori M, Ishguro SI. Analysis of Complexation Equilibria of Polyacrylic

Acid by a Donnan-Based Concept. J Colloid Polym Sci 1997;187:259–66.[5] Nordmeier E. Studies of Polyelectrolyte Solutions V. Effect of Counterion Binding

by Polyions of Varying Charge Density and Constant Degree of Polymerisation.Polym J 1994;26:539–50.

[6] Pochard I, Foissy A, Couchot P. Conductometric and microcalorimetric analysis ofthe alkaline-earth/alkali-metal ion exchange onto polyacrylic acis. Colloid PolymSci 1999;277:818–27.

[7] Pochard I, Couchot P, Foissy A. Potentiostatic and conductometric analysis of thebinding of Barium ions with alkali polyacrylate. Colloid Polym Sci 1998;276:1088–97.

[8] Blaul J, Wittemann M, Ballauff M, Rehahn M. Osmotic Coefficient of a SyntheticRodlike Polyelectrolyte in Salt-Free Solution as a Test of the Poisson-BoltzmannCell Model. J Phys Chem B 2000;104:7077–81.

[9] Porcar L, Yun L, Verduzco R, Kunlun H, Butler PD, Magid LJ. Structural Investiga-tion of PAMAM Dendrimers in Aqueous Solutions Using Small-Angle NeutronScattering: Effect of Generation. J Phys Chem B 2008;112(47):14772–8.

[10] Hinderberger D, Spiess HW, Jeschke G. Spearation of Polyelectrolyte Chain Dy-namics and Dynamics of Counterion attachment by EPR spectroscopy. MacromolSymp 2004;211:71–86.

[11] Hinderberger D, Spiess HW, Jeschke G. Radial Counterion distributions in poly-electrolyte solutions determined by EPR spectroscopy. Europhys Lett 2005;70:102–8.

[12] Grass K, Holm C. Polyelectrolytes in electric fields: measuring the dynamicaleffective charge and effective friction. Soft Matter 2009;5:2079–92.

[13] Ballauff M, Jusufi A. Anomalous small-angle X-ray scattering: analyzing correla-tions and fluctuations in polyelectrolytes. Colloid Polym Sci 2006;284:1303–11.

••[14] Stuhrmann HB. Resonance Scattering in Macromolecular Structure Research. Adv

Polym Sci 1985;67:123–63.[15] Oberthür RC. Lösungsmittelumgebung und gegenseitige Wechselwirkung

hochsymmetrischer Teilchen in Lösung und Lösungsmittel: dargestellt amBeispiel von Hexamethylentetramin in Wasser und Alkali-DNS in wässerigerAlkalihalogenidlösung. Thesis, University of Mainz, 1974.

[16] Stuhrmann HB. Anomalous small angle scattering. Q Rev Biophys 1981;14:433–62.

[17] Sabbagh I, Delsanti M, Lesieur P. Ionic distribution and polymer conformation,near phase separation, in sodium polyacrylate/divalent cations mixtures: smallangle X-ray and neutron scattering. Eur Phys J B 1999;12:253–60.

[18] Guilleaume B, Ballauff M, Goerigk G, Wittemann M, Rehahn M. Correlations ofcounterions with rodlike macroions as assessed by anomalous small-angle x-rayscattering. Colloid Polym Sci 2001;279:829–35.

[19] Guilleaume B, Blaul J, Ballauff M, Wittemann M, Rehahn M, Goerigk G. The distri-bution of counterions around synthetic rod-like polyelectrolytes in solution. EurPhys J E 2002;8:299–309.

[20] De Robillard Q, Guo X, Dingenouts N, Ballauff M. Application of Anomalous SmallAngle X-Ray Scattering to Spherical Polyelectrolyte Brushes. Macromol Symp2001;164:81–90.

[21] Le Bret M, Zimm B. Distribution of Counterions Around a Cylindrical Polyelectrolyteand Mannig's Condensation Theory. Biopolymers 1984;23:287–312.

[22] Das R, Mills TT, Kwok LW, Maskel GS, Millet IS, Doniach S. Counterion Distributionaround DNA Probed by Solution X-Ray Scattering. Phys Rev Lett 2003;90:188103-1–4.

[23] Patel M, Rosenfeldt S, Dingenouts N, Pontoni D, Narayanan T, Ballauff M. Analysisof the correlation of counterions to rod-like macroions by anomalous small-angleX-ray scattering. Phys Chem Chem Phys 2004;6:2962–7.

••[24] Dingenouts N, Patel M, Rosenfeldt S, Pontoni D, Narayanan T, Ballauff M. Counterion

Distribution around a Spherical Polyelectrolyte Brush Probed by Anomalous Small-Angle X-ray Scattering. Macromolecules 2004;37:8152–9.

•[25] Goerigk G, Schweins R, Huber K, Ballauff M. The distribution of Sr2+ counterions

around polyacrylate chains analyzed by anomalous small-angle X-ray scattering.Europhys Lett 2004;66(3):331–7.

[26] Cromer DT, Liberman D. Relativistic Calculation of Anomalous Scattering Factorsfor x Rays. J Chem Phys 1970;53:1891–8.

[27] Cromer DT, Liberman D. Anomalous dispersion calculations near to and on thelong-wavelength side of an absorption edge. Acta Crystallogr 1981;A37:267–8.

73K. Huber, U. Scheler / Current Opinion in Colloid & Interface Science 17 (2012) 64–73

[28] Schweins R, Goerigk G, Huber K. Shrinking of anionic polyacrylate coils inducedby Ca2+, Sr2+ and Ba2+: A combined light scattering and ASAXS study. EurPhys J E 2006;21:99–110.

•[29] Goerigk G, Huber K, Schweins R. Probing the extent of the Sr2+ ion condensation

to anionic polyacrylate coils: A quantitative anomalous small-angle x-ray scatter-ing study. J Chem Phys 2007;127:154908.

[30] Porod G, Kolloid Z. Die Röntgenkleinwinkelstreuung von dichtgepackten kolloidenSystemen I. Teil. 1951;124:83-114 (Chapter I.6), see also Spalla O. In: Lindner P,Zemb Th, editors. Neutrons, X-rays and Light: Scattering Methods Applied to SoftMatter; 2002. p. 49–71. [North-Holland].

[31] Horkay F, Hecht AM, Rochas C, Basser PJ, Geissler E. Anomalous small angle x-rayscattering determination of ion distribution around a polyelectrolyte biopolymerin salt solution. J Chem Phys 2006;125:234904.

[32] Horkay F, Basser PJ, Hecht AM, Geissler E. Comparative Study of Scattering andOsmotic Properties of Synthetic and Biopolymer Gels. Macromol Symp 2007;256:80–7.

[33] Sugiyama M, Mitsui T, Sato T, Akai Y, Soejima Y, Orihara H, et al. StructuralAnalysis of Polyelectrolyte Film Absorbing Metal Ion by SAXS Utilizing withX-ray Anomalous Dispersion Effect. J Phys Chem B 2007;111:8663–7.

[34] Pabit SA, Meisburger SP, Li L, Blose JM, Jones CD, Pollack L. Counting Ions aroundDNA with Anomalous Small-Angle X-ray Scattering. J Am Chem Soc 2010;132:16334–6.

[35] Förster S, Schmidt M. Polyelectrolytes in Solution. Adv Polym Sci 1995;120:51–133.[36] Ernst RA, Bodenhausen G, Wokaun A. Principles of Nuclear Magnetic Resonance in

One and Two Dimensions. Oxford: Oxford University Press; 1991.[37] Schmidt-Rohr K, Spiess HW. Multidimensional Solid-State NMR and Polymers.

London: Academic Press Limited; 1994.[38] Scheler U. NMR on Polyelectrolytes, Current opinion on colloid and interface.

Science 2009;14:212–5.[39] Callaghan PT. Principles of Nuclear Magnetic Resonance Microscopy. Oxford:

Oxford University Press; 1991.[40] Stejskal EO, Tanner JE. J Chem Phys 1964;42:282.[41] Gottwald A, Kuran P, Scheler U. Separation of velocity distribution and diffusion

using PFG NMR. J Magn Reson 2003;162:364–70.[42] Holz M. Electrophoretic NMR. Chem Soc Rev 1994;23:165–74.[43] Johnson Jr CS, He Q. Adv Magn Reson 1989;13:131.

[44] Scheler U. in Encyclopedia of Magnetic Resonance. John Wiley & Sons, Ltd.; 2012.doi:10.1002/9780470034590.emrstm0154.pub2.

[45] Stilbs P, Furó I. Electrophoretic NMR. Curr Opin Colloid Interface Sci 2006;11:3–6.

[46] Godefroy S, Callaghan PT. 2D relaxation/diffusion correlations in porous media.Magn Reson Imaging 2003;21:381–3.

[47] Wong S, Scheler U. Electrophoresis of macromolecules in solution detected byelectrophoresis-NMR. Colloids and Surfaces 2001;A 195:253–7.

[48] Grass K, Böhme U, Scheler U, Cottet H, Holm C. Importance of hydrodynamicshielding for the dynamic behavior of short polyelectrolyte chains. Phys RevLett 2008;100:096104 [4pp].

[49] Frank S, Winkler RG. Mesoscale hydrodynamic simulation of short polyelectro-lytes in electric fields. J Chem Phys 2009;131.

[50] Muthukumar M. Theory of counter-ion condensation on flexible polyelectrolytes:Adsorption mechanism. J Chem Phys 2004;120:9343–50.

[51] Netz RR. Polyelectrolytes in electric fields. J Phys Chem B 2003;A107:8208–17.

•[52] Böhme U, Scheler U. Counterion Mobility and Effective Charge of Polyelectrolytes

in Solution. Macromol Symp 2004;211:87–92.

••[53] Böhme U, Scheler U. Counterion condensation and effective charge of poly(styre-

nesulfonate). Adv Coll Interf Sci 2010;158:63–7.[54] Böhme U, Scheler U. Effective charge of poly(styrenesulfonate) and ionic strength

- an electrophoresis NMR investigation. Colloids Surf, A Physicochem Eng Asp2003;222:35–40.

[55] Böhme U, Scheler U. Hydrodynamic size and electrophoretic mobility of poly(-styrene sulfonate) versus molecular weight. Macromol Chem Phys 2007;208:2254–7.

[56] Böhme U, Scheler U. Effective charge of poly(styrenesulfonate) and ionic strength- an electrohoresis NMR investigation. Coll Surf A 2003;222:35–40.

[57] Böhme U, Scheler U. Effective charge of polyelectrolytes as a function on thedielectric constant of a solution. J Colloid Interface Sci 2007;309:231–5.

[58] Böhme U, Hänel B, Scheler U. Influence of the counterions on the behaviour ofpolyelectrolytes. Prog Colloid Polym Sci 2011;138:45–8.

[59] Böhme U, Vogel C, Meier-Haack J, Scheler U. Determination of charge and molecularweight of rigid-rod polyelectrolytes. J Phys Chem 2007;B 111:8344–7.

[60] Böhme U, Klenge A, Hänel B, Scheler U. Counterion condensation and effectivecharge of PAMAM dendrimers. Polymers 2011;3:812–9 [Open Access].