Distribution of Counterions around Lignosulfonate Macromoleculesin Different Polar Solvent MixturesUlla Vainio,*,† Rolf A. Lauten,‡ Sylvio Haas,§ Kirsi Svedstrom,⊥ Larissa S. I. Veiga,∥ Armin Hoell,#

and Ritva Serimaa⊥

†HASYLAB at DESY, Notkestr. 85, D-22607 Hamburg, Germany‡Borregaard Lignotech, P. O. Box 162, N-1701 Sarpsborg, Norway§Humboldt-Universitat zu Berlin, Institut fur Chemie, Brook-Taylor-Strasse 2, D-12489 Berlin, Germany⊥Department of Physics, P. O. Box 64, FI-00014 University of Helsinki, Finland∥Instituto de Fisica ″Gleb Wataghin″, UNICAMP, CP 6165, 13083-970 Campinas, Brazil#Helmholtz-Zentrum Berlin fur Materialien und Energie, Institute of Applied Materials, Hahn-Meitner-Platz 1, D-14109 Berlin,Germany

*S Supporting Information

ABSTRACT: Lignosulfonate is a colloidal polyelectrolyte that is obtained as a sideproduct in sulfite pulping. In this work we wanted to study the noncovalentassociation of the colloids in different solvents, as well as to find out how the chargedsulfonate groups are organized on the colloid surface. We studied sodium andrubidium lignosulfonate in water−methanol mixtures and in dimethyl formamide.The number average molecular weights of the Na- and Rb-lignosulfonate fractionswere 7600 g/mol and 9100 g/mol, respectively, and the polydispersity index for bothwas 2. Anomalous small-angle X-ray scattering (ASAXS) was used for determiningthe distribution of counterions around the Rb-lignosulfonate macromolecules. Thescattering curves were fitted with a model constructed from ellipsoids of revolution of different sizes. Counterions were takeninto account by deriving an approximative formula for the scattering intensity of the Poisson−Boltzmann diffuse double layermodel. The interaction term between the spheroidal particles was estimated using the local monodisperse approximation and theimproved Hayter−Penfold structure factor given by the rescaled mean spherical approximation. Effective charge of thepolyelectrolyte and the local dielectric constant of the solvent close to the globular polyelectrolyte were followed as a function ofthe methanol content in the solvent and lignosulfonate concentration. The lignosulfonate macromolecules were found toaggregate noncovalently in water−methanol mixtures with increasing methanol or lignosulfonate content in a specific directionalmanner. The flat macromolecule aggregates had a nearly constant thickness of 1−1.4 nm, while their diameter grew whencounterion association onto the polyelectrolyte increased. These results indicate that the charged groups in lignosulfonate aremostly at the flat surfaces of the colloid, allowing the associated lignosulfonate complexes to grow further at the edges of thecomplex.

■ INTRODUCTIONLignosulfonate (LS) is a colloidal polyelectrolyte, which isproduced in the sulfite pulping process of wood. Duringpulping, lignin is separated from cellulose and made soluble inwater by sulfonate groups which get grafted onto theamorphous aromatic polymer. Lignosulfonate is used widelyas a dispersant in many industrial processes,1 and recently,sodium lignosulfonate has been found to be a superb dispersantfor carbon nanotubes.2 Unlike many dispersants which formwell-defined association colloids, lignosulfonate is a branchedpolyelectrolyte, a colloid by itself. Lignosulfonate is known tobe surface active, particularly at high concentrations.3 Longchain alcohols have been shown to improve slightly the surfaceactivity of calcium lignosulfonate,3 and it has been suggestedthat lignosulfonate forms mixed surfactant systems with anionicsurfactants.4 Both the ability to form mixed surfactant systemsand association with long chain alcohols, in addition to the

possible self-association, are related to the shape anddistribution of charged as well as uncharged groups on themacromolecular surface.The shape of lignosulfonate macromolecules has been

studied with various methods. Thicknesses from about 1.3 to3.5 nm were observed for different lignins on liquid surfaces inthe beginning of the 1970s.5 Remarkably, the thickness did notdepend on the molecular mass of lignin, indicating that themolecules adopt a flat conformation on the interface. Later, theresult was corroborated by electron microscopy studies, whichshowed a 2 nm thickness for lignosulfonate macromolecules onsurfaces.6 In solution, the shape may be different, but in thepast years, solute low molar mass macromolecules of lignin

Received: November 14, 2011Revised: December 20, 2011Published: December 22, 2011

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derivatives have mostly been described as being flat.7,8

However, the shape of the lignin particle has been in mostcases determined using solutions where electrostatic inter-actions between macromolecules have not been present. Byusing pure water and methanol−water mixtures as solvents forlignosulfonate and by using several different concentrations oflignosulfonate, we attempt here to find structural informationon both the shape and charge of the macroion under differentconditions. Instead of screening the electrostatic interactions,these interactions are included in the structural model that isfitted to the experimental data.Anomalous small-angle X-ray scattering (ASAXS) has been

recently applied to study the distribution of counterions aroundspherical and rod-like colloids,9−11 including DNA,12 while arelated method called resonant X-ray reflectivity has been usedto study counterion distribution close to flat lipid surfaces.13

Applying the ASAXS method to the study of lignosulfonate isnot straightforward due to the polydisperse and random natureof this natural product. High statistical accuracy is needed forthe ASAXS data, and therefore we decided to apply it toconcentrated solutions in order to get enough scattering signalin a reasonable amount of time. The theoretical handling ofconcentrated solutions of nonspherical charged or evennoncharged particles relies very much on approximations, anda robust analytical way to approach the problem is still missing.In this study we develop and test a model which can be appliedto study the counterion condensation in ellipsoidal macroionsystems.

■ EXPERIMENTAL SECTIONTwo lignosulfonate salts prepared from one sample of carefullyfractionated lignosulfonate were used in this work. The preparation offractionated sodium lignosulfonate is described elsewhere.8 To obtainrubidium lignosulfonate, an ion-exchange resin (Amberlite IR120) wasused to convert the sodium salt to lignosulfonate acid. Thelignosulfonate acid was neutralized with an aqueous solution ofRbOH (Aldrich). The metal content, i.e., rubidium and sodium, wasdetermined using inductively coupled plasma spectrometry (ICP) toverify that treatment with the ion-exchange resin removed the metalcounterions and that rubidium replaced sodium as counterion duringneutralization. Molecular weight was determined with a gel permeationchromatography (GPC) technique described elsewhere.14 Thelignosulfonate salts with, respectively, sodium or rubidium ascounterions differ in molecular weight and the difference betweenthe two fractions corresponds to the difference in the counterionatomic weight. Table 1 summarizes the characteristics of the twolignosulfonate salts. The polydispersity of the fraction was 2.07.

Sodium and rubidium lignosulfonate concentration series wereprepared in water and in water−methanol mixtures with 25 vol % and50 vol % methanol. Additionally, one sodium lignosulfonateconcentration series was prepared in dimethyl formamide, DMF.The water was from a Milli-Q unit, the methanol, and DMF were ofp.a. quality from Merck.The Rb content in Rb lignosulfonate was 17.5 wt %, so if we assume

that each counterion is attached to one charged group, using the

knowledge of the molecular weight of the molecules as well as theatomic weight of Rb, we can calculate that Rb and Na lignosulfonatemolecules with the number average molecular weight given in Table 1contain about 19 charged groups, which corresponds to an effectivesurface charge density of about 1.16 e−/nm2 (16 μC/cm2), whenassuming a spherical shape with volume equivalent radius 1.1 nm forthe particles and no counterion association.

By measuring the density of solutions of sodium and rubidiumlignosulfonate dissolved in water and using the knowledge of theconcentrations of the solution, the densities of sodium and rubidiumlignosulfonate were calculated to be 2.02 g/cm3 and 2.44 g/cm3 (error2%), respectively, the ratio of which is the same as that of themolecular weights. It is noteworthy that the measured density forsodium lignosulfonate fraction is larger than the density 1.4 g/cm3 thatwe used for the same fraction in our previous publication,8 where wesimply assumed the density to be the same as the literature value15 forlignin. The main results of that publication, such as the particle shapedetermination, are not affected, but unfortunately the volume fractionmight be systematically wrong by a scaling factor.

Small-angle X-ray scattering (SAXS) was measured from allconcentration series of sodium and rubidium lignosulfonate. Selectedrubidium lignosulfonate samples were also measured at several photonenergies using anomalous small-angle X-ray scattering (ASAXS). Inthis method the energy of the X-rays is tuned below the absorptionedge of an element in the sample in order to achieve resonantscattering effects in the SAXS pattern.

The SAXS and ASAXS experiments were carried out at two ASAXSdedicated experimental stations 7T-MPW SAXS at BESSY synchro-tron in Berlin and B1 at DORIS III in Hamburg.16,17 The samplesolutions were prepared a few weeks before the measurement and letto incubate in room temperature. At 7T-MPW SAXS, the size of thebeam on the sample was 0.5 mm × 0.5 mm. The sample solutionswere injected into glass mark tubes (Hilgenberg) that had about 50-μm-thick walls and inner diameter of 3.9 mm. The thickness of eachtube was measured separately, because the standard deviation of thethickness of the tubes was 0.6 mm. The mark tubes were sealed withTeflon, or for long measurements also glued tight, and placed verticallyto the sample holder. A two-dimensional multiwire proportionalcounter was placed at two different distances (3749 mm and 1404mm) behind the sample to detect the SAXS patterns and to cover asuitable scattering vector range. One SAXS pattern was typicallymeasured in 8 min. ASAXS measurements from samples were made bymeasuring SAXS at 14803 (E1), 15085 (E2), and at 15166 eV (E3) nearthe Rb K-edge, which is at 15200 eV. A typical ASAXS measurementof one sample took a couple of hours in total. All measurements weremade at (22 ± 1) °C. The obtained SAXS data was corrected for deadtime, normalized by detector flatfield, incoming photon flux, as well astransmission and thickness of the sample, integrated over the azimuthangle, calibrated to absolute intensity units (1/cm) using a glassycarbon reference sample, and corrected for solvent background whiletaking into account the volume fraction of the polyelectrolyte.8

Half a year later, ASAXS measurements of one Rb lignosulfonatesample (ϕ = 0.17) in water were repeated at beamline B1 using similarexperimental parameters as at 7T-MPW SAXS. Furthermore, SAXSmeasurements of samples with DMF as the solvent were measured atan energy of 11021 eV at B1, and these samples were placed in quartzcapillaries of 2 mm in diameter.

■ THEORY

Small-angle X-ray scattering is a popular way to study indirectlystructures at the nanometer scale. The method can be mademore powerful with contrast variation by using the resonanteffect near the absorption edge of an element. This contrastvariation method is called anomalous small-angle X-rayscattering. At the absorption edge of an element the atomicscattering factor of the element changes rapidly. The atomicscattering factor can be written at small angles in the form

Table 1. Lignosulfonate Salt Molecular Weight, Density ρm,and Puritya

typeMn

(g/mol)Mw

(g/mol)ρm

(g/cm3) Na Ca Rb

NaLS 7600 18 000 2.02 4.7% <0.001% -RbLS 9100 21 000 2.44 0.09% 0.001% 17.5%

aIn weight percent.

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= + ′ + ″f E Z f E if E( ) ( ) ( ) (1)

where Z is the atomic number of the element (37 forrubidium), and f ′ and f ″ are the anomalous scattering factors.The real part, f ′, is typically negative, and the imaginary part, f ″,which is related to absorption of X-rays, is typically small belowthe absorption edge in case of K-absorption edges.Our Rb lignosulfonate system is composed of three

components: Rb counterions (1), lignosulfonate (2), andsolvent (3). The SAXS intensity I(q,E) from a three-phasesystem can be understood as a sum of six components:18

= + +

+ + +

I q E I q E I q I q E

I q E I q E I q

( , ) ( , ) ( ) ( , )

( , ) ( , ) ( )11 22 33

12 13 23 (2)

where 1 represents here the phase containing the element thathas its absorption edge close to energy E. The other phases forwhich the scattering intensity is independent of energy arerepresented by 2 and 3. We define the magnitude of thescattering vector as q = 4π sin θ/λ, where θ is half of thescattering angle and λ is the wavelength. In the most simplecase, if each phase would contain only one type of atoms,I11(q, E) would depend on the atomic scattering factor ofelement 1 squared (|f1

2|), while I12(q, E) depends on f1*f 2, I22(q)on |f 2|

2, and so on. Now, let us vary the energy of the incidentphotons near the absorption edge of the anomalous element.The intensities from phases 2 and 3 are assumed not to change,as is the case for lignosulfonate and the solvent near the Rb K-edge, where only the scattering of Rb atoms is altered. Then thedifference between two measurements made at differentphoton energies is

Δ = α + β + γI q E I q E I q E I q E( , ) ( , ) ( , ) ( , )11 12 13 (3)

where α, β, and γ are constants. Often the anomalouscomponent I11(q, E) is found by solving a set of equationsfrom measurements at three or more energies, but otherapproaches also exist. For example, Torkkeli et al.19 fitted ananalytical model simultaneously to several intensity curves andtheir differences, while recently Haas et al.20 fitted an analyticalmodel to intensity curves measured at different absorptionedges. Here we follow the idea of Torkkeli et al. and fit ananalytical model simultaneously to the total intensity in eq 2and to the difference in eq 3. The latter we will call “separatedintensity”.Figure 1 shows a simple sketch of how phase 1 (dots) could

be dispersed in a dispersion medium, phase 3 (whitebackground), containing also phase 2 (circles filled withlines). The dots here represent the counterions and the circlerepresents a lignosulfonate particle. In the unphysical case inwhich the ion phase would be distributed homogeneously inthe dispersion medium (Figure 1a), the cross term I12(q, E) ineq 2 vanishes, because in such a two-phase system only thecontrast between the two phases matters, and we may treat oneof the phases as having scattering length density of zero. Ananalogous but inverse situation is depicted in Figure 1d, whereall the ions would have condensated inside the lignosulfonatephase. The phases are now completely correlated and the crossterm of the two phases is of the same shape as I11(q, E). So, forcases (a) and (d), total intensity I(q, E) and the separatedintensity ΔI(q, E) would have the same shape. The schemespresented in (b) and (c) of Figure 1 represent situations wheredifferent fractions of counterions are condensed onto thepolyelectrolyte. For these cases the total intensity curve and the

separated curve have different shape, because now the crossterm is important. By comparing shapes of I(q, E) and ΔI(q,E), we can get information on where rubidium is in thelignosulfonate solution, or if it is associated to the sulfonategroups inside the molecule. If the rubidium counterions arehomogeneously distributed inside lignosulfonate particles, thetotal and separated intensities have the same shape. However,with an increasing dissociation of rubidium counterions theshape of the separated intensity ΔI(q, E) becomes increasinglydifferent from the total intensity I(q, E) due to the cross term(eq 3).An analytical formula can be derived for the form factor of

spherical particles and their counterions. The amplitude is given

∫ = ρ − · A q E V r E iq r( , ) d ( , ) exp( )(4)

where

ρ =ρ <

ρ −κ − ≥<

>⎪⎪⎧⎨⎩

r EE r a

E a r r a r a( , )

( )

( )( / ) exp[ ( )]

r a

r a (5)

The first row in eq 5 applies to a homogeneous scatteringlength density within a sphere with radius a and the lowerequation comes from the potential (ψ = ψ0(a/r) exp[−κ(r −a)]) of a diffuse layer of counterions around the sphere fromthe Poisson−Boltzmann equation by using the Debye−Huckelapproximation.21 The Debye−Huckel approximation holdsonly for low potentials ψ for which |eψ| < kT, where e is theelectric charge of one electron, k is the Boltzmann constant,and T is the temperature. Here ψ is the electric potential whichan ion must overcome to move from bulk solution onto thesurface of the polyelectrolyte. In eq 5 we have approximated thepotential and the counterion cloud to have approximately thesame shape, while in fact, according to the Boltzmann equationcounterion density is proportional to exp(−eψ/kT), for valence+1 ions.22 Note that the electric charge of an electron e isnegative. However, the series expansion of the exponentialfunction (exp(x) = 1 + x + (x2/2!) ... ≈ 1 + x, if |x|≪ 1) tells usthat, if the Debye−Huckel approximation is valid and thus thepotential is low enough, we may safely approximate thescattering density length of counterions compared to solventρr>a(E) to have the same shape as the potential (whileneglecting a constant background). The surface charge densityof our polyelectrolyte may be high enough to violate the

Figure 1. Two-dimensional illustrations of counterions (dots) can bedispersed differently into the particle (circle with lines) and in adispersion medium (white background), representing different stagesof counterion condensation.

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assumption |eψ| < kT, depending on degree of counterionassociation. However, we are interested only in an approximateshape of the ion cloud around the polyelectrolyte and actuallyalso assume that the associated counterions are foundthroughout the polymer instead of forming a compactcounterion layer on the surface. If the Debye−Huckelapproximation is valid, an approximate thickness for theelectrical double layer is then given by the Debye screeninglength 1/κ, where we now define the Debye−Huckel parameterκ as23

κ = ϕεεze

V k T

2

p 0 B (6)

Here z is the electric charge number and Vp is the volume ofthe polyelectrolyte particle, ϕ is the volume fraction ofpolyelectrolyte in solution, ε is the dielectric constant of thesolvent, and ε0 is the vacuum permittivity. Any contributionfrom background electrolyte is not included in this formula.Regarding the scattering contrast that depends on energy

ρr>a(E) in eq 5, for the special case of Rb lignosulfonate thisscattering length density of ions compared to solvent, changesas a function of energy E near the absorption edge of Rb due tothe change in the anomalous scattering factors of the ions. Ifsome ions are located inside the particles the scattering lengthdensity ρr<a(E) inside the particles will depend on energy. ForRb lignosulfonate, ρr<a(E) is the sum of the scattering lengthdensity of lignosulfonate without ions compared to the solvent(ρr<a,LS) and the scattering length density of rubidium inside thelignosulfonate compared to the solvent (ρr<a,Rb). Figure 2

illustrates the definitions of the different scattering lengthdensities with realistic values specific to the case of Rblignosulfonate.From eq 4, by employing q ·r = |q||r|cos θ and by integrating

in spherical coordinates, we get the intensity which is thesquare of the amplitude

= | |

= | + |

I q E A q E

F q E F q E

( , ) ( , )

( , ) ( , )

2

sphere ions2

(7)

The well-known form factor amplitude of a sphere is

= ρ ×

−<

⎡⎣⎢⎢

⎤⎦⎥⎥

F q E E V

qr qr qr

qr

( , ) ( )

3sin( ) cos( )

( )

r asphere

3(8)

where r is the radius of the sphere (here r = a), and V is thevolume of the sphere (4πa3/3). Integration of eq 4 byemploying the second row in eq 5 gives us the form factoramplitude for the diffuse double layer

= ρ ×

κ +κ +

>⎡⎣⎢⎢

⎤⎦⎥⎥

F q E E V

r qr qr qr

q r qr

( , ) ( )

3sin( ) cos( )

( )

r aions

2 3 3(9)

where again r = a and V is the volume of the sphere.By approximating the diffuse double layer model to apply

also on a spheroid surface, we propose an equation for the formfactor of a spheroid particle with a diffuse double layer bymodifying the intensity equation for a spheroid24 by adding theform factor amplitude of counterions to it

∫= | α

+ α | α α

πP q E F q E r

F q E r

( , ) ( , , ( ))

( , , ( )) sin d0

/2sphere

ions2

(10)

Here r(α) = a(sin2 α + ν2 cos2 α)1/2, as in the case of a simpleellipsoid of revolution, i.e., a spheroid with semiaxes a, a, andνa.24,25 The volume V in eqs 8 and 9 is in this case the volumeof an ellipsoid (4πνa3/3). Even though this analytical solutionis possible by using the Debye−Huckel approximation, it isimportant to note that the Debye−Huckel approximation is nota very good approximation for spheroids. However, in the caseof lignosulfonates, we presume that the polydispersity and theshape deviations of the colloidal polyelectrolyte from a spheroidcause more error to the fit results than the approximations usedfor the shape of the counterion distribution. Thus we haveselected the simplest possible shape for the studied colloidalparticles believing this will still capture any changes.Furthermore, we need to take into account the interactions

between the particles in the rather concentrated solutions. Thisis done by using the rescaled mean spherical approximation(RMSA) structure factor S(q), which is also known as theHayter−Penfold structure factor.26,27 Strictly, the RMSAstructure factor applies only to spherical monodisperseparticles, but it is also used for nonspherical systems. We usethe volume equivalent radius of a sphere as an effective radiusof the ellipsoid. The intermolecular interactions dependstrongly on the charges of the particles. We let the effectivecharge zeff of the macromolecule vary with the size of theparticle, choosing the effective surface charge density, σeff = zeff/A, to be our fitting parameter. The surface area, A, is calculatedusing the volume equivalent radius. A constant surface chargedensity is a reasonable assumption based on diffusionexperiments of lignosulfonate in aqueous solutions withmonovalent NaCl salt.28 The routine to calculate the RMSAstructure factor was adapted to Matlab language from the IgorPro SANS package.23 Furthermore, when calculating thestructure factor, the effect from the so-called penetratingbackground, which depends on the volume fraction ϕ ofpolyelectrolyte, was taken into account by redefining the charge

Figure 2. Scattering length density model (thick solid line) for alignosulfonate particle of an effective radius a and its diffuse layer ofcounterions calculated with the approximative formula in eq 5, havinga Debye screening length 1/κ. ρbg is the scattering length density of thesolvent, and the other parameters are explained in the text.

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z = zeff/(1 − ϕ) and the screening constant κ = κeff − 2ϕ1/3 ln(1− ϕ).29,30

The macroscopic differential cross section (∑(q))/(dΩ),noted here as I(q, E) for simplicity, is given by

∑= −I q E V N r P q E r S q E r( , ) ( ) ( , , ) ( , , )r

irr1

(11)

where Virr is the irradiated volume of the sample and thenumber size distribution N(r) can be obtained from gelpermeation chromatography measurements. In this formula thelocal monodisperse approximation has been used.24

A multivariable regression was done by fitting the model(Itot,model(q, E) and IΔ,model(q, E)) to the experimental totalintensity Itot(q, E) and to the separated intensity IΔ(q, E) =Itot(q, E1) − Itot(q, E3) in Matlab using the Nelder-Meadsimplex search method, by minimizing

∑ ∑χ =−

+ −

= Δ =

Δ

⎡⎣⎢⎢

⎤⎦⎥⎥K

I q E I q E

s q E

P P p

( , ) ( , )

( , )

/( )

k i

P

kk i k i

k i

2

tot, 1

,model2

tot

k

where sk(qi, E) is the standard deviation of intensity Ik(qi, E), Pkis the number of points in the q-scale and p is the number offitted parameters. Since the separated curve has larger errorsthan the total curve, the minimization routine tends to fit wellonly the total curve if the multiplication factors Ktot and KΔ areboth 1. Therefore, for the separated intensity KΔ = 30 was usedto increase the weight of the separated curve while fitting.When calculating the final χ2 that is presented in Table 2, bothmultiplication factors were set to 1.In summary, ten variables were fitted for each volume

fraction of lignosulfonate and each solvent composition:thickness d of oblate lignosulfonate ellipsoids and their effectivesurface charge density σeff, scattering length density oflignosulfonate without counterions compared to solventρr<a,LS, scattering length density of free counterions comparedto solvent ρr>a,Rb(E), scattering length density of ions insidelignosulfonate compared to solvent ρr<a,Rb(E), the localdielectric constant of the solvent εeff, and a scaling factor KMfor the mass scale of the molar mass distribution which hadbeen obtained using GPC. The volume equivalent effectiveradius of the particles was calculated from the scaled molarmass distribution and it was used in the structure factorcalculations. Furthermore, we added constant backgrounds B1and B2 to curves Itot,model(q, E) and IΔ,model(q, E), respectively.Separated intensity in the model was defined IΔ,model(q, E) =Imodel(q, E1) − Imodel(q, E3). The model intensity Imodel(q, E3)was calculated otherwise exactly like Imodel(q, E1), but forImodel(q, E3) the scattering length densities ρr>a,Rb(E) and

ρr<a,Rb(E) were multiplied with fitted constant Kf, whichrepresents the change in the atomic scattering factor ofrubidium between the measured energies E1 and E3.

■ RESULTSLignosulfonate dispersions were prepared with sodium andrubidium counterions in different solvents with nearly identicalvolume fractions of lignosulfonate to enable comparisonbetween scattering curves of the two lignosulfonate salts atidentical conditions. The SAXS curves shown in Figures 3 and

4 show the volume fraction normalized scattering for the

different samples in MeOH solvent mixtures. Figure 5 shows

the SAXS curves for Na lignosulfonate in different concen-

trations in DMF. Data from Rb lignosulfonate in DMF is not

Table 2. ASAXS Fit Results for Rb Lignosulfonatea

vol % MeOH ϕ d (nm) σeff (e−/nm2) εeff KM χ2

25 0.037 1.43 ± 0.14 0.47 ± 0.04 41 ± 13 0.76 ± 0.10 8250 0.037 1.27 ± 0.08 0.23 ± 0.05 21 ± 15 0.97 ± 0.15 510 0.170 1.43 ± 0.05 0.20 ± 0.02 57 ± 8 1.16 ± 0.06 198*0 0.170 1.32 ± 0.05 0.20 ± 0.04 56 ± 19 1.39 ± 0.07 13225 0.170 1.4 ± 0.3 0.10 ± 0.01 43 ± 14 1.47 ± 0.07 24150 0.170 1.0 ± 0.2 0.06 ± 0.01 50 ± 13 2.4 ± 0.2 178

aThe different parameters are explained in the text. The vol % MeOH is the volume fraction of methanol in the methanol−water solvent mixture andϕ is the nominal volume fraction of Rb lignosulfonate. The errors are given as 1σ standard deviation. * refers to results from exactly the same samplemeasured 1/2 year later at a different ASAXS instrument.

Figure 3. SAXS curves of different volume fractions of Nalignosulfonate (ϕ = 0.01, 0.04, 0.08, 0.11, 0.14, 0.17) in water−methanol mixtures (0, 25, and 50 vol % of MeOH), normalized by ϕ.

Figure 4. SAXS curves of different volume fractions of Rblignosulfonate (ϕ = 0.01, 0.04, 0.08, 0.11, 0.14, 0.17) in water−methanol mixtures (0, 25, and 50 vol % of MeOH), normalized by ϕ.

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included because Rb lignosulfonate was not fully soluble inDMF at the used concentrations.An approximate way to study the average distance of

macromolecules in solution is to determine the Bragg distancefrom the position of a maximum. The distance betweenmacromolecules depends on their aggregation state. If macro-molecules cluster into bigger aggregates, the distance betweenthe aggregates is bigger than the distance between singlemacromolecules. As is apparent from Figures 4 and 5, it is notpossible to find a clear peak in all of the scattering curves. Toovercome this limitation, the intensities were multiplied by q,and via this approach it was possible to find a peak and itsposition on the q scale, but naturally we also shift the positionsof peaks to larger q values. The corresponding results aresummarized in Figure 6. Even though this characteristic length

should not be interpreted as a true average distance betweenthe macromolecules in solution, from this data we are able todeduce some trends and make conclusions about theaggregation of state the polyelectrolyte at each concentrationand in each solvent.In the methanol−water mixtures the dielectric constant is

decreasing slowly with increasing methanol content. DMF hasthe lowest dielectric constant of the studied solvent mixturesand it is aprotic, while methanol and water are both proticsolvents. Based upon analysis of the characteristic length

determined from the scattering curves, the dielectric constant ofthe solvent strongly affected the dispersion of lignosulfonatemolecules, water being the best solvent for both lignosulfonatesalts. As expected, solubility of Rb lignosulfonate was worsethan that of Na lignosulfonate in all solvents.Based on the characteristic lengths in Figure 6, both Na and

Rb lignosulfonate are in a similar aggregation state at the lowestvolume fraction (ϕ = 0.01) in all water−methanol solventmixtures, but around ϕ = 0.04 differences appear. Ifmacromolecules stick together to form small clusters, thecharacteristic length and thus also the distance between theclusters is bigger than if the macromolecules do not formclusters. However, the distance also depends on the volumefraction of lignosulfonate and it should decrease with increasingvolume fraction if we assume that the macromolecules arehomogeneously dispersed in the solvent. By comparing datameasured at same volume fractions but different solvents, wesee for example that addition of methanol causes moreaggregation in case of Rb lignosulfonate than in the case ofNa lignosulfonate once the volume fraction of lignosulfonatesalt becomes higher than 0.04.The aggregation seen here can be either association of only a

few macromolecules into small clusters or sedimentation. Wedid not see significant decrease in the scattered intensity, so thefirst explanation seems more correct. Curiously, Na lignosul-fonate in DMF has a much larger characteristic lengthcompared to Na lignosulfonate in all the other solvents alreadyat very low concentrations of lignosulfonate. On the otherhand, at low concentrations of lignosulfonate all the H2O−MeOH solutions have the same characteristic lengths,indicating similar aggregation states for Rb and Na lignosulfo-nate in the studied H2O−MeOH solvents. The upturn in SAXSintensity curves at low q values is often interpreted as a sign ofsample aggregation. In the Rb lignosulfonate data, this upturn ismuch less pronounced than in the corresponding Nalignosulfonate data, even though there is an upturn in thescattering curves of Na lignosulfonate in all studied solventmixtures. The average distance between macromolecules issignificantly larger in DMF. This indicates that it is difficult toguess the extent of aggregation simply by looking at the upturnat low q values.Contrast variation in small-angle X-ray scattering can be

made either by ASAXS or by ion exchange method. While inASAXS the contrast variation is achieved by changing theincident photon energy, in the ion exchange method thecontrast variation is achieved by replacing the counterions ofthe polyelectrolyte by other ions with the same valence, forexample, by replacing Na with Rb. Figure 7 compares ASAXSand ion exchange contrast variation for the most concentratedsolutions in water and in water−methanol. Comparison of thescattering profiles obtained with the two methods is somewhatcomplex. In our comparison, we focus on the shape of theprofiles, where one can see that at high q the scattering profilesin water have a similar plateau. At intermediate q values theydemonstrate similar dependencies of q before a divergence atlow q. In contrast, the scattering profiles in 50 vol % MeOHdisplay similar features only at low q. At both intermediate andhigh q the behavior is different. At small q values the ionexhange and separated ASAXS curves are completely different,implying that the ion exchange method fails here to give us thetrue contrast variation and shows simply that there is adifference in the interactions between the lignosulfonatemacromolecules with different counterions. On the other

Figure 5. SAXS curves of different volume fractions of Nalignosulfonate (ϕ = 0.01, 0.05, 0.08, 0.11, 0.15, 0.18) in DMF,normalized by ϕ.

Figure 6. Fitted characteristic length as a function of Na and Rblignosulfonate volume fraction ϕ in DMF, in pure water (0 vol %MeOH), and in water−methanol mixtures with 25 and 50 vol % ofmethanol. The line shown in both graphs is a fit to the three lowestvolume fraction points of Na lignosulfonate in pure water and itfollows the law characteristic length ∝ ϕ−2.0.

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hand, at large q, which corresponds to interactions on very localscale, near the surface of the macromolecule, the differentcontrast variation methods give similar results, but only whenthe solvent is pure water.With increasing concentration of methanol or lignosulfonate

in solution we also see increasing similarity of the total intensityand the separated intensity. When the curves are more similar,rubidium is dispersed into the macromolecules morehomogeneously. This is demonstrated by eqs 2 and 3. Oneexplanation for the increasing similarity of separated and totalintensity with increase in methanol or lignosulfonateconcentration is that the counterions are more and moreembedded inside the macromolecules, as is expected forsystems where counterion condensation occurs, but this canhappen also due to aggregation of the macromolecules.The next step in understanding the lignosulfonate system in

different solvents is to fit the ASAXS data with a model and seehow the fitting parameters change with increasing methanolconcentration. After testing different models, we decided to usethe molar mass distribution obtained from GPC as the sizedistribution to keep the model as simple as possible. One axis ofthe ellipsoids (the thickness d) was kept constant for allellipsoids, while being a fitting parameter, so the two other axesof the ellipsoids were calculated from the molar mass that hadbeen weighted with fitting parameter KM, and by using theknowledge of the measured density of 2.44 g/cm3 forlignosulfonate. This approach leads to an axis ratio ν whichvaries with molar mass.Figure 8 shows an example of fitted ASAXS data. The

residual shows that this fit is not perfect and the model doesnot fully describe the system. However, the parameters derivedfrom the model have true physical meaning and they arereproducible. The scaled number size distribution, thecalculated axis ratio, and the Debye screening length for thisspecific fit are shown in Figure 9. Tables 2 and 3 show therelevant fitted parameters as a function of Rb lignosulfonateconcentration and methanol content in the solvent. The datafor each sample was fitted with different starting parameters 25times, with the whole procedure taking in total several monthson a normal desktop computer. The fitting routine did notconverge to any specific one result, so out of those 25 fits, weselected a number of fits that had χ2 below a certain threshold

and considered them as the best fits. The number of acceptedbest fits is given in column ″Fits″ in Table 3 and the average χ2

of those fits is given in Table 2. Therefore, the values given forthe parameters are the mean values obtained from averaging theresults from the best fits and the errors are the standarddeviations. Figure 10 collects some of the results together ingraphical form.From the fitted parameters we derived several parameters

which are given in Table 4. The mean axis ratio of the spheroidνn was calculated from the fitted thickness (d) for particleshaving molecular weight equal to MnKM, where Mn = 9100g/mol, and in the same way the effective mean charge numberzn was calculated from the effective surface charge σeff and thevolume of the average-sized particle. Correspondingly, theapproximate Debye screening length for average lignosulfonateparticle sizes 1/κn was calculated using eq 6 from the fittingparameters for molecular aggregates having the same molecularweight of KnMn.Using the method described by Pabit et al.31 we can also

determine the total number of Rb ions related to onelignosulfonate particle. The method requires knowledge ofthe concentration of lignosulfonate particles in the solution.The values reported in Table 4 were obtained by converting theform factor part P(q, E) (units 1/cm) of the fitted curves to

Figure 7. Comparison of contrast variation using ion exchange andASAXS for lignosulfonate (ϕ = 0.17) in pure water (0 vol % MeOH)and in a water−methanol mixture. The symbols give the separatedASAXS curves (I(q, E1) − I(q, E3)), while the black full lines are thecorresponding curves obtained from subtracting the SAXS signal of Nalignosulfonate from the SAXS signal of otherwise identical Rblignosulfonate solutions. The curves have been scaled for clarity.

Figure 8. Upper plot shows the SAXS intensity Itot(q) at E1 and theseparated intensity IΔ(q) for 17 vol % Rb lignosulfonate in water. Fulllines show the fits and dashed lines show just the form factor part ofthe fit without the Hayter−Penfold structure factor. The plot on thebottom shows the residuals (Kk[Ik(qi) − Ik,model(qi)]/sk(qi)) of the fitsof both curves. Here, Ktot = 1 and KΔ = 30.

Figure 9. Number size distribution N(r), axis ratio ν, and Debyescreening length (1/κ) as a function of the diameter of the particle Das derived from the fit in Figure 8.

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electron units (e2)

=I q EP q E M K

r cN( , )

( , )e

n M

e2

A (12)

where Mn is 9100 g/mol, KM is given in Table 2, re is theclassical electron radius (cm), c is the concentration oflignosulfonate in solution (g/cm3), and NA is the Avogadronumber. The normalization is only approximate because of thepolydisperse nature of the studied lignosulfonate. After the unitconversion, the form factors were extrapolated to q = 0 usingthe Guinier approximation for E = E1 and E = E3. Finally, thenumber of excess ions surrounding one lignosulfonate particlecan be calculated as31

=−

′ − ′N

I E I E

f E f E

(0, ) (0, )

( ) ( )ionse 1 e 3

1 3 (13)

where in our case we use the theoretical values f ′(E1) = −3.10and f ′(E3) = −5.53 given by the Cromer−Liberman calculationfor a single Rb atom in vacuum using the programHephaestus.32

■ DISCUSSION

To study the distribution of the sulfonate groups inlignosulfonate molecules in different solvents, we made contrastvariation SAXS from lignosulfonate salts using Na and Rbcounterions. Even though the method worked for Das et al.12

for dilute solutions of DNA, a similar approach failed with

Table 3. ASAXS Fit Results for Rb Lignosulfonate

vol % MeOH ϕ Kf ρr>a,LS (1010 cm−2) ρr>a,Rb (10

10 cm−2) ρr>a,Rb (1010 cm−2) fits

25 0.037 0.968 ± 0.004 5.1 ± 0.6 1.9 ± 0.2 0.9 ± 0.5 650 0.037 0.965 ± 0.006 5.2 ± 0.4 1.4 ± 0.1 2.2 ± 0.6 60 0.170 0.969 ± 0.003 4.4 ± 0.2 1.4 ± 0.2 0.5 ± 0.6 7*0 0.170 0.954 ± 0.004 3.8 ± 0.2 1.2 ± 0.2 0.07 ± 0.05 525 0.170 0.973 ± 0.001 3.0 ± 0.7 1.0 ± 0.2 1.7 ± 0.9 450 0.170 0.978 ± 0.002 3.2 ± 0.8 0.9 ± 0.4 2.1 ± 0.4 3

Figure 10. Some of the fitted parameters and those derived from them. Filled symbols represent results for lignosulfonate volume fraction of ϕ =0.17 while open symbols are for ϕ = 0.04. The * marks the data for the sample with ϕ = 0.17 that was measured 1/2 year later at a different ASAXSinstrument. The error bars show 1σ standard deviation. See text and Tables 2−4 for more details of the meaning of the parameters.

Table 4. The Average Aspect Ratio νn, Charge Number zn, the Total Number of Rb Ions in and around an Average-SizedLignosulfonate Particle Nions, and Approximate Debye Screening Length 1/κn for Rb Decorated Lignosulfonate Aggregatesa

vol % MeOH ϕ νn zn Nions 1/κn (nm)

25 0.037 0.57 ± 0.07 −6.5 ± 0.6 18.9 ± 1.0 1.03 ± 0.0250 0.037 0.42 ± 0.04 −3.7 ± 0.7 22 ± 2 1.01 ± 0.020 0.170 0.46 ± 0.02 −3.7 ± 0.3 15.8 ± 0.4 0.95 ± 0.01*0 0.170 0.38 ± 0.02 −4.0 ± 0.7 22.9 ± 1.2 0.97 ± 0.0125 0.170 0.38 ± 0.07 −2.2 ± 0.2 21.2 ± 0.4 1.20 ± 0.0150 0.170 0.19 ± 0.04 −1.8 ± 0.3 33.8 ± 1.1 1.81 ± 0.02

aDerived from the measured parameters given in Table 1 and the fitted parameters of Tables 2 and 3.

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semidilute solutions of lignosulfonate and quantitative analysiscould not be carried out. The most obvious difference betweenSAXS curves obtained for sodium and rubidium lignosulfonatesolutions is seen in the beginning of the SAXS curves. Eachcurve from sodium lignosulfonate solutions shows an upturn,while for all Rb-lignosulfonate solutions the upturn is eithermissing or is at smaller q values, where we are limited by theexperiment and therefore have no data.Recently, Qiu et al.33 proposed that sodium lignosulfonate

would form loose hollow micelles in water at lowconcentrations of lignosulfonate. Such big aggregates couldactually explain an upturn in the SAXS curves of Nalignosulfonate at the small q values. However, the upturn wasonly observed for Na lignosulfonate, and we have seen inFigure 6 that there is no difference in the aggregation states ofNa and Rb lignosulfonate in water and water−methanolmixtures at low lignosulfonate concentrations according to thecharacteristic length determination. The difference between thebehavior of Na and Rb ions may be attributed to be due to thelarger affinity of sodium ions to water, but it is not clear howthis could explain the upturn at the beginning of the scatteringcurves.The dielectric constant near a charged polyelectrolyte is

different from that of the bulk solvent. A lowered dielectricconstant is observed, for example, in the grooves of DNAmolecules.34 Here we have shown experimental data for thelocal dielectric constant εeff near the surface of a globularbranced polyelectrolyte at different polyelectrolyte andmethanol concentrations. Both εeff and the average molecularcharge is determined simultaneously even though parameter κin eq 6 is proportional to both parameters. This is facilitated bythe polydispersity of lignosulfonate, where each macromoleculehas different charge even if the charge density σeff is assumedconstant. εeff does not depend upon the size of the molecules.The fitted local dielectric constant εeff of lignosulfonate is lowerthan that expected for the measured water−methanol mixturesin bulk, which is in good agreement with previous observationsfor other molecules. The error of the fit is still so large, that nosolid conclusions can be drawn on what kind of trend εeffactually follows. Adding methanol to the solvent mixture seemsto decrease εeff and there are no big differences in εeff indifferent concentrations of lignosulfonate at the same methanolcontent. In simulations it may be possible to replace constantεeff with a distance-dependent function close to thepolyelectrolyte, but so far unphysical results have been obtainedwith the proposed models for spherical particles.35 Furtheranalysis is unfortunately beyond the scope of this study.However, it might be interesting to use the ASAXS method tostudy the variation in local dielectric constant as a function ofdistance from the particles with monodisperse and sphericalpolyelectrolyte particles.By measuring the SAXS intensities on absolute intensity

scale, we could fit also the scattering length densities oflignosulfonate and Rb in different solvents. The theoreticalscattering length density of water is 9.5 × 1010 cm−2 and ofmethanol 7.6 × 1010 cm−2. Because the scattering length densityof methanol is lower than that of water, we would expect thescattering contrast, i.e., the scattering length density oflignosulfonate (without Rb) compared to the solvent ρr<a,LS,to increase when methanol is added to the solution because thescattering length density of lignosulfonate is higher than that ofthe solvents. Such an increase is seen only for the lowconcentration of lignosulfonate, while at the high concentration

the opposite trend might be caused by formation of largeraggregates of the polyelectrolyte. When these aggregates arelarge enough, they will not be observed by SAXS, and thereforesome of the signal is missing.Our measurements show that the effective surface charge and

the charge number of lignosulfonate particles decreases withmethanol volume fraction as well as concentration oflignosulfonate. The thickness of the lignosulfonate macro-molecule is not observed to change dramatically although somedecrease with increasing methanol content is observed,indicating a change in the particle shape from rather sphericalto more flat. The average axial ratio νn decreases significantly ina similar manner as the effective surface charge. The decrease inνn and thus increasingly oblate shape of the polyelectrolytecolloids could be partially explained by increasing aggregationthat is happening only at the edges of the macromolecules andnot on the flat sides. Especially at the highest concentration oflignosulfonate and methanol, the parameter KM which is relatedto sizes of the lignosulfonate particles nearly doubles comparedto other concentrations, giving an aggregation number slightlylarger than two compared to the situation in the leastaggregated state. The same behavior is observed whenobserving the total number of Rb ions per lignosulfonatemacromolecule aggregate Nions as shown in Table 4. Thenumber of ions per macromolecule aggregate fits closely to thetheoretical value of 19 ions calculated based on the molecularweight of Rb and Rb lignosulfonate as explained in theExperimental section. The method of Pabit et al.31 is sensitiveto calibration of the instrument and this might be the reason forthe discrepancy of the measurement made at the differentASAXS instrument (marked with * in the tables and figures)from the rest of the series. The error reported in Table 4 doesnot account for systematic errors. However, the possibility thatthe discrepancy comes from time-dependent aggregationbehavior cannot be completely excluded. Figure 11 shows a

schematic presentation of the interpretation of the SAXS and

ASAXS results for Rb lignosulfonate. Na lignosulfonate behaves

Figure 11. Schematic presentation of Rb lignosulfonate (Rb LS)macromolecules in different conditions from 100 vol % water to 50 vol% MeOH solution and in DMF solvent. On the left side of the figure isschematically shown the orientation of one structural unit inside thelignosulfonate macromolecule.

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in a similar manner but the aggregation behavior is lesspronounced, and it does not, for example, precipitate in DMF.These results explain perfectly why sometimes macromoleculesof lignin derivatives are observed to have different aspect ratios,ranging from prolate to spherical and all the way oblate shapes.However, it must be noted that the results presented in ourstudy do not cover any dynamical aspects of associationbehavior of the lignosulfonate macromolecules and we can onlydescribe a sort of an average situation after a longer stabilizationtime.The first study to address the topic of association of lignin

macromolecules in solutions was that of Benko,36 whoproposed that lignosulfonic acid forms noncovalently bondedassociation colloids in solutions. Benko reported that inalcohols and DMSO solutions lignosufonic acid was lessassociated than in aqueous 0.1 N KCl. Later, Sarkanen et al.37

studied association in kraft and organosolv lignin solutions as afunction of time by measuring the molecular weight withdifferent methods and proposed a perpendicular disposition forthe aromatic rings with respect to the surface in alkali lignin.The results shown in the current study definitely support theidea that the lignosulfonate macromolecule has its aromaticrings oriented antiparallel to the flat colloid surface also whenfully solvated, a finding which corroborates the interpretation ofLuner and Kempf5 about monomolecular lignin layers spreadon liquid surfaces. In fact, molecular aggregation in alkali ligninhas been proposed to be caused by π−π interaction betweenaromatic rings. In a recent study by Deng et al.38 aboutaggregation of alkali lignin in THF, the π−π interactionbetween aromatic rings of lignin was adjusted with iodine.Their estimate is that about 18 mol % of the aromatic groups inalkaline lignin form π−π aggregates and that both inter- andintramolecular π−π aggregates exist in alkali lignin solutions.Similar π−π interactions are possible between the aromaticrings in lignosulfonate. With the aromatic rings antiparallel tothe surface of the molecular aggregate, the charged −OH,−COOH, and −SO3

− groups have a chance to be located at theflat surfaces of the colloid. The localization of charges on theflat surfaces would promote aggregation at the edges of thecolloids where the charge is lower, thus explaining the observedresult where the thickness of the aggregate remains unchangedwhile the other dimensions grow.

■ CONCLUSIONSWe developed a model and an analytical formula for ASAXS toquantitatively fit many local parameters relevant to colloidalpolyelectrolyte solutions. With this method, the effectivesurface charge of the macromolecules was determined veryprecisely in different solvents and different concentrations ofpolyelectrolyte. With the help of the element specific ASAXSmethod we have shown here that low-molecular-weightlignosulfonate does not only form association colloids indifferent solvent mixtures but that the degree of association alsochanges as a function of lignosulfonate concentration. Theaggregation state of lignosulfonate was found to be verysensitive to the counterion (Na or Rb), to the concentration oflignosulfonate, as well as to the composition and dielectricconstant of the solvent. For Na lignosulfonate someaggregation was seen in water−methanol mixtures, but it wasnot as strong as that of Rb lignosulfonate due to the greateraffinity of the Na ion to water. In water−methanol mixtures,when the dielectric constant of the solvent is decreased, the flatpolyelectrolyte colloids prefer to orient and attach to each other

in such a way that the resulting aggregates remain flat. Thethickness of the flat polyelectrolyte macromolecules and theiraggregates was determined to be about 1−1.4 nm. Based on theresults presented here and previous studies on different lignins,we may conclude that charged groups of lignosulfonate arelocated mostly at the flat sides of the lignosulfonatemacromolecules.

■ ASSOCIATED CONTENT

*S Supporting InformationMatlab code used for calculating the RMSA structure factorthat had been corrected for the penetrating background. Thismaterial is available free of charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: ulla.vainio@desy.de. Phone: +49-40-8998 5474. Fax:+49-40-8998 2787.

■ ACKNOWLEDGMENTSWe thank BESSY for providing the access to the 7T-MPWSAXS instrument and BMBF for financial support in the travelexpenses, DESY for beamtime at the B1 instrument andBorregaard Lignotech for providing the samples.

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