harry reynaerslight scattering study of polyelectrolyte ...chemistry, laboratory of macromolecular...

88 FIBRES & TEXTILES in Eastern Europe January/December 2003, Vo.11. No. 5 (44)

1. IntroductionPolyelectrolytes are long-chain moleculescarrying a large number of ionisablegroups. For every charged group on themacromolecule, there is, somewhere inthe vicinity, a small mobile counter-ionof opposite charge. A polyelectrolytesolution may, and generally does, alsocontain other low-molar mass electro-lytes, which ionise to give small ions.The small ions that carry a charge of thesame sign as the �macro�-ion are calledco-ions [1,2].

Biopolymeric polyelectrolytes, in con-trast to synthetic polyelectrolytes, are asubstantial part of the components ofliving cells. Such polyelectrolytes mayexist in solution either in a linear openconformation, as do many nucleic acidsfor example, or in a tertiary structure,as do proteins. Apart from these twowell-known groups of charged biopoly-mers, there is another very importantclass of naturally occurring polyelec-trolytes, namely acidic polysaccha-rides, such as cellulose sulphate,hyaluronan, carrageenans, alginates,xanthan, etc. [2].

The physical-chemical polyelectrolytebehaviour has been the subject of manyinvestigations [3]. Their propertiesstrongly differ from simple electrolytes,and a great many theoretical and ex-perimental studies have been per-formed in order to understand theirbehaviour in solutions, as they playsuch an important role in nature. Thepresent work focuses on the associatingbehaviour of a polyelectrolytic polysac-charide: more specifically, iota- (ι-) car-rageenan has been investigated bymeans of light scattering studies to elu-cidate its conformation-induced net-work-forming abilities.

Apart from the relevance of any funda-mental study of the molecularity ofbiopolymeric polyelectrolytes, the po-tentials of biopolymeric engineering, asclarified in Figure 1 stress the importanceof studies of the physical-chemical

Light scattering study of polyelectrolytepolysaccharides - the carrageenans

Harry ReynaersCatholic University of Leuven, Department of

Chemistry, Laboratory of MacromolecularStructural Chemistry,

Celestijnenlaan 200F, 3001 Heverlee, Belgium;e-mail: [email protected]

Abstract

The results of light scattering investigations on the conformational/associative properties of ι-carrag-eenan in aqueous solution are presented. The effect of different salts (NaCl, NaI, LiCl) has been inves-tigated. The experimental data has been treated according to the Open Association Model by Elias.The results fit into the current model for the intermolecular association (i.e. the lateral junction ofsingle-helical stretches) of the polysaccharide. The importance of preparative procedures in order toobtain reproducible light scattering data from aqueous solutions of (κ- and ι-) carrageenan solutionsis discussed.

Key words: polyelectrolytes, carrageenans, light scattering, association

behaviour of polysaccharide polyelec-trolytes in solution [4]. The present workintends to contribute to the better under-standing of gelling polysaccharides.Both in biological and artificial systems,a gel appears to be a most common statefor polysaccharides. The polymer chainsform a three-dimensional interconnectednetwork immobilising a large quantityof solvent (water) with or without otherwater-soluble components. Apart fromtheir biological function, polysaccharidegels have widespread commercial usesin the food, cosmetic, pharmaceuticaland medical industries [5].

1.1. ApplicationsLow costs and high quality are commer-cially important. Phycocolloids such asagar, carrageenans and alginates areespecially good at fulfilling these con-ditions. Therefore they are widespreadin the food industry, as they are in ad-dition non-toxic and have unique rheo-logical properties [6]. Their viscosity, gelstrength and stability in aqueous mix-tures, solutions and emulsions makethem common additives in �modern�food such as ice-cream, pudding,sauces, dressings, canned food, syrups,jams, artificially whipped toppings, etc.The special ability of carrageenans tointeract with and stabilise milk proteinsmeans they are widely used in dairyproducts. Because they are well toler-ated by the human organism, they canbe safely used in skin applications.Therefore they are also employed in thecosmetic and pharmaceutical industry,where they are used as emulsifiers,stabilisers and thickeners in creams, lo-tions, dentifrice, shampoos, soaps, etc.

Their softening and refreshing proper-ties, due to the evaporation of the waterthey incorporate, are also valued prop-erties [4]. The mechanism of gel forma-tion related to the behaviour of thephycocolloids upon heating and cooling,pH changes and the addition of monova-lent and divalent cations, differentiatetheir applications in various products.

1.2. CarrageenansCarrageenans are sulphated linearpolysaccharides extracted from red al-gae of the class Rhodophyceae [7]. Carra-geenans have many different functionsin seaweed. For instance, they play arole as a cation-exchange barrier and asa sea-water reservoir to prevent desic-cation. Moreover, they also supply themechanical protection necessary for cellsto withstand the force of the waves.

Genetic information

↓Biosynthesis

↓Primary Structure

↓Conformation

↓Physical Properties

↓ ↓Biologicalfunctions

Technical uses

Figure 1. The localisation of the importance ofconformational and physical studies in biopoly-mer engineering.

STRU

CTUR

E-PR

OPER

TIES

INVE

STIG

ATIO

NS O

F PO

LYM

ERS

BY X

-RAY

SCA

TTER

ING

AND

RELA

TED

MET

HODS

FIBRES & TEXTILES in Eastern Europe January/December 2003, Vo.11. No. 5 (44) 89

On the basis of their chemical composi-tion, the carrageenans can be differenti-ated into three major groups, kappa- (κ-)carrageenan, iota- (ι-) carrageenan andlambda- (λ-) carrageenan. Present andformer studies have been mainly re-stricted to κ- and ι-carrageenan, whichdiffer only in their sulphate content:κ-carrageenan is formed of alternatingβ-(1-3)-D-galactose 4-sulphate andα-(1-4) 3,6-anhydro-D-galactose units,(see Figure 2a), and ι-carrageenan of al-ternating β-(1-3)-D-galactose 4-sulphateand α-(1-4)-3,6-anhydro- D-galactose-2-sulphate units (see Figure 2b).

One of their basic properties is the abilityto induce thickening or gelation of solu-tions. The most characteristic features arerelated to their ability to form ionotropicgels, in particular in the presence of alka-line metal counter-ions (K+, Rb+, Cs+). Thisprocess appears to be thermoreversiblewith or without hysteresis, and has beenthe object of numerous studies [8-10].Detailed molecular interpretation of thegelation mechanism has, however, beencontroversial for a long time.

It is now universally accepted that in thefirst step of the gelation mechanism ofcarrageenans at moderate polymer con-centration, a thermoreversible, ion-in-duced conformational transition takesplace from a disordered (coil) to an or-dered (helix) form. This is followed byaggregation and network formationwhen the polymer concentration and/or

the ionic strength are increased and/orthe temperature is decreased. The dis-agreement between the research teamsinvolved is at the level of the nature ofthe disorder-to-order transition and thedegree of helix formation. Several inves-tigators favour a coaxial double helix asthe fundamental ordered conformation[11-15]: upon reducing the temperatureor increasing the salt concentration,double-helical stretches will be formed ina thermoreversible process. Afterwards,these double helices will pair and formthe physical junctions responsible for thegelling behaviour of carrageenans (seeFigure 3). Other groups of researcherspresent evidence of a single helix as thefundamental ordered state of the carrag-eenans prior to association and gel for-mation. In the latter context, it is acceptedthat single-helical regions along the poly-mer chain associate into larger stretchesupon increasing salt concentration and/or polymer concentration and/or de-creasing temperature without levelling ofat a special integral number, such as twofor a double and three for a triple helix[16-22] (see Figure 4).

It is well-known that carrageenan, es-pecially ι-carrageenan, easily under-goes association even in low salt con-centrations and at moderate tempera-tures. Therefore the light scattering datahas been analysed within the frame ofthe Open Association Model of Elias,which has already been shown in thepast to be the best model available with

respect to the type of association, i.e. openor closed [23,24].

A comprehensive study of the literatureon κ-carrageenan clarifies the two mod-els and the ongoing debate. However, therecent findings of three different groupsstrongly support the single helical strandas the candidate for association into largersegments. Moreover, it was shown thatthe Open Association Model of Eliascould be applied to describe the associa-tion process. In order to probe whetherthe latter approach has more general va-lidity, a research project was started onthe polyelectrolyte behaviour of iota- (ι-)carrageenan in saline solutions.

2. Light scattering study ofιιιιι- carrageenan in salinesolutions

Wide-angle laser light scattering is usedto explore the influence of both ionicstrength and temperature on the weight-average molar mass, on the polymer-solvent interactions, and on the chaindimensions of ι-carrageenan in dilutedsalt solutions. For more information onthe static light scattering technique, thereader is referred to classical textbooks,where the basic principles are explained.It is well known that careful and accu-rate sample preparation plays a crucialrole in performing reproducible and re-liable light scattering measurements[23,25]. Particularly in the case of

a)

O O

OSO3-

OO

O

OH

CH2OHOSO3

-b)

O

O

OH

CH2OHOSO3

-

O

OH

O

O

Figure 2. The chemical structure of a) κ-carrageenan and b) ι-carrageenan co-polymeric repeating unit.

Figure 4. Gelation model for carrageenans according to Smidsrod [17,18]Figure 3. Gelation model for carrageenans according to Rees [5]


ι-carrageenan, it has been found that(depending on the preparation method)there is a time dependence on the inten-sity measurements, which strongly af-fects the molecular parameters obtainedfrom light scattering [23]. Hence, an ex-act description of the experimental con-ditions of the light scattering measure-ments on polysaccharides is necessary.The present light scattering study onnative ι-carrageenan samples shows thatunder all experimental conditions, i.e.during the conformational transitionfrom the disordered to the ordered form,intermolecular association is induced asindicated by an increase of M w. At suf-ficiently high salt concentrations anddepending on the nature of the addedsalt, a concentration-dependent upwardbending in the curve of the inverse ofthe reduced Rayleigh ratio at zero-angleis clearly observed. This behaviour isthermoreversible. The upward bendingof the zero-angle light scattering data isespecially striking in chloride-contain-ing solutions, whereas the presence ofiodide ions seems to partially impede theassociation. At low polymer concentra-tions, both in NaCl, NaI and LiCl, theoccurrence of the ordered state inι-carrageenan is not accompanied bychain dimerisation. Hence the possibil-ity that the fundamental ordered confor-mation of ι-carrageenan is a doublestranded helix can a priori be ruled out.

A minimum in the zero-angle scatteringintensity as a function of the polymer con-centration was already reported in thelight scattering study on the dilute andsemi-dilute solutions of, e.g., poly(γ-benzyl-L-glutamate), PBLG, in 1,2-dichloroethane by Wissenburg et al [26].This behaviour was attributed to an ag-gregation effect of the semi-flexiblePBLG chains. Upon increasing the tem-perature from 25°C to 50°C, the upwardcurvature became less pronounced. Thishas been ascribed to the universalbehaviour of the PBLG chains in vari-ous solvents (DMF, 1,2-dichloroethane).

Here we focus on interpreting the asso-ciate behaviour of the experimental lightscattering data at zero-angle in the frameof the Open Association Model of Elias[27]. Particular attention is paid to themost fundamental molecular parameter,i.e. the weight-average molar mass, M w,because it is felt that a sound assessmentof this fundamental property is a man-datory prerequisite for any furtheranalysis, like that of the angular depen-dence of the scattering function or thestudy of dynamic light scattering.

The study of macromolecular associa-tion was performed by analysing the

concentration dependence of the scatter-ing at zero angle (extrapolated WALLSdata). Hence the second part of this workis restricted to interpretation of the an-gular dependence of the data. In orderto achieve this goal, a non-trivial andself-consistent method is developed indetail, and the obtained knowledge onthe weight average molar mass and thesecond virial coefficient under associat-ing conditions is used to calculate theradius of gyration and the persistencelength of ι-carrageenan.

2.1. Concentration dependence ofassociation phenomena

2.1.1. Sample preparation and handling

Polysaccharide solutions are particularlydifficult to handle due to their high vis-cosity and their tendency to form en-tanglements, aggregates and other formsof interchain association [28]. Moreover,it is common knowledge that themethod for preparing polysaccharidesolutions is very critical: the way inwhich the solutions are prepared, the fil-tration procedure, the thermal treatmentand the procedure of cleaning thesample cells may strongly affect the ul-timate values of the molecular param-eters obtained [29,30]. Since the lightscattering method is, by its own nature,very sensitive to the quality of the solu-tions, the presence of small amounts ofdust or aggregates can strongly influencethe calculated values of the weight av-erage molar mass. Therefore it is neces-sary to achieve �safe� protocols and toreport all the experimental details con-cerning the sample preparation, in or-der to guarantee the reproducibility ofthe measurements and to allow for com-parison with data obtained by others.

When κ-carrageenan solutions [31] areprepared, starting from either an aque-ous κ-carrageenan solution of knownconcentration or from a weighed amountof lyophilised κ-carrageenan sample dis-solved in pure water, reliable light scat-tering data are obtained. In both casesthe correct amount of salt is then addedto obtain the desired molarity of the fi-nal solutions. Light scattering measure-ments reveal poor reproducibility and ahigher molar mass when freeze-driedκ-carrageenan is dissolved directly intoan aqueous salt solution [15].

If light scattering measurements are car-ried out starting from given salt solu-tions to which an aqueous ι-carrageenansolution of known concentration isadded, and a dilution series is obtainedby addition of increasing amounts of the

salt solutions, unusual Zimm-plots areobtained [23]. There appears to be a gen-eral tendency towards an increase in themolar mass as a function of increasing saltconcentration, and all the experimentalobservations point to an irreversible as-sociation-aggregation effect as mani-fested by unusual Zimm-plots and bythe presence of time-effects [23,32]. Fil-ter clogging was frequently observed,and the �routine� laboratory experimentsresult in non-reproducible molar massvalues. Moreover, the importance of theseresults is fully manifested only if a largenumber of experiments are performed,since one single occasional experimentcan yield apparently reliable data.

When starting from an aqueous carrag-eenan solution of known concentration,to which a fixed amount of a salt is slowlyadded while stirring and filling up thevolumetric flask with pure water to thegraduation mark in order to obtain thedesired polymer and salt concentrations,good results are obtained only if the so-lutions are stored for 24 hours at roomtemperature before measurement. Thereason why the solutions are stored for24 hours before measuring becomes clearas a rapid increase of the light scatteringintensity is observed during the first twohours after the filtration, followed by aslower increase, up to constant valuesafter approximately 5 hours. Such find-ings point to a non-equilibrium state ofthe ι-carrageenan solutions at higher saltconcentrations immediately after prepa-ration. The perturbing effect of the shearduring the filtration step must certainlybe taken seriously into account. Anotherpoint of interest is the fact that in the pres-ence of NaI the effect is less pronounced,which illustrates the influence of the an-ion in the process.

Therefore experiments are always per-formed on ι-carrageenan solutions inthermodynamic equilibrium, i.e. 24hours after following this �safe� prepa-ration protocol.

2.1.2. Application of the OpenAssociation Model of Elias

To obtain the reduced scattered intensityat zero angles, KCp/R0, from wide-anglelaser light scattering data, the reducedscattered intensity values, KCp/Rθ, are fit-ted with a function of sin2(θ/2). K is theoptical constant, R0 the Rayleigh ratio atangle zero and Cp the polymer concen-tration. The intercepts then give the val-ues of KCp/R0. For all cases, a polyno-mial function of the second order, i.e.F(sin2(θ/2)) = B0 + B1sin2(θ/2) +B2(sin2(θ/2))2, yields almost the same re-sults as the linear one, and the B2 values


are always very low, and therefore neg-ligible. So, in order to obtain KCp/R0,preference is given to the linear extrapo-lation procedure.

At higher salt concentrations and lowertemperatures, chain association is ob-served: the light-scattering concentrationdependence passes through a minimum.The trend of the KCp/R0 versus Cp plotsis markedly curved in a very asymmet-ric way, and much more so than one ex-pects for associating systems. Hence thefamiliar classical light scattering equa-tion for non-associating systems gives avery poor fit for these systems, as isclearly evidenced from the results in ref-erence [24].

Some improvement is obtained by us-ing a virial expansion including a thirdterm: Eq 1.

Here A2 and A3 are respectively the sec-ond and third virial coefficient, and( M w)0 stands for the weight-averagemolar mass of the fundamental molecu-lar form undergoing association. Thecurve obtained by this quadratic fit hasa lower value of χ2, which points to abetter fit. A negative value of the sec-ond virial coefficient, A2, is expected foran associating system. Nevertheless, theintrinsic symmetry of Equation 1 doesnot seem to provide the best shape forfitting the experimental data. Empiri-cal hyperbolic functions of the type ofEquations 2 and 3 appear to be moreadequate, as shown by their fittingcurves in reference [24]. Eq 2., Eq 3.

whereα and β are empirical fitting constants.The progressive improvement of the fit-tings on passing from the classical lightscattering equation at zero angle toEquation 2 and Equation 3, as indicatedby a decrease in χ2, is accompanied by adecrease ( M w)0. It is anyway evident thata linear function for the treatment of thezero-angle intensity data erroneouslyoverestimates the value of ( M w)0 for ι-carrageenan in associating conditions.

The general equation for light scatter-ing of associating systems in solutionis mostly written as follows: Eq 4.

where( M w)app is the apparent weight-averagemolar mass (which is a function of Cp)and A2,0(Cp), the second virial coefficientof the associating system, which could bea function of ( M w)app, and therefore of Cp.

Associating systems can be analysed byvarious physical models. The onewhich by general consensus seems to

give the best results is the so-called�Open Association Model� as describedby Elias [27]. In this model, a continu-ous distribution of the associated formsof the basic non-associated form, the�unimer�, into dimers, trimers, tetram-ers etc., is assumed. Each step is de-scribed by an equilibrium constant, K1,K2, K3, K4,�, and all steps of the openassociation are assumed to be thermo-dynamically equivalent, i.e. they all haveidentical equilibrium constants, K.

Since an intermolecular association pro-cess is always accompanied by a changeof particle mass, association can bestudied by means of the concentrationdependence of ( M w)app. Elias [27] de-fines ( M w)app as the molar mass calcu-lated from experimental data at finiteconcentrations using an equation validfor infinite dilution only. The initial de-crease observed of (( M w)app)-1 with in-creasing Cp is due to an increase in thedegree of association. For higher valuesof Cp, the curve increases again to theprevailing effect of the second virialcoefficient. The expression for the ap-parent weight-average molar mass ac-cording to Elias is given by Equation 5.

Here ( M w)0/( M n)0 is the weight-averagemolar mass over the number-averagemolar mass of the non-associating sys-tem, or, in other words, the polydisper-sity of the system, and Kapp is the appar-ent association constant. In the presentwork, it was conservatively decided toconsider it as an �apparent� associationconstant, in as much as it will be treatedas a phenomenological parameter of ther-modynamic nature whose absolute value

and whose dependence on temperature,ionic strength and other physical-chemi-cal parameters will not be calculated apriori on the basis of any particular model.By substitution of ( M w)app into Equa-tion 4, one obtains: Eq 6.

The analytical form of Equation 6 is thesame as that of Equation 3. AlthoughEquation 6 is a three-parameter equation,namely of ( M w)0, Kapp and A2(Cp) whenthe same polymer sample is used in bothnon-associating and associating condi-tions, the value of (Mw)0 must remain con-sistently the same for all cases. Therefore,when ( M w)0 is fixed to the value of thenon-associating form, Equation 6 canthen be simplified to a two-parameterequation, i.e. of Kapp and A2(Cp).

2.1.3. Determination of (M w)0

As it has been clearly pointed out in theprevious paragraphs that ι-carrageenanundergoes interchain association, it isvery important to obtain a sound estimateof ( M w)0. The series of experiments per-formed with Na+ as a counter-ion at con-stant temperature (T=25°C) and variablesalt concentrations, Cs, are consideredfirst. To introduce even the slightest formof bias into the data treatment, a qua-dratic-type of fitting function is chosenfor fitting the scattered data for Cs valueslarger than 0.06 M. At this concentrationthe major deviation from linearitybegins,even if the hyperbolic-types have beendemonstrated to be much better. In allother cases a linear extrapolation is used.The series of M w values as a function ofCs obtained in this way can be best de-scribed by a linear correlation with Cs, i.e.

(1)( )

22 3

0 0

12 3p

p p

W

KCA C A C

R M= + +

( ) 20 0

12p

p

W p

KCA C

R M Cα= +

+

( )21

2 20

0

12p

p

W p

KCA C

RM Cβ

= + +

( ) ( )1

2,00

2pW p papp

KCM A C C

R

−= +

( ) ( ) ( )( ) ( )0

0 00

4000W

W W app W pappn

MM M K M C

M= +

( ) ( )( )21

2 20

0 0

12

4000 . .

pp p

W app W p

KCA C C

RM K P I M C

= + +

(2)

(3)

(4)

(5)

(6)

Equation 1-6


M w,(Cs) = M w,(Cs=0) + a1.Cs, with a positive

slope, a1=0.54; this is a further indicationthat a concentration-dependent associa-tion indeed occurs. (See Figure 5.) Thevalue of M w,(Cs=0) = 203 000 ± 18 000 is areasonable estimate for the upper bound-ary of ( M w)0 in Na+-containing solutions.

The results of applying the same proce-dure to the data obtained, as well as theweight-average molar mass of Na+-ι-car-rageenan in NaCl 0.08 M as a functionof temperature are also reported in Fig-ure 5. In this case, a quadratic fit is usedfor temperature values lower than 35°C.The linear correlation, as can be seen inthe salt concentration series, is absent. In-stead, if the values of the two lowest tem-perature measurements are left out, theseries can be fitted nicely with a polyno-mial fit resulting levelling off at the samevalue of 200 000 as found in the salt-con-centration dependent series.

By comparing the two sets of indepen-dent data, it can be concluded that avalue of 200 000 for ( M w)0 of the Na+-ι-carrageenan form can be very safely as-sumed for subsequent calculations.

For the Li+-salt, similar results are ob-tained: a linear correlation is foundwith the value of the interceptM w,(Cs=0)=197 000 ± 31 000 and the slopea1=0.53. Having taken ( M w)0 =200 000for the Na+-salt form, with some logicalpriority and by straightforward applica-tion of stoichiometric calculations to takeinto account the difference in atomicmass between the (two/repeating unit)Na+-and the Li+-ions, one obtains for Li+-ι-carrageenan ( M w)0=187 000. This valuelies well within the experimental errorof the above-cited upper bound of

197 000, and it will therefore be used inthe following calculations.

2.1.4. Concentration dependence of A2

Interchain association might in principlebe accompanied by changes in chain di-mensions, and an effect on the values ofthe second virial coefficient cannot apriori be excluded. Three different casesare analysed and compared. In the firstcase, no concentration dependence onthe second virial coefficient is taken intoaccount for ι-carrageenan, i.e.:

( )2 2pA C A= (7)

A linear dependence is considered as asecond possibility: Eq. 8.

In the third case, the logarithmic depen-dence as found for κ-carrageenan [36]is written as: Eq. 9.

Kapp, A2, or A2,0 and B2, are calculated forthe data of ι-carrageenan in 0.08M NaCland for those in 0.09M NaCl both at25°C with Equation 6, using the threedifferent concentration dependencyforms of A2. The molar mass is alwaysfixed at 200 000 and the polydispersityis 2. The results are given in Table 1.

It can be seen that there is no clear-cutdependence of A2 on ( M w)app in theinvestigated Cp regime. In fact, in thecases where a correction is used, theerrors and the chi-square factors# χ2

become larger, indicating that the fitting

Figure 5. The weight-average molar mass of Na+-ι-carrageenan in 0.08 M NaCl as a function of the temperature (solid squares), and as a function of NaCl,(open squares) and NaI (solid circles), concentration at 25°.

( ) ( )( )( )2 2,0 2 , 01 /p W app WA C A B M M= − −

( ) ( )( )2 2,0 2 , 0log /p W app WA C A B M M= −

Equation 8,9

(8)

(9)

Cs (M) Dependence Kapp ⋅⋅ 10-5 A2,0⋅⋅103,

mL⋅⋅mol⋅⋅g-2 B2⋅⋅103,

mL⋅⋅mol⋅⋅g-2 χχ² ⋅⋅ 1014

Constant, Equation 7

6.24 ± 0.68 2.95 ±0.20 -- 1.416

Linear, Equation 8 6.89 ± 2.37 3.58 ± 2.03 0.24 ± 0.72 1.830

0.08

Logarithmic, Equation 9 7.09 ± 3.11 4.02 ± 3.64 1.84 ± 5.98 1.832

Constant, Equation 7 4.89 ± 0.24 2.36 ±0.90 -- 0.264

Linear, Equation 8 4.43 ± 0.69 1.75 ± 0.89 -0.30 ± 0.47 0.305

0.09

Logarithmic, Equation 9 4.34 ± 0.86 1.43 ± 1.44 -1.88 ± 3.06 0.311

Table 1. Dependence of the second virial coefficient on polymer concentration for ι-carrageenan inaqueous NaCl solutions: comparison between different concentration dependency forms in the OpenAssociation Model of Elias.

# The standard way of defining the best fit is to choose parameters so that the sum of the squares of the deviations of the theoretical curve(s) from the experimentalpoints for a range of independent variables, χ², is at its minimum.


is less good. There is no need to intro-duce any specific correction for the sec-ond virial coefficient dependence on( M w)app, and hence on Cp, without af-fecting the reliability of the derived pa-rameters.

2.2. Angular dependence andconformational properties

2.2.1. Theoretical approach

The tedium involved in collectingmulti-angle data and the often hazard-ous extrapolation in the case of non-lin-ear Zimm-plots, as is the case for asso-ciating systems, are probably the rootcause of the scarce data sets which canbe retraced in the current literature. Thisis especially true for ι-carrageenan asoriginally shown by Vanneste [23]. It is,however, worthwhile and rewarding toconsider the information content of theform factor, Pθ, as it enables the extrac-tion of the z-average root mean squareradius of gyration, (<Rg2>z)1/2, an im-portant parameter related to the aver-age shape of the macromolecules in so-lutions as given by Equation 11,

where:λ is the wavelength of light and θ thescattering angle.

Whether or not the second virial coeffi-cient is affected by an angular depen-dence contribution has long been andstill is a matter of debate. Different equa-tions have been proposed to introduce

Pθ in the angular independent light scat-tering equation. The two most simpleand widely used forms are: Eq. 12, 13

Equation 12 represents the classical ap-proach, whereas in Equation 13 the in-termolecular effects are related to ther-modynamic properties. In betweenthese two extreme cases, several othercases have been proposed and dis-cussed [10,33,34]. It is known from thisliterature data that intermolecular inter-actions more readily affect A2 than themolecular dimensions as reflected by<Rg2>z. The theoretically derived equa-tions, however, are rather complex anddifficult to use in practice [10].

For the present purpose, the angulardependent light scattering data hasbeen initially fitted to derive the radiusof gyration with several empirical func-tions. From preliminary calculations, itis found that when no thermodynamicinteractions are accounted for in theangular dependence of the light scat-tering data, inconsistencies arise in fit-ting wide-angle laser light scatteringdata in the coil-to-helix transition re-gion by using Equation 12. There is con-siderable scatter in the values of the re-sulting radii of gyration, and stronglydeviating results appear under the ex-perimental conditions at which the con-formational transition takes place. Pθ,measured as a function of scatteringangle and polymer concentration, canalso be said to depend on both vari-ables, as stated in Equation 13. Never-theless, in treating the light scatteringdata of macromolecular solutions

showing some degree of association, thestraightforward use of the latter Equa-tion 12 also leads to incorrect values of<Rg0

2>z in the coil-to-helix transition.Hence when analysing the angular scat-tering data in the coil-to-helix transitionregion, it seems that one has to accountin some way for thermodynamic inter-actions. A best set of fitted results is ob-tained using the following empiricalfunction, essentially based on the prin-ciples of the Open Association Model ofElias [27]. Eq. 14.

with ( M w)0 being the molar mass of thenon-associating species, i.e. 200 000, andM w,app the apparent molar mass, T1 andT2 are thermodynamic parameters re-lated to intermolecular interactions. Theparameter a1 in the power of the ratioof the apparent molar mass over themolar mass of the non-associating sys-tem accounts for the molar mass depen-dence of the radius of gyration for as-sociating systems.

Hydrodynamic and conformational pa-rameters of similar macromolecules, dif-fering only in their molar mass, are of-ten related to Ma [35]. The value of �a�depends on the conformational charac-teristics of the polymer. The slope of adouble logarithmic plot of the radius ofgyration versus the weight average mo-lar mass gives for a1 the theoretical value1 for rod-like molecules, 0.6 for flexiblecoils in good solvents and 0.33 forspheres [1]. Due to the high number ofparameters in Equation 14, it was pre-ferred not to use a1 as a completely in-dependent parameter, but rather to fixit. In fact, it was found that the effect ofa1 on the fitting and, in particular, on thevalue of Rg0, varied between the follow-ing extremes. For low extent of associa-tion (e.g. at low I) there was no effect of a1

on the quality of the fitting, as monitoredby the sum of the squares of the residu-als, χ2, and on Rg0. For high values of as-sociation, a slight dependence of χ2 andof Rg0 on a1 was noticed. In particular, Rg0

slightly decreased upon increasing a1

from 0.33 to 0.85. However, the accom-panying decrease of χ2 was small enoughto justify the assumption of a constantvalue of a1 for all values of I. In the ab-sence of additional independent infor-mation, the simplest theoretical choicewas obviously that of the sphere. In ad-dition, the latter model seems to be themost plausible one for the intermolecu-lar associate, highly branched and likelyto be star-shaped because of the highelectrostatic repulsion.

Another important parameter in thecharacterisation of a polymer is thepersistence length Lp, which characterises

( )2sin3

161lim

222

21

0

θλπ

θθ z

RgP +=−

→

20

1 12p

pW

KCA C

R P Mθ

= +

( ) ( )( )

( ) ( )

+

⋅

−−+=

2sin3

161

2sin2sin121

2

2

0

,202

2

2212

,

1

θλπ

θθθ

a

W

appW

z

pappW

p

M

MRg

TTCAMR

KC

20

12p

pW

KCA C

R M Pθ

= +

−+−=

bN

L1

bN

L21

L3

bNLRg

n

p

w

p

p

z2pz

20

(11)

(12)

(13)

(14)

(15)

Equation 11-15


the flexibility of the chain. The persistencelength is defined as the integral of theaverage projections of chain elements ofthe infinitely long chain on its initial di-rection. From the radius of gyration, thepersistence length of a wormlike chaincan be calculated using the equation ofBenoit & Doty [36] as modified for poly-disperse samples [37] (see Equation 15).

Here N z, N w and N n are the averagenumber of chain elements, with the in-dex indicating the appropriate kind of av-erage and b is the monomeric projectionlength. Again assuming a Schulz-Zimmdistribution (as indicated by previousanalyses of the sample) [23], the persis-tence length Lp can be calculated. For b,

the following position was used:b(x)=bc(xc)+bh(1-xc), where bc and bh were1.09 nm and 0.867 nm, and the indexes cand h refer to �coil� and �helix� respec-tively. xc is the molar fraction of segmentsin the disordered (�coil�) conformation, asgiven by optical activity measurements.

2.2.2. Analyses of the conformationalparameters in differentexperimental conditions

Wide-angle laser light scattering experi-ments are performed in NaCl, NaI andLiCl solutions at different salt concen-trations, from 0.02 M to 0.10 M. In thisconcentration range, a conformational

transition takes place in all studied saltsfrom a disordered (coil) to an ordered(helix) conformation, as indicated byoptical activity measurements. The se-ries in LiCl has been repeated after a longtime interval to test the reproducibilityof the measurements [10,24]. The angu-lar dependence of ι-carrageenan solu-tions is also studied using a constantmolarity of salt, 0.08M NaCl, as a func-tion of temperature.

In Table 2 and Table 3, (<Rg02>z)1/2 is

given for NaCl and NaI, and for LiClrespectively as calculated from Equation14 using a Levenberg-Marquardtoptimisation in the non-linear leastsquares fit. M w, Kapp and A2 are used asconstants, obtained as described in part2.1 of this work. The error on thez-average mean square root radius ofgyration of the non-associated form isalways rather small. The values of T1 arein most cases between 0 and 1, and T2

oscillates around 0 (data not shown here).

In all four cases there is a clear distinc-tion between the radii of gyration above0.05M and below 0.05M, as can be seenfrom the plots of the data in Figure 6and Figure 7, where the radius of gyra-tion is plotted against the inversesquare root of the ionic strength.

This phenomenon is no pure coincidence.In the behaviour of the second virial co-efficient and the apparent association con-stant, as demonstrated and explainedbefore, similar conclusions were found.Therefore it was decided to fit the tworegions separately. In particular the analy-sis of the data in NaCl and NaI solutionindicates that there is no statistical rea-son for not making the simple assump-tion that, in the range of Cs from 0.02M to0.05M, a single average value of Rg0 holdsfor both salts and all concentrations. Thehigh ionic strength regime, i.e. above

Table 3. The radius of gyration of ι-carrageenan as calculated from Equation 14 in LiCl salt solutions(Series I and Series II) at different molarities at 25°C.

Cs (M) (<Rg02>z)

1/2, nm (<Rg02>z)

1/2 , nm

Series I Series II 0.02 121.60 ± 1.10 110.40 ± 8.19 0.03 113.00 ± 6.44 125.60 ± 1.09 0.04 116.70 ± 7.79 129.80 ± 1.26 0.05 108.60 ± 6.92 122.00 ± 6.17

0.06 -- 112.60 ± 4.40 0.07 97.25 ± 6.99 109.20 ± 5.04 0.08 93.70 ± 8.89 90.25 ± 5.43 0.09 -- 94.83 ± 4.28

Cs (M) (<Rg02>z)

1/2, nm (<Rg02>z)

1/2 , nm

NaCl NaI 0.02 115.30 ± 1.57 110.20 ± 1.74 0.03 122.70 ± 1.72 107.70 ± 1.63 0.04 98.93 ± 7.83 107.60 ± 2.02 0.05 105.30 ± 10.28 113.00 ± 6.53 0.06 105.00 ± 10.30 104.50 ± 4.02 0.07 107.90 ± 9.00 104.40 ± 4.11 0.08 97.57 ± 7.95 96.69 ± 3.67 0.09 88.44 ± 5.74 90.80 ± 3.44

Table 2. The radius of gyration of ι-carrageenan as calculated from Equation 14 in NaCl and NaI saltsolutions at different molarities at 25°C.

Figure 6. The radius of gyration for ι-carrageenan in NaCl and NaI solutionsas obtained from Equation 14; the solid lines represent the linear fit in the highionic strength range and the mean value in the low ionic strength range.

Figure 7. The radius of gyration for ι-carrageenan in LiCl solutions as ob-tained from Equation 14; the solid lines represent the linear fit in the highionic strength range and the mean value in the low ionic strength range.

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

(<Rgc=0

2>

z)1/2

/ nm

NaI

NaCl

I-0.5

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

(<Rgc=0

2>

z)1/2

/ nm

Series I

Series II

I-0.5

, ,


0.05M, where the dependence of the ra-dius of gyration on the ionic strength islarge, is fitted with a linear function. Thefitted values of the radius of gyration arereported in Table 4 and Table 5 for Na+

and Li+ respectively as the counter-ion.The plots of the fitted values of Rg0 as afunction of I are reported in Figure 8. Theuse of I on the abscissa allows for aclearer identification of the two regionsof the ionic strength � and hence of theconformational transition � in which thebehaviour differs. From Figure 8 one cannote that, beyond about 50% of confor-mational change, the average chain di-mensions are the same for the sodiumand the lithium forms, whereas for lowerionic strength values the Li+-form ismore expanded than the Na+-form. Thisis in perfect agreement with the resultsfound for the second virial coefficientand the apparent association constant inthe previous part of this chapter.

From the radius of gyration obtainedafter fitting, the persistence length of theunassociated chain is calculated usingthe Doty-Benoit equation modified forpolydisperse systems, i.e. Equation 15.The results are also reported in Table 4and Table 5, and a plot is given in Figure9. The trend of the values of the persis-tence length as a function of the ionicstrength is similar to that of the radiusof gyration. More or less similar valuesfor the two counter-ions, Na+ and Li+, arefound in the high salt ranges, whereas amarked difference is noted in the lowersalt region of the first part of the confor-mational transition. In the latter rangethe values of the persistence length forthe Li+-form are very high (Lp>100 nm),almost reaching the values for the per-

sistence length of a rod with the samecontour-length as the disorderedLi+-ι-carrageenan conformation. Thismeans that the calculated values of Lp areat the limit of validity of the Doty-Benoitequation, and that a concept of a �loosehelix� is much more valid for describingthe conformation of ι-carrageenan in thedisordered form and in the first stages ofits transition to the regular helical con-formation. The overall behaviour as afunction of increasing ionic strength canbe qualitatively explained in this way: atfirst the �loose helix� is very extended dueto the low screening effect of the ionic at-mosphere. During the conformationaltransition one can envisage two oppositephenomena: a progressive stiffening ofthe backbone, which should lead to amore expanded conformation, and, onthe other hand, a progressive shieldingof the polyelectrolyte charges due to theincrease in ionic strength. The calculatedbehaviour of the persistence length in-dicates that the first phenomenon pre-vails in the first part of the conforma-tional transition, whereas the secondpredominatesbeyond 50% of transfor-mation. This explains the pseudo-biphasic behaviour of the persistencelength as a function of the ionic strength.For mere comparison purposes, in Fig-ure 10 the dependence on I-0.5 of the per-sistence length for I values correspond-ing to conformational ordering ≥ about50% have been reported. In Figure 10 aline was drawn through those points tothe intercept at infinite ionic strength(Lp∞) calculated for the helical conforma-tion of the accompanying moleculeκ-carrageenan (26 nm) [31]. This opera-tion is only aimed at indicating that thecalculated values of Lp for the ordered

conformation of ι-carrageenan arecompatible with that of the Lp∞ value,thereby giving some confidence to thecalculation procedure used. Finally, itshould be mentioned that we at-tempted to calculate Lp values for theordered conformation of ι-carrageenanassuming a double-helical conforma-tion. The only conformation of thistype compatible with the experimen-tal observation that the disorder-to-or-der transition is an intramolecular one,is that of a perfectly matched hairpin-like double-helix. However, in all casesit was not even possible to calculate Lpvalues of Rg0. This seems to be a clear-cut evidence that the only ordered con-formation of ι-carrageenan which iscompatible with the experimental datais that of a single-stranded helix.

Cs (M) (<Rg02>z)

1/2, nm xcoil Lp, nm

0.02 110.09 1.00 78.90 0.03 110.09 0.95 80.62 0.04 110.09 0.90 84.47 0.05 110.09 0.85 86.67 0.06 107.76 0.55 93.42 0.07 101.43 0.35 84.05 0.08 96.33 0.10 85.13 0.09 92.10 0.00 74.21

Cs (M) (<Rg02>z)

1/2, nm xcoil Lp, nm

0.02 118.46 1.00 101.98 0.03 118.46 0.95 105.87 0.04 118.46 0.90 111.02 0.05 118.46 0.85 120.96 0.06 111.29 0.56 115.74 0.07 102.64 0.30 92.16 0.08 95.67 0.05 83.15 0.09 89.90 0.00 68.26

Table 4. The fitted radius of gyration and calculated persistence length for ι-carrageenan inNa+-solutions of the two co-ions, Cl- and I- at 25°C.

Table 5. The fitted radius of gyration and calculated persistence length for ι-carrageenan inLi+-solutions at 25°C.

Figure 10. The persistence length of ι-carrageenanas a function of the inverse of the ionic strengthfor the high ionic strength regime in Na+. The dot-ted line represents the extrapolation to Lp∞ as cal-culated for the helical conformation of the accom-panying molecule κ-carrageenan.

0 1 2 3 4 50

20

40

60

80

100

120

140

Lpc=0

/ nm

I-0.5

Figure 8. The fitted values of the radius of gyrationfor ι-carrageenan in both Na+ and Li+-solutions.

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.100

20

40

60

80

100

120

140

(<Rgc=0

2>

z)1/2

/ nm

Na+

Li+

I

Figure 9. The persistence length for ι-carrageenanin both Na+ and Li+-solutions.

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.100

20

40

60

80

100

120

140

Lpc=0

/ nm

Na+

Li+

I

,

,

,


For the dependence of the radius of gy-ration against the temperature in 0.08 MNaCl solution, similar results are found.When the radius of gyration is calculatedin the same way as for the ionic strengthdependence, one finds a more or lesscontinuous increase with temperature(see Figure 11). The results of <Rg02>1/2

are presented in Table 6. After fitting thecalculated radius of gyration against thetemperature with a linear fitting func-tion, new, smoothed, radii of gyrationcan be calculated, which are also givenin Table 6. From these results, the per-sistence length is calculated using theDoty-Benoit equation. During the con-formational transition a break in the lin-earity of the persistence length is clearlyobserved, indicating the presence of twoopposing mechanisms. The results areplotted in Figure 12.

2.3. Conclusionsι-Carrageenan solutions are studied bywide-angle laser light scattering as afunction of ionic strength in NaCl, NaIand LiCl solutions at constant tempera-ture and as a function of temperature in0.08 M NaCl in order to explore the in-fluence of these factors on coil dimen-sions, helical conformation and poly-mer-solvent interactions. The radius ofgyration is obtained by fitting the lightscattering data with an empirical func-tion, accounting for some thermody-namic interaction in the particle-scatter-

ing function. The function permits us tocalculate the radius of gyration using theLevenberg-Marquardt optimisationmethod when the weight-average mo-lar mass in non-associating conditions,the apparent association constant andthe second virial coefficient are known.

The radius of gyration depends on theconformation of the ι-carrageenanchains, and a break is observed in thelinear dependence of (<Rg2>z)1/2 on I-0.5

when changing from a coil to a helix.A similar phenomenon is noticed in theintrinsic viscosity measurements duringthe conformational transition.

q

AcknowledgementsThis article is part of the doctoral thesis of K. Bongaerts

( Leuven 1999). The author is indebted to Prof. S. Pao-

letti for his continuous interest, support and discus-

sion. The work was supported financially by KULeuven,

FWO-Vlaanderen and INTAS-2000.

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q Received 14.01.2003, Revieved 05.05.2003

T, °C

(<Rg02>z)

1/2, Calc, nm

(<Rg02>z)

1/2, Fit, nm xcoil

Lp, nm

11 86.34 ± 4.10 91.20 0.00 71.65

15 93.11 ± 3.20 95.67 0.00 75.93

20 103.78 ± 6.05 94.50 0.10 76.10

25 89.12 ± 5.21 96.33 0.14 79.67

30 103.77 ± 7.38 98.17 0.69 64.08

35 93.70 ± 2.16 100.00 0.90 62.32

45 109.97 ± 1.74 103.37 1.00 65.82

50 107.64 ± 1.96 105.51 1.00 69.68

55 101.96 ± 2.43 107.37 1.00 73.19

Table 6. The radius of gyration of ι-carrageenan in 0.08 M NaCl solution at different temperatures.

Figure 11. The radius of gyration as a functionof the temperature in 0.08M NaCl solutions; theline is a guide to the eye.

Figure 12. The persistence length as calculatedby the equation of Doty-Benoit for ι-carrageenanin 0.08M NaCl solution at different temperatures.

0 10 20 30 40 50 60 700

20

40

60

80

100

120

140

(<Rgc=0

2>

z)1/2

/ nm

T /°C

0 10 20 30 40 50 60 700

20

40

60

80

100

120

140

Lpc=0

/ nm

T / °C

, ,

harry reynaerslight scattering study of polyelectrolyte ...chemistry, laboratory of macromolecular...

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