Polyelectrolyte chain dimensions and concentration

JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 7 15 AUGUST 2003

Polyelectrolyte chain dimensions and concentration fluctuationsnear phase boundaries

V. M. Prabhu and M. Muthukumara)

Department of Polymer Science and Engineering, Materials Research Science and Engineering Center,University of Massachusetts, Amherst, Massachusetts 01003

G. D. Wignall and Y. B. MelnichenkoOak Ridge National Laboratory, Condensed Matter Sciences, Oak Ridge, Tennessee 37831-6393

~Received 20 March 2003; accepted 23 May 2003!

We have measured the temperature~T! dependence of the correlation length~j! for concentrationfluctuations in aqueous solutions of sodium–poly~styrene sulfonate! with a fixed level of addedbarium chloride salt. Apparent critical behavior is observed upon lowering the temperature toprecipitation phase boundaries that complements our earlier work on salt-dependent behavior. Weinterpret experimental deviations fromj22 versusT21 as crossover from the mean field to the Isinguniversality class. We also measured the radius of gyration (Rg) of labeled chains andj forsemidilute polyelectrolyte solutions at low ionic strengths. We recovered the familiar result ofjscaling with polymer concentration (Cp) and degree of polymerization~N!, such thatj5(7369)N0Cp

20.4860.03 @Å #, and using SANS high concentration labelingRg5(400628)Cp20.2460.01 @Å #

~for N5577) and Rg5(2.862.1)N0.660.1 @Å # ~for Cp5206 gL21), respectively. The indicesrecovered are in agreement with theoretical predictions for low ionic strength semidilute solutions.Such experiments offer insight into relatively unexplored phase behavior in chargedmacromolecular solutions. ©2003 American Institute of Physics.@DOI: 10.1063/1.1592496#

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I. INTRODUCTION

Equilibrium properties of polyelectrolytes remain a sinificant challenge both experimentally and theoretically. Eperimental techniques such as static and dynamic light stering measure the collective behavior of the static strucand relaxation modes.1 Specific to polyelectrolytes, these eperiments reveal a long-time scale relaxation mode coupto a short-time scale relaxation,2–4 the slow and fast modesrespectively. The origin of the slow mode has been attributo a static multichain aggregate, while the fast mode istributed to the coupled diffusion of polyion and counterionThe aggregation behavior has been explored and new rehave revealed that the size scale is significantly greaterthe single chain dimensions using a combination of expmental techniques such as light and neutron scatterin5,6

However, the molecular origins that lead to such aggregaprocesses in low ionic strength solutions and connectionthermodynamic polyelectrolyte phase diagrams remain laing. Many experiments have demonstrated phase diagrfor polyelectrolytes in the presence of added salts.7–14 Theseprecipitation diagrams, typically plotted as added salt ccentration versus polymer concentration show similar ftures for a variety of polyelectrolytes including flexible ansemiflexible polyelectrolytes. Strong theoretical efforts habeen put forth to understand the effect of multivalent ionsthe phase diagrams of flexible polyelectrolytes.7,15 However,the theories do not consider the specific influence of ad

a!Author to whom correspondence should be addressed. [email protected]

4080021-9606/2003/119(7)/4085/14/$20.00

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salts on the configurational properties. Yet, some progresmade regarding the general shapes and trends of phasegrams in understanding experiments.7

The role of temperature on aggregation and precipitatbehavior has not received sufficient attention. This is surping considering inverse temperature times molecular wedetermines the state of miscibility for neutral polymers in tform of xN, wherex is the Flory–Huggins interaction parameter andN the degree of polymerization. Such behavishould also be observed with polyelectrolytes. Howevermay be masked by the proximity of the phase diagramrelatively small experimental window of temperature thaqueous solutions are able to be examined, as opposepolymer melts.

In this paper, we study aqueous solutions of sodiumpoly~styrene sulfonate! ~NaPSS! with the added salt bariumchloride. We explore the homogeneous phase by investiing the effects of added salt concentration versus polymconcentration (Cs–Cp) and temperature versus polymconcentration (T–Cp). To address the collective propertiewe have utilized small-angle neutron scattering~SANS! tomeasure the correlation length~j! and osmotic compressibility, proportional to the scattered intensity extrapolatedzero angle,} I (0), asfunctions of degree of polymerizatio~N!, polymer concentration (Cp), added salt concentratio(Cs), and temperature. The experiments focus in the limitboth low ionic strength and high ionic strength. Comparisof the experimental data forj is made with respect to theanticipated scaling behavior ofj(Cp ,N) in the low ionicstrength regime andj(Cs ,Cp ,T) andI (0)(Cs ,Cp ,T) in thehigh ionic strength regime. To address the configuratioproperties, we use the high concentration labeling techniil:

5 © 2003 American Institute of Physics

license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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4086 J. Chem. Phys., Vol. 119, No. 7, 15 August 2003 Prabhu et al.

to measure the size of labeled chains in semidilute soluas functions of molecular weight, polymer concentration, aadded salt concentration. The expectation of strong coil ctraction with added multivalent salts is compared within tlimits of extended-chain conformations and Gaussian coglobular dimensions are not experimentally achieved. Tscaling behavior ofRg (Cp ,N) is determined for the case ono added salt, and the results are consistent with scalaws16 and the double screening theory17 for semidilute so-lutions.

The rest of the paper is organized as follows: Sectiondescribes our experimental protocols covering the SAmethodology in Sec. II A, sample preparation in Sec. IImethodology for determining theCs–Cp and T–Cp phasediagrams in Sec. II C. Experimental results are given in SIII. Collective properties and labeled chain behavior are dcussed in Secs. IV and V, respectively. Section VI contamajor conclusions.

II. EXPERIMENT

Here we will briefly describe the origin of the samplethe protocol to prepare samples for the neutron scattermethod of determining the temperature-dependent phasegrams, and the SANS high concentration labeling methodogy.

A. Small-angle neutron scattering

We refer to the main results of the high concentratmethodology for the two component solutions uspreviously,14,18,19

I ~q!5I s~q!1I t~q!, ~1!

I s~q!5KnN2Ss~q!, ~2!

I t~q!5LnN2St~q!, ~3!

with prefactors,

K5@bh2bd#2xh~12xh!,

L5@bhxh1bd~12xh!2bs8#2.

The absolute differential coherent scattering cross stion, I (q), in units of cm21, in this method is composed ofsum of two types of scattering; scattering associated wintrachain monomer–monomer correlations (I s) and totalscattering from all monomer–monomer correlations (I t),both intrachain and interchain.bh and bd are the scatteringlengths of the protonated and deuterated monomers, restively. bs8 is the scattering length of a solvent molecule nmalized via the ratio of the specific volume of the monomand solvent molecule. The average scattering length ofsolvent,bs8 , may be adjusted by using a mixture of H2O andD2O such thatbs85yhbH2O1(12yh)bD2O, whereyh is themole fraction of H2O andbH2O andbD2O are the pure com-ponent scattering lengths.n is the number of polymer molecules per unit volume of solution. The essential physconcerning the configurational properties remains insingle-chain structure factor (Ss), and the collective properties in the total scattering structure factor (St). The singlechain scattering may be directly measured by making


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prefactorL to the intensity of total scattering go to zero. Wwill refer to the matching point as the condition such that taverage scattering length density20 of the monomer matchethat of the solvent which givesL50.

The SANS experimental results reported here were pformed at three different neutron scattering facilities; tW. C. Koehler SANS facility at the Oak Ridge NationLaboratory ~ORNL!,21 the National Institute of Standardand Technology~NIST! Center for Neutron Research facilit~NCNR!,22 and at the FRJ-2 research reactor of the Fochungzentrum Ju¨lich, Germany.23

At the W. C. Koehler SANS facility at ORNL, measurements were made using a neutron wavelength of 4.75 Å wwavelength distribution ofDl/l'5%. The scattered neutrons were collected with a two-dimensional area detecusing a sample-detector distance of 5.8 m. The neutron crate was corrected for instrumental backgrounds and deteefficiency on a cell-by-cell basis prior to radial averagiresulting in aq range of 0.008,q,0.1 Å21, whereq is thescattering wave vector defined byq54pl21 sin(u/2), whereu is the scattering angle. The net intensities were conveto an absolute~64%! differential cross section per unsample volume~in units of cm21! by comparison with precalibrated secondary standards, based on the measuremebeam flux, vanadium incoherent cross section, the scattefrom water and other reference materials.24 Procedures forcalculating the incoherent background, arising largely frothe protons in the sample, have been described previous25

At the NG3 30 m SANS facility at the NCNR, measurments were made using a neutron wavelength of 6.0 Å wwavelength distribution ofDl/l'15%. Two configurations,a 10 m and 3.8 m sample-detector distance, were used ling to an overlappingq-range of 0.005 to 0.0595 Å21 and0.0218 to 0.337 Å21, respectively. The measured neutrocount rate, across a two-dimensional area detector, wasmalized per 108 monitor counts and corrected for detectefficiency, cadmium blocked beam background, and emcell scattering on a cell-by-cell basis in accord with NISTstandard procedure. These net intensities were thenverted to an absolute differential cross section per usample volume~in units of cm21! using a light water refer-ence material.

At the KWS-II 40 m SANS beam line, FRJ-2 researreactor of the Forschungzentrum Ju¨lich, Germany, measurements were conducted using a neutron wavelength of 7Three separate configurations were used: 20 m, 8 m, anm. The scattered intensity data were reduced in a mansimilar to that at ORNL as the scattered neutrons werelected across a two-dimensional area detector and thetron count rate was corrected for detector sensitivity, cmium blocked beam background, and empty cell scatterprior to radial averaging. The net intensities were converto an absolute~64%! differential cross section per unsample volume~in units of cm21! by comparison with precalibrated secondary standards, based on the measuremebeam flux, vanadium incoherent cross section, the scattefrom water and other reference materials.24


eight.

4087J. Chem. Phys., Vol. 119, No. 7, 15 August 2003 Polyelectrolyte chain dimensions and concentration fluctuations

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TABLE I. Polymer characteristics.a

Name

Parent PS Sulfonated NaPSS

Mn

@g mol21# PDI

Nn

Fmonomers

chain GNSANS

Fmonomers

chain G %S~E.A.! SANS Facility

h-NaPSS-82 8565(2) 1.03 82 97 ORNLh-NaPSS-234 24 350(3) 1.02 234 100 ORNLd-NaPSS-274 28 500(2) 1.03 274 280614 92 ORNLh-NaPSS-287 29 800(1) 1.02 287 280614 96 ORNLd-NaPSS-563 63 000(2) 1.05 563 626630 86 NISTh-NaPSS-577 60 000(3) 1.02 577 626630 89 NISTd-NaPSS-958 107 300(2) 1.04 958 979660 86 FRJ-2h-NaPSS-962 100 085(3) 1.05 962 979660 88 FRJ-2h-NaPSS-1990 207 000(2) 1.05 1990 FRJ-2

aNSANS are averaged over all polymer concentrations and salt concentrations for each molecular wM1

d-PS5112 g mol21 andM 1h-PS5104 g mol21. Parent PS samples were obtained from~1! Polymer Labs,~2!

Polymer Source, or~3! MRSEC-University of Massachusetts.

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B. Sample preparation

We have prepared several molecular weights of deuated and protonated sodium–poly~styrene sulfonate! by sul-fonating poly~styrene!s from Polymer Source, PolymeLaboratories, and MRSEC facilities at the UniversityMassachusetts–Amherst. We have outlined the typsample preparation in our earlier publication for sulfonatof low polydispersity parent poly~styrene!s.14 However, wemake one modification here; following the sulfonation aseparation of the aqueous polymer rich phase the solutiopassed through a column of 400 cm3 of Dowex MR-3 mixedbed ion-exchange resin, purchased from Sigma–Aldrich,lowed by an equivalent amount of Milli-Q UF quality wateThe ion-exchange method was preferred over dialysis duthe large volumes of fluids encountered and rapid purifition. The ion-exchange resin was rinsed thoroughly wdeionized water and agitated to remove any low molar mimpurities, after which the effluent water is checked for loconductance and light scattering to ensure removal of ctaminants. After ion exchange of the polymer solution,clear acidic solution is then filtered through a 0.45mm cel-lulose acetate filter unit manufactured by Corning CosSubsequent neutralization of the acidic protons was achieby titrating with a sodium hydroxide solution purchasfrom Fisher Scientific, certified 0.201–0.199N. At this powe checked the concentration of polyelectrolyte by UV sptroscopy and compared this to the concentration estimatetitration, achieving quantitative agreement. This solution wthen filtered through a 0.22mm cellulose acetate filter unimanufactured by Corning Costar followed by freezing alyophillization. After lyophillization, the samples were futher dried under a purge of dry N2 in an oven at 105 °C for1th. Following this procedure, thermogravimetric analy~TGA! was performed and repeatedly reveal a reversmass loss, attributed to water.

Table I contains the information regarding the protonaand deuterated molecular weight matching required forhigh concentration labeling experiments as well as the cacterization details regarding the parent poly~styrene!s andsulfonation levels determined by elemental analysis~EA!.

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We provide the degree of polymerization obtained froSANS as well as the neutron scattering facility indicatedORNL, NIST, or FRJ-2. We will refer to the degree of polymerization~N! rather than molecular weight in order to bconsistent from sample to sample as the degree of sulfotion varies leading to an uncertainty in molecular weigSamples related to correlation length measurements wilrefered to as the correspondingh-NaPSS-N, whereas labeledchain experiments,d-NaPSS-N, where N is the degree ofpolymerization.

The general scheme for SANS sample preparation isuse protonated Na–poly~styrene sulfonate! ~h-NaPSS! indeuterium oxide solutions for the total scattering expements. For the high concentration labeling experimentsuse a fixed mass fraction ofh-NaPSS of 0.40 (xh50.40),and the balance deuterated NaPSS. Then we mix thesethe solvent mixture of D2O and H2O such that the volumefraction of H2O is 0.32 (yh50.32). For barium chloride salcontaining experiments, we always use stock salty solutiowhich can be prepared with accuracy. The H2O used in thisstudy is of Milli-Q UF Plus System quality with resistivity o18 MV cm. The D2O is from Cambridge Isotope Laboratories, Inc.,~99.9%!. The barium chloride salt, purchased froFisher-Scientific, used in this study is a di-hydrate (BaC2

•2H2O) under ambient conditions, but was dried to remowater by heating to 140 °C for 24 h.

To prepare samples, a predetermined mass of polyetrolyte, accounting for adsorbed water, is added to placentrifuge vials to facilitate easy transfer, as the solutionsnot wet these containers, thus minimizing loss of materFor the case of samples which do not exhibit precipitationroom temperature, we add an appropriate volume of salt-or salty stock solution to the vials and then agitate withvortexer. These samples may then be transferred usingEppendorf Pipet-man into the required quartz cuvettesSANS. However, for samples which exhibit precipitation wuse an alternative method. A fixed volume of salt-free sotion is added to the dried polyelectrolyte. Then an amounthis polymer solution equal to half the volume of the cuveis added to the cuvette. We then add a concentrated s


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solution of equivalent volume directly to the quartz cuvetsuch that precipitation occurs only within the controlled vume. This is then subsequently heated and mixed withaid of a vortexer. The reason for this protocol is that it is npossible to transfer a precipitated solution with reliabilitythe final concentrations. As an example, to prepare a soluwith final barium chloride concentration of 0.25 M anNaPSS concentration of 103 gL21, 0.325 mL of D2O wasadded to 73.65 mg, this was then mixed and 0.300 mL wtransferred to a quartz cuvette with path length 2 mm adiameter 2 cm. Then 0.300 mL of 0.50 M barium chlorisalt containing solution is added directly into the SANS cvette. Identical sample preparation was used for the labchain studies, except that we usexh50.40 andyh50.32. Toinvestigate polymer concentrations greater than 206 gL21,we account for the absorbed H2O as adjusting the D2O/H2Omixture. Thus we prepare specially a stock solvent accoing for this mass of water.

C. Phase diagrams

The miscibility of polyelectrolytes with added salts cabe represented on many types of phase diagrams. Thefrequent plot is salt concentration (Cs) versus polymer con-centration (Cp).7–11 We have determined these precipitatidiagrams for samplesh-NaPSS-82,h-NaPSS-287,h-NaPSS-577, andh-NaPSS-962 at 294 K. These diagrams are cstructed by preparing a series of stock salt-containing stions and salt-free polymer solutions, and then mixing inglass vials in a systematic fashion to furnish the librarystates. The distinction of a homogeneous solution versuscipitated solutions was determined visually for a clear sotion, or a white precipitate. The concentration range of thdiagrams spans the dilute to semidilute and will be discusin Sec. III A. We have probed the homogeneous phby SANS to determine the collective and configurationproperties.

The temperature dependence of polyelectrolyte phdiagrams has been explored far less.10,26–28Here we providethe background as to how our phase diagrams on theT–Cp

and T–CS coordinate planes are determined. We obserthat upon heating the precipitated salty polymer solutiothey would clear leaving a clear solution. This was foundbe true with exception to low polymer concentrations whigh salt concentrations as the boiling point of the wapreceded the clearing temperature. To provide an estimatthe phase boundaries, which would be subsequently proby SANS, we used small-angle light scattering~SALS! andvisual inspection. We observed that the SALS intensity nthe phase transition became strong indicating the correlaton the order of micron length scale. To perform these expments we used a home built SALS apparatus and hot st

The hot-stage used for the temperature studies accequartz cuvettes identical to those used for SANS. The hstage was made from two aluminum parts. Each part halarge volume milled such that when the two parts were ftened together, with machine screws and a paper gasketvolume served as the reservoir for continuous flow of acirculating fluid for the required temperature control, facitated by drilled ports for fluid inlet and outlet. We used


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PGC Scientifics heating-cooling recirculating bath withinternal fluid temperature stability of 0.1 K, using eith50:50 ethanol:water or deionized water. The sample cell wmounted into a small fitting which was accepted by the hstage and had a pilot hole for which a resistive thermalvice ~RTD! was inserted for monitoring the temperaturethe sample. The stability of the sample temperature waster than 0.5 K over a 24 hour period with insulation. The hstage had a hole drilled which allowed for the laser ligto pass through the stage and thermostatically controquartz cuvette. The temperature of the bath was chanmanually and the sample temperature recorded from the Rcontroller.

The majority of the samples investigated precipitateroom temperature. The precipitation temperature upon cing was determined by heating the sample in the hot-stuntil no evidence of precipitate remained upon visual insption and SALS. However, we observe a contributionSALS even in the apparent high-temperature homogenephase, for a givenCs and Cp . These reported precipitatiotemperature determined by light scattering were determiover a range of temperatures. We monitored the SALS astemperature was lowered in decrements of 2 to 0.5 K, firesolution in anticipation of precipitation. A minimum of 1minutes equilibration time was used at each temperatureapproach to the phase boundary. At each temperatureSALS intensity was recorded by a Princeton EG&G CCcamera in the form of the two-dimensional~2D! scatteringimage, or observed visually for an increase in the intensEither method would give an estimate for the precipitatitemperature range.

One feature which we observed was that the SALStensity would grow rapidly indicating the onset of the kineics of precipitation. The time scales of this process ofwere too fast for most of the samples investigated and wobe a strong function of the quench depth, polymer andconcentrations. We restrict our interpretation of the prectation temperature from the light scattering as an approprrange of temperatures over which the system crossed fone in which the system in a nonprecipitated homogenephase to one in which the kinetics are important. Thus bylight scattering method we do not obtain the spinodal teperatures by extrapolation methods as with SANS, but bquery of whether the system is in the region of stabilityevolving.

We provide an example forh-NaPSS-962 where thepolymer concentration is 51.5 gL21 and salt concentration0.19 M. In this case we were able to record a significnumber of data points capturing the increase in the scattintensity upon decrements in temperature. First with a coaapproach to determine the approximate range of temptures and second with a longer equilibration waiting time afiner resolution. The coarse approach was equilibration75.4 °C for 15 minutes, followed by equilibration at 68.6 °for equal time followed by a change to 66.0 °C. At 66 °C tscattered intensity observed visually could be seen tocrease leading to precipitation. We followed this with a moelaborate approach as shown in Fig. 1. In Fig. 1~a! we dem-


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FIG. 1. ~a! SALS observations for afixed NaPSS concentration of 51.gL21, barium chloride salt concentration of 0.19 M andN5962. ~b! Inte-grated scattered intensity versus temperature, same sample. The large riin intensity at 66.5 °C is typical andindicates the phase boundary asguide for further experiments.

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onstrate on a linear plot of SALS intensity versus wave vtor for several temperatures on approach to precipitation.increase in intensity is easily observed, as well as the surping scattering in the high temperature limit. To gain an emate for the phase boundary we plot the integrated intenversus temperature in Fig. 1~b! and notice the sharp increasat 339.65 K, which is marked as the phase boundary duthe onset of the kinetics. This provides empirical knowledas to the important ranges of temperature to performSANS experiments. In the following figures the precipitatitemperatures are extrapolated from SALS and visual insption as the range of temperatures above the last temperexamined in which the kinetics are important. Two excetions are the samples of polymer concentration 0.2 gL21 and154.5 gL21 both with salt concentration 0.19 M. For the 0gL21 sample the sample precipitation temperature couldbe determined by cooling from the high temperature phanor with the hot stage due to the high temperatures involvThus, this particular sample was heated on a hot platevisually inspected for clearing, leading to the 98.9 °C99.8 °C result. For the sample withCp5154.5 gL21 we vi-sually observed precipitation within the SANS spectromeat the final observation temperature of 281.15 K.

III. EXPERIMENTAL RESULTS

A. Phase diagrams

TheCs–Cp diagrams are shown in Fig. 2 for the systeof added barium chloride (BaCl2) concentration versus polyelectrolyte monomer concentration, in molar units for degof polymerization~a! 82, ~b! 287, ~c! 577, and~d! 962. Thefilled circles represent a precipitated solution, the opcircles homogeneous clear solutions. We have already sha portion of the results forh-NaPSS-287 previously,14 buthere we have explored higher concentrations. Upon insption, we find a weakN dependence in these precipitatiodiagrams with regards to the salting-out curve. The lowmolecular weight requires higher added salt to induce pcipitation, a trend that is observed fromN582 to 962, withthose limits being 0.02 M and 0.008 M, respectively.


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In addition, we have explored the role of temperaturethe phase behavior on several of these solutions and hdiscovered that the role of temperature and salt concentraare complementary. We have determined the polymer ccentration dependence of the miscibility temperature,given by the solid squares in Fig. 3, for a fixed salt conctration (Cs50.19 M) andN ~962!. The range of polymerconcentration examined was 0.2 to 154 gL21. We did notprobe the thermal dependence for lower polymer concention due to the high temperatures involved. The solid dmonds are the spinodal temperatures determined by SANbe discussed below.

This phase diagram is also a strong function of the sconcentration. Fixing the polymer concentration at 103 gL21

we examined the influence of added salt concentration onprecipitation temperatures as shown in Fig. 3~b!. We ob-served that with increasing levels of added salt the prectation temperature increases significantly over a small raof salt concentration, 28 K for a 0.075 M increment in sconcentration. This would suggest, in particular, for biolocal processes mediated by salt ions that a small change inconcentration, or gradients, influence the thermal stabilitycharged macromolecules. The competing roles of iostrength and temperature was demonstrated in a transpmanner via the Flory–Huggins treatment in our previopaper.14

In Figs. 3~a! and 3~b!, we also present the SANS resulfor the phase boundary, determined by the divergence inextrapolated scattered intensity to zero angle,I (0). Thesedata points appear within the observation range of temptures estimated by SALS and visual inspection. The methology for extracting theI (0) is the subject of the followingsection, but to clarify the results forh-NaPSS-287 andh-NaPSS-577 samples, we present the SANS determphase boundaries as well in Fig. 4. These phase diagramon theT–Cs planes and indicate the role of increasing sleading to an increase in the precipitation temperatures.following results, examined by SANS total scattering, E~3!, probe the density–density correlations from the hi



FIG. 2. Precipitation phase diagram at 294 K for several degree of polymerizations~a! 82, ~b! 287, ~c! 577, and~d! 962 NaPSS with added salt bariumchloride.

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B. Low ionic strength: Collective properties

To build upon our previous work in which the saldependent phase behavior was explored we have returnthe case of low ionic strength solutions. The density–dencorrelations in low ionic strength polyelectrolyte solutiohave long ago been measured.29 However, this is the natura


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reference state before exploring the influence of added sand temperature dependence. Thus we have explored focase of no added salt, the density correlations as a functiodegree of polymerization and polymer concentration.

Polyelectrolyte solutions at low ionic strength exhibitpeak in the total scattering structure factor at a finite wavector qÞ0. This feature is due to the presence of lonranged electrostatic interactions and is absent in neutralmopolymer solutions which are dominated by the sho

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FIG. 3. ~a! Influence of temperaturefor a fixed added salt concentration o0.19 M, examined forN5962. Thephase boundary by SALS~solidsquares! were determined upon observation of the kinetics of the phaseseparation.~b! Influence of salt con-centration on the precipitation temperature for a fixed polymer concentration ofCp5103 gL21 examined forN5962. The cloud point temperaturedetermined by SALS~solid squares!are systematically at lower temperatures than the SANS data~solid dia-monds!.


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FIG. 4. Influence of extrapolated spinodal temperature for a fixed polymeconcentration of 206 gL21, examinedfor N5287 ~a! and 577 ~b!. The m

symbols are determined from thSANS experiment where theI (0) areextrapolated within the Ising regimeThe j symbols are determined by extrapolation in the Ising regime, whilethel is using the extrapolation in themean field limit.

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ranged interactions. Theory and simulation have predicthis polyelectrolyte peak.30–36

This scattering peak was investigated for flexible poelectrolytes including poly~styrene sulfonate!,29,37,38,40

poly~N-methyl-2-vinyl pyridine chloride!,5 and biopolymerssuch as xanthan,39 BSA,41 and DNA.42 The data of Kajiet al.38,40 demonstrate that the inverse of the scattering wvector (qmax

21 ) at the peak intensity scales with the polymconcentration asCp

21/3 then crosses over toCp21/2, consistent

with the crossovers from dilute to semidilute polyelectrolysolutions, respectively. At even higher concentrations a nrow experimental window shows a second crossover frCP

21/2 to Cp21/4 consistent with the crossover from semidilu

to concentrated regimes consistent with the double screetheory.17 An example of the polyelectrolyte peak can be oserved in the total scattering intensity at highq shown in Fig.5~a! for the no added salt case forh-NaPSS-577 and polymeconcentration 206 gL21.

At the maximum intensity of the polyelectrolyte pea(qÞ0) the wave vectorqmax

21 ([j) is defined and plottedversusCp in Fig. 6 for all N and concentrations examineThese experimental data demonstrate the semidilute na


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of the solutions, without added salts, since the corration length appears independent of the molecular weand varies with polymer concentration asqmax

21 5(7369)Cp

20.4860.03. This also suggests the proper experimenranges in which configurational properties may be invegated in the semidilute limit. However, with the additionsalt the calculated overlap concentration (C!) would changeas the chain dimensions decrease, due to screening, andsibly shift the conditions for semidilute solution to highpolymer concentrations.

Since each molecular weight has a different value foroverlap concentration, the experimental data can be reduon a dimensionless plot ofj/LC versusC/C! consistent withthe following scaling ansatz:

j;NnRgS Cp

C!D y

, ~4!

where

C!53Mw

4pRg3NA

, ~5!

frn

ts

FIG. 5. ~a! Typical scattered intensityversus wave vector as a function obarium chloride salt concentration foa fixed NaPSS polymer concentratioof 206 gL21 of N5577 and fixed tem-perature of 298 K.~b! CorrespondingOrnstein–Zernike plot from whichjand I (0) are extracted avoiding the fifrom the downturn inverse excesscattering.


eUs-

e

e

nd

,ionbp

tdeioesiusunlu

rth-

f tla

-cor-to

ed

ex-Thesev-

ns

n-cede-be-

nr-

a-

wasc-tedis

tn


and y represents the concentration scaling exponent,NA

Avogadro’s number, andnRg the scaling exponent for thmolecular weight dependence of the radius of gyration.ing a value fornRg of 1.0 we estimateC! using an extendedchain conformation~rodlike limit! for the radius of gyrationand nondimensionalize the abscissa. The ordinate, hencdimensionless by dividingj by the contour length (LC) lead-ing to the inset of Fig. 6. The fit over the entire range giv(qmax

21 /LC)5(0.06360.003)(Cp /C!)20.4960.01 over the rangein degree of polymerization~82 to 1990! and concentration~18 to 300! gL21 consistent with the scaling ansatz ascreening in semidilute polyelectrolyte solutions.

C. High ionic strength: Collective properties

The data in Fig. 6 are results without any added saltsa fixed temperature of 298 K and are far from precipitatphase boundaries; these solutions can be heated to nearing and cooled to freezing without any onset of phase seration. In a previous study, Boue´ et al.43 demonstrated thathis polyelectrolyte peak had only a weak temperaturependence. This is expected since the range of correlatwill increase significantly only in the proximity of a phasboundary. Hence, the correlations remain on the mesh-length scale of semidilute solutions, familiar to that of netral polymers,16 but with different scaling properties. In thipaper we have explored how the presence of phase boaries leads to long-range correlations in polyelectrolyte sotions with added salt.

We have found that with the addition of salt,j increasesconsistent with the results of Iseet al.37 and qualitative pre-dictions from theory.32,33,35The experimental data taken fodegree of polymerization 234 are shown in Fig. 7, wherevariation in qmax

21 with salt concentration and polymer concentration is shown. These data complement the values ocorrelation length taken from the Ornstein–Zernike corre

FIG. 6. Plot of inverse polyelectrolyte scattering peak (j[qmax21 ) versus

polymer concentration (Cp), with no added salt (T5298 K). Regressionover all raw data yields (j)5(7369) Cp

20.4860.03. Inset, dimensionless ploof j/LC versusCp /C!, usingn of 1.0 to calculate the overlap concentratio(C!). Regression yields (j/LC)5(0.06360.003)(Cp /C!)20.4960.01.


-

, is

s

at

oil-a-

-ns

ze-

d--

e

he-

tions under high salt conditions.14 We now proceed to address the temperature dependence of the density–densityrelations as the phase boundary is approached,complement the role of added salt.

D. Temperature dependent collective properties

Our earlier data, for varying ionic strengths at a fixtemperature, showed the divergence ofj and I (0) as theprecipitation phase boundary was approached.14 This reliedon fitting the data to the Ornstein–Zernike formula,

St~q!215St~0!21~11j2q2! ~6!

for wave vectors larger than those dominated by strongcess scattering, occurring at the lowest wave vectors.presence of this excess scattering has been verified foreral molecular weights and polymer concentrations.

The new results probe the concentration fluctuatioquantified by a divergence ofj and I (0) with decreasingtemperature. We examined for a fixed level of salt concetration the scattering behavior as the temperature is reduto the point of precipitation. Similarities are observed btween the temperature dependence and the salt-inducedhavior. We have explored the behavior for samplesh-NaPSS-287, h-NaPSS-577, andh-NaPSS-962. We report the mairesults forh-NaPSS-287 which directly complement our ealier results for the salt-induced precipitation.14 The data forthe remainingN will follow in the discussion section.

For N5287, with a fixed barium chloride salt concentrtion of 0.38 M and fixed polymer concentration of 206 gL21,the scattered intensity was measured as the systemcooled from the high temperature limit to the point of maroscopic precipitation following the phase diagram presenin Fig. 4~a!. The scattered intensity versus wave vector

FIG. 7. Variation in inverse polyelectrolyte peak position (qmax21 ) with added

salt concentration for three polymer concentrations and fixedN5234 (T5298 K).


fer

r


FIG. 8. ~a! Typical scattered intensityversus wave vector as a function odecreasing temperature to phasboundary for a fixed NaPSS polymeconcentration of 206 gL21 and bariumchloride salt concentration 0.38 M foN5287. ~b! Corresponding Ornstein–Zernike plot from whichj and I (0)are extracted avoiding the fit to thedownturn inverse excess scattering.

–

th

d

hthes

nsle

ledreee.re-thet,

inehe

two

shown in Fig. 8~a! as well as the corresponding OrnsteinZernike plot in Fig. 8~b!. In Fig. 8~b!, data for only a fewtemperatures are presented for clarity. Correlations ofOrnstein–Zernike type are seen by the linearity ofI (q)21

with q2 and the excess scattering deviations are observethe low wave vectors. The systematic deviation at lowqbecomes negligible as the temperature approaches the pboundary. In this system, even under high salt conditions,excess scattering remains at correspondingly high temptures. This suggests that large scale fluctuations are preeven under strong electrostatic screening. A comparisoFigs. 5 and 8 demonstrates clearly the complementaryand temperature effects on the phase behavior of polyetrolyte solutions.


e

at

asee

ra-entofaltc-

E. Configurational properties

We have measured the radius of gyration of labechains as functions of the polymer concentration and degof polymerization, without and with added barium chloridExamining the scattering behavior without added salt pcludes the effect of the specific interactions mediated bydivalent barium ions. At the condition of the matching poinsuch that the coefficientL50, we measured the single chastructure factor (Ss) from the small-angle scattering of thdeuterium labeled chains. To illustrate the quality of telimination of the total scattering, we present in Fig. 9~a! acomparison of the measured scattered intensity for thetypes of experiments that measureI t(q) and I s(q) in the

FIG. 9. Absolute intensity of totalscattering ~symbol m! and single-chain scattering~symbol j, N5563,CP5206 gL21). ~a! Cs50.0 M. ~b!Cs50.30 M. Data are not correctedfor incoherent background.


len

sf

ne-


FIG. 10. ~a! For a fixedN5563 with-out added salt, the measured singchain scattered intensity as a functioof wave vector is shown for four dif-ferent polymer concentrations.~b!Fixed polymer concentration of 103gL21 and varying salt concentrationas indicated in the legend. Radius ogyration and degree of polymerizatioare extracted from fits to the Debystructure factor with incoherent background, shown as the dark lines.

omleai,b

lsinth

wityeo

g-

nngfa

heter

dneuthave

ion

a

is-ente

l to-

is-michey to

tralcs

a

asethemeh

ged.

limits of no added salt. It can be seen that contributions frthe qÞ0 polyelectrolyte peak are not present in the labechain experiment, suggesting an elimination of interchcorrelations. Our data on the total scattering experimentsthe lowq region, show the excess scattering not predictedcurrent theories or simulations. For comparison we ashow in Fig. 9~b! the case with high added salt of 0.30 Mwhich the total scattered intensity rises significantly ashigh-q data ofI t surpasses that ofI s .

The results from four polymer concentrations are shoin Fig. 10~a! for d-NaPSS-563 on a plot of scattered intensversus wave vector and in Fig. 10~b! the results for the sammolecular weight, but at a fixed polymer concentration103 gL21 with varying levels of added salt. From these fiures we extract theZ-averaged radius of gyration (Rg) by fitsto the Debye structure factor,SD(q),

SD~q!52

q4Rg4 ~e2q2Rg

2211q2Rg

2!. ~7!

Clearly, the fits are excellent over most of theq-range stud-ied, with exception to the lowestq data. The fits from theDebye structure factor, including an incoherent backgroureveal that the extrapolated scattered intensity at zero a~and hence molecular weight! is consistent with the degree opolymerization from the GPC on the parent polystyrenesummarized in Table I. The deviation at the lowestq tohigher intensities is not within the experimental error of texperiment, but reflects a contribution from the total scating. Attempts to fit the entireq range including the three ofour lowest data points with Eq.~7! were unsuccessful. Wenote that experiments performed with lowerq resolutionwould yield perfect fits to Eq.~7!, but proceeding to lowerqmay reveal deviations. The advantage with using a moform factor is that the size of the labeled chains was obtaiover a wide wave vector range. Since, we did not measthe very highq data, we do not extract the persistence lengSeveral groups44,45 have chosen this route, but rely onwormlike chain model to extract the persistence length, ein the semidilute condition.


dninyo

e

n

f

d,le

s

r-

eld

re.

n

IV. DISCUSSION: TEMPERATURE DEPENDENTCOLLECTIVE PROPERTIES

We now proceed with the discussion and interpretatof the SANS results. It was demonstrated earlier14 that amean field model for polyelectrolyte solutions leads tomodified Flory–Huggins interaction parameter such that,

xeff5x02wc

k2 . ~8!

x0 is the Flory–Huggins parameter for the chemical mmatch modeling the neutral hydrophobic monomer–solvshort-ranged interactions,wc represents the strength of thscreened Coulombic interaction between monomers, andk2

is the inverse-square Debye screening length, proportionasalt concentration.xeff is the effective Flory–Huggins interaction parameter, representing the sum total chemical mmatch. Based upon this model the mean field thermodynabehavior was predicted to have the following form for tcorrelation length and the extrapolated scattered intensitzero angle:

j22;S 1

Ts2

1

T1

wc

k D 2n

and ~9!

St~0!21;S 1

Ts2

1

T1

wc

k2D g

.

Ts is the spinodal temperature of the corresponding neupolymer in solution, without the influence of electrostatiandn andg are the critical indices. If mean field values ofnandg were valid, a plot ofj22 @or St(0)21] versus 1/k2 willyield a straight line with a positive slope. Equivalently, forfixed 1/k2 a plot ofj22 @or St(0)21] versus 1/T will yield astraight line with a negative slope. Deviation near the phtransition is attributed to the crossover from mean field toIsing universality class as the role of fluctuations becoimportant. This is the expected behavior in the limit of higsalt where the electrostatic interactions become short ran


r

s-d-

,ne

n-e


FIG. 11. Results for a fixed polymeconcentration of 206 gL21 and saltconcentration 0.38 M forN5287 withdecreasing temperature.~a! Using amean field plot, the solid line indicatethe linear region of mean field behavior and extrapolated to the mean fielspinodal temperature; and the curvature, indicated by the dotted lineshows the interpreted Ising fluctuatioregion; the inset shows the divergencof j with decreasing temperature.~b!Similar mean field plot for theextrapolated-inverse scattered intesity at zero angle; the inset shows thdivergence ofI (0). ExtrapolatedTmf

5337.94 K andTfluct5333.34 K.

dasiula

ina

aln,

ngÅ

-o

ea

ysde-ra-

to

larser

ndi-setola-

r in

ets.ys-po-the

Returning to the case ofh-NaPSS-287 we have probethe phase diagram shown in Fig. 4 that lead to the mresults in Fig. 8. Now we have examined the density–dencorrelations quantified by the Ornstein–Zernicke formproviding thej andI (0) in the insets of Fig. 11~a! and 11~b!,respectively for the fixedCs50.38 M andCp5206 gL21. Itis clearly observed that the correlation length steadilycreases from 35 to near 195 Å close to the phase boundThis significant increase was studied before in the sinduced precipitation study. We do not have the correspoing temperature dependence of the radius of gyration datawe can not make a comparison betweenRg andj. Neverthe-less we note that at high temperatures the correlation leis smaller than the theta-coil and rodlike predictions, 42and 200 Å, respectively.

The mean field plot provided by Eq.~9! is demonstratedin Fig. 11 in which a linear region in 1/T is clearly observedindicating a region of mean field applicability. This, however, is followed by the departure due to the importancefluctuations closer to the phase transition. The fit to the mfield region is withn50.5 andg51.0 and the curved line


inty

-ry.t-d-so

th

fn

that deviates from the mean field region is fit usingn50.63 andg51.26, the result for the Ising universalitclass.46 From Fig. 11~b!, the mean field extrapolation givethe spinodal temperature of 337.94 K. By the systematicviation from the mean field behavior the spinodal tempeture is depressed to 333.34 K, a 4.6 K depression duefluctuations.

We observed this behavior for the remaining molecuweights as well. Forh-NaPSS-962, we examined the phadiagram displayed in Fig. 3~a! on the temperature-polymeconcentration plane forCp5154 gL21 andCs50.19 M. TheOrnstein–Zernike plots display the same features as icated for the lower molecular weight. Far from the phatransition the low-q deviation is apparant, but then closerthe transition the deviation is less important as the corretion length increases. The mean field to Ising crossoveshown in Fig. 12~a! and 12~b!, for j and I (0), respectively.The divergence in these quantities is given in the insAgain, we observed the mean field region followed by stematic deviation close to the phase transition. The extralated mean field temperature is 289.00 K and by using

-

FIG. 12. Mean field fluctuation re-gimes, for j ~a! and I (0) ~b! Cp

5154 gL21, Cs50.19 M for N5962.Solid line extrapolates to Tmf

5289.00 K and dotted Ising fluctuation regime curve extrapolates toTfluct5284.77 K.


-

h

f

--

-


FIG. 13. ~a! Divergence in the corre-lation length for polymer concentration 206 gL21 and four different addedsalt concentrationsj, 0.32 M;c, 0.31M; * , 0.30 M; andl, 0.275 M. SANSexperiments were performed belowambient conditions at 283 K. It wasobserved visually that samples witsalt concentration 0.32 M and 0.31 Mreached precipitation, but at the end othe experiment 0.30 M and 0.275 Mwere clear as reflected in the correlation lengths extracted from the resulting Ornstein–Zernike plots.~b! Meanfield fluctuation crossover. Cp

5206 gL21, Cs50.31 M for N5577.Solid line extrapolates to Tmf

5288.01 K and dotted Ising fluctuation regime curve extrapolates toTfluct5284.69 K.

gehe-2,aK

ra

en

th

bpticsainuo

iot,ne

o

oli

hee

irndthatob-te

ntfsi-ns

on-

s a

re-

ingolu-

fhensentn,ex-rpen-

ent6.tionnot

gth,

g-tfor

Ising exponent we measure the spinodal temperature to284.77 K, a 4.2 K depression.

We examined several solutions which were homoneous at 294 K, but with sufficient levels of salt near tsalting-out line in Fig. 2 forh-NaPSS-577 with polymer concentration 206 gL21 and varied salt concentrations 0.30.31, 0.30, and 0.275 M. We show the results for the increin the correlation length with lowering temperature to 283in Fig. 13~a!. Two of the samples,Cs50.32 and 0.31 M,were observed to precipitate, within the allowable tempeture limits, leading to divergingI (0) and j. Figure 13~b!demonstrates the same analysis of Eq.~9! fitting the inverseisothermal compressibilities for the sample with salt conctration 0.31 M. The extrapolated mean field temperature288.01 K, while the spinodal temperature upon assumingIsing indicies leads to 284.69 K, a 3.3 K depression.

These experimental data clearly show the crossoverhavior for high salt polyelectrolyte solutions. The data suport the idea that under conditions of screened electrostapolyelectrolyte solutions fall into the same universality claas neutral polymer solutions. The examination of criticphenomena in low ionic strength polyelectrolyte solutionsstill elusive, as the appropriate experimental system hasbeen found. For electrolyte solutions, experimental data sgest a distinction between Coulombic criticality and solvphobic criticality.47,48 For solvophobic criticality the systemis dominated by short-range nearest neighbor interactand displays Ising criticality very close to the critical poinas in neutral solutions. For Coulombic criticality, experimetal data on different systems show either the mean ficriticality49,50 or a crossover from mean field to Ising.48,51–53

Our results on polyelectrolyte phase behavior clearly demstrate deviations from mean field behavior.

V. DISCUSSION: CONFIGURATIONAL PROPERTIES

A. Low ionic strength

The influence of added salts of type 1:1, 1:2, 1:3, etc.,the configurational properties of polyelectrolytes is compcated by specific interactions among charged species. Tinteractions will contribute to the phase behavior of charg


be

-

se

-

-ise

e--s,

slsotg--

ns

-ld

n-

n-sed

macromolecules, in particular via formation of ion-pabridging among monomers mediated by multivalent ions acondensation mechanisms. It was demonstratedmultivalent-ion-mediated interactions are necessary toserve salient features in multivalent salt-polyelectrolyphase diagrams,11,12,54,55in particular the observed re-entraor salting-in phase behavior.7,8 However, the interpretation othese effects on the configurational properties for intrincally flexible polyelectrolytes in nondilute solutions remaiunsolved and thus the experimental data need to be demstrated to assist in this problem.

The main results of the measured radius of gyration afunction of polymer concentration are shown in Fig. 14~a!for d-NaPSS-563. The four data points were fit with thegression result ofRg5(400628)Cp

20.2460.01 @Å #. This re-sult is in excellent agreement with the predictions of scaltheory and the double screening results for semidilute stions. We also examined the influence ofN for a fixed poly-mer concentration of 206 gL21 as shown in Fig. 14~b!. Thisleads toRg5(2.862.1)N0.660.1 @Å#, where the exponent o0.660.1 is in comparison to the predicted value of 1/2. Texponent of 0.660.1 suggests that the polyelectrolyte chaiare quite flexible in the semidilute solution and is consistwith the single chain density–density correlation functiomodeled by the Debye structure factor. However, moreperiments are needed to confirm this result over a wideNrange. These two results combine to give the general dedence ofRg5(9.569.8)Cp

20.2460.01N0.660.1 @Å #. We do notobserve deviation from the semidilute predictions, consistwith the results from the correlation length study of Fig.Further experiments are needed to examine the distincbetween the semidilute and concentrated regimes. We didanalyze the structure factor in terms of a persistence lensince our data do not extend to such highq.

B. High ionic strength

Consistent with our earlier investigations, we find sinificant coil contraction with the addition of the multivalensalt barium chloride. This coil contraction was measured


f


FIG. 14. Dependence ofRg on poly-mer concentration for a fixed degree opolymerization 563~a! and molecularweight for fixed polymer concentra-tion of 206 gL21 ~b!.

r

ontsveth

teelieera--44-aie

raon

for

76thsmer

llyre-in-rac-islarofbyvol-paree ef-ledtly

d-NaPSS-274,d-NaPSS-563, andd-NaPSS-958 and polymeconcentrations between 50 and 350 gL21, all within the se-midilute regime, demonstrated earlier. Figure 15 demstrates the main results by plottingRg andj versus added salconcentration forN5563 and three polymer concentration51, 103, and 206 gL21 as indicated in the legend. We obsera coil contraction for all concentrations and an increase incorrelation length with added salt concentration, consiswith our earlier report.14 For the no added salt conditions thsemidilute solution is observed consistent with the earcorrelation length scaling result and also by using the expmentally determinedRg to calculate the overlap concentrtion leading toC/C* .1. As the salt concentration is increased the labeled chains undergo a change by 48%,and 24% for 51, 103, and 206 gL21, respectively. The estimatedRg values for the theta condition and extended chconformation are 62 Å and 420 Å, respectively. In all casthe chain dimensions are well below the rodlike configutions and under high salt conditions the theta-coil dimensiare not achieved. Figure 16 shows the result ford-NaPSS-

FIG. 15. Dependence ofRg ~filled! and j ~open! on Cs for samples ofmatched molecular weight,h-NaPSS-577 and d-NaPSS-563. Cp

551.5 gL21 ~j andh!, 103 gL21 ~d ands!, and 206 gL21 ~b andv!.


-

,

ent

ri-

%,

ns-s

958. Again, the observed coil contraction is observedpolymer concentrations of 51, 103, and 206 gL21 as 26%,22%, and 36.1%, respectively. The estimatedRg values forthe theta condition and extended chain conformation areÅ and 710 Å, respectively. The observed correlation lengalso increase as a function of added salt for each polyconcentration, in agreement with the earlier findings.

In the present investigation, we observe experimentano signatures of a coil–globule transition. Precipitation pcedes the expected ideal chain configuration with thecreased levels of added salt which should screen the intetion. Although, this point was made earlier, we confirm thwith a wider range of polymer concentration and molecuweight which concludes that even under the conditionadded multivalent salt, polyelectrolytes remain swollenexcluded volume. To address these issues of excludedume and electrostatic screening it is necessary to comthese experimental data with a theory that addresses thfect of multivalent added salt on the configuration of labechains in semidilute solutions. Such a theory is curren

FIG. 16. Dependence ofRg and j on Cs for samplesd-NaPSS-958 andh-NaPSS-962. Symbols are the same as in Fig. 15.


u

othth

thioyr

-onesini-hthththetan

seitic

Gfor.

-3l-thwialoeyo

er

y

nd

S

ll,

im,

ring,

and

on-O

hys.

ans.

J.

o-

Y.

-

rs,

W.


unavailable. We hope that the present experimental reswill stimulate such a theory.

VI. CONCLUSIONS

Evidence for screening in low ionic strength polyelectrlyte solutions is found as demonstrated by the scaling ofcorrelation length, taken from the inverse wave vector ofqÞ0 polyelectrolyte peak, with polymer concentration~ingL21! leading to the result of (j[qmax

21 )5(7369)Cp

20.4860.03. The exponent of 0.48 is in comparison withe scaling result of 1/2. Under the identical concentratand molecular weight ranges, the measured radius of gtion of labeled chains recovers the following results,Rg

5(400628)Cp20.2460.01 and Rg5(2.862.1)N0.660.1, using

N5577 andCp5206 gL21, respectively. Here, the exponents forN and polymer concentration are in compariswith the theoretical values of21/4 and 1/2, respectively. Whave measured the crossover from the mean field to Icriticality from the correlation length and susceptibility dvergence with decreasing temperature. This crossover beior was anticipated due to the high salt solution in whichrange of the interactions is short-ranged and falls intosame universality class as neutral polymer solutions inlimit of high salt. It must be remarked that we have assumthe existence of this crossover, although we have not eslished the existence of a critical point in this multicomponesystem. The mean field and Ising critical indices were uas an input to estimate the depression of the apparent crtemperatures due to fluctuations.

ACKNOWLEDGMENTS

The authors are grateful to Professor D. Hoagland,Carri, and K. Ghosh for stimulating discussions, P. Butlerassistance with the SANS measurements at NIST and DSchwahn for neutron beam time and hospitality in Ju¨lich,Germany, and C. Stafford for synthesis of parent poly~sty-rene!s (N5234, 577, and 962!. V.M.P. acknowledges the financial support from a National Research Service Award TGM08515 from the National Institutes of Health. Acknowedgment is made to NSF Grant No. DMR 9970718 andMRSEC at the University of Massachusetts. We acknoedge the sponsored research by the Division of MaterScience and the Laboratory Directed Research and Devement Program of Oak Ridge National Laboratory, managby UT-Battelle, LLC, for the U.S. Department of Energunder Contract No. DE-AC05-00OR22725. ExperimentsY.B.M. in Julich were supported in part by the Alexandvon Humboldt-Stiftung, Germany.

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