Flory exponent of the chain of the expanding polyion gelShigeo Sasaki, Hiroaki Ojima, Kumi Yataki, and Hiroshi Maeda Citation: The Journal of Chemical Physics 102, 9694 (1995); doi: 10.1063/1.468788 View online: http://dx.doi.org/10.1063/1.468788 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/102/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Binary hard chain mixtures. I. Generalized Flory equations of state J. Chem. Phys. 105, 7669 (1996); 10.1063/1.472550 Generalized Flory equations of state for hard heteronuclear chain molecules J. Chem. Phys. 104, 5220 (1996); 10.1063/1.471149 The excludedvolume expansion in polymer chains: Evaluation of the Flory exponent in the Gaussianapproximation J. Chem. Phys. 87, 1817 (1987); 10.1063/1.453194 Equation of state for chain molecules: Continuousspace analog of Flory theory J. Chem. Phys. 85, 4108 (1986); 10.1063/1.450881 On the validity of the Flory–Huggins approximation for semiflexible chains J. Chem. Phys. 74, 2596 (1981); 10.1063/1.441332

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Flory exponent of the chain of the expanding polyion gelShigeo Sasaki, Hiroaki Ojima, Kumi Yataki, and Hiroshi MaedaDepartment of Chemistry, Faculty of Science, Kyushu University 33, Hakozaki, Higashi-ku, Fukuoka812, Japan

~Received 15 November 1994; accepted 20 March 1995!

A theory to describe the swelling behavior of polyion gels is presented on the basis of de Gennestheory for the conformational entropy of chains under traction and ideal Donnan approximationcombined with the counterion condensation of polyelectrolytes for the osmotic expansion force. TheFlory exponentn is regarded as a parameter in theory to describe the effect of the intrachaininteraction on the fractal nature of the chain conformation. The volume of poly~acrylic acid! gel wasmeasured as a function of the ionization degree, the salt concentration, and the polymerizationdegree of chains between crosslinks. The analysis of the expansion behavior showed that thenvalues varied from about 0.8 to about 0.6 for the gels having long chains when the salt concentrationchanged from 10 to 100 mM, while for the gels consisting of short chains the exponent was kept toabout 0.8, irrespective of the salt concentration change. The results indicate that the Flory exponentof the expanding polyion gel increases with the strength of the electrostatic interaction betweenionized groups. The volume of a copolymer gel of fully neutralized maleic acid and styreneconsisting of the short chains was also measured as a function of the salt concentration. The analysisof the expansion behavior gaven50.71 in the salt concentration range between 3 and 320 mM incontrast with a value of 0.8 for the poly~acrylic acid! gel under similar conditions. This resultsuggests that the hydrophobic interaction between the phenyl groups reduces then value. © 1995American Institute of Physics.

I. INTRODUCTION

It has been found by Kuhnet al.1 that ionic polymer gelsexpand tremendously at high ionization degrees and that theexpansion degree depends strongly on the salt concentrationin the gel. High osmotic pressures due to the counterionssurrounding the polyions have been considered to expand thegel.2–4 However, the effect of the electrostatic interaction onthe contractile force of the chain has been scarcely taken intoaccount in the prevailing theories.2–4 The electrostatic forcecostrains the chain to expand and reduces the conformationalentropy. As reported previously,5 the observed expansion ofcarboxymethyldextran gels consisting of stiff chains hasbeen successfully explained by the effect of the constraint.

The existing theories2–4 have been based on the statisti-cal mechanics of the infinitely long chain. However, the de-gree of the polymerization of the chains between cross link-age in the gel are not necessarily long enough for the theoriesto be applied. Long range effects of the electrostatic forcemake the statistical mechanics complex and difficult to bedealt with. We consider the problem in the following way.The end-to-end distanceRe of a chain is related to the con-tour length of the chainl by Re; l n because of the fractalnature of the random flight chain. Flory has theoreticallyfound thatn is 0.6 for the self-avoiding random flight of longchains.6 The Flory exponentn is 0.5 for an ideal randomflight chain and 1 for a rod. Whenl is as short as a few stepslength of the random flight, the trajectory of the flight ap-pears to be a rod and then value in this case might be veryclose to 1. The random flight of the chain bears analogy tothe Brownian movement. The fractal dimension of the move-ment trajectory~which corresponds ton21! has been found tovary continuously from 1 to 2 with changing the observationscale from a length smaller than the mean free path~which

corresponds to one step of the random flight! to that muchgreater than the latter.7 Electrostatic repulsive force mightlengthen one step of the random flight, that is, it might in-crease the number of monomeric units in one statistical seg-ment. If it does, the number of segments in one chain or thenumber of steps in the random flight decreases and thenvalue of chain increases. A computer simulation8 of molecu-lar dynamics for linear polyelectrolytes in the salt free con-dition has demonstrated that the Flory exponentn varies withthe chain length. For short chains, it is about 0.8 which isgreater than 0.6, the value predicted by Flory.6 The result ofthe simulation encourages us to describe the expansion be-havior of the polyion gels in terms of the Flory exponent.

We have proposed a theory5,9 to describe the expansionbehavior of gels on the basis of de Gennes theory for theconformational entropy of chains under traction and idealDonnan approximation combined with the counterion con-densation of polyelectrolyte for the osmotic contribution. Inthe theory,5,9 the dimensionalityd of the space for the ran-dom flight of the chain was regarded as a variable. We usedthe relationn5~3/d12! for estimatingd. The low d valuesreported previously would be more properly interpreted asthe large Flory exponent of the expanding chains. In thepresent paper, a refined theory will be described in terms ofthe Flory exponentn as a parameter. The theory gives theexpansion volume as a function ofn and is applied to ana-lyze the expansion data of the poly~acrylic acid! gel ~PAAgel! and the copolymer gel of maleic acid and styrene~STMA gel!. The expansion behavior of the PAA gel wasmeasured as a function of the ionization degree~a! of ioniz-able group under different NaCl concentrations (Cs) and fordifferent degrees of polymerization (Dp) of cross linkedchains. The expansion behavior is well described by adjust-

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ing only one parametern. The expansion behavior of thefully ionized STMA gel was also measured as a function ofCs for examining the effect of the hydrophobic interaction onthe Flory exponent.

II. THEORY

The force balance between Donnan expanding osmoticforcePos and the contractile elastic forcePel is written as

Pos1Pel50. ~1!

In Eq. ~1! the mixing entropy contribution is neglected sincethe polymer concentration encountered in the present studywas small. In this section, we will describe the formulationof Pos andPel .

A. Osmotic pressure

Under the approximation of ideal Donnan equilibriumcombined with the counterion condensation,2,10 the Donnanosmotic pressure is given by11,12

Pos5kT2CsH F11S aeCpm

2CsD 2G1/221J , ~2!

wherekT, ae , andCpm are the Boltzmann factor, the effec-tive ionization degree of the ionizable group, and the poly-mer concentration on the ionizable group basis. According tothe theory of Katchalsky,12 ae is given by

ae512b2

2l0, ~3!

wherel0 is the ratio of the Bjerrum length to the distancebetween neighboring ionizable groupsb, that is,l05(e0

2/bekT) andb is given by12

al0512b2

11b coth@2 12 ln~pba2Cpm!#

. ~4!

Heree0, e, anda are the elementary charge, dielectric con-stant of solvent, and the radius of polymer skeleton, respec-tively.

B. Elastic force of cross linked chains

The elastic forcePel is given by Eq.~5! in terms of theconformational free energyFconf and the gel volumeVt :

Pel52]Fconf

]Vt5Cpm2

Nm

]Fconf

]Cpm, ~5!

where

Vt5Nm

Cpm. ~6!

HereNm denotes the total number of ionizable group in thegel. According to the theory of de Gennes,13 Fconf for thetractile chain is described as

Fconf5npkTS RRFD d

, ~7!

wherenp , R, andRF denote the number of polymer chains,average end-to-end distances of chains in the gel and in theFlory state defined by de Gennes13 ~in the absence of tractileforce!, respectively.

In the Flory state, the chains are free from the tractileforce caused by the osmotic pressure. It is important to note,however, that this does not necessarily require the absence ofthe electrostatic interaction between any pair of charges. Forexample, electric repulsion between the neighboring seg-ments may not contribute to the expansion of the chain. Gen-erally, we should expect the effect of electrostatic interactionto remain in the Flory state. In the Flory state, the fractalnature of the random flight holds and gives the followingscaling law forRF ~Refs. 14 and 15! in terms of the length ofa monomeric unitb:

RF5bDpn . ~8!

The ratio of the contour lengthL to the one step length of therandom flight determinesn in Eq. ~8!. For the gel,Dp isgiven by

Dp5L

b S 5Nm

npD . ~9!

In the present study, the dependence ofn on Cs , a, andDp

has been experimentally examined.According to the theory of de Gennes, the Flory expo-

nent is related to the parameterd as13

d5~12n!21. ~10!

Combining Eqs. ~7! and ~5! and using the relationVt5AVR

3np ~AV is a constant!, we obtainPel as

Pel52kTd

3AV

2d/3b2dDp22d/3Cpm

12~d/3! . ~11!

The relation describing the expansion of the ionic gel, Eq.~1!, is finally cast into the following relation:

Pos

Cpm5kT

d

3AV

2d/3b2dDp22d/3Cpm

2d/3. ~12!

Equation~12! has successfully described the expansion be-havior of carboxymethyl dextran gels.5 This means that theshort-range effect on the conformation of polyelectrolytescan be well described with the Flory exponent.

III. EXPERIMENT

PAA gels were prepared by radical copolymerization inaqueous solutions of acrylic acid~10 vol %! andN,N8-methylenebis~acrylamide!. The polymerization wasinitiated by ammonium peroxydisulfate and was carried outin an oven at 70 °C. The STMA gel was prepared by radicalcopolymerization in tetrahydrofuran~THF! solutions of ma-leic anhydride~2 M/L!, styrene~2 M/L!, and divinylbenzene~0.04 M/L!. The polymerization was initiated bya,a8-azobis~isobutyronitrile! and was carried out at 60 °C. Val-ues ofDp , the average numbers of carboxyl groups in the gelchain, were assumed to be

Dp5@acrylic acid#/~2•@bis~acrylamide!#!, ~13a!

9695Sasaki et al.: Expanding polyion gel

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Dp5~2•@maleic anhydride#!/~2•@divinylbenzene#!,~13b!

where @acrylic acid#, @bis~acrylamide!#, @maleic anhydride#,and@divinylbenzene# were the concentrations of acrylic acid,N,N8-methylenebis~acrylamide!, maleic anhydride, and divi-nylbenzene, respectively. TheDp values of PAA gels were60, 110, and 490 and that of STMA gel was 50. The PAAgels were synthesized in a plate form 1 mm thick. TheSTMA gel synthesized in a glass tube~3 mm diameter! andcut into a rod form~3 mm length! were rinsed thoroughlywith THF solution. The hydrolysis of maleic anhydride wascarried out by soaking the gel into the mixture of THF~100mL! and ammonia water~10 mL! for 1 day. The gels wererinsed thoroughly with distilled water after soaking in 1 MHCl solutions for 12 h. Then the gels were freeze-dried andused. All chemicals used were of reagent grade.

Gel swelling experiments were carried out at 2560.5 °C.A small piece of dry gel~a few mg! was suspended in avolume ~20 cm3! of NaCl solution of a given concentration.Neutralization of the gels was carried out with NaOH solu-tion. The PAA gels were partially neutralized, while theSTMA gels were fully neutralized. The gels were equili-brated with the solution phase for more than a week. It tookat least several days for equilibrium to be attained. Values of

a for the PAA gel were calculated from the added amount ofNaOH andpH of the solutions on the basis of the neutralitycondition. That is,

a5~@Na1# t1@H# t!/@COOH# t , ~14!

where @Na1#t , @H1#t , and @COOH#t are the total moles ofadded NaOH to the solution, the total mole of H1 ~in thesolution1the gel! estimated from thepH measured and thetotal moles of carboxyl groups in the gel, respectively. Thetotal moles of carboxyl group in the gels for the PAA gel andfor the STMA gel were estimated, respectively, from the drygel weightWd and from thepH titration with assuming thatthepH increment pera was the maximum ata50.5. A kindof jump of pH at a50.5 has been commonly observed forthe polymer composed of maleic acid.16,17For the estimationof @H1#t , the activity coefficients of H1 were approximatedwith that in the absence of the gel. In Eq.~14! the contribu-tion from OH2 to a was neglected. The gel volumeVt wasdetermined by the relationVt5Wg/r, whereWg andr werethe weight and the density of the gel and the latter was as-sumed to be identical with the density of the outer solution.Gel densities of low volume fraction of polymer chains~0.02at most! can be approximated to that of the outer solutionphase. The liquid attached on the surface of the gel wascarefully removed and then the gel was weighed to deter-mineWg . Values ofVt were in the range of 0.0005–0.002dm3 depending onCs anda. Cpm was determined using therelation

Cpm5Wd /~Ma•Vt!, ~15!

whereMa andWd were the molecular weight per carboxylicacid of the gel polymer and the weight of dry gel, respec-tively.

IV. RESULTS AND DISCUSSION

Dependencies of the PAA gel volume per mole of mo-nomeric unitV ~51/Cpm! on a are shown in Figs. 1~a!–1~c!for different Cs . The volumeV increases witha but de-

FIG. 1. Salt concentration dependencies of the PAA gel volumes per mono-meric unit as function ofa. Dp values of gels: ~a! 60; ~b! 110;~c! 490. Thedata forDp5110 were taken from the previous study~Ref. 9!.

FIG. 2. Salt concentration dependencies of the STMA gel volume ata51.

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creases withCs . Under a given set ofa andCs , V increaseswith Dp . This fact suggests that the estimated values ofDp

based on the feeding ratio of the monomer to the cross-linking agent parallel the average degrees of polymerizationof gel chain. It should be noted that the increment ofV witha is reduced whena is greater than about 0.3. This is con-sistent with the prediction of the counterion condensationtheory that the increment ofae with a becomes very smallwhen al0 is greater than 1.10,12 In the present case,l0 atroom temperature is about 2.8 forb50.25 nm on the basis ofthe chemical structure. TheV values at highera strongly

depend onCs as shown in Figs. 1~a!–1~c!. This fact indicatesthat the reduced increment ofV at the higha is not causedby the full extension of the chains. It is to be noted that thebehavior was commonly observed for allDp of the PAA gelexamined.

The dependence of fully neutralized STMA gel volumeV on Cs is shown in Fig. 2. The dependence is very similarto that of PAA gel. Theae value of the STMA gel in thepresent study is considered to be constant, sinceal0 isgreater than 1 for thea value of the fully neutralized STMAgel.

FIG. 3. Salt concentration dependencies of the power relations ofPos/Cpm againstCpm for PAA gel.Dp values of gels: ~a! 60; ~b! 110; ~c! 490. The datafor Dp5110 were taken from the previous study~Ref. 9!. The solid lines are the least-squares fit of the data to Eq.~12!.

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We expect the power relation betweenPos/Cpm andCpm

from Eq.~12!. Values ofPos/Cpm are plotted againstCpm forthree samples in Figs. 3~a!–3~c!, which show the power re-lation. Here, we assumed thata50.6 nm~Ref. 18! for PAA.Values ofd andAV were obtained by the least mean squarefit of the data to Eq.~12!. The Flory exponents for the chainof swollen gels were evaluated from thed values through Eq.~10!. They are shown in Table I and Fig. 4. Then values tendto increase asCs decreases. It is interesting that the expo-nents depend on bothDp andCs . At high Cs , the n valuestend to increase with decreasingDp . Reasonable values ofAV ~about 1! were obtained as shown in Table I. It should benoted that small systematic deviations of the data points fromthe calculated values are observed in Figs. 3~a!–3~c!. Thedeviations seem comparable with the experimental errors.Thus we can tentatively say that thea dependence ofn isvery small in comparison with the dependencies ofn onDp

and Cs in the ranges ofa measured~a50.1–0.9! and Cs

used~Cs51–100 mN!. However, it should be noted that thedeviations are relatively larger at a lowCs value such as 1mN. This might suggest thatn depends ona in a salt freecondition.

It should be emphasized that an important finding in thepresent study is then value is greater than 0.6 as shown inTable I and Fig. 4. According to the theory of Flory,6 n is the

universal constant of 0.6 for long polymer chains in a goodsolvent. This is not necessarily the case for short chains. Aconstantn value should give the same slope in log–log plotof Pos/Cpm andCpm . The difference of the slopes in Figs.3~a!–3~c! is attributed to the variation of then value in thepresent study. There could be another way to explain thedifference. According to Eq.~12!, a different slope couldappear ifAV and/orb would vary withCpm

h . If this is thecase, the slope smaller than25/6, the value corresponding ton50.6, is attributed to the positive value ofh. The positivevalueh means thatAV and/orb decreases with the gel vol-ume. There is no reason to consider that the swelling forcedecreasesAV and/or b. In the framework of the presenttheory, it is most plausible that the variation ofn value givesthe different slopes in Figs. 3~a!–3~c!.

It might be worth mentioning the relation betweenCs

dependencies of then values and the persistence length ofthe polyelectrolyte. Recent light scattering studies19,20 haveelucidated that the persistence length of flexible polyelectro-lyte decreases with increasingCs . According to the result ofForster et al.,19 the total persistence length~the sum of in-trinsic and electrostatic contributions! reduces from about 5to 2 nm with increasingCs from 50 to 100 mN while itchanges from about 10 to 8 nm with increasingCs from 1 to10 mN. The number of steps in the random flight of thesegment increases with the ratio, the contour length to the

FIG. 4. Salt concentration dependencies of the Flory exponentn for the PAAgel. The data forDp5110 were taken from the previous study~Ref. 9!. Asolid line shows the case of the long chain limit~n50.6!. The dotted linesare drawn as a guide to the eye.

FIG. 5. Power relation ofPos/Cpm againstCpm for STMA gel. A solid lineis the least-squares fit of the data atCs53;320 mN to Eq.~12!.

TABLE I. Cs andDp dependencies ofd and n values. Thed, n, andAV values were obtained from theleast-squares fit of the data to Eq.~12!.

Dp Cs 1 mN 5 mN 10 mN 25 mN 50 mN 100 mN

490 d 4.260.3 4.160.1 3.060.2 2.460.1 2.660.3n 0.7660.02 0.7660.01 0.6760.02 0.5860.02 0.6260.04AV 0.360.1 0.260.1 0.660.2 1.260.5 0.960.3

10 d 5.360.4 4.960.5 3.860.2 3.560.2 3.160.1 2.460.2n 0.8160.02 0.8060.01 0.7460.01 0.7160.01 0.6860.01 0.5860.03AV 2.860.5 2.560.4 4.061.0 3.661.2 4.561.0 9.165.0

60 d 5.360.3 5.060.1 4.060.2n 0.8160.1 0.8060.01 0.7560.1AV 1.260.8 1.060.6 1.160.6

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persistence length of the chain. Then value decreases withthe number of steps or the ratio. The result ofCs dependen-cies of then value shown in Fig. 4 can be qualitatively ex-plained by the ratio. The ratio for the gel ofDp5490 ~thecontour length of the cross linked chain5122.5 nm! in-creases from 15 to 25 with increasingCs from 10 to 50 mNand correspondingly the observedn value decreases from0.76 to 0.58. The observedn value changes from 0.74~Cs510 mN! to 0.58~Cs5100 mN! for the gel ofDp5110~the contour length527.5 nm! corresponds to the change ofthe ratio from 3 to 14. The observedn value for the gel ofDp560 ~the contour length515 nm! is substantially constantabout 0.8 while the ratio increases from 1.5~Cs51 mN! to7.5 ~Cs5100 mN!. The transitional ratio from the highnvalues to that of the long chain limit~n;0.6! seems to in-crease withDp . The statistical mechanics of long chains inthe gel is similar to that of the chain in the solution. Decreas-ing Dp , the effect of cross links on the statistical mechanicsof the chain is considered to increase.

The power relation betweenPos/Cpm and Cpm is alsoobserved for the STMA gel atCs between 3 and 320 mM asshown in Fig. 5. In the estimation ofPos, we assumeda50.6nm anda51. We obtainedn50.71 andAV50.85 from thepower relation. It is interesting that then value obtained forthe STMA gel is smaller than that of the PAA gel~n50.75–0.81! of the corresponding condition~Dp560 andCs51;100 mN!. This might be due to the hydrophobic ef-fect of phenyl groups in the STMA gel. The interaction be-tween the hydrophobic phenyl groups along the chain mightpartially counterbalance the electrostatic repulsive force.Thus, the number of monomeric units in a step of the randomwalk of the chain decreases, that is, the number of steps inone chain increases and then value decreases accordingly. Itis to be noted that the estimatedPos values atCs5600 and1000 mM are larger than the expected values from the powerrelation ofn50.71 as shown in Fig. 5. This means thatPel isgreater than that expected by Eq.~11! with n50.71. That is,Pel at Cs5600 and 1000 mM might be given by Eq.~11!with n less than 0.71~d less than 3.4!.

In previous papers5,9 we ascribed thed values higherthan 2.5 to the chain dimensionalityd lower than 3. Thisdimensionality will be better interpreted as the dimensional-ity related to the fractal dimension rather than the dimensionof real space. The Flory exponent is more appropriate fordescribing the fractal nature of the polymer chain. The pre-vious results5,9 will be better understood when they are ana-lyzed in terms of the Flory exponents instead of the dimen-sionality. It might be worthwhile to mention that theu state~n50.5! is realized for the cased54.15 Poor solvents confinethe configuration of the polymer segment within a domaincorresponding compact conformations, in the opposite direc-tion to the effect of the electrostatic interaction. On the other

hand, the electrostatic repulsive interaction distributes thesegments in a more expanded space. This could be said todecrease the dimensionality.

In the present study, the force contribution from mixingthe solvent and the chain segments is neglected because thisterm is very small as compared withPel andPos in Eq. ~1!owing to very low volume fractions of the chain segment inhighly expanded polyion gels. The maximum volume frac-tion encountered in the present study was estimated to be0.02. It has been reported for the compressional osmoticmodulus of PAA gels that at a volume fraction of about 0.08the neutral network contribution is much smaller than thatfrom free counterions.21 This finding clearly indicates thatany contribution other than the electrostatic one to the os-motic pressure is quite small in polyion gel systems.

ACKNOWLEDGMENT

This work was partially supported by a Grant-in-Aid~No. 02403004! from the Ministry of Education, Science,and Culture of Japan.

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