Static scattering from polyelectrolyte solutions

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Static scattering from polyelectrolyte solutions

Ulrike Genz, Rudolf Klein, Mustapha Benmouna

To cite this version:

Ulrike Genz, Rudolf Klein, Mustapha Benmouna. Static scattering from polyelectrolyte solutions.

Journal de Physique, 1989, 50 (4), pp.449-460. �10.1051/jphys:01989005004044900�. �jpa-00210928�

Static scattering ^from polyelectrolyte ^solutions

Ulrike Genz (1), Rudolf Klein (1) ^and Mustapha ^Benmouna (2) (1) Fakultdt für Physik, Universität Konstanz, 7750 Konstanz, West Germany (2) INES Sciences Exactes, Physics department, Tlemcen BP 119, Algeria (Reçu le 28 décembre 1987, révisé le 16 août 1988, accepté ^{le 17 octobre} 1988)

Résumé. 2014 On étudie théoriquement l’intensité diffusée de manière statique par

système composé ^de polyions filiformes chargés, de contre-ions et de co-ions. La configuration ^des polyions ^est décrite par

facteur de forme de chaîne aléatoire

longueur ^de persistance

et

paramètre de volume exclu qui ^sont tirés de la théorie d’Odijk. ^{On tient} compte ^{de la} condensation des contre-ions d’une part pour les interactions ^entre segments ^de polymère appartenant à des macromolécules distinctes, ^d’autre part pour la configuration ^{d’un seul} polymère. ^En représentant ^les segments ^de polymère (monomères)

^des sphères ^dures chargées,

peut exprimer les corrélations entre segments de chaines différentes et ions libres

fonction des tailles et charges des constituants. Dans cette description ^{à trois} composants,

obtient des résultats pour la diffusion par des polyions ^{et leurs} contre-ions ;

examine leur rapport

^données expérimentales.

Abstract. 2014 The static scattering intensity ^from

system composed ^of charged, ^coiled polyions,

counterions and coions is investigated theoretically. ^The polyion configuration is described by

wormlike chain form factor having

persistence length ^and

excluded volume parameter obtained from Odijk’s theory. Counterion condensation is accounted for in both, the interactions between polymer segments

^different polymers ^{and free ions, and the} single polymer ^form.

Modelling ^the polymer segments,

monomers,

charged ^hard spheres ^{allows to} express the correlations between segments

different chains and free ions in terms of sizes and charges ^of

the constituents. Within the three-component description ^results

^the scattering ^from polyions

and from counterions

obtained and their relation to experimental data is discussed.

Classification

Physics ^Abstracts

61.12Bt

61.25Hq

^82.70Dd

1. Introduction.

Static properties ^of aqueous solutions containing charged polymers ^{have been a} subject ^of

intensive research [1]. Polyelectrolytes, ^which may carry charges ^on every ^{monomer, as} for instance NaPSS, ^{or which} may be less strongly charged, e.g. ionomer solutions, ^differ considerably from neutral polymers ^with respect ^{to their} scattering properties. ^{This can be}

attributed to long-ranged electrostatic interactions. For salt-free systems scaling theory [2, ^3, 4] ^{has been} applied ^to give ^some theoretical understanding ^{of the} complex ^{behavior. An} approach ^{due to} Koyama [5] is also aimed at salt-free systems. ^The approach described here differs from these theories in several respects : Direct correlation functions between

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005004044900

monomers, ^or segments ^on different chains are introduced to describe interchain correlations in terms of interactions between _monomers, which are characterized by charges and sizes of the segments ^under consideration. It is straightforward ^to employ a multicomponent description here, ^{so that the} polyion, counterion and coion species ^can be treated on the same

basis. A further advantage ^{of this} type ^{of a} multicomponent description ^{arises from the} possibility ^to study separately the various contributions to the total scattered intensity. ^As

demonstrated experimentally [6] ^{it is} possible ^to investigate ^the partial contribution arising

from the scattering by counterions, ^{which can then be} compared with theoretical results from

our multicomponent theory. ^The approach may be viewed as an approximation ^{to the more} general theory presented ^{in reference} [7]. ^A somewhat similar approach ^{has been} applied by

Benmouna and Grimson [8, 9]. The relative simplicity ^{of the} calculation results from an

approximation : ^{It was} shown in reference [7] ^{that the} segment-segment correlations on one

polymer ^are coupled ^{to those on different} polymers. ^This coupled problem of self and distinct correlations is simplified by constructing ^{the self} part (which ^{is the} single-polymer ^form factor) separately, taking the influence of other polymers and of salt ions into account in an

approximate way. By ^this procedure ^the theory developed ^{in reference} [7] loses its self-

consistency, ^{but, as the} problem ^is very complex, this is considered to be a good starting point

to understand polyelectrolyte scattering ^{in terms} of interchain interactions. Because nematic interaction is not accounted for, ^the approach is restricted to coiled polymers, ^{which means}

that either the polymer concentration or the salt concentration are supposed ^{to be in an order}

of magnitude ^{to allow for a more or less coiled} configuration.

The _paper is organized ^as follows : section 2 briefly sketches the theory. ^{In section 3 we will}

summarize how the single-polymer configuration is treated : Manning’s [10] theory ^{will be} employed ^{to account for} counterion condensation, Odijk’s [3, 11] theory will be used to

obtain the radius of gyration ^{of the} polyion ^{and a} wormlike chain form factor [12] ^{is assumed}

to describe the polymer shape. ^{Section 4} deals with the results obtained for the polyion- polyion partial intensity, ^{which has been studied} in many experiments. ^{In section 5 some}

results on other partial intensities are discussed.

2. Theoretical description.

We consider a system containing polyions, counterions and coions. The solvent provides ^a

uniform background. ^The scattering intensity, ^which may be measured in a neutron scattering experiment, ^{can be} decomposed ^into partial intensities I,,,,e (q ) :

Here, species a ^{and /3 are} polymer segments ( a = 1 ), non-condensed counterions

(a

2 ) and coions (a = 3). ao: denotes the scattering length ^{of a unit} belonging ^to species

a. As long ^as their sizes are smaller than the inverse scattering ^vector q-1 ^the q-dependence of ^{can be} neglected. The number density ^of condensed ions is taken from Manning’s theory [10]. ^{Because a more concise} knowledge ^{of their} spatial distribution is not easily available, ^it

seems to be justifiable, ^at least for small and intermediate values of the scattering vector, ^to treat them as a part ^{of the} polyion ^{and thus} reducing ^{the bare} charge ^{to an effective} charge.

Therefore, ^{in the} case, when all monomers are charged and counterion condensation occurs,

a polymer segment ^{is chosen to be} a monomer including ^a corresponding fraction of condensed counterions. For a weakly charged polyion ^this problem ^{does not} arise and a

segment ^{is assumed to be a} part of the chain carrying ^{a unit} charge. This choice has the

advantage ^that pair interactions between units on different polymers ^{are identical.}

In (I, Eq. (1.9)) partial ^{structure factors} Sa/3 (q ) ^were introduced by

where N is the number of segments per polyion, cl

n, N/V ^the segment ^number concentration, and C2, c3 ^{are number} concentrations of non-condensed counterions (2) ^and

coions (3). P (q ) is the form factor resulting ^{from the} segment distribution on an individual

polymer :

where rn and r. denote the positions ^of segments n ^{and m on the same} polymer. ^The Sa/3 ^{are related to «} particle » total correlation functions, ÎIa13’ ^{(see I, Eq. (1.10) and (I.36))}

This _may be regarded ^as definition of Îlaf3. Under certain approximations ^{described in} (I),

« direct correlation functions » ê al3 ^{can be found, so that the} relation between Îl al3 ^and ê af3 ^{is formally} identical tô an Ornstein-Zernike (OZ) equation

It is straightforward ^{to solve} equations (4) ^and (5) ^for Saf3 ^{in terms of} é p ^{by using the fact}

that the matrix S, having ^elements SaP, ^{is the} inverse of the matrix with elements

[8af3 - (ca Cf3)1/2 C a b ] . ^The êaf3 ^{are related to} direct correlation functions C (d ) between units

belonging ^{to different} particles ^of species a ^and 8. ^{If a =} f3

1, these units are segments ^on different polymers, having ^{indices n, m to} indicate their position ^{on a} polyion, or, if

« =

1, 3 ~ 1, these units are segments ^at position n ^{and free} ions, ^or if a =1= 1, 8 = 1, these units are free ions. The C ab ^{can be expressed as (see I, Eq. (1.22), (I.33))}

Neglecting ^the dependence ^of the direct correlation functions on the segment ^index

equation (6) simplifies ^to

where cab (a, ^b

^{1, 2,} 3 ) denotes the direct correlation between arbitrary ^{units of} species

a

and 3. These units are now modelled as charged ^hard spheres. ^The polymer segments ^have

a diameter and an effective charge Zeff . e, (Zeff>- 0), ^{and the} (free) ^{ions have a diameter}

d and charge ^{± e.} The direct correlation functions cab (q) ^are approximated by ^direct

correlation functions known for charged ^hard spheres. ^{If u}

d, the Waisman Lebowitz [13]

solution of the mean-spherical approximation (MSA) ^{has been} employed. ^{For the more} general ^case

=A d ^the solution of the MSA obtained by ^Blum [14] is used. Since some

analytical approximations will be derived for zero-angle scattering, ^{we note in} particular ^that

these MSA expressions ^{reduce to}

for vanishing q. Here, Q = e 2/ ( ekB T ) ^{is the} Bjerrum length (-7 Â), e ^{the dielectric}

constant of the solvent and kB T the Boltzmann factor.

The procedure of the calculation of the partial scattering intensities is the following : ^direct sphere correlation functions caf3 (q ) ^{and the} polyion form factor (which ^{will be discussed in the}

next Sect.) ^{are inserted in} equation (7) ^to give ê af3 (q). ^Knowing ê af3 (q), equations (4) ^and (5) ^{can be} solved for Saf3 ^{in terms of} Thé ^partial intensities las ^{can than} be obtained from equation (2).

The relation to other theories becomes more transparent ^when expressing Sll ^{in the form}

where ceff (q) ^is independent of the form factor, ^{but it} depends ^on concentrations and interactions between the subunits. For a salt-free system ^{one obtains}

while for a salt-containing system ^the corresponding expression ^{is more} complicated, but may also be calculated from equations (4), (5), (7). Inserting (9) ⁱⁿ (2) ^the polyion partial scattering intensity takes the form

If, instead of using equation (10), Ceff (q) ^is approximated by - f3 Ueff (q), ^which corresponds

to the MSA on the level of a description including ^the polyions alone, ^{and if} Ueff ^{is taken to be}

the Debye-Hückel interaction, equation (11) ^{reduces to a} result discussed by ^Jannink [1]. ^This procedure ^{is not followed} here ; ^{we use} equation (10) with small-ion correlations including

finite size effects. Therefore, ^{we can relate} Sll, ^{and other} partial ^{structure factors as} well, ^to pair interactions between segments ^and/or (free) ^{ions. Because salt ions can be} accounted for in a three-component description, information about salt-containing ^{solutions are also}

obtained.

3. Single polymer ^behavior.

An important ingredient ^{of the} expression ^{for the} scattering functions is the single polymer

form factor, ^{as can be} seen, for instance, ^from equation (11). As discussed earlier, ^{the form}

factor should be obtained from the self-consistent formulation, ^which couples ^{the self-}

correlations to the distinct ones. In lieu of a solution of this coupled problem ^{we will now}

construct the form factor separately, following ^known results. The form factor as defined in

equation (3), describes the configuration ^{of the N} segments ^{of a} polymer. ^One requirement

on P (q ) obviously ^is

It was assumed that the polyion ^{has a coiled} shape. ^{Therefore we assume that} P (q ) ^{has a} Debye ^{form for} small q :

where u

(qR g )2 ^and Rg is the radius of gyration. Still, ^{as the} polyion ^is charged, ^it may be stiff on the length ^{scale of several monomers} leading ^to

for intermediate q-values. ^The requirements ^of equations (13) ^and (14) ^{can be met} by ^a

wormlike chain form factor which has been calculated by Koyama [12]. ^For q-vectors probing

distances smaller than the segment size correlations between segments ^become unimportant

and the decrease of the scattered intensity, equation (1), ^{is due to the structure of an}

individual segment which is characterized by ^its scattering length. ^{For a} large ^{number of}

monomers N equations (12)-(14) ^can be combined to

where the 1/N ^{term is} irrelevant for intermediate q-vectors ^{but is} kept ^to ascertain the correct

limiting behavior for a chain composed ^{of a} finite number of segments. ^The explicit ^{form of}

the wormlike chain form factor can be obtained from reference [12]. ^In the calculation we

employ equation (15) ^to incorporate ^the dominant features expected for the distribution of the segments, ^{or monomers.}

The radius of gyration Rg characterizes the size of the polymer, ^{which was found to} depend strongly ^{on its} surrounding [15, 16], ^{that is, on the} polyion and salt concentration. For wormlike chains Rg is calculated in terms of a persistence length PP ^{and a chain expansion}

parameter as. To have an estimate on their magnitude Odijk’s theory ^is employed. ^The persistence length [3, 17] is described as a sum of an intrinsic persistence length fo, ^{and an} electrostatic contribution ^which depends ^on polymer and salt concentration. It

was given by Odijk [3] ^as

and

where o- denotes the contour distance between two charges, ^{which for} simplicity is assumed to be equal ^{to the} segment diameter. K -1 is the Debye screening length ^{due to} non-condensed counterions and coions :

According ^to Manning [10] condensation occurs if the separation ^{between two} charges ^{on the}

polyion chain o- is smaller than the Bjerrum length Q. ^If

Q, ^a charge of the order of

magnitude (1 - /Q ) e per segment, having a ^bare charge ^{+ e, cannot be diluted} away from the polymer and thus reduces the charge ^{of a} segment ^{to an effective value} Ze ff e

ue/Q. The number density of non-condensed counterions, ^{which are} dissociated from either the polyion ^{or the salt} molécules, is obtained from the electroneutrality ^condition

The chain expansion ^can be described by ^an excluded volume parameter Zep ^{which was} given by Odijk and Houwaart [11] ^as

where L

N eT denotes the contour length. Koene und Mandel [15] employ a ^different expression ^for Zee, but whether we use their expression ^or equation (19) ^{was found to have a} negligible ^{effect on} the results given ^{in the} following sections. The excluded volume parameter determines the chain expansion factor as by the Yamakawa-Tanaka [18]

expression

The radius of gyration is then calculated from

with

where y

f p/ L. ^{We want to point out that} equations (16)-(21) merely ^{serve to} give ^an

estimate of Rg ^{here, which is necessary to get some numerical results.} Experimental ^results [15, 16] indicate that it is insufficient to neglect the influence of polyion and salt concentration

on the size of the polymer ^{and that} Odijk’s theory gives ^a guide to account for these influences.

4. Polyion-polyion partial intensity.

Many scattering experiments [1, ^{6, 19,} 20] have been aimed at the investigation ^{of the} polyion structure. Therefore we will first discuss results for Il, (q ). ^In figure 1, Il, (q )/cl ^{in a}

salt-free system is shown for various polymer concentrations. Our model system consists of

polyions having ^N

^{1 000 monomers with a} diameter o-

2.5 Â and counterions of the same

size. According ^to Manning’s theory ^{the bare} charge ^{of the} monomers, + e, is reduced to an

effective charge Zeff. e

Q/ u . e ’" ^{0.36 e.} The intrinsic persistence length Qo ^{was assumed to}

be 10 Â. With the results from equations (16)-(21) the radius of gyration ^{was obtained as}

-

470 Â for the lowest and - 220 Â for the highest concentration _cl, shown in figure ^1.

Therefore the form factor exhibits a 1 /q dependence ^{in the} q-range of the maximum

intensity. The value of I11 (q

0 )/cl is, ^{to a} very good approximation, independent ^of

cl. From the behavior of the direct correlation functions for small _q, equation (8), ^{the zero-} angle scattering intensity ^{can be} approximately calculated as

where 0 denotes the volume fraction of the solute, ^{that is} polyions and counterions in this

case. The independence ^of I11 (q

0 )/cl ^on cl agrees with experimental findings [1]. ^The

Fig. ^1.

lu (q )/C1 ^for

^salt-free system ^is shown for various polyion concentrations : (a) ^0.01 monomol/l, (b) ^0.02 monomoln, (c) ^0.03 monomoln, (d) ^0.05 monomolll, (e) ^{0.07 monomoln,} (f) ^0.1 monomol/1, (g) ^0.2 monomolll. The parameters

^N = 1 000, Zeff

^{0.36, o- = d}

^2.5 Á, fo = 10 ^Â.

reason why ^{the effective} charge determines the q - ^{0 behavior can be} understood from the fact that the long-ranged ^{part of} the interactions are govemed by this effective charge, ^{and not}

the bare charge.

The curves of figure ^{1 show a maximum. The} position ^of this maximum shifts to higher q- values for increasing polyion concentration in qualitative agreement ^with experimental

observations [1, 19]. One may notice that the peak position ^is approximately ^{in the} proper order of magnitude, ^{but the} scaling ^{behavior is} rather qmax’- c 13 in this example ^than Cl/2, ^{which was obtained in the} experiments. ^The c 12 dependence ^is typical for rodlike systems.

It has been well confirmed in measurements by Nierlich et al. [19] ^for polyions having ^{a lower}

number of monomers per chain and thus being ^{closer to} the rod limit. In the theory ^nematic

interaction ^was neglected, which may possibly ^{be a reason for this} discrepancy.

One further notices that the height ^{of the} peak decreases with increasing cl, which has also been observed experimentally [1, 19]. This behavior can qualitatively be understood on the basis of equation (11). For q = q.ax, P (q) ^oc q 1 is independent of cl. Since

and Ceff(qmax) ^{0, we have} Sll (qmax ) ^{1 and} Il, (qmaX )

^{cl NP} (qmax). ^By increasing

cl ^we increase qmax and therefore the value of P (qmaX ) will further decrease. Therefore,

IOqmaX )cl decreases with increasing cl. Here the relative height ^{of the} peak compared ^to 11 ( )I 1 ^{is found to} be somewhat low. The value of the effective charge ^{has a} considerable influence on the position ^and therefore the height ^{of the} peak ^{in such a} way that the peak ^is

shifted to lower q-values ^{for a smaller} charge leading ^{to a} higher ^{value of} III (qmax). ^{One is}

tempted ^to fit the effective charges ^{in order to obtain a better} description ^of experimental

results, ^{but this} procedure contradicts the attitude to consider the effective charges ^as physical quantities.

In the next example ^a system ^with salt is considered. Using ^{the same} parameters ^{as in the}

preceding figure, IU(q)/Cl ^{is shown in} figure ² for various salt concentrations and fixed

polyion concentration. An increase of zero-angle scattering intensity ^{can be observed.}

Qualitatively ^{this can} be understood as a consequence of the screening ^of polyion-polyion

interactions by salt ions and is in agreement ^with experimental findings [19]. According ^to

section 3 the radius of gyration varies form - 350 À to - 200 À in this example. Neglecting

excluded volume effects the zero-angle scattering intensity ^{can be} approximately calculated as

where s

C3/ (Cl Zeff). ^{In the limit s}

⁰ equation (24) ^{reduces to} equation (23), ^{while for} large s, ^{which means} c3 > cl NZ2

the result corresponding ^to non-interacting polymers ^is

obtained. It can be observed from figure ^{2 that a small amount of salt} hardly influences the

peak position ^{in this} example, ^{as it was observed} experimentally ^{as well} [19]. ^{Here one should} keep ^{in mind that} the influence of salt on both, the form factor and the intermolecular interaction, ^{has been taken into account.}

Fig. ^2.

111 (q )/cl ^{for fixed} polyion concentration 0.03 monomol/1 is shown for

system ^which contains salt. The concentrations of the 1:1 electrolyte

(a)

salt, (b) ^0.003 mol/l, (c) ^0.01 mol/l,

(d) ^0.015 mol/1, (e) 0.03 mol/l, (f) 0.04 mol/1, (g) 0.06 mol/1. The parameters N, Zeff, ^{u, d and}

Po

^{the same}

indicated in figure ^1.

As a next example (Fig. 3) ^a moderately charged system ^was investigated. ^{The reason to} study ^this example is twofold : one aspect ^{is to} show that this scheme _may also be used for less

strongly charged polymers ^{if the} segments ^{can be chosen in a} proper way. The second aspect ^is

to show a comparison between the strongly charged system ^{described in} figure ^{1 and a}

moderately charged system. ^{To draw some} comparison ^a polymer consisting ^{of N}

³⁶⁰

Fig. ^3. - I,,(q)lc, is shown for

salt-free system for various polyion concentrations. The system parameters

^N

360,

¹⁰ Â, fo

¹⁰ Â, Zl

Zeff ^{= 1} (no condensation), d

^{2.5 Â. The} polyion particle concentrations correspond ^{to the} particle concentrations in figure ^{1 :} (a) ^3.6

10- 3 monomol/l, (b) ^7.2

^{10- 3} monomol/1, (c) ^1.08

10- 2 monomol/l, (d) ^1.8

^{10- 2} monomol/l, (e)

2.5

10- 2 monomovl, (f) ^3.6

^{10- 2} monomol/l, (g) ^7.2

^{10- 2 monomol/l.}

segments ^with charge ^{+ e, size o-}

10 A and persistence length fo

^{10 A is} considered.

Condensation should not occur here so that the total charge, 360 e, corresponds ^{to the total}

effective charge ^{in the} preceding examples. ^{The radius of} gyration ^was calculated as indicated in section 3 and _ranges from - 515 A - 260 À for the concentrations considered. The form

amplitudes of the individual segments, which are assumed to be spherical, ^{are included to} give figure 3, but this is of minor importance. ^{Here one} may compare ^curves of figure ^{1 and} figure ³ having ^{the same number of} polymer molecules per unit volume, nI/V, ^{but different}

monomer concentration _cl

_n, N/V. The result is quite ^{similar to} figure 1, ^if the relative intensities normalized to I11 (0) ^are compared. ^The absolute values differ mainly because the number of scattering units is smaller in figure ^3. Equation (22) still holds approximately, ^but

excluded volume effects, ^{which are} neglected ⁱⁿ equation (22), ^{have a} stronger ^influence.

When comparing the results given ^{here to} experimental ^{data one should} keep ^{in mind the}

various assumptions ^{which are} inherent in this approach. ^{It cannot be} claimed that a full

understanding ^of polyelectrolyte scattering ^is provided, but various characteristic trends of the experimental ^{results are} reproduced.

5. Other partial ^{structure functions.}

Other partial intensities, e.g. the counterion intensity, ^{have also} been studied in neutron

scattering experiments [6, 21]. ^{The scheme} presented here enables us to calculate partial

structure factors Sa03B2 ^{related to the} polyion, non-condensed counterion and coion species.

Figure 4 shows the result for a system having parameters N, ^(T, Zeff ^{and d as in the} example

given ⁱⁿ figure ^{1 and} containing ^{a small amount of salt.} One notices that the functions

Fig. ^{4. - The structure factors} SaP (q)

^{shown for}

^system containing

^{small amount} of salt : The

polyion concentration is 0.03 monomol/l, the salt concentration is 0.003 mol/1. Parameters for the

polyions and free ions

the same

in figure ^1.

S22 ^and S33, describing correlations between the small ions, ^{are close to} unity ^{and do not}

exhibit a pronounced structure, ⁱⁿ contrast to the experimental observation on the counterion

partial intensity [6]. ^{This shows that the} experimental findings ^{cannot be} explained by ^the scattering ^{from non} condensed counterions alone. S12 ^{is found to be} positive ^{in this} q-region,

but may become negative ^for larger q. 811 ranges from a small value at q

^{0 to} unity ^for large

q. In other examples containing ^{small amounts of salt} the results are qualitatively ^similar.

We will now investigate the total scattered intensity 1 (q) ^{in more detail} taking ^the possibility ^{into account that} scattering from both free and condensed counterions contributes.

In order to include the effects of condensed ions we formally assign ^a scattering length

al

amon ⁺ (1 - Ze ff ) a, ^to each polyion segment, ^where amon is ^{the monomer} scattering length and a, ^the scattering length ^{of a} counterion. Except ^{for low} q-values ^this approximation

may be poor, but ^{as a more} realistic description of condensed counterions is not available, these results _{may be of} interest. Figure 5 shows the results for I (q )/cl, equation (1), ^for

amon = 1, aI = 0 ^and amon = 0, al

^{1. Thus,} the first case corresponds ^to pure polyion scattering ^as investigated ^{earlier, whereas the second case} represents ^the intensity, ^if only scattering ^from counterions, ^{both free and condensed} ones, is observed. The counterion

partial intensity then also shows a peak ^{at a} position being closely ^{related to the maximum of} Ill. ^The experimental ^data [6] ^{also shows a maximum at this} position, ^{but in} contrast to our

results they ^{exhibit a rather} sharp decrease for values of q ^above max’

Some analytical results for the zero-angle scattering intensity may be of interest. Neglecting

excluded volume effects the following expression ^can be derived similarly ^as equation (22),

In the salt-free case, s

0, letting al

aman ⁺ (1 - Zeff) a,, ^one obtains from equation (25)

Fig. ^5. 1 (q )/c1 is shown for the two cases a..

1, al

^0, only ^the polyions ^scatter (---), ^and

amon

0, al

^1, free and condensed counterions scatter (-). ^{The monomer} concentration of the salt-free system is 0.03 monomol/l. Parameters for the polyions and free ions

the

in figure ^1.

indicating ^{that the} charge-charge ^{structure factor, which} is obtained by letting amon

-

ai, vanishes at q

^{0 in} agreement ^with experimental observations [21].

The existence of condensed ions, which may influence the experimental ^results strongly,

poses ^a major problem ^when investigating counterions partial structure, charge-charge

structure or number-number structure factors theoretically [22]. ^The results for these

quantities, from which we only ^{showed the} counterion intensity, ^{seem to be in} qualitative

agreement ^with experiment for small _{q, but} it can be clearly ^{seen that for} intermediate q the approximations ^are insufficient.

6. Conclusions.

The theoretical description ^{for the static} properties ^of polyelectrolyte solutions has been

given ^{in terms of a} multicomponent formalism. The form factor of a single polymer ^{in the}

presence of other polymers and small ions has been taken from Odijk’s theory including ^the possibility ^of condensation. This is used as an input ^to calculate distinct correlations and the scattered intensity. ^{The results are} compared ^{to various} experiments, including scattering from counterions. The trends observed experimentally ^{are well} reproduced by ^this theory, ^but

various details are not found to be in full agreement. ^{This has to be} expected considering ^the complexity ^of polyelectrolytes ^{and the} approximations ^{we had to introduce.}

The description developed ^{here treats the} polymer ^{as a} collection of connected charged

segments interacting among each other and interacting ^with counterions, ^{salt ions and} segments ^{on other} polymers. ^These charged objects ^are of finite size. The values of sizes and

charges assigned ^{to these} objects ^{can be chosen} according ^{to the} particular system ^under

investigation. ^This flexibility allows for the consideration of possible counterion condensation

effects.

Most of the simplifying assumptions, ^{which had to be made, are} particularly ^{reflected in the}

results for scattering wavenumbers q around qmaX. ^{These are} the polyion ^form factor, ^the assumption ^{of condensed} ions, ^the disregard of nematic interactions and the modeling ^{of the} segments ^as spheres. ^{In the} long wavelength limit the results are less sensitive to these

assumptions except ^{for the} magnitude of the effective charge. ^{The main} problem, ^which

remains to be solved, is the self-consistent treatment of the self and distinct correlations as

described by ^the coupled ^{set of} equations (1.29) ^and (1.30) ^{in I. This has to be concluded from}

the missing quantitative agreement ^{of our results with} experimental ^{data. The use of} equation (5) together ^{with current results for the} single-polyion ^{form factor P} (q ) ^{seems to be}

insufficient for a quantitative theory.

Acknowledgments.

M. Benmouna would like to thank the Physics Department ^{of the} University ^of Konstanz for

appointing ^{him as a} guest professor ^{for two months} during ^{the summer of 1987} during ^which

this work was started. Financial assistance from the Deutsche Forschungsgemeinschaft (SFB 306) ^is gratefully acknowledged.

It is a pleasure ^{to thank our} colleagues ^{Dr. G.} Nâgele, ^{Prof. Dr. M.} Medina-Noyola ^and

Prof. Dr. Z. Akcasu for helpful ^and stimulating discussions.

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