# Studies of Electrolytic Conductivity of Some Polyelectrolyte Solutions: Importance of the Dielectric Friction Effect at High Dilution

Authors: Anis Ghazouani; Sondes Boughammoura; Jalel M'Halla
Journal: Journal of Chemistry
 (2013)

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## Abstract

We present a general description of conductivity behavior of highly charged strong polyelectrolytes in dilute aqueous solutions taking into account the translational dielectric friction on the moving polyions modeled as chains of charged spheres successively bounded and surrounded by solvent molecules. A general formal limiting expression of the equivalent conductivity of these polyelectrolytes is presented in order to distinguish between two concentration regimes and to evaluate the relative interdependence between the ionic condensation effect and the dielectric friction effect, in the range of very dilute solutions for which the stretched conformation is favored. This approach is illustrated by the limiting behaviors of three polyelectrolytes (sodium heparinate, sodium chondroitin sulfate, and sodium polystyrene sulphonate) characterized by different chain lengths and by different discontinuous charge distributions.


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## Body

## 1. Introduction

Conductivity is a powerful technique of high accuracy allowing the qualitative and the quantitative detection of ionic species even at low concentrations. On the other hand, it is well known that for electrolytes (or for electrolytes mixtures), it exists a universal linear limiting law relating the equivalent conductivity at high dilution to the square root of ionic strengthI1/2. This concentration effect is caused by two sorts of ionic frictions: the electrophoretic effect and the ionic relaxation effect. This limiting law has been extended in the case of semidilute simple electrolytes [1–4]. In contrast, a completely satisfactory theory to describe the dynamic behavior in general and particularly the electrolytic conductivity of dilute flexible polyelectrolytes in aqueous solution is not yet available despite some interesting progress toward this objective [5–16]. This difficulty arises from the complex interdependence between polyion conformation, ionic condensation, screening effect, and frictional forces. Moreover, it is important to underline that these different attempts have ignored the influence of the translational dielectric friction on moving polyions as well as its dependence on concentration [17, 18]. The present paper is a supplementary contribution toward this goal in order to propose a general formal limiting equation expressing the influence of the effects cited above on the equivalent conductivity of polyelectrolytes, and applicable in the range of very dilute solutions for which the stretched conformation is favored and for which the electrophoretic effects and ionic relaxation effects are negligible. This approach will be illustrated by the behaviors of three polyelectrolytes (sodium heparinate, sodium chondroitin sulfate, and sodium polystyrene sulphonate) characterized by different chain lengths and by different discontinuous charge distributions.


## 2. Theoretical Model and Parameters

### 2.1. Presentation of the Cylindrical Model

For dilute polyelectrolyte solutions the long chains of ionized polymers are generally assumed to be completely stretched [5–13], so that each chain can be modeled as a cylinder of RC radius, LS structural length (LS=AB) and ZSe structural charge. On the other hand, as the linear distribution of the total charge ZSe is uniform, then the distance of separation between two successive ionizable groups is bS=LS/|ZS|. Because of the ionic condensation effect, these charged groups are, in absence of additional salt (salt-free), partially neutralized only by counterions “i” of Zie charge and Ri effective radius. According to Manning’s rod-like model (RC/LS→0) [6–12], the degree of ionic condensation (1-αM) is independent on the counterions concentration Ci so that αM=bS/|Zi|LB, where LB=e2/εokBT is the Bjerrum length, kB is the Boltzmann constant, and εo is the permittivity of the solvent. In fact, experimental conductivity results are not in general in conformity with the current polyelectrolyte theories [10, 11, 14, 15]. In particular, we have proved that for more realistic polyelectrolyte models (rod-like or cylindrical models with finite length and models having ellipsoidal conformations of A, B focuses with AB=L<LS) [9–11], the corresponding degree of dissociation α obeys in general to the Ostwald’s principle of dilution and consequently α→1 when Ci→0. The degree α was calculated on the basis of the two-state model [10] (double layer) proposed by Sélégny, Manning, Dobrynin, and Rubinstein [6–8], but in a different way. According to this approach, the condensed counterions are distributed on an ellipsoidal first layer of A, B focuses and R minor axis, whereas the free counterions constitute the ionic atmosphere which can be represented by an ellipsoidal second layer having the same focuses A, B and a dP minor axis (see Figure 1). The thickness dP is a function of R, α and Ci via the Debye-MSA screen parameter ΓPMSA [1] and the Debye length χD-1. Notice that both first and second layers are equipotential surfaces of total charges equal, respectively, to -(1-α)ZSe and -αZSe. In the case of the cylindrical model (AB=LS, R≅RC and α≡αC), calculation of αC needs the resolution of the following ensemble of implicit equations:
(1)(1-αC)αC=(1-VPCP)-1VPCP×[g(RC,LS)g(dP,LS)]{(αC+|Zi/ZS|)/αM},(2)g(x,LS)=[(4x2+LS2)1/2+LS][(4x2+LS2)1/2-LS],(3)dP=RC+12ΓPMSA,(4)2ΓPMSA=[-1+(1+4χDRi)1/2]2Ri;χD=(4πLBZi2αCCi)1/2.Ci and CP are the microscopic concentrations (number of particles/Å3) of, respectively, the counterions and the polyions. VP=πRC2LS is the structural volume of the cylindrical polyion. g(RC,LS)  and g(dP,LS) are “the configuration functions” which depend on the conformation of the polyion. Notice first that (1) is a generalization to cylindrical polyions, of the Fuoss expression relative to the ionic association of simple electrolytes; and second, that in the restrictive case of Manning’s model (RC/LS→0) and for dilute solutions (RC/dP≪1), αC increases very slowly with dilution so that αC remains approximately constant in a large range of low concentrations. However, αC approaches toward αM only for some particular polyelectrolytes [10].Figure 1
Representation of a polyelectrolyte according to the cylindrical model.Notice also that (1) can be applied for ellipsoidal polyions of any shape (i.e.,  L≠LS,  R≠RC and for all VP).


### 2.2. Theoretical Conductivity of Cylindrical Polyelectrolytes in Dilute Solutions

In practice we measure the specific conductanceχ of the polyelectrolyte solution in S·cm−1. χ is related to the equivalent ionic conductivities λP and λi, respectively, of the polyion and the counterion as follows:
(5)χ=10-3[αC|ZS|λPCP°+αC|Zi|λiCi°],
where CP°  is the molar concentration of the polyions and Ci°  is the total molar concentration of the counterions so that αCCi°  is the molar concentration of the free counterions, and (αC|ZS|e) is the apparent (effective) charge |Zap|e of the polyion partially neutralized by the condensed counterions. The electroneutrality condition implies that |ZS|CP°=|Zi|Ci°. The equivalent conductivity ΛPoly of the polyelectrolyte is therefore defined by
(6)ΛPoly=1000χ|Zi|Ci°=αC(λP+λi).λP  and λi depend on the concentration of the free counterions because of the brake effects on the moving ions (or on polyions) due to their ionic atmosphere. In general one distinguishes two different ionic friction effects [2–4, 9, 12]: (a) the electrophoretic effect which is a hydrodynamic friction on the ionic atmosphere transmitted to the central ion (or polyion) via solvent molecules, (b) the ionic relaxation effect due to the perturbation of the charge distribution of the ionic atmosphere by the external electrical field E. This polarization effect induces on the moving central ion (or polyion) a local field ΔXir opposed to E. Quantitatively, these two effects appear in the expression of the ionic conductivity λi of simple ions (in our case, the counterions) via the Δλiel corrective term and the βiir relaxation term as follows:
(7)λi=(λi°-Δλiel)(1+βiir),(8)Δλiel=(|Zi|Fe6πηdi);di=Ri+12ΓiMSA,(9)ΓiMSA2≅πLB{((Zi)2αCCi[1+2ΓiMSARi]2)+(ZS2αC2CP[1+2ΓiMSA〈RP〉]2)},(10)〈RP〉=LSLn[g(RC,LS)].λi° is the ionic equivalent conductivity of the counterion at infinite dilution which expresses both the hydrodynamic friction, due to the viscosity η of the solvent (Robinson and Stokes [2]), and the dielectric friction effect (Zwanzig [19]). λi° is determined experimentally by linear extrapolation at infinite dilution and according to the Debye-Onsager limiting equation [2], of the equivalent conductivity ΛiX (√I) (with the square root of the ionic strength I) of any corresponding simple electrolyte (e.g., if i≡Na+, we can choose ΛiX≡ΛNaCl; extrapolation of ΛNaCl in water at 25°C leads to λNa+°=50.1cm2·Ω-1·eqvNa+-1). The term F is the Faraday, Ri is the effective radius of the solvated counterion “i”, di is the radius of its ionic atmosphere, and ΓiMSA is its corresponding Debye-MSA screen parameter. Notice that ΓiMSA differs from the screen parameter ΓPMSA relating to the polyion, because considering the high repulsion between polyions, we have assumed that the ionic atmosphere of the polyions is constituted only by free counterions; on the contrary, the ionic atmosphere of a counterion encloses both polyions and counterions.  〈RP〉 is the mean radius of the polyion (analog to the radius of gyration) which is also equal to the electrostatic capacitance CAP (in c.g.s.u.e units) of the ellipsoidal (or cylindrical) polyion [17, 20]. Finally, the explicit expression of the ionic relaxation term βiir will be examined at the end of this section because of its interdependence with the βPir term relating to the polyions.The expression of the equivalent conductivityλP of the polyion is more complex because its ionic equivalent conductivity at infinite dilution λP° expressing both hydrodynamic friction and dielectric friction effect is experimentally inaccessible. Indeed, in contrast with simple electrolytes, ionic transport behavior of polyelectrolytes is not governed by any universal limiting law [5] allowing the determination of λP° by an extrapolation method at infinite dilution.  For this reason we decomposed the λP  expression as follows [9]:
(11)λP=λPHenry(1+βPir+βPdf)=(λP°Hyd-ΔλPel)(1+βPir+βPdf).
The justification of the above equation is the following: the external electric field E acting on the polyion polarizes its ionic atmosphere as well as its surrounding solvent molecules, which gives place to an ionic relaxation field ΔXir and to a dielectric relaxation field ΔXdf slowing down the movement of the polyion. The velocity v of the polyion can thus be written in two manners:
(12)v=uPE=uPHenry(E+ΔXir+ΔXdf)≡uPHenryE(1+βPir+βPdf).uP is the actual electrical mobility of the polyion and uPHenry is the Henry mobility [9, 21] so that
(13)λP=FuP;λPHenry=FuPHenry.βPir and βPdf express, respectively, the ionic relaxation effect and the dielectric friction effect acting on the polyion. Equation (12) implies that
(14)βPir=|ΔXir||E+ΔXir+ΔXdf|,βPdf=|ΔXdf||E+ΔXir+ΔXdf|.
The explicit expression of the Henry equivalent conductivity λPHenry of the polyion  is, [9],
(15)λPHenry=(αCFe6πηbS)Ln[g(RC,LS)g(dP,LS)].
Accordingly, we can separate λPHenry into two contributions: the purely hydrodynamic term λP°Hyd due to the viscosity η of the solvent, and the so-called electrophoretic term ΔλPel, so that
(16)λP°Hyd=(αCFe6πηbS)Ln[g(RC,LS)],ΔλPel=(αCFe6πηbS)Ln[g(dP,LS)].
It is remarkable to underline that the expression of λP°Hyd coincides with the Hubbard-Douglass general relation [20] expressing the hydrodynamic mobility: uP°HD≡λP°HD/F of an arbitrarily shaped unspecified macroion of ZSe charge, in terms of its capacitance CAP (generalization of Stokes’ law):
(17)uP°HD=|ZS|e6πηCAP,(18)CAP=LSLn[g(RC,LS)]=〈RP〉.
We can therefore generalize the Hubbard-Douglass relation given by (17) to the Henry equivalent conductivity λPHenry as follows [17]:
(19)λPHenry=αC|ZS|Fe6πηCAP′.CAP′ is now the electrostatic Gouy capacitance (in c.g.s.u.e units) of the ellipsoidal (or cylindrical) capacitor constituted by the polyion and by its ionic atmosphere of mean radius 〈dP〉:
(20)(CAP′)-1=[〈RP〉-1-〈dP〉-1],〈dP〉=LSLn[g(dP,LS)].
Notice that the relative importance of the electrophoretic effect can be evaluated by the ratio:
(21)ΔλPelλP°Hyd=〈RP〉〈dP〉.
This last equation implies that the electrophoretic effect vanishes in the range of highly dilute solutions, that is, when 〈dP〉→∞. It is also interesting to notice that certain authors [22] have described the electrophoretic mobility of polyions in polyelectrolyte solutions by means of the Debye-Onsager-MSA approach using the mean spherical approximation for the coil conformation of the polyion chain. The corresponding spherical hydrodynamic radius RH was evaluated according to Stokes-Einstein relation: RH=kBT/6πηDP°, where DP° is the self diffusion coefficient of the polyion at high dilution. The same spherical approximation could be used for the calculation of the ionic relaxation effect using Onsager relation applicable to spherical simple ions [3], therefore
(22)βPir≅αC|ZiZS|LBqr2[3(1+qr)〈dP〉]
with: qr≅1, if we assume that the ionic atmosphere is free of polyions. However, Manning has demonstrated that for infinite rod-like model (LS≫dP), βPir remains sensibly constant, equal to 0.13.  In order to conciliate the two results into a general expression one of the authors has proposed the following relation [9, 10]:
(23)βPir=αC|ZSZi|LB(3dP2+LS2/4)[18(dP2+LS2/4)3/2],
so that when LS/dP→0, then βPir→αC|ZiZS|LB/(6dP); this limiting expression converges toward the Debye-Onsager relation concerning spherical ions. In contrast, for polyions of large length, LS≫dP, βPir→1/9 for all Ci° if αC→αM.On the other hand, according to linear irreversible thermodynamics (T.I.P) the differentβkir relaxation terms of all the “k” species (ions or polyions) in solution are interdependent via the general relation [9, 14]: ∑kZkCkβkir=0, with ∑kZkCk=0. This means that in the case of our binary system the two relaxation terms βPir and βiir, respectively, of the polyions and the counterions are equal:
(24)βPir=βiir.
Lastly, because of the importance of the dielectric friction effect on a stretched polyion (which is the main subject of this paper) the friction term βPdf will be discussed in detail in the next paragraph.


### 2.3. Importance of the Dielectric Friction Effect on a Stretched Polyion

The aim of the present paragraph is to evaluate succinctly the frictional force on a slowly moving polyion due to dielectric loss in its surrounding medium. In fact, this dielectric friction effect depends on the conformation (shape) of the polyion. In order to show the link with previous works, we will start by presenting the general formal treatment adopted in all cases; hence we will recall the computation results relating to the spherical and ellipsoidal models. Then, we will treat, without going into the mathematical details, the specific case of a stretched polyion modeled as a chain of|ZS| identical charged spheres, each one having a charge qn=αCe and a radius  Rg (a linear discontinuous distribution of ionized groups).The general mechanism of dielectric friction is the following: when a sphere of chargeq and radius RS is submitted to a moderate external alternating field E along the z axis, it acquires a velocity v=uE, where u is its electrical mobility. We indicate by rq(t)=vt the position of the center of the sphere at time t. During its movement the charge q induces at each point M of the dielectric medium (solvent) defined by its radius-vector r(x,y,z,t) a time-dependent polarization P(r,t) which is proportional to the displacement field D(r,t-t1) created by q at different anterior times (t-t1):
(25)D(r,t-t1)=qr′(r′)3,r′=[r(x,y,z,t)-rq(t-t1)].
The module r′ is the distance between the point M and the center of the sphere at time (t-t1). This noninstantaneous response results from the fact that each solvent molecule needs a relaxation time τ to be oriented along the radial field D. Mathematically, the linear relation between P(r,t) and D(r,t-t1) is given by the following convolution integral [17, 19]:
(26)P(r,t)=(14π)∫0∞γ(t1)D(r,t-t1)dt1
with
(27)γ(t)=[1-(1ε∞)]δ(t)+[(ε0-ε∞)(τε∞2)-1exp(-ε0tε∞τ)].γ(t) is the after effect function which depends on the delta function δ(t) representing electronic relaxation and on  the permittivities  ε0 and ε∞,  respectively, the static and the high-frequency dielectric constants of the solvent. For water at 25°C, ε0=78.3 and ε∞=1.77. Note that we have set the upper limit of the above integral to ∞, because in general the dielectric relaxation time τ is small by comparison to t so that γ(t)→0 (vanishes rapidly) when t>τ.In turn, this induced polarizationP(r,t) exerts back on the charged sphere a resulting dielectric frictional force Fdf=qΔXdf where ΔXdf is the so called dielectric relaxation field having a direction opposed to the external field E. The general integral relation between the  z component ΔXzdf of ΔXdf and the Dx, Dy, Dz components of D(r,t-t1) via P(r,t) and therefore γ(t1) is, [17],
(28)ΔXzdf=(14π)∫0∞γ(t1)Iz(t1)dt1.Iz(t1) is the so called key integral defined by
(29)Iz(t1)=∭[-Dz(t-t1)r′3+3Z2Dz(t-t1)r′5+3xZDx(t-t1)r′5+3yZDy(t-t1)r′5]×dxdydz,
where Z=z-vt. Integration is taken over the whole volume except the finite region including the charged sphere (or the polyion in general) from which the dielectric medium is excluded.It is obvious that no dielectric friction occurs,ΔXzdf→0 when v→0 (immobile sphere) or when τ→0 (instantaneous response so that γ(t)→0). In other words, the delay effect (τ≠0) causes a perturbation of the equilibrium distribution of solvent molecules around the moving sphere and therefore leads to a nonsymmetrical polarization responsible of the resulting dielectric relaxation field: ΔXzdf≠0. Consequently, linearity between causes and response implies that ΔXdf~(vτ). On the other hand, as the dimension of the electrical relaxation force Fdf=qΔXdf is ~(q)2/[length]2, scaling analysis yields to Fdf~-(q)2(vτ)/RS3.More rigorous derivations of the expression of the dielectric frictional force on a charged sphere were performed successively by Zwanzig [19], Hubbard and Onsager [23], and Wolynes [24].  In particular, if the charged sphere of large radius RS is assumed to be a conductor then, hydrodynamic effects become small and all theories reduce to Zwanzig’s original result [19] which can be derived from (27) and (28) following the substituting of the explicit expression of  D(r,t-t1) given by (25) into (29):
(30)ΔXdf=±(2q/3)[1-ε∞/εo](vτ)(εoRS3).
We can use the above equation to compute the dielectric friction effect on a spherical polyion of effective charge q=eαSZS and radius RS. Indeed, according to (12) and (14), its velocity v is given by v=uPHenry(E+ΔXir+ΔXdf)  with uPHenry=λPHenry/F  and the relative dielectric friction effect is defined by βPdf=|ΔXdf|/|E+ΔXir+ΔXdf|, therefore (30) leads to βPdf=(2eαS|ZS|/3F)[1-ε∞/εo](λPHenryτ)/(εoRS3). If we explicit the expression of λPHenry according to (19) and after replacing the relaxation time τ by its Debye expression in terms of the solvent radius Rw(RH2O=1.4Å) and the viscosity η so that τ=6πηRw3/kBT, we obtain
(31)βPdf=(23)(αC|ZS|)2(RwRS)3(LBCAP′)[1-ε∞εo].
More recently, authors of [17] have demonstrated that this last expression remains valid even in the case of an ellipsoidal polyion of minor axis R, interfocuses distance L, and effective charge q=eαEZS but with the proviso of replacing the spherical radius RS by an apparent ray Rapp which is a function of the eccentricity γ=L/2R so that Rapp≈〈R〉 for γ<1 and Rapp≈L/2 for γ>1.Now, in the case of stretched chain of|ZS| successive charged spheres of charge qn=-eαC  and of radius Rn=Rg (n=1, |ZS|), each sphere n moving along direction z with velocity vp undergoes from the polarized solvent molecules a dielectric frictional force Fndf=qnΔXndf, where ΔXndf is the local dielectric relaxation field. Because of the axial symmetry of the system around the z axis, only the z component  ΔXn,zdf of ΔXndf is different from zero:
(32)ΔXn,zdf=(14π)∫0∞γ(t1)In,z(t1)dt1.
The corresponding key integral In,z(t1) is defined by
(33)In,z(t1)=∭[-Dz(t-t1)rn′3+3Zn2Dz(t-t1)rn′5+3xZnDx(t-t1)rn′5+3yZnDy(t-t1)rn′5]×dxdydz.x, y, and Zn are the components of the vector rn′ separating a point M(x,y,z)  of the dielectric medium and the position of  the charge qn at time (t-t1). The principal difference between (25)–(29) relating to a spherical polyion and (32) and (33), comes primarily from the fact that the expression of the displacement field D(r,t-t1) created by the stretched chain of charged spheres at M(x,y,z) is now
(34)D(r,t-t1)=qnrn′(rn′)3+∑i≠n|ZS|-1Di(r,t-t1),Di(r,t-t1)=qiri′(ri′)3.
This equation results from the principle of superposition so that (34) differs from (25) by the sum ∑Di(r,t-t1) expressing the interference of the displacement field Dn(r,t-t1) created at M by qn at an anterior time (t-t1), with the different Di(r,t-t1) due to the (|ZS|-1) charges qi of the moving chain. Notice however that, if the distance: bS=LS/|ZS|, between two successive charged groups, is sufficiently large (bS≫2Rg, i.e., f=bS/Rg≫2) so that: ri′>rn′, then the dielectric friction undergone by the sphere qn is essentially due to polarized solvent molecules of its entourage. We could therefore neglect the interference effect and D(r,t-t1) is reduced thus to Dn(r,t-t1)=qnrn′/(rn′)3; in other words, each charged sphere of the moving polyion of radius Rg and charge qn=-eαC behaves as if it were alone to polarize the dielectric medium. Consequently, simple direct application of Zwanzig’s original result leads by analogy to (31) to the following obvious equation:
(35)βPdf=(23)αC2(RwRg)3(LBbS)(LSCAP′)[1-ε∞εo].
Recent calculations [18] based on (14), (27), and (32)–(34) including the interference effect lead to the following corrected expression βPdfi in replacement to the above expression of βPdf:
(36)βPdfi=(23f3)αC2(RwRg)3(LBbS)(LSCAP′)×[1-ε∞εo]|(f3-4.808)+6.58|ZS||,f=bSRg.
Notice that (35) and (36) are valid only in the case of slowly moving polyions so that vPτ≤Rg. The evaluation of the interference effect in terms of the “interference parameter” f=bS/Rg≥2 is quantified by the ratio between βPdfi in presence of interference and βPdf in absence of interference:
(37)βPdfiβPdf=|(1-4.808f3)+6.58f3|ZS||≤1,f≥2.
This equation shows that for a discontinuous charge distribution, that is, bS≫Rg(f≫2), the interference effect becomes negligible. In contrast, it is maximal for a chain of tangent spheres (f=2) and increases with |ZS|.It is interesting to note thatβPdfi increases with dilution as αC2 and it reaches its maximal value βP°dfi at infinite dilution, that is, when αC=1 (Ostwald) and CAP′=〈RP〉:
(38)βP°dfi=(23)(RwbS)3(LBbS)(LS〈RP〉)[1-ε∞εo]×|(f3-4.808)+6.58|ZS||.
Finally it is important to underline the singular case of Manning’s polyions (Rg≅0 and LS→∞, |ZS|→∞). Because of the infinite length of its moving chain, the structural state of the polyion (distribution of charges, distribution of solvent molecules, and therefore P(r,t)) varies periodically with time with a period equal to: θ=bS/vP. As for slowly moving polyions, vPτ≈bS, therefore θ≈τ, that is, the solvent molecules have not sufficient time to reorient themselves toward the new field D(r,t) during the periodic variation. Consequently, the polyion seems to be immobile (vP=0) and thus surrounded by its initial symmetrical cylindrical distribution of solvent molecules. This conservation of the equilibrium symmetry implies the absence of any resulting dielectric relaxation field (ΔXdf=0). It is the reason for which the dielectric relaxation effect is completely absent in the restrictive case of the Manning’s model.


## 2.1. Presentation of the Cylindrical Model

For dilute polyelectrolyte solutions the long chains of ionized polymers are generally assumed to be completely stretched [5–13], so that each chain can be modeled as a cylinder of RC radius, LS structural length (LS=AB) and ZSe structural charge. On the other hand, as the linear distribution of the total charge ZSe is uniform, then the distance of separation between two successive ionizable groups is bS=LS/|ZS|. Because of the ionic condensation effect, these charged groups are, in absence of additional salt (salt-free), partially neutralized only by counterions “i” of Zie charge and Ri effective radius. According to Manning’s rod-like model (RC/LS→0) [6–12], the degree of ionic condensation (1-αM) is independent on the counterions concentration Ci so that αM=bS/|Zi|LB, where LB=e2/εokBT is the Bjerrum length, kB is the Boltzmann constant, and εo is the permittivity of the solvent. In fact, experimental conductivity results are not in general in conformity with the current polyelectrolyte theories [10, 11, 14, 15]. In particular, we have proved that for more realistic polyelectrolyte models (rod-like or cylindrical models with finite length and models having ellipsoidal conformations of A, B focuses with AB=L<LS) [9–11], the corresponding degree of dissociation α obeys in general to the Ostwald’s principle of dilution and consequently α→1 when Ci→0. The degree α was calculated on the basis of the two-state model [10] (double layer) proposed by Sélégny, Manning, Dobrynin, and Rubinstein [6–8], but in a different way. According to this approach, the condensed counterions are distributed on an ellipsoidal first layer of A, B focuses and R minor axis, whereas the free counterions constitute the ionic atmosphere which can be represented by an ellipsoidal second layer having the same focuses A, B and a dP minor axis (see Figure 1). The thickness dP is a function of R, α and Ci via the Debye-MSA screen parameter ΓPMSA [1] and the Debye length χD-1. Notice that both first and second layers are equipotential surfaces of total charges equal, respectively, to -(1-α)ZSe and -αZSe. In the case of the cylindrical model (AB=LS, R≅RC and α≡αC), calculation of αC needs the resolution of the following ensemble of implicit equations:
(1)(1-αC)αC=(1-VPCP)-1VPCP×[g(RC,LS)g(dP,LS)]{(αC+|Zi/ZS|)/αM},(2)g(x,LS)=[(4x2+LS2)1/2+LS][(4x2+LS2)1/2-LS],(3)dP=RC+12ΓPMSA,(4)2ΓPMSA=[-1+(1+4χDRi)1/2]2Ri;χD=(4πLBZi2αCCi)1/2.Ci and CP are the microscopic concentrations (number of particles/Å3) of, respectively, the counterions and the polyions. VP=πRC2LS is the structural volume of the cylindrical polyion. g(RC,LS)  and g(dP,LS) are “the configuration functions” which depend on the conformation of the polyion. Notice first that (1) is a generalization to cylindrical polyions, of the Fuoss expression relative to the ionic association of simple electrolytes; and second, that in the restrictive case of Manning’s model (RC/LS→0) and for dilute solutions (RC/dP≪1), αC increases very slowly with dilution so that αC remains approximately constant in a large range of low concentrations. However, αC approaches toward αM only for some particular polyelectrolytes [10].Figure 1
Representation of a polyelectrolyte according to the cylindrical model.Notice also that (1) can be applied for ellipsoidal polyions of any shape (i.e.,  L≠LS,  R≠RC and for all VP).


## 2.2. Theoretical Conductivity of Cylindrical Polyelectrolytes in Dilute Solutions

In practice we measure the specific conductanceχ of the polyelectrolyte solution in S·cm−1. χ is related to the equivalent ionic conductivities λP and λi, respectively, of the polyion and the counterion as follows:
(5)χ=10-3[αC|ZS|λPCP°+αC|Zi|λiCi°],
where CP°  is the molar concentration of the polyions and Ci°  is the total molar concentration of the counterions so that αCCi°  is the molar concentration of the free counterions, and (αC|ZS|e) is the apparent (effective) charge |Zap|e of the polyion partially neutralized by the condensed counterions. The electroneutrality condition implies that |ZS|CP°=|Zi|Ci°. The equivalent conductivity ΛPoly of the polyelectrolyte is therefore defined by
(6)ΛPoly=1000χ|Zi|Ci°=αC(λP+λi).λP  and λi depend on the concentration of the free counterions because of the brake effects on the moving ions (or on polyions) due to their ionic atmosphere. In general one distinguishes two different ionic friction effects [2–4, 9, 12]: (a) the electrophoretic effect which is a hydrodynamic friction on the ionic atmosphere transmitted to the central ion (or polyion) via solvent molecules, (b) the ionic relaxation effect due to the perturbation of the charge distribution of the ionic atmosphere by the external electrical field E. This polarization effect induces on the moving central ion (or polyion) a local field ΔXir opposed to E. Quantitatively, these two effects appear in the expression of the ionic conductivity λi of simple ions (in our case, the counterions) via the Δλiel corrective term and the βiir relaxation term as follows:
(7)λi=(λi°-Δλiel)(1+βiir),(8)Δλiel=(|Zi|Fe6πηdi);di=Ri+12ΓiMSA,(9)ΓiMSA2≅πLB{((Zi)2αCCi[1+2ΓiMSARi]2)+(ZS2αC2CP[1+2ΓiMSA〈RP〉]2)},(10)〈RP〉=LSLn[g(RC,LS)].λi° is the ionic equivalent conductivity of the counterion at infinite dilution which expresses both the hydrodynamic friction, due to the viscosity η of the solvent (Robinson and Stokes [2]), and the dielectric friction effect (Zwanzig [19]). λi° is determined experimentally by linear extrapolation at infinite dilution and according to the Debye-Onsager limiting equation [2], of the equivalent conductivity ΛiX (√I) (with the square root of the ionic strength I) of any corresponding simple electrolyte (e.g., if i≡Na+, we can choose ΛiX≡ΛNaCl; extrapolation of ΛNaCl in water at 25°C leads to λNa+°=50.1cm2·Ω-1·eqvNa+-1). The term F is the Faraday, Ri is the effective radius of the solvated counterion “i”, di is the radius of its ionic atmosphere, and ΓiMSA is its corresponding Debye-MSA screen parameter. Notice that ΓiMSA differs from the screen parameter ΓPMSA relating to the polyion, because considering the high repulsion between polyions, we have assumed that the ionic atmosphere of the polyions is constituted only by free counterions; on the contrary, the ionic atmosphere of a counterion encloses both polyions and counterions.  〈RP〉 is the mean radius of the polyion (analog to the radius of gyration) which is also equal to the electrostatic capacitance CAP (in c.g.s.u.e units) of the ellipsoidal (or cylindrical) polyion [17, 20]. Finally, the explicit expression of the ionic relaxation term βiir will be examined at the end of this section because of its interdependence with the βPir term relating to the polyions.The expression of the equivalent conductivityλP of the polyion is more complex because its ionic equivalent conductivity at infinite dilution λP° expressing both hydrodynamic friction and dielectric friction effect is experimentally inaccessible. Indeed, in contrast with simple electrolytes, ionic transport behavior of polyelectrolytes is not governed by any universal limiting law [5] allowing the determination of λP° by an extrapolation method at infinite dilution.  For this reason we decomposed the λP  expression as follows [9]:
(11)λP=λPHenry(1+βPir+βPdf)=(λP°Hyd-ΔλPel)(1+βPir+βPdf).
The justification of the above equation is the following: the external electric field E acting on the polyion polarizes its ionic atmosphere as well as its surrounding solvent molecules, which gives place to an ionic relaxation field ΔXir and to a dielectric relaxation field ΔXdf slowing down the movement of the polyion. The velocity v of the polyion can thus be written in two manners:
(12)v=uPE=uPHenry(E+ΔXir+ΔXdf)≡uPHenryE(1+βPir+βPdf).uP is the actual electrical mobility of the polyion and uPHenry is the Henry mobility [9, 21] so that
(13)λP=FuP;λPHenry=FuPHenry.βPir and βPdf express, respectively, the ionic relaxation effect and the dielectric friction effect acting on the polyion. Equation (12) implies that
(14)βPir=|ΔXir||E+ΔXir+ΔXdf|,βPdf=|ΔXdf||E+ΔXir+ΔXdf|.
The explicit expression of the Henry equivalent conductivity λPHenry of the polyion  is, [9],
(15)λPHenry=(αCFe6πηbS)Ln[g(RC,LS)g(dP,LS)].
Accordingly, we can separate λPHenry into two contributions: the purely hydrodynamic term λP°Hyd due to the viscosity η of the solvent, and the so-called electrophoretic term ΔλPel, so that
(16)λP°Hyd=(αCFe6πηbS)Ln[g(RC,LS)],ΔλPel=(αCFe6πηbS)Ln[g(dP,LS)].
It is remarkable to underline that the expression of λP°Hyd coincides with the Hubbard-Douglass general relation [20] expressing the hydrodynamic mobility: uP°HD≡λP°HD/F of an arbitrarily shaped unspecified macroion of ZSe charge, in terms of its capacitance CAP (generalization of Stokes’ law):
(17)uP°HD=|ZS|e6πηCAP,(18)CAP=LSLn[g(RC,LS)]=〈RP〉.
We can therefore generalize the Hubbard-Douglass relation given by (17) to the Henry equivalent conductivity λPHenry as follows [17]:
(19)λPHenry=αC|ZS|Fe6πηCAP′.CAP′ is now the electrostatic Gouy capacitance (in c.g.s.u.e units) of the ellipsoidal (or cylindrical) capacitor constituted by the polyion and by its ionic atmosphere of mean radius 〈dP〉:
(20)(CAP′)-1=[〈RP〉-1-〈dP〉-1],〈dP〉=LSLn[g(dP,LS)].
Notice that the relative importance of the electrophoretic effect can be evaluated by the ratio:
(21)ΔλPelλP°Hyd=〈RP〉〈dP〉.
This last equation implies that the electrophoretic effect vanishes in the range of highly dilute solutions, that is, when 〈dP〉→∞. It is also interesting to notice that certain authors [22] have described the electrophoretic mobility of polyions in polyelectrolyte solutions by means of the Debye-Onsager-MSA approach using the mean spherical approximation for the coil conformation of the polyion chain. The corresponding spherical hydrodynamic radius RH was evaluated according to Stokes-Einstein relation: RH=kBT/6πηDP°, where DP° is the self diffusion coefficient of the polyion at high dilution. The same spherical approximation could be used for the calculation of the ionic relaxation effect using Onsager relation applicable to spherical simple ions [3], therefore
(22)βPir≅αC|ZiZS|LBqr2[3(1+qr)〈dP〉]
with: qr≅1, if we assume that the ionic atmosphere is free of polyions. However, Manning has demonstrated that for infinite rod-like model (LS≫dP), βPir remains sensibly constant, equal to 0.13.  In order to conciliate the two results into a general expression one of the authors has proposed the following relation [9, 10]:
(23)βPir=αC|ZSZi|LB(3dP2+LS2/4)[18(dP2+LS2/4)3/2],
so that when LS/dP→0, then βPir→αC|ZiZS|LB/(6dP); this limiting expression converges toward the Debye-Onsager relation concerning spherical ions. In contrast, for polyions of large length, LS≫dP, βPir→1/9 for all Ci° if αC→αM.On the other hand, according to linear irreversible thermodynamics (T.I.P) the differentβkir relaxation terms of all the “k” species (ions or polyions) in solution are interdependent via the general relation [9, 14]: ∑kZkCkβkir=0, with ∑kZkCk=0. This means that in the case of our binary system the two relaxation terms βPir and βiir, respectively, of the polyions and the counterions are equal:
(24)βPir=βiir.
Lastly, because of the importance of the dielectric friction effect on a stretched polyion (which is the main subject of this paper) the friction term βPdf will be discussed in detail in the next paragraph.


## 2.3. Importance of the Dielectric Friction Effect on a Stretched Polyion

The aim of the present paragraph is to evaluate succinctly the frictional force on a slowly moving polyion due to dielectric loss in its surrounding medium. In fact, this dielectric friction effect depends on the conformation (shape) of the polyion. In order to show the link with previous works, we will start by presenting the general formal treatment adopted in all cases; hence we will recall the computation results relating to the spherical and ellipsoidal models. Then, we will treat, without going into the mathematical details, the specific case of a stretched polyion modeled as a chain of|ZS| identical charged spheres, each one having a charge qn=αCe and a radius  Rg (a linear discontinuous distribution of ionized groups).The general mechanism of dielectric friction is the following: when a sphere of chargeq and radius RS is submitted to a moderate external alternating field E along the z axis, it acquires a velocity v=uE, where u is its electrical mobility. We indicate by rq(t)=vt the position of the center of the sphere at time t. During its movement the charge q induces at each point M of the dielectric medium (solvent) defined by its radius-vector r(x,y,z,t) a time-dependent polarization P(r,t) which is proportional to the displacement field D(r,t-t1) created by q at different anterior times (t-t1):
(25)D(r,t-t1)=qr′(r′)3,r′=[r(x,y,z,t)-rq(t-t1)].
The module r′ is the distance between the point M and the center of the sphere at time (t-t1). This noninstantaneous response results from the fact that each solvent molecule needs a relaxation time τ to be oriented along the radial field D. Mathematically, the linear relation between P(r,t) and D(r,t-t1) is given by the following convolution integral [17, 19]:
(26)P(r,t)=(14π)∫0∞γ(t1)D(r,t-t1)dt1
with
(27)γ(t)=[1-(1ε∞)]δ(t)+[(ε0-ε∞)(τε∞2)-1exp(-ε0tε∞τ)].γ(t) is the after effect function which depends on the delta function δ(t) representing electronic relaxation and on  the permittivities  ε0 and ε∞,  respectively, the static and the high-frequency dielectric constants of the solvent. For water at 25°C, ε0=78.3 and ε∞=1.77. Note that we have set the upper limit of the above integral to ∞, because in general the dielectric relaxation time τ is small by comparison to t so that γ(t)→0 (vanishes rapidly) when t>τ.In turn, this induced polarizationP(r,t) exerts back on the charged sphere a resulting dielectric frictional force Fdf=qΔXdf where ΔXdf is the so called dielectric relaxation field having a direction opposed to the external field E. The general integral relation between the  z component ΔXzdf of ΔXdf and the Dx, Dy, Dz components of D(r,t-t1) via P(r,t) and therefore γ(t1) is, [17],
(28)ΔXzdf=(14π)∫0∞γ(t1)Iz(t1)dt1.Iz(t1) is the so called key integral defined by
(29)Iz(t1)=∭[-Dz(t-t1)r′3+3Z2Dz(t-t1)r′5+3xZDx(t-t1)r′5+3yZDy(t-t1)r′5]×dxdydz,
where Z=z-vt. Integration is taken over the whole volume except the finite region including the charged sphere (or the polyion in general) from which the dielectric medium is excluded.It is obvious that no dielectric friction occurs,ΔXzdf→0 when v→0 (immobile sphere) or when τ→0 (instantaneous response so that γ(t)→0). In other words, the delay effect (τ≠0) causes a perturbation of the equilibrium distribution of solvent molecules around the moving sphere and therefore leads to a nonsymmetrical polarization responsible of the resulting dielectric relaxation field: ΔXzdf≠0. Consequently, linearity between causes and response implies that ΔXdf~(vτ). On the other hand, as the dimension of the electrical relaxation force Fdf=qΔXdf is ~(q)2/[length]2, scaling analysis yields to Fdf~-(q)2(vτ)/RS3.More rigorous derivations of the expression of the dielectric frictional force on a charged sphere were performed successively by Zwanzig [19], Hubbard and Onsager [23], and Wolynes [24].  In particular, if the charged sphere of large radius RS is assumed to be a conductor then, hydrodynamic effects become small and all theories reduce to Zwanzig’s original result [19] which can be derived from (27) and (28) following the substituting of the explicit expression of  D(r,t-t1) given by (25) into (29):
(30)ΔXdf=±(2q/3)[1-ε∞/εo](vτ)(εoRS3).
We can use the above equation to compute the dielectric friction effect on a spherical polyion of effective charge q=eαSZS and radius RS. Indeed, according to (12) and (14), its velocity v is given by v=uPHenry(E+ΔXir+ΔXdf)  with uPHenry=λPHenry/F  and the relative dielectric friction effect is defined by βPdf=|ΔXdf|/|E+ΔXir+ΔXdf|, therefore (30) leads to βPdf=(2eαS|ZS|/3F)[1-ε∞/εo](λPHenryτ)/(εoRS3). If we explicit the expression of λPHenry according to (19) and after replacing the relaxation time τ by its Debye expression in terms of the solvent radius Rw(RH2O=1.4Å) and the viscosity η so that τ=6πηRw3/kBT, we obtain
(31)βPdf=(23)(αC|ZS|)2(RwRS)3(LBCAP′)[1-ε∞εo].
More recently, authors of [17] have demonstrated that this last expression remains valid even in the case of an ellipsoidal polyion of minor axis R, interfocuses distance L, and effective charge q=eαEZS but with the proviso of replacing the spherical radius RS by an apparent ray Rapp which is a function of the eccentricity γ=L/2R so that Rapp≈〈R〉 for γ<1 and Rapp≈L/2 for γ>1.Now, in the case of stretched chain of|ZS| successive charged spheres of charge qn=-eαC  and of radius Rn=Rg (n=1, |ZS|), each sphere n moving along direction z with velocity vp undergoes from the polarized solvent molecules a dielectric frictional force Fndf=qnΔXndf, where ΔXndf is the local dielectric relaxation field. Because of the axial symmetry of the system around the z axis, only the z component  ΔXn,zdf of ΔXndf is different from zero:
(32)ΔXn,zdf=(14π)∫0∞γ(t1)In,z(t1)dt1.
The corresponding key integral In,z(t1) is defined by
(33)In,z(t1)=∭[-Dz(t-t1)rn′3+3Zn2Dz(t-t1)rn′5+3xZnDx(t-t1)rn′5+3yZnDy(t-t1)rn′5]×dxdydz.x, y, and Zn are the components of the vector rn′ separating a point M(x,y,z)  of the dielectric medium and the position of  the charge qn at time (t-t1). The principal difference between (25)–(29) relating to a spherical polyion and (32) and (33), comes primarily from the fact that the expression of the displacement field D(r,t-t1) created by the stretched chain of charged spheres at M(x,y,z) is now
(34)D(r,t-t1)=qnrn′(rn′)3+∑i≠n|ZS|-1Di(r,t-t1),Di(r,t-t1)=qiri′(ri′)3.
This equation results from the principle of superposition so that (34) differs from (25) by the sum ∑Di(r,t-t1) expressing the interference of the displacement field Dn(r,t-t1) created at M by qn at an anterior time (t-t1), with the different Di(r,t-t1) due to the (|ZS|-1) charges qi of the moving chain. Notice however that, if the distance: bS=LS/|ZS|, between two successive charged groups, is sufficiently large (bS≫2Rg, i.e., f=bS/Rg≫2) so that: ri′>rn′, then the dielectric friction undergone by the sphere qn is essentially due to polarized solvent molecules of its entourage. We could therefore neglect the interference effect and D(r,t-t1) is reduced thus to Dn(r,t-t1)=qnrn′/(rn′)3; in other words, each charged sphere of the moving polyion of radius Rg and charge qn=-eαC behaves as if it were alone to polarize the dielectric medium. Consequently, simple direct application of Zwanzig’s original result leads by analogy to (31) to the following obvious equation:
(35)βPdf=(23)αC2(RwRg)3(LBbS)(LSCAP′)[1-ε∞εo].
Recent calculations [18] based on (14), (27), and (32)–(34) including the interference effect lead to the following corrected expression βPdfi in replacement to the above expression of βPdf:
(36)βPdfi=(23f3)αC2(RwRg)3(LBbS)(LSCAP′)×[1-ε∞εo]|(f3-4.808)+6.58|ZS||,f=bSRg.
Notice that (35) and (36) are valid only in the case of slowly moving polyions so that vPτ≤Rg. The evaluation of the interference effect in terms of the “interference parameter” f=bS/Rg≥2 is quantified by the ratio between βPdfi in presence of interference and βPdf in absence of interference:
(37)βPdfiβPdf=|(1-4.808f3)+6.58f3|ZS||≤1,f≥2.
This equation shows that for a discontinuous charge distribution, that is, bS≫Rg(f≫2), the interference effect becomes negligible. In contrast, it is maximal for a chain of tangent spheres (f=2) and increases with |ZS|.It is interesting to note thatβPdfi increases with dilution as αC2 and it reaches its maximal value βP°dfi at infinite dilution, that is, when αC=1 (Ostwald) and CAP′=〈RP〉:
(38)βP°dfi=(23)(RwbS)3(LBbS)(LS〈RP〉)[1-ε∞εo]×|(f3-4.808)+6.58|ZS||.
Finally it is important to underline the singular case of Manning’s polyions (Rg≅0 and LS→∞, |ZS|→∞). Because of the infinite length of its moving chain, the structural state of the polyion (distribution of charges, distribution of solvent molecules, and therefore P(r,t)) varies periodically with time with a period equal to: θ=bS/vP. As for slowly moving polyions, vPτ≈bS, therefore θ≈τ, that is, the solvent molecules have not sufficient time to reorient themselves toward the new field D(r,t) during the periodic variation. Consequently, the polyion seems to be immobile (vP=0) and thus surrounded by its initial symmetrical cylindrical distribution of solvent molecules. This conservation of the equilibrium symmetry implies the absence of any resulting dielectric relaxation field (ΔXdf=0). It is the reason for which the dielectric relaxation effect is completely absent in the restrictive case of the Manning’s model.


## 3. Results and Discussion

In order to emphasize the importance of the dielectric friction on stretched polyions at high dilution we studied the conductivity behaviors of the following polyelectrolytes: sodium heparinate of high molecular weight (RB21055), sodium chondroitin sulfate, and sodium polystyrene sulphonate (NaPSS). Details of the experimental protocols of conductivity measurements are given in previous papers [9–11]. Notice that conductivity results concerning (NaPSS) are those published by Vink [14]. Comparisons for each polyelectrolyte, between experimental equivalent conductivities ΛPolyexp=1000χ/CNa+° and theoretical equivalent conductivities ΛPolyth=αC(λP+λi)  calculated in absence or in presence of dielectric friction and also in absence or in presence of interference effect, are given in Tables 1–3 and Figures 2, 3, and 4. In each table we show the different molar total concentrations CNa+° of Na+ counterions, the experimental equivalent conductivities ΛPolyexpincm2·Ω-1·eqvNa+-1, the degrees of Na+ condensation (1-αC) on polyions, the apparent charge numbers Zap=αCZS of the polyions, the theoretical equivalent conductivities ΛHy,El,R in absence of dielectric friction (i.e., only: hydrodynamic, electrophoretic, and ionic relaxation effects), the group radius Rg, the theoretical equivalent conductivities ΛPoly in absence of interference, the theoretical equivalent conductivities ΛPolyi in presence of interference, the % of the dielectric friction effect βdfi (36) in presence of interference, and the ratios βPdfi/βPdf  (37) expressing the relative importance of the interference effect (βdf is the dielectric friction effect  in absence of interference).Table 1
Variations with counterions concentrationCNa+° of the experimental ΛNaHRBexp and theoretical ΛNaHRB, ΛNaHRBi equivalent conductivities, of the degree of Na+ condensation (1-αC), of the apparent charge number Zap, and of the dielectric frictions in absence (βdf) and in presence (βdfi) of interference, in the case of sodium heparinate RB21055 in water at 25°C.
C
N
a
+
°  mol·L−1
Λ
NaHRB
exp  cm2·Ω-1·eqvNa+-1
1
-
α
C
Z
ap
Λ
Hy,El,R  cm2·Ω-1·eqvNa+-1
R
g  (Å)
Λ
NaHRB  cm2·Ω-1·eqvNa+-1
Λ
NaHRB
i  cm2·Ω-1·eqvNa+-1
100
β
dfi
β
P
dfi
/
β
P
df
5.1083 × 10−5
82.61
0.244
−37.80
117.58
1.691
59.62
82.68
78.03
0.30
7.4666 × 10−5
80.73
0.262
−36.91
111.18
1.70
58.19
80.67
70.03
0.30
1.4152 × 10−4
74.72
0.291
−35.45
100.68
1.693
55.06
74.73
66.14
0.30
2.4088 × 10−4
69.51
0.314
−34.29
92.31
1.678
52.18
69.15
66.08
0.32
5.0027 × 10−4
61.87
0.344
−32.82
81.54
1.656
48.29
61.96
65.73
0.35
9.3208 × 10−4
56.72
0.366
−31.72
73.27
1.643
45.29
56.79
62.98
0.36
1.7299 × 10−3
52.04
0.385
−30.77
66.05
1.627
42.51
52.08
60.87
0.37
3.3723 × 10−3
48.29
0.402
−29.94
59.47
1.623
40.16
48.26
54.60
0.38
5.1303 × 10−3
46.34
0.410
−29.55
55.92
1.625
38.95
46.31
49.80
0.38Table 2
Variations with counterions concentrationCNa+° of the experimental ΛNaChondroexp and theoretical ΛNaChondro, ΛNaChondroi equivalent conductivities, of the degree of Na+ condensation (1-αC), of the apparent charge number Zap, and of the dielectric frictions in absence (βdf) and in presence (βdfi) of interference, in the case of sodium chondroitin sulfate in water at 25°C.
C
N
a
+
°  mol·L−1
Λ
NaChond
exp  cm2·Ω-1·eqvNa+-1
1
-
α
C
Z
ap
Λ
Hy,El,
R  cm2·Ω-1·eqvNa+-1
R
g  (Å)
Λ
NaChondro  cm2·Ω-1·eqvNa+-1
Λ
NaChondro
i  cm2·Ω-1·eqvNa+-1
100
β
dfi
β
P
dfi
/
β
P
df
1.3868 × 10−4
78.54
0.148
−63.93
112.76
2.15
73.47
78.69
92.3
0.76
1.5944 × 10−4
77.93
0.152
−63.62
110.80
2.15
72.75
77.85
90.6
0.76
1.7674 × 10−4
76.99
0.155
−63.40
109.38
2.14
71.97
76.91
91.1
0.76
2.1070 × 10−4
75.74
0.160
−63.03
107.04
2.14
71.11
75.91
88.9
0.76
2.6553 × 10−4
74.02
0.166
−62.57
104.11
2.12
69.55
74.04
89.4
0.77
3.1443 × 10−4
73.16
0.171
−62.24
102.09
2.12
68.82
73.19
87.3
0.77
3.2445 × 10−4
72.86
0.172
−62.19
101.72
2.11
68.45
72.74
88.5
0.77
4.0603 × 10−4
71.42
0.176
−61.79
99.20
2.11
67.55
71.69
85.7
0.77
4.4241 × 10−4
70.85
0.178
−61.65
98.29
2.09
66.77
70.73
87.8
0.77
5.0362 × 10−4
69.92
0.181
−61.43
96.62
2.08
66.08
69.90
87.7
0.78
5.1437 × 10−4
69.95
0.182
−61.40
96.71
2.08
66.00
69.81
87.4
0.78
5.1480 × 10−4
70.06
0.182
−61.40
96.70
2.09
66.23
70.09
85.8
0.78
6.7844 × 10−4
68.38
0.187
−60.98
93.97
2.08
65.07
68.72
83.8
0.78
6.9613 × 10−4
67.94
0.188
−60.94
93.72
2.08
64.77
68.35
85.1
0.78
1.0479 × 10−3
65.88
0.195
−60.39
89.96
2.05
63.11
66.36
82.8
0.79
1.1149 × 10−3
65.29
0.196
−60.31
89.41
2.02
62.31
65.39
86.6
0.80Table 3
Variations with counterions concentrationCNa+° of the experimental ΛNaPSSexp and theoretical ΛNaPSS, ΛNaPSSi equivalent conductivities, of the degree of Na+ condensation (1-αC), of the apparent charge number Zap, and of the dielectric frictions in absence (βdf) and in presence  (βdfi) of interference, in the case of sodium polystyrene sulphonate in water at 25°C. Distinction between Manning concentration regime and Ostwald concentration regime.
C
N
a
+
°  mol·L−1
Λ
NaPSS
exp  cm2·Ω-1·eqvNa+-1
1
-
α
C
Z
ap
Λ
Hy,El,R  cm2·Ω-1·eqvNa+-1
R
g  (Å)
Λ
NaPSS  cm2·Ω-1·eqvNa+-1
Λ
NaPSS
i  cm2·Ω-1·eqvNa+-1
100
β
dfi
β
P
dfi
/
β
P
df
1× 10−10
0.395
−1754
216.9
1.21
42.13
54.55
698
0.45
1× 10−9
0.447
−1605
182.4
1.21
39.21
51.16
558
0.45
1× 10−8
0.497
−1460
149.75
1.21
36.08
47.40
459
0.45
1× 10−7
0.543
−1328
118.56
1.21
32.55
43.03
373
0.45
1
×
10
-
6
0.576
-
1229
91.22
1.21
29.01
38.40
303
0.45
1 × 10−5
0.587
−1196
78.03
1.21
28.03
36.65
252
0.45
1 × 10−4
0.579
−1221
70.55
1.21
28.45
36.45
213
0.45
1.159 × 10−3
35.52
0.559
−1279
60.98
1.21
28.52
35.53
172
0.45
2.151 × 10−3
35.37
0.553
−1298
58.41
1.212
28.55
35.38
159
0.45
2.416 × 10−3
35.33
0.551
−1302
57.92
1.212
28.55
35.33
157
0.45
3.526 × 10−3
35.16
0.547
−1314
56.31
1.214
28.54
35.16
149
0.45
5.142 × 10−3
34.88
0.543
−1326
54.64
1.214
28.49
34.88
142
0.45Figure 2
Comparison between variations withCNa+° of experimental (ΛNaHRBexp, blue ◊ points) and theoretical equivalent conductivities (ΛHy,El,R  without dielectric friction,  ΛNaHRB with dielectric friction in absence of interference, ΛNaHRBi with dielectric friction in presence of interference) of sodium heparinate RB21055 in water at 25°C.Figure 3
Comparison between variations withCNa+° of experimental (ΛNaChondroexp, blue ◊ points) and theoretical equivalent conductivities (ΛHy,El,R without dielectric friction,  ΛNaChondro with dielectric friction in absence of interference, ΛNaChondroi with dielectric friction in presence of interference) of sodium chondroitin sulfate in water at 25°C.Figure 4
Comparison between variations withCNa+° of experimental (ΛNaPSSexp, blue ◊ points) and theoretical equivalent conductivities (ΛHy,El,R without dielectric friction,  ΛNaPSS with dielectric friction in absence of interference, ΛNaPSSi with dielectric friction in presence of interference) of sodium polystyrene sulphonate in water at 25°C.


### 3.1. Conductivity of Sodium Heparinate (RB21055)

The biological polyelectrolyte sodium heparinate (RB21055) is a linear polysaccharide, well known for its anticoagulant activity. Its monomer unity is a hexasaccharide, in which each disaccharide consists in a glucosamine followed by an uronic acid. The charged groups areSO3- (NSO3- or OSO3-) and COO- with a ratio OSO3-/COO-≈7/3. This heparin is provided by Sigma as a sodium salt extracted from pork stomach. The physical characteristics of the Sodium heparinate (RB21055) are as follows:MW≈10000 g·mol-1 is the average molecular weight of the Sodium Heparinate (RB21055).ZS=-53±2 at pH≈7 is the structural charge number.LS=160±8 Å is the structural length.bS=3.2±0.2 Å is the charge-to-charge distance.RC=10±1 Å is the cylindrical radius of the polyion chain.Table1 shows that the variation of the degree of dissociation αC of Na+ from heparin in the concentration range: 5×10-5M<CNa+°<5×10-3M is in conformity with the dilution principle so that αC increases with dilution from 0.59 to 0.76 and it differs from its Manning’s value αM=bS/|Zi|LB=0.44. Consequently, the apparent charge number Zap=αCZS varies with the concentration from −37.8 for CNa+°=5.11×10-5M to −29.5 for CNa+°=5.13×10-3M. Notice that this last value seems to be different from the value Zap′=-18 obtained from electrophoretic mobility using Nernst-Einstein relation: uP/DP=|Zap′|e/kBT with DP≈7×10-7 cm2·s−1 [22]. In fact Zap′ differs from Zap because it depends at the same time on ionic condensation (αC) and on ionic friction effects via uP. Table 1 and Figure 2 show that the experimental conductivity ΛNaHRBexp of sodium heparinate (RB21055) decreases sharply from 82.6 to 61.9cm2·Ω-1·eqvNa+-1 in the low concentration range: 5×10-5M<CNa+°<5×10-4M. Theoretically, the hydrodynamic contribution (at infinite dilution) ΛNaHRB°HD to ΛNaHRBexp is obtained in absence of ionic condensation (αC=1), of ionic frictions (ΔλPel=0, βPir=0) and in absence of dielectric friction (βPdf=0). According to (6), (10), (16), and (17), ΛNaHRB°HD=(λHRB°HD+λNa+°) with λNa+°=50.1, λHRB°HD=|ZS|Fe/6πη〈RP〉.  The mean radius 〈RP〉 of heparinate (RB21055) is calculated from (10). We found 〈RP〉=28.8Å, λHRB°HD=159.84, and ΛNaHRB°HD=210cm2·Ω-1·eqvNa+-1. Table 1 shows that the dielectric friction (βPdfi>50%) is the most significant retarding effect by comparison to the electrophoretic effect and the ionic relaxation effect even when taking into account the interference of the local displacement fields. Now, as βPdfi is proportional to (αC)2 then βPdfi≈(αC)2βP°dfi, which means that βPdfi increases with dilution toward its maximal value βP°dfi at infinite dilution (αC=1). According to (38), βP°dfi depends on the interference factor f=bS/Rg. Adjustment between experimental conductivities ΛNaHRBexp and theoretical conductivities ΛNaHRBi of heparinate (RB21055) leads to a group radius equal to Rg=1.65±0.05Å so that f≈2 (a succession of tangent charged spheres). The maximal value βP°dfi   is therefore equal to βP°dfi=(2/3)(Rw/bS)3(LB/bS)(LS/〈RP〉)[1-ε∞/εo]|(f3-4.808)+6.58/|ZS||=2.25, for bS=3.2Å, f≈2, LS=160Å and Rw=1.4Å. Consequently, the correct limiting conductivity at infinite dilution ΛNaHRB° must take into account the limiting dielectric friction effect βP°dfi  in addition to the hydrodynamic friction as follows: ΛNaHRB°=λNa+°+λHRB°HD/(1+βP°dfi)=50.1+49.2=99.3cm2·Ω-1·eqvNa+-1. This ΛNaHRB° is experimentally inaccessible because of the nonexistence of a universal limiting law allowing a rigorous extrapolation of ΛHRBexp at infinite dilution. This impossibility is due primarily to ionic condensation effect. However, according to (6), (7), and (11), and after neglecting ionic friction effects, we can derive the following approximate complex relation between ΛNaHRBexp and αC applicable in the range of very dilute solutions for which the stretched conformation is favored and for which the electrophoretic effects and ionic relaxation effects are negligible:
(39)ΛNaHRBexp≈αC[λNa+°+αCλHRB°HD(1+αC2βP°dfi)]≈αC[50.1+159.84αC(1+2.25αC2)].
This expression can be used as an indirect method to evaluate experimentally the degree of ionic condensation (1-αC) from experimental measurements of the equivalent conductivity of the polyelectrolyte at high dilution. Calculation shows that for CNa+°<10-4M, αC≥0.8 in conformity with theoretical values ((1)–(4)) but incompatible with the Manning’s value αM=bS/LB≈0.44. It is important to notice at this stage that the constancy of the condensation parameter at high dilution with αC=αM would imply that ΛNaHRBexp increases first with dilution then attains a “palier” (platform) in the range of very low concentrations in which αC obeys to the Manning’s model. This kind of behavior is not in general experimentally observed [14].In order to reinforce the hypothesis of the stretched conformation at high dilution we will compare the equivalent conductivity of sodium heparinateΛNaHRBSph calculated in the case of coiled conformation (quasi-spherical) having a mean radius RS≈〈RP〉≈kBT/6πηDP≈30Å, to its corresponding experimental equivalent conductivity ΛNaHRBexp. According to Zwanzig-Frank notation [19, 25] the expression of the equivalent conductivity λHRBSph of the spherical polyion at very high dilution is λHRBSph≈AαSph|ZS|/[RS+(αSph2ZS2C/RS3)], with A=92, C=10 at 25°C in water, RS=30 Å, and |ZS|=53. The equivalent conductivity of the (spherical) sodium heparinate is therefore ΛNaHRBSph≈αSph(λNa+°+λHRBSph). Now, according to the ionic association theory of Fuoss [3] the degree of dissociation: αSph>0.68 for CNa+°<10-4 M, therefore ΛNaHRBSph>100cm2·Ω-1·eqvNa+-1. This expected result is at least twenty percent larger (20%) than the experimental equivalent conductivity ΛNaHRBexp  (~83cm2·Ω-1·eqvNa+-1), and it therefore invalidates the hypothesis of spherical conformation for heparin RB21055 polyions at high dilution. Finally it is important to underline the following principal conclusions.(i)
The interference effect decreases by 70%–60% the dielectric friction on the stretched polyion. However, despite this important attenuation, the resulting dielectric friction remains the principal frictional effect (βPdfi decreases with concentration from 78% to 50%) in comparison to ionic relaxation effect and electrophoretic effect.(ii)
The adjustment of the experimental equivalent conductivitiesΛNaHRBexp with the theoretical conductivities ΛHRBi leads to a group radius Rg sensibly constant, of about 1.65±0.05 Å, that is, equal to the half of the distance bS=3.2±0.2 Å between two successive spherical charged groups, so that the coherent condition: f=bS/Rg=constant≥2 is verified in the studied concentration range.(iii)
The ionic relaxation effectβir on polyions and counterions becomes important (10%–20%) for CNa+°>10-4 M, while βir is <6% for highly dilute solutions. This result is not in conformity with the Manning prediction: βir→13%.(iv)
The electrophoretic effect is relatively weak forNa+ counterions (<6%), while it decreases as expected with dilution from 28% to 4% in the case of stretched heparin polyions.(v)
The degree of ionic condensation (1-αC) of Na+ on heparin RB21055 increases in % with the concentration from 24% to 41% and differs from the Manning value: 56%.(vi)
The sharp increasing with dilution of the equivalent conductivity of sodium heparinate proves that both thermodynamics behavior and electrolytic conductivity behavior of this polyelectrolyte are governed by the Ostwald concentration regime despite the stretched conformation of heparin polyions.


### 3.2. Conductivity of Sodium Chondroitin Sulfate

The biological polyion chondroitin sulfate is a large linear polysaccharide composed of repeating disaccharide units altering an amino sugarN-acetyl-β-galactosamine-4-sulfate and an β-glucuronic acid. The sulfate groups as well as the uronic acids result in linear chains having a negative charge.  Chondroitin sulfate is provided by Sigma as a sodium salt from bovine trachea. The physical characteristics of the macroion are [10, 11] as follows:MW≈21430g·mol-1  is the average molecular weight of the used polyelectrolyte.ZS=-75±3 is the structural charge number.LS=435±15 Å is the structural length.bS=5.8±0.2 Å is the charge-to-charge distance.RC=6±0.5 Å is the cylindrical radius of the polyion chain.This polyion is therefore about three times longer than heparin RB, and regarding its significant charge separationbS, it presents a more discontinuous linear charge distribution. On the other hand, according to Manning’s theory, we expect a weaker degree of condensation (1-αC) despite the relative importance of its structural charge number. Table 2 shows that the degree of dissociation αC of Na+ from chondroitin increases slightly with dilution from 0.804 to 0.852 in the concentration range: 1.38×10-4M<CNa+°<1.11×10-3M. Sodium chondroitin sulfate is one of peculiar polyelectrolytes for which the behavior of ionic condensation in aqueous solution is compatible at the same time with the model of Manning and with the principle of dilution [10, 11]. Indeed, the Manning’s value of the condensation parameter is αM=bS/|Zi|LB=0.81. Consequently, the apparent charge number Zap=αCZS varies slightly with the concentration from −63.9 for CNa+°=1.38×10-4M to −60.3 for CNa+°=1.11×10-3M.Table2 and Figure 3 show that the experimental conductivity of sodium chondroitin ΛNaChondroexp increases with dilution in a monotonous way from 65.3 to 78.54 cm2·Ω-1·eqvNa+-1 in the concentration range:  1.38×10-4M<CNa+°<1.11×10-3M, (i.e., without the appearance of any palier). The hydrodynamic contribution (at infinite dilution) ΛNaChondro°HD to ΛNaChondroexp is obtained in absence of ionic condensation and in absence of other frictional effects from (6), (10), (16), and (17), so that ΛNaChondro°HD=(λChondro°HD+λNa+°) with λNa+°=50.1 with λChondro°HD=|ZS|Fe/6πη〈RP〉.   The mean radius 〈RP〉 of the chondroitin sulfate is equal to 50.8 Å, therefore λChondro°HD=136.1 and ΛNaChondro°HD=186.2cm2·Ω-1·eqvNa+-1. Table 2 shows also  that the dielectric friction (βPdfi>85%) is the most significant retarding effect by comparison to the electrophoretic effect and the ionic relaxation effect even when taking into account the interference of the local displacement fields. βPdfi depends on the interference factor f=bS/Rg.  Adjustment between experimental and theoretical conductivities of chondroitin sulfate leads to a group radius equal to Rg=2.14±0.2 Å so that f=bS/Rg=2.71 (a discontinuous linear charge distribution), the maximal value βP°dfi   is, according to (38), equal to βP°dfi=(2/3)(Rw/bS)3(LB/bS)(LS/〈RP〉)[1-ε∞/εo]|(f3-4.808)+6.58/|ZS||=1.47 for bS=5.8Å, f≈2.7, LS=435Å and Rw=1.4Å. Consequently, the new limiting ΛNaChondro° must take into account the limiting dielectric friction effect βP°dfi in addition to the hydrodynamic friction as follows: ΛNaChondro°=λNa+°+λChondro°HD/(1+βP°dfi)=50.1+36.7=86.8cm2·Ω-1·eqvNa+-1. This ΛNaChondro° is experimentally inaccessible. However, according to our previous analysis and to (6), (7), and (11), we can proceed in the same manner that for Heparin RB21055 in order to derive the following approximate relation between the experimental equivalent conductivity of chondroitin sulfate ΛNaChondroexp and the degree of dissociation αC in the range of very dilute solutions for which the stretched conformation is favored and for which the electrophoretic effects and ionic relaxation effects are negligible, we obtain
(40)ΛNaChondroexp≈αC[λNa+°+αCλChondro°HD(1+αC2βP°dfi)]≈αC[50.1+136.13αC(1+1.47αC2)].
In conclusion we can underline the following points.(i)
The interference effect decreases by 25% the dielectric friction on stretched chondroitin sulfate polyions. However, despite this attenuation, the resulting dielectric friction remains the principal frictional effect (βPdfi decreases with concentration from 92% to 86%) in comparison to ionic relaxation effect and electrophoretic effect.(ii)
The adjustment of experimental equivalent conductivitiesΛNaChondroexp with the theoretical conductivities ΛNaChondroi leads to a group radius Rg sensibly constant of about 2.13±0.2 Å so that the coherent condition: f=bS/Rg=2.7≥2 is verified in the studied concentration range.(iii)
The ionic relaxation effectβir on chondroitin polyions and Na+ counterions remains approximately constant (~15%) in the studied range of relatively low concentrations: 1.38×10-4M<CNa+°<1.11×10-3 M in conformity with the Manning prediction.(iv)
The electrophoretic effect is relatively weak for theNa+ conterions (<4%), while it normally decreases with dilution from 27% to 12% for stretched chondroitin sulfate polyions.(v)
The degree of ionic condensation (1-αC) of Na+ on stretched chondroitin sulfate polyions increases in % slowly with the concentration from 15% to 19.6%. These values are close to the Manning value: (1-αM)=0.19.(vi)
The weak increasing with dilution of the equivalent conductivity of sodium chondroitin sulfate proves that both thermodynamics behavior and electrolytic conductivity behavior of this polyelectrolyte at very high dilution are intermediate between the Ostwald concentration regime and the Manning concentration regime.


### 3.3. Conductivity of Sodium Polystyrene Sulphonate (NaPSS)

Polystyrene sulphonate is a linear chain of 4-styrenesulfonic acid groups. The linear formula of NaPSS is –(C8H7NaO3S)n–; it is provided by Aldrich Chemicals as a sodium salt. The physical characteristics of the studied sodium polystyrene sulphonate are as follows:M
W
≈
600000 is the average molecular weight in g·mol−1 of the used polyelectrolyte.Z
S
=
-
2900
±
120 is the structural charge number.L
S
=
7250
±
300
Å is the structural length.b
S
=
2.5
±
0.1 Å is the charge-to-charge distance.R
C
=
6.9
±
0.5 Å is the cylindrical radius of the polyion chain.Table3 shows that the degree of dissociation αC of Na+ from PSS decreases very slightly with dilution from 0.441 to 0.457 in the concentration range: 1.16×10-3M<CNa+°<5.14×10-3 M. This behavior seems in contradiction with the Ostwald regime. However this slight variation is not very significant in regard of uncertainties and we can assume that αC remains sensibly constant. In fact, theoretical formal calculations according to (1) show that αC presents a little minimum (0.413) for CNa+°≈10-5M, then it increases slowly according to the principle of dilution until 0.605. We can therefore distinguish between two concentration ranges or regimes: (a) the Manning regime:  1×10-5M<CNa+°<5.1×10-3M, in which αC remains practically constant, and (b) a formal Ostwald concentration regime: CNa+°≪1×10-5M, in which αC increases with dilution. Because of the large length of PSS (7250 Å) it is not possible to experimentally observe highly diluted regime via the increasing of conductivity. Only the Manning concentration regime was experimentally observed via the constancy of the equivalent conductivity of NaPSS during dilution in the studied concentration range. Note that the Manning regime has also been observed for NaPSS polyelectrolytes of molar masses Mw between 8000 g·mol−1 and 360000 g·mol−1 [16]. However, despite the large length of PSS and despite the constancy of its degree of dissociation, the actual value ~0.45 of αC is in fact different from its Manning’s value αM=bS/|Zi|LB=0.35. Consequently, the apparent charge number Zap varies with the concentration from −1326 for CNa+°=5.14×10-3M to −1279 for CNa+°=1.16×10-3M.Table3 and Figure 4 show that the experimental conductivity ΛNaPSSexp decreases very slowly with concentration in a monotonous way from 35.5 to 34.9 cm2·Ω-1·eqvNa+-1 in the indicated concentration range. The mean radius 〈RP〉 of the polyion is equal to 521.2 Å; therefore according to (16)–(18) the hydrodynamic mobility λPSS°HD of the PSS polyion is equal to λPSS°HD=|ZS|Fe/6πη〈RP〉=512.8 cm2·Ω-1·eqvNa+-1. Adjustment between experimental and theoretical conductivities of NaPSS polyelectrolyte leads to a constant group radius equal to Rg≅1.21 Å so that f=bS/Rg=2.06 (a quasi-continuous linear charge distribution of tangent spheres); the maximal value of βP°dfi is therefore βP°dfi=(2/3)(Rw/bS)3(LB/bS)(LS/〈RP〉)[1-ε∞/εo]|(f3-4.808)+6.58/|ZS||=17.9 for bS=2.5 Å, f≈2, LS=7250 Å and Rw=1.4 Å.  Using our previous analysis and (6), (7), and (11), we can proceed in the same manner that for Heparin and Chondroitin in order to derive the following approximate relation between ΛNaPSSexp   and αC, in the range of dilute solutions for which the stretched conformation is favored, we obtain
(41)ΛNaPSSexp≈αC[λNa+°+αCλPSS°HD(1+αC2βP°dfi)]≈αC[50.1+512.8αC(1+17.9αC2)].Now, asαC undergoes weak variation in the Manning region, ΛNaPSSexp remains sensibly constant (palier of ΛNaPSSexp~35 cm2·Ω-1·eqvNa+-1), in the concentration range: 1×10-3M<CNa+°<5×10-3M.  We can however formally expect the emergence of highly diluted regime for CNa+°<10-6 M. It is also important to underline that the experimental observation of Manning behavior proves the veracity of the stretched conformation. Indeed, it is well known that for spherical conformation the degree of dissociation αS increases normally with dilution (Fuoss behavior) so that we will not be able to observe in this case any palier of conductivity.In conclusion we can underline the following points.(i)
The interference effect decreases by 55% the dielectric friction on PSS polyions. However, despite this attenuation, and particularly in this case of very long chains, the dielectric friction remains the principal frictional effect (βPdfi decreases with concentration from 172% to 142%) in comparison to ionic relaxation effect and electrophoretic effect.(ii)
The adjustment of experimental equivalent conductivitiesΛNaPSSexp with the theoretical conductivities ΛNaPSSi leads to a group radius Rg sensibly constant of about 1.21 Å so that the coherent condition: f=bS/Rg=2.06≥2 is verified in the studied concentration range.(iii)
The ionic relaxation effectβir on PSS polyions and Na+ counterions is relatively important and remains sensibly constant (~14%) in the studied range of relatively low concentrations: 1.16×10-3M<CNa+°<5.14×10-3M. The reason of the constancy of the relaxation effect is the large chain length of PSS (7250 Å) by comparison to the minor axis of the ionic atmosphere dP. Indeed, dP=199 Å for CNa+°=1.16×10-3M and dP=98 Å for CNa+°=5.14×10-3M. Therefore, dP/LS<0.03 and according to (23), βPir≈(αC/αM)(1/9)≈0.14 for all CNa+°.(iv)
The electrophoretic effect(ΔλPel/λP°Hyd=〈RP〉/〈dP〉) on PSS polyions is more important than βir in the studied concentration range; it decreases slowly with dilution from 61.9% to 51.7%. The electrophoretic effect on PSS becomes formally negligible (<5%) for highly dilute solution: CNa+°<10-6M.(v)
The observed conductivity variation of NaPSS in the concentration range:1.16×10-3M<CNa+°<5.14×10-3M reveals a behavior compatible with the Manning concentration regime because of the large length of its stretched chain.


## 3.1. Conductivity of Sodium Heparinate (RB21055)

The biological polyelectrolyte sodium heparinate (RB21055) is a linear polysaccharide, well known for its anticoagulant activity. Its monomer unity is a hexasaccharide, in which each disaccharide consists in a glucosamine followed by an uronic acid. The charged groups areSO3- (NSO3- or OSO3-) and COO- with a ratio OSO3-/COO-≈7/3. This heparin is provided by Sigma as a sodium salt extracted from pork stomach. The physical characteristics of the Sodium heparinate (RB21055) are as follows:MW≈10000 g·mol-1 is the average molecular weight of the Sodium Heparinate (RB21055).ZS=-53±2 at pH≈7 is the structural charge number.LS=160±8 Å is the structural length.bS=3.2±0.2 Å is the charge-to-charge distance.RC=10±1 Å is the cylindrical radius of the polyion chain.Table1 shows that the variation of the degree of dissociation αC of Na+ from heparin in the concentration range: 5×10-5M<CNa+°<5×10-3M is in conformity with the dilution principle so that αC increases with dilution from 0.59 to 0.76 and it differs from its Manning’s value αM=bS/|Zi|LB=0.44. Consequently, the apparent charge number Zap=αCZS varies with the concentration from −37.8 for CNa+°=5.11×10-5M to −29.5 for CNa+°=5.13×10-3M. Notice that this last value seems to be different from the value Zap′=-18 obtained from electrophoretic mobility using Nernst-Einstein relation: uP/DP=|Zap′|e/kBT with DP≈7×10-7 cm2·s−1 [22]. In fact Zap′ differs from Zap because it depends at the same time on ionic condensation (αC) and on ionic friction effects via uP. Table 1 and Figure 2 show that the experimental conductivity ΛNaHRBexp of sodium heparinate (RB21055) decreases sharply from 82.6 to 61.9cm2·Ω-1·eqvNa+-1 in the low concentration range: 5×10-5M<CNa+°<5×10-4M. Theoretically, the hydrodynamic contribution (at infinite dilution) ΛNaHRB°HD to ΛNaHRBexp is obtained in absence of ionic condensation (αC=1), of ionic frictions (ΔλPel=0, βPir=0) and in absence of dielectric friction (βPdf=0). According to (6), (10), (16), and (17), ΛNaHRB°HD=(λHRB°HD+λNa+°) with λNa+°=50.1, λHRB°HD=|ZS|Fe/6πη〈RP〉.  The mean radius 〈RP〉 of heparinate (RB21055) is calculated from (10). We found 〈RP〉=28.8Å, λHRB°HD=159.84, and ΛNaHRB°HD=210cm2·Ω-1·eqvNa+-1. Table 1 shows that the dielectric friction (βPdfi>50%) is the most significant retarding effect by comparison to the electrophoretic effect and the ionic relaxation effect even when taking into account the interference of the local displacement fields. Now, as βPdfi is proportional to (αC)2 then βPdfi≈(αC)2βP°dfi, which means that βPdfi increases with dilution toward its maximal value βP°dfi at infinite dilution (αC=1). According to (38), βP°dfi depends on the interference factor f=bS/Rg. Adjustment between experimental conductivities ΛNaHRBexp and theoretical conductivities ΛNaHRBi of heparinate (RB21055) leads to a group radius equal to Rg=1.65±0.05Å so that f≈2 (a succession of tangent charged spheres). The maximal value βP°dfi   is therefore equal to βP°dfi=(2/3)(Rw/bS)3(LB/bS)(LS/〈RP〉)[1-ε∞/εo]|(f3-4.808)+6.58/|ZS||=2.25, for bS=3.2Å, f≈2, LS=160Å and Rw=1.4Å. Consequently, the correct limiting conductivity at infinite dilution ΛNaHRB° must take into account the limiting dielectric friction effect βP°dfi  in addition to the hydrodynamic friction as follows: ΛNaHRB°=λNa+°+λHRB°HD/(1+βP°dfi)=50.1+49.2=99.3cm2·Ω-1·eqvNa+-1. This ΛNaHRB° is experimentally inaccessible because of the nonexistence of a universal limiting law allowing a rigorous extrapolation of ΛHRBexp at infinite dilution. This impossibility is due primarily to ionic condensation effect. However, according to (6), (7), and (11), and after neglecting ionic friction effects, we can derive the following approximate complex relation between ΛNaHRBexp and αC applicable in the range of very dilute solutions for which the stretched conformation is favored and for which the electrophoretic effects and ionic relaxation effects are negligible:
(39)ΛNaHRBexp≈αC[λNa+°+αCλHRB°HD(1+αC2βP°dfi)]≈αC[50.1+159.84αC(1+2.25αC2)].
This expression can be used as an indirect method to evaluate experimentally the degree of ionic condensation (1-αC) from experimental measurements of the equivalent conductivity of the polyelectrolyte at high dilution. Calculation shows that for CNa+°<10-4M, αC≥0.8 in conformity with theoretical values ((1)–(4)) but incompatible with the Manning’s value αM=bS/LB≈0.44. It is important to notice at this stage that the constancy of the condensation parameter at high dilution with αC=αM would imply that ΛNaHRBexp increases first with dilution then attains a “palier” (platform) in the range of very low concentrations in which αC obeys to the Manning’s model. This kind of behavior is not in general experimentally observed [14].In order to reinforce the hypothesis of the stretched conformation at high dilution we will compare the equivalent conductivity of sodium heparinateΛNaHRBSph calculated in the case of coiled conformation (quasi-spherical) having a mean radius RS≈〈RP〉≈kBT/6πηDP≈30Å, to its corresponding experimental equivalent conductivity ΛNaHRBexp. According to Zwanzig-Frank notation [19, 25] the expression of the equivalent conductivity λHRBSph of the spherical polyion at very high dilution is λHRBSph≈AαSph|ZS|/[RS+(αSph2ZS2C/RS3)], with A=92, C=10 at 25°C in water, RS=30 Å, and |ZS|=53. The equivalent conductivity of the (spherical) sodium heparinate is therefore ΛNaHRBSph≈αSph(λNa+°+λHRBSph). Now, according to the ionic association theory of Fuoss [3] the degree of dissociation: αSph>0.68 for CNa+°<10-4 M, therefore ΛNaHRBSph>100cm2·Ω-1·eqvNa+-1. This expected result is at least twenty percent larger (20%) than the experimental equivalent conductivity ΛNaHRBexp  (~83cm2·Ω-1·eqvNa+-1), and it therefore invalidates the hypothesis of spherical conformation for heparin RB21055 polyions at high dilution. Finally it is important to underline the following principal conclusions.(i)
The interference effect decreases by 70%–60% the dielectric friction on the stretched polyion. However, despite this important attenuation, the resulting dielectric friction remains the principal frictional effect (βPdfi decreases with concentration from 78% to 50%) in comparison to ionic relaxation effect and electrophoretic effect.(ii)
The adjustment of the experimental equivalent conductivitiesΛNaHRBexp with the theoretical conductivities ΛHRBi leads to a group radius Rg sensibly constant, of about 1.65±0.05 Å, that is, equal to the half of the distance bS=3.2±0.2 Å between two successive spherical charged groups, so that the coherent condition: f=bS/Rg=constant≥2 is verified in the studied concentration range.(iii)
The ionic relaxation effectβir on polyions and counterions becomes important (10%–20%) for CNa+°>10-4 M, while βir is <6% for highly dilute solutions. This result is not in conformity with the Manning prediction: βir→13%.(iv)
The electrophoretic effect is relatively weak forNa+ counterions (<6%), while it decreases as expected with dilution from 28% to 4% in the case of stretched heparin polyions.(v)
The degree of ionic condensation (1-αC) of Na+ on heparin RB21055 increases in % with the concentration from 24% to 41% and differs from the Manning value: 56%.(vi)
The sharp increasing with dilution of the equivalent conductivity of sodium heparinate proves that both thermodynamics behavior and electrolytic conductivity behavior of this polyelectrolyte are governed by the Ostwald concentration regime despite the stretched conformation of heparin polyions.


## 3.2. Conductivity of Sodium Chondroitin Sulfate

The biological polyion chondroitin sulfate is a large linear polysaccharide composed of repeating disaccharide units altering an amino sugarN-acetyl-β-galactosamine-4-sulfate and an β-glucuronic acid. The sulfate groups as well as the uronic acids result in linear chains having a negative charge.  Chondroitin sulfate is provided by Sigma as a sodium salt from bovine trachea. The physical characteristics of the macroion are [10, 11] as follows:MW≈21430g·mol-1  is the average molecular weight of the used polyelectrolyte.ZS=-75±3 is the structural charge number.LS=435±15 Å is the structural length.bS=5.8±0.2 Å is the charge-to-charge distance.RC=6±0.5 Å is the cylindrical radius of the polyion chain.This polyion is therefore about three times longer than heparin RB, and regarding its significant charge separationbS, it presents a more discontinuous linear charge distribution. On the other hand, according to Manning’s theory, we expect a weaker degree of condensation (1-αC) despite the relative importance of its structural charge number. Table 2 shows that the degree of dissociation αC of Na+ from chondroitin increases slightly with dilution from 0.804 to 0.852 in the concentration range: 1.38×10-4M<CNa+°<1.11×10-3M. Sodium chondroitin sulfate is one of peculiar polyelectrolytes for which the behavior of ionic condensation in aqueous solution is compatible at the same time with the model of Manning and with the principle of dilution [10, 11]. Indeed, the Manning’s value of the condensation parameter is αM=bS/|Zi|LB=0.81. Consequently, the apparent charge number Zap=αCZS varies slightly with the concentration from −63.9 for CNa+°=1.38×10-4M to −60.3 for CNa+°=1.11×10-3M.Table2 and Figure 3 show that the experimental conductivity of sodium chondroitin ΛNaChondroexp increases with dilution in a monotonous way from 65.3 to 78.54 cm2·Ω-1·eqvNa+-1 in the concentration range:  1.38×10-4M<CNa+°<1.11×10-3M, (i.e., without the appearance of any palier). The hydrodynamic contribution (at infinite dilution) ΛNaChondro°HD to ΛNaChondroexp is obtained in absence of ionic condensation and in absence of other frictional effects from (6), (10), (16), and (17), so that ΛNaChondro°HD=(λChondro°HD+λNa+°) with λNa+°=50.1 with λChondro°HD=|ZS|Fe/6πη〈RP〉.   The mean radius 〈RP〉 of the chondroitin sulfate is equal to 50.8 Å, therefore λChondro°HD=136.1 and ΛNaChondro°HD=186.2cm2·Ω-1·eqvNa+-1. Table 2 shows also  that the dielectric friction (βPdfi>85%) is the most significant retarding effect by comparison to the electrophoretic effect and the ionic relaxation effect even when taking into account the interference of the local displacement fields. βPdfi depends on the interference factor f=bS/Rg.  Adjustment between experimental and theoretical conductivities of chondroitin sulfate leads to a group radius equal to Rg=2.14±0.2 Å so that f=bS/Rg=2.71 (a discontinuous linear charge distribution), the maximal value βP°dfi   is, according to (38), equal to βP°dfi=(2/3)(Rw/bS)3(LB/bS)(LS/〈RP〉)[1-ε∞/εo]|(f3-4.808)+6.58/|ZS||=1.47 for bS=5.8Å, f≈2.7, LS=435Å and Rw=1.4Å. Consequently, the new limiting ΛNaChondro° must take into account the limiting dielectric friction effect βP°dfi in addition to the hydrodynamic friction as follows: ΛNaChondro°=λNa+°+λChondro°HD/(1+βP°dfi)=50.1+36.7=86.8cm2·Ω-1·eqvNa+-1. This ΛNaChondro° is experimentally inaccessible. However, according to our previous analysis and to (6), (7), and (11), we can proceed in the same manner that for Heparin RB21055 in order to derive the following approximate relation between the experimental equivalent conductivity of chondroitin sulfate ΛNaChondroexp and the degree of dissociation αC in the range of very dilute solutions for which the stretched conformation is favored and for which the electrophoretic effects and ionic relaxation effects are negligible, we obtain
(40)ΛNaChondroexp≈αC[λNa+°+αCλChondro°HD(1+αC2βP°dfi)]≈αC[50.1+136.13αC(1+1.47αC2)].
In conclusion we can underline the following points.(i)
The interference effect decreases by 25% the dielectric friction on stretched chondroitin sulfate polyions. However, despite this attenuation, the resulting dielectric friction remains the principal frictional effect (βPdfi decreases with concentration from 92% to 86%) in comparison to ionic relaxation effect and electrophoretic effect.(ii)
The adjustment of experimental equivalent conductivitiesΛNaChondroexp with the theoretical conductivities ΛNaChondroi leads to a group radius Rg sensibly constant of about 2.13±0.2 Å so that the coherent condition: f=bS/Rg=2.7≥2 is verified in the studied concentration range.(iii)
The ionic relaxation effectβir on chondroitin polyions and Na+ counterions remains approximately constant (~15%) in the studied range of relatively low concentrations: 1.38×10-4M<CNa+°<1.11×10-3 M in conformity with the Manning prediction.(iv)
The electrophoretic effect is relatively weak for theNa+ conterions (<4%), while it normally decreases with dilution from 27% to 12% for stretched chondroitin sulfate polyions.(v)
The degree of ionic condensation (1-αC) of Na+ on stretched chondroitin sulfate polyions increases in % slowly with the concentration from 15% to 19.6%. These values are close to the Manning value: (1-αM)=0.19.(vi)
The weak increasing with dilution of the equivalent conductivity of sodium chondroitin sulfate proves that both thermodynamics behavior and electrolytic conductivity behavior of this polyelectrolyte at very high dilution are intermediate between the Ostwald concentration regime and the Manning concentration regime.


## 3.3. Conductivity of Sodium Polystyrene Sulphonate (NaPSS)

Polystyrene sulphonate is a linear chain of 4-styrenesulfonic acid groups. The linear formula of NaPSS is –(C8H7NaO3S)n–; it is provided by Aldrich Chemicals as a sodium salt. The physical characteristics of the studied sodium polystyrene sulphonate are as follows:M
W
≈
600000 is the average molecular weight in g·mol−1 of the used polyelectrolyte.Z
S
=
-
2900
±
120 is the structural charge number.L
S
=
7250
±
300
Å is the structural length.b
S
=
2.5
±
0.1 Å is the charge-to-charge distance.R
C
=
6.9
±
0.5 Å is the cylindrical radius of the polyion chain.Table3 shows that the degree of dissociation αC of Na+ from PSS decreases very slightly with dilution from 0.441 to 0.457 in the concentration range: 1.16×10-3M<CNa+°<5.14×10-3 M. This behavior seems in contradiction with the Ostwald regime. However this slight variation is not very significant in regard of uncertainties and we can assume that αC remains sensibly constant. In fact, theoretical formal calculations according to (1) show that αC presents a little minimum (0.413) for CNa+°≈10-5M, then it increases slowly according to the principle of dilution until 0.605. We can therefore distinguish between two concentration ranges or regimes: (a) the Manning regime:  1×10-5M<CNa+°<5.1×10-3M, in which αC remains practically constant, and (b) a formal Ostwald concentration regime: CNa+°≪1×10-5M, in which αC increases with dilution. Because of the large length of PSS (7250 Å) it is not possible to experimentally observe highly diluted regime via the increasing of conductivity. Only the Manning concentration regime was experimentally observed via the constancy of the equivalent conductivity of NaPSS during dilution in the studied concentration range. Note that the Manning regime has also been observed for NaPSS polyelectrolytes of molar masses Mw between 8000 g·mol−1 and 360000 g·mol−1 [16]. However, despite the large length of PSS and despite the constancy of its degree of dissociation, the actual value ~0.45 of αC is in fact different from its Manning’s value αM=bS/|Zi|LB=0.35. Consequently, the apparent charge number Zap varies with the concentration from −1326 for CNa+°=5.14×10-3M to −1279 for CNa+°=1.16×10-3M.Table3 and Figure 4 show that the experimental conductivity ΛNaPSSexp decreases very slowly with concentration in a monotonous way from 35.5 to 34.9 cm2·Ω-1·eqvNa+-1 in the indicated concentration range. The mean radius 〈RP〉 of the polyion is equal to 521.2 Å; therefore according to (16)–(18) the hydrodynamic mobility λPSS°HD of the PSS polyion is equal to λPSS°HD=|ZS|Fe/6πη〈RP〉=512.8 cm2·Ω-1·eqvNa+-1. Adjustment between experimental and theoretical conductivities of NaPSS polyelectrolyte leads to a constant group radius equal to Rg≅1.21 Å so that f=bS/Rg=2.06 (a quasi-continuous linear charge distribution of tangent spheres); the maximal value of βP°dfi is therefore βP°dfi=(2/3)(Rw/bS)3(LB/bS)(LS/〈RP〉)[1-ε∞/εo]|(f3-4.808)+6.58/|ZS||=17.9 for bS=2.5 Å, f≈2, LS=7250 Å and Rw=1.4 Å.  Using our previous analysis and (6), (7), and (11), we can proceed in the same manner that for Heparin and Chondroitin in order to derive the following approximate relation between ΛNaPSSexp   and αC, in the range of dilute solutions for which the stretched conformation is favored, we obtain
(41)ΛNaPSSexp≈αC[λNa+°+αCλPSS°HD(1+αC2βP°dfi)]≈αC[50.1+512.8αC(1+17.9αC2)].Now, asαC undergoes weak variation in the Manning region, ΛNaPSSexp remains sensibly constant (palier of ΛNaPSSexp~35 cm2·Ω-1·eqvNa+-1), in the concentration range: 1×10-3M<CNa+°<5×10-3M.  We can however formally expect the emergence of highly diluted regime for CNa+°<10-6 M. It is also important to underline that the experimental observation of Manning behavior proves the veracity of the stretched conformation. Indeed, it is well known that for spherical conformation the degree of dissociation αS increases normally with dilution (Fuoss behavior) so that we will not be able to observe in this case any palier of conductivity.In conclusion we can underline the following points.(i)
The interference effect decreases by 55% the dielectric friction on PSS polyions. However, despite this attenuation, and particularly in this case of very long chains, the dielectric friction remains the principal frictional effect (βPdfi decreases with concentration from 172% to 142%) in comparison to ionic relaxation effect and electrophoretic effect.(ii)
The adjustment of experimental equivalent conductivitiesΛNaPSSexp with the theoretical conductivities ΛNaPSSi leads to a group radius Rg sensibly constant of about 1.21 Å so that the coherent condition: f=bS/Rg=2.06≥2 is verified in the studied concentration range.(iii)
The ionic relaxation effectβir on PSS polyions and Na+ counterions is relatively important and remains sensibly constant (~14%) in the studied range of relatively low concentrations: 1.16×10-3M<CNa+°<5.14×10-3M. The reason of the constancy of the relaxation effect is the large chain length of PSS (7250 Å) by comparison to the minor axis of the ionic atmosphere dP. Indeed, dP=199 Å for CNa+°=1.16×10-3M and dP=98 Å for CNa+°=5.14×10-3M. Therefore, dP/LS<0.03 and according to (23), βPir≈(αC/αM)(1/9)≈0.14 for all CNa+°.(iv)
The electrophoretic effect(ΔλPel/λP°Hyd=〈RP〉/〈dP〉) on PSS polyions is more important than βir in the studied concentration range; it decreases slowly with dilution from 61.9% to 51.7%. The electrophoretic effect on PSS becomes formally negligible (<5%) for highly dilute solution: CNa+°<10-6M.(v)
The observed conductivity variation of NaPSS in the concentration range:1.16×10-3M<CNa+°<5.14×10-3M reveals a behavior compatible with the Manning concentration regime because of the large length of its stretched chain.


## 4. Conclusions

In this work we have used electrolytic conductivity as a powerful tool to investigate, in dilute aqueous solutions at 25°C, the behavior of three kinds of flexible polyelectrolytes of different chain lengthsLS and different charge distances bS   and without added salt: sodium heparinate (LS=160Å, bS=3.2Å), sodium chondroitin sulfate (LS=435 Å, bS=5.8Å), and sodium polystyrene sulphonate (LS=7250Å, bS=2.5Å). A strong influence on their equivalent conductivities ΛPoly has been shown to exist due to the interdependent effects of Na+ condensation and of dielectric friction and particularly when their chains adopt a stretched conformation at high dilute solutions.  In contrast, the ionic friction effects on counterions and polyions due to the electrophoretic effects (ΔλNa+el, ΔλPel) and to the ionic relaxation effect (βir) are less important and become negligible at high dilution. The correlation between ΛPoly and, respectively, the degree of condensation (1-αC), the limiting equivalent conductivity λNa+°, the hydrodynamic equivalent conductivity of the polyion λP°HD, the limiting dielectric friction coefficient (βP°df), and the ionic friction terms (ΔλNa+el, ΔλPel, βir)  can be expressed by the following general expression:
(42)ΛPoly≈αC(1+βir)-1×[(λNa+°-ΔλNa+el)+(αCλP°HD-ΔλPel)(1+αC2βP°dfi)].
The limiting dielectric friction coefficient (βP°df) is very sensitive to the conformation of the polyion, to the charge distance bS, and to the radius Rg of charged groups. Indeed, (βP°df) is weak for coiled conformations and vanishes when vPτ>Rg, where vP is the velocity of the polyion and τ  is the dielectric relaxation time of the solvent molecules. In contrast, for a stretched chain of charged spheres, (βP°df) is in general larger than 50% and increases with the interference term f3=(bS/Rg)3 as follows:
(43)βP°dfi=(23)(RwbS)3(LBbS)(LS〈RP〉)[1-ε∞εo]×|(f3-4.808)+6.58|ZS||~Ln(LSRC).
On the basis of our data and of our analysis, we can provisionally conclude that it is not possible to propose a universal model in order to describe the electrolytic conductivity behavior of polyelectrolytes having whichever chain lengths and distances between charged groups. However, two concentration regimes could be identified: (a) the Ostwald concentration regime (it is the case of heparinate of sodium) for which αC and ΛPoly increase with dilution, while ΔλNa+el, ΔλPel, and βir→0; (b) the Manning concentration regime (it is the case of sodium polystyrene sulphonate) for which αC, βir, and ΛPoly remain sensibly constant with dilution, while ΔλNa+el and ΔλPel decrease with concentration. It is important to underline that the ideal Manning behavior (LS/RC→∞) implies the restrictive conditions: αC=αM=bS/LB, βir=1/9, and βP°dfi=0 (no dielectric friction: vPτ>Rg), while less restrictive conditions imply αC=constant, βir=(αC/9αM), and βP°dfi≠0. Only the sodium chondroitin sulfate presents an intermediate behavior between Ostwald concentration regime and Manning concentration regime with αC≈αM=0.81 and βir≈0.13. This mixed behavior is characterized by a slowly increasing of its equivalent conductivity with dilution.



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*Source: 852752-2013-02-06.xml*