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

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.is 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.


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 strength  1/2 .is concentration effect is caused by two sorts of ionic frictions: the electrophoretic effect and the ionic relaxation effect.is limiting law has been extended in the case of semidilute simple electrolytes [1][2][3][4].In contrast, a completely satisfactory theory to describe the dynamic behavior in general and particularly the electrolytic conductivity of dilute �exible polyelectrolytes in aqueous solution is not yet available despite some interesting progress toward this objective [5][6][7][8][9][10][11][12][13][14][15][16].is 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 in�uence of the translational dielectric friction on moving polyions as well as its dependence on concentration [17,18].e present paper is a supplementary contribution toward this goal in order to propose a general formal limiting equation expressing the in�uence 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.is 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.

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][6][7][8][9][10][11][12][13], so that each chain can be modeled as a cylinder of   radius,   structural length (  = ) and    structural charge.On 2Γ MSA = −1 + 1 + 4 and   are the microscopic concentrations (number of particles/Å 3 ) of, respectively, the counterions and the polyions.  =  2    is the structural volume of the cylindrical polyion.(  ,    and (  ,    are "the con�guration functions" which depend on the conformation of the polyion.Notice �rst 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 (  /  → 0) and for dilute solutions (  /  ≪ 1),   increases very slowly with dilution so that   remains approximately constant in a large range of low concentrations.However,   approaches toward   only for some particular polyelectrolytes [10].
Notice also that (1) can be applied for ellipsoidal polyions of any shape (i.e.,     ,     and for all   ).

eoretical Conductivity of Cylindrical Polyelectrolytes in Dilute
Solutions.In practice we measure the speci�c conductance  of the polyelectrolyte solution in S⋅cm −1 . is related to the equivalent ionic conductivities   and   , respectively, of the polyion and the counterion as follows:  = 10 −3         ∘  +         ∘   , where  ∘  is the molar concentration of the polyions and  ∘  is the total molar concentration of the counterions so that    ∘  is the molar concentration of the free counterions, and (  |  |) is the apparent (effective) charge | ap | of the polyion partially neutralized by the condensed counterions.e electroneutrality condition implies that |  | ∘  = |  | ∘  .e equivalent conductivity Λ Poly of the polyelectrolyte is therefore de�ned by and   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 �eld .is polarization effect induces on the moving central ion (or polyion) a local �eld  ir opposed to .Quantitatively, these two effects appear in the expression of the ionic conductivity   of simple ions (in our case, the counterions) via the Δ el  corrective term and the  ir  relaxation term as follows: ∘  is the ionic equivalent conductivity of the counterion at in�nite dilution which expresses both the hydrodynamic friction, due to the viscosity  of the solvent (Robinson and Stokes [2]), and the dielectric friction effect (Zwanzig [19]).

𝜆𝜆 ∘
is determined experimentally by linear extrapolation at in�nite dilution and according to the Debye-�nsager limiting equation [2], of the equivalent conductivity Λ iX () (with the square root of the ionic strength ) of any corresponding simple electrolyte (e.g., if   Na + , we can choose Λ iX  Λ NaCl ; extrapolation of Λ NaCl in water at 25 ∘ C leads to  ∘ Na + = 50.1 cm 2 ⋅ Ω −1 ⋅ eqv Na + −1 ).e term  is the Faraday,   is the effective radius of the solvated counterion "",   is the radius of its ionic atmosphere, and Γ MSA is its corresponding Debye-MSA screen parameter.Notice that Γ MSA differs from the screen parameter Γ MSA 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.⟨  ⟩ is the mean radius of the polyion (analog to the radius of gyration) which is also equal to the electrostatic capacitance  AP (in c.g.s.u.e units) of the ellipsoidal (or cylindrical) polyion [17,20].Finally, the explicit expression of the ionic relaxation term  ir  will be examined at the end of this section because of its interdependence with the  ir  term relating to the polyions.e expression of the equivalent conductivity   of the polyion is more complex because its ionic equivalent conductivity at in�nite dilution  ∘  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  ∘  by an extrapolation method at in�nite dilution.For this reason we decomposed the   expression as follows [9]: e �usti�cation of the above equation is the following: the external electric �eld  acting on the polyion polarizes its ionic atmosphere as well as its surrounding solvent molecules, which gives place to an ionic relaxation �eld  ir and to a dielectric relaxation �eld  df slowing down the movement of the polyion.e velocity  of the polyion can thus be written in two manners: It is remarkable to underline that the expression of  ∘Hyd  coincides with the Hubbard-Douglass general relation [20] expressing the hydrodynamic mobility:  ∘HD    ∘HD  / of an arbitrarily shaped unspeci�ed macroion of    charge, in terms of its capacitance  AP (generalization of Stokes' law): We can therefore generalize the Hubbard-Douglass relation given by (17) to the Henry equivalent conductivity  Henry  as follows [17]: ′ AP 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 ⟨  ⟩: Notice that the relative importance of the electrophoretic effect can be evaluated by the ratio: is last equation implies that the electrophoretic effect vanishes in the range of highly dilute solutions, that is, when ⟨  ⟩ → ∞.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.e corresponding spherical hydrodynamic radius   was evaluated according to Stokes-Einstein relation:   =   6 ∘  , where  ∘  is the self diffusion coefficient of the polyion at high dilution.e same spherical approximation could be used for the calculation of the ionic relaxation effect using Onsager relation applicable to spherical simple ions [3], therefore with:   ≅ 1, if we assume that the ionic atmosphere is free of polyions.However, Manning has demonstrated that for in�nite rod-like model (  ≫   ),  ir  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]: so that when     → 0, then  ir  →   |    |  (6  ); this limiting expression converges toward the Debye-Onsager relation concerning spherical ions.In contrast, for polyions of large length,   ≫   ,  ir  → 19 for all  ∘  if   →   .On the other hand, according to linear irreversible thermodynamics (T.I.P) the different  ir  relaxation terms of all the "" species (ions or polyions) in solution are interdependent via the general relation [9,14]: ∑       ir  = 0, with ∑      = 0. is means that in the case of our binary system the two relaxation terms  ir  and  ir  , respectively, of the polyions and the counterions are equal: 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  df  will be discussed in detail in the next paragraph.

Importance of the Dielectric Friction Effect on a Stretched
Polyion.e 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.en, we will treat, without going into the mathematical details, the speci�c case of a stretched polyion modeled as a chain of |  | identical charged spheres, each one having a charge   =    and a radius   (a linear discontinuous distribution of ionized groups).
e general mechanism of dielectric friction is the following: when a sphere of charge  and radius   is submitted to a moderate external alternating �eld  along the  axis, it acquires a velocity  = , where  is its electrical mobility.We indicate by   () =  the position of the center of the sphere at time .During its movement the charge  induces at each point  of the dielectric medium (solvent) de�ned by its radius-vector (, , , ) a time-dependent polarization (, ) which is proportional to the displacement �eld (, −  1 ) created by  at different anterior times ( −  1 ): e module  ′ is the distance between the point  and the center of the sphere at time ( −  1 ).is noninstantaneous response results from the fact that each solvent molecule needs a relaxation time  to be oriented along the radial �eld .Mathematically, the linear relation between (, ) and (,  −  1 ) is given by the following convolution integral [17,19]: with is the aer effect function which depends on the delta function  representing electronic relaxation and on the permittivities  0 and  ∞ , respectively, the static and the highfrequency dielectric constants of the solvent.For water at 25 ∘ C,  0 = 78.3 and  ∞ = 1.77.Note that we have set the upper limit of the above integral to ∞, because in general the dielectric relaxation time  is small by comparison to t so that   0 (vanishes rapidly) when   .
In turn, this induced polarization   exerts back on the charged sphere a resulting dielectric frictional force  df =  df where  df is the so called having a direction opposed to the external �eld .e general integral relation between the  component Δ df  of  df and the   ,   ,   components of     1  via   and therefore  1  is, [17], 1  is the so called key integral de�ned by where  =   .Integration is taken over the whole volume except the 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, Δ df   0 when   0 (immobile sphere) or when   0 (instantaneous response so that   0).In other words, the delay effect (  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 �eld: Δ df   0. Consequently, linearity between causes and response implies that  df ∼ .On the other hand, as the dimension of the electrical relaxation force  df =  df is ∼  2 /[length] 2 , scaling analysis yields to  df ∼  2 / 3   .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   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    1  given by ( 25) into (29): We can use the above equation to compute the dielectric friction effect on a spherical polyion of effective charge  =     and radius   .Indeed, according to ( 12) and ( 14), its velocity  is given by  =  ( Finally it is important to underline the singular case of Manning's polyions (  ≅ 0 and Because of the in�nite length of its moving chain, the structural state of the polyion (distribution of charges, distribution of solvent molecules, and therefore  )) varies periodically with time with a period equal to:  =   /  .
As for slowly moving polyions,       , therefore   , that is, the solvent molecules have not sufficient time to reorient themselves toward the new �eld  ) during the periodic variation.Consequently, the polyion seems to be immobile (  = 0) and thus surrounded by its initial symmetrical cylindrical distribution of solvent molecules.is conservation of the equilibrium symmetry implies the absence of any resulting dielectric relaxation �eld ( df = 0).It is the reason for which the dielectric relaxation effect is completely absent in the restrictive case of the Manning's model.

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][10][11].Notice that conductivity results concerning (NaPSS) are those published by Vink [14].Comparisons for each polyelectrolyte, between experimental equivalent conductivities Λ exp Poly = 1000/ ∘ Na + and theoretical equivalent conductivities Λ th Poly =     +   ) 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  ∘ Na + of Na + counterions, the experimental equivalent conductivities Λ exp Poly in cm 2 ⋅Ω 1 ⋅eqv Na + 1 , the degrees of Na + condensation (1    ) on polyions, the apparent charge numbers  ap =     of the polyions, the theoretical equivalent conductivities Λ Hy,El,R in absence of dielectric friction (i.e., only: hydrodynamic, electrophoretic, and ionic relaxation effects), the group radius   , the theoretical equivalent conductivities Λ Poly in absence of interference, the theoretical equivalent conductivities Λ  Poly in presence of interference, the % of the dielectric friction effect  d� (36) in presence of interference, and the ratios  d�  / df  (37) expressing the relative importance of the interference effect ( df is the dielectric friction effect in absence of interference).

Conductivity of Sodium Heparinate (RB21055)
. e 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.e charged groups are SO 3 − (NSO 3 − or OSO 3 − ) and COO − with a ratio OSO 3 − /COO − ≈ 7/3.is heparin is provided by Sigma as a sodium salt extracted from pork stomach.e physical characteristics of the Sodium heparinate (RB21055) are as follows: ≈ 10 000 g⋅mol −1 is the average molecular weight of the Sodium Heparinate (RB21055).
= 10 ± 1 Å is the cylindrical radius of the polyion chain.
Table 1 shows that the variation of the degree of dissociation   of Na + from heparin in the concentration range: 5 × 10 −5 M <  ∘ Na + < 5 × 10 −3 M is in conformity with the dilution principle so that   increases with dilution from 0.59 to 0.76 and it differs from its Manning's value   =   /|  |  = 0.44.Consequently, the apparent charge number  ap =     varies with the concentration from −37.8 for  ∘ Na + = 5.11 × 10 −5 M to −29.5 for  ∘ Na + = 5.13 × 10 −3 M. Notice that this last value seems to be different from the value  ′ ap = −18 obtained from electrophoretic mobility using Nernst-Einstein relation:   /  = | ′ ap |/   with   ≈ 7 × 10 −7 cm 2 ⋅s −1 [22].In fact  ′ ap differs from  ap because it depends at the same time on ionic condensation (  ) and on ionic friction effects via   .Table 1 and Figure 2 show that the experimental conductivity Λ exp NaHRB of sodium heparinate (RB21055) decreases sharply from 82.6 to 61.9 cm 2 ⋅ Ω −1 ⋅ eqv Na + −1 in the low concentration range: 5 × 10 −5 M <  ∘ Na + < 5 × 10 −4 M. eoretically, the hydrodynamic contribution (at in�nite dilution) Λ ∘HD NaHRB to Λ exp NaHRB is obtained in absence of ionic condensation (  = 1), of ionic frictions (Δ el  = 0,  ir  = 0) and in absence of dielectric friction ( df  = 0).According to (6), ( 10), (16), and (17), Λ ∘HD NaHRB = ( ∘HD HRB +  ∘ Na + ) with  ∘ Na + = 50.1, ∘HD HRB = |  |/6  ⟩. e mean radius   ⟩ of heparinate (RB21055) is calculated from (10).We found   ⟩ = 28.8Å,  ∘HD HRB = 159.84,and Λ ∘HD NaHRB = 210 cm 2 ⋅ Ω −1 ⋅eqv Na + −1 .Table 1 shows that the dielectric friction ( d�  > 50%) is the most signi�cant retarding effect by comparison to the electrophoretic effect and the ionic relaxation effect even when taking into account the interference of the local displacement �elds.Now, as  d�  is proportional to (  ) 2   then  d�  ≈ (  ) 2  ∘d�  , which means that  d�  increases with dilution toward its maximal value  ∘d�  at in�nite dilution (  = 1).According to (38),  ∘d�  depends on the interference factor  =   /  .Adjustment between experimental conductivities Λ exp NaHRB and theoretical conductivities Λ  NaHRB of heparinate (RB21055) leads to a group radius equal to   = 1.65 ± 0.05 Å so that  ≈ 2 (a succession of tangent charged spheres).e maximal value  ∘d�  is therefore equal to  ∘d�  = (2/3)(  /  ) 3  NaHRB must take into account the limiting dielectric friction effect  ∘d�  in addition to the hydrodynamic friction as follows: NaHRB is experimentally inaccessible because of the nonexistence of a universal limiting law allowing a rigorous extrapolation of Λ exp HRB at in�nite dilution.is impossibility is due primarily to ionic condensation effect.However, according to ( 6), (7), and (11), and aer neglecting ionic friction effects, we can derive the following approximate complex relation between Λ exp NaHRB 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: ( is expression can be used as an indirect method to evaluate experimentally the degree of ionic condensation (1−  ) from experimental measurements of the equivalent conductivity of the polyelectrolyte at high dilution.Calculation shows that for  ∘ Na + < 10 −4 M,   ≥ 0.8 in conformity with theoretical values ((1)-( 4)) but incompatible with the Manning's value   =   /  ≈ 0.44.It is important to notice at this stage that the constancy of the condensation parameter at high dilution with   =   would imply that Λ exp NaHRB increases �rst with dilution then attains a "palier" (platform) in the range of very low concentrations in which   obeys to the Manning's model.is 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 heparinate Λ Sph NaHRB calculated in the case of coiled conformation (quasi-spherical) having a mean radius   ≈   ⟩ ≈   /6  ≈ 30 Å, to its corresponding experimental equivalent conductivity Λ  ).Now, according to the ionic association theory of Fuoss [3] the degree of dissociation:  Sph > 0.68 for  ∘ Na + < 10 −4 M, therefore Λ Sph NaHRB > 100 cm 2 ⋅ Ω −1 ⋅ eqv Na + −1 .is expected result is at least twenty percent larger (20%) than the experimental equivalent conductivity Λ exp NaHRB (∼ 83 cm 2 ⋅ Ω −1 ⋅ eqv Na + −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.

𝐂𝐂 ∘
(i) e 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 ( d�  decreases with concentration from 78% to 50%) in comparison to ionic relaxation effect and electrophoretic effect.
(ii) e adjustment of the experimental equivalent conductivities Λ exp NaHRB with the theoretical conductivities Λ  HRB leads to a group radius   sensibly constant, of about 1.65 ± 0.05 Å, that is, equal to the half of the distance   = 3.2 ± 0.2 Å between two successive spherical charged groups, so that the coherent condition:  =   /  = constant ≥ 2 is veri�ed in the studied concentration range.
(iv) e electrophoretic effect is relatively weak for Na + counterions ( < 6%), while it decreases as expected with dilution from 28% to 4% in the case of stretched heparin polyions.
(v) e degree of ionic condensation (1 −   ) of Na + on heparin RB21055 increases in % with the concentration from 24% to 41% and differs from the Manning value: 56%.
(vi) e 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.

Conductivity of Sodium Chondroitin
Sulfate.e biological polyion chondroitin sulfate is a large linear polysaccharide composed of repeating disaccharide units altering an amino sugar -acetyl--galactosamine-4-sulfate and an glucuronic acid.e 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.e physical characteristics of the macroion are [10,11] as follows: ≈ 21430 g⋅mol −1 is the average molecular weight of the used polyelectrolyte.
= 6 ± 0.5 Å is the cylindrical radius of the polyion chain.
is polyion is therefore about three times longer than heparin RB, and regarding its signi�cant charge separation   , 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 −   ) despite the relative importance of its structural charge number.Table 2 shows that the degree of dissociation   of Na + from chondroitin increases slightly with dilution from 0.804 to 0.852 in the concentration range: 1.38 × 10 −4 M <  ∘ Na + < 1.11 × 10 −3 M. 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   =   /|  |  = 0.81.Consequently, the apparent charge number  ap =     varies slightly with the concentration from −63.9 for  ∘ Na + = 1.38 × 10 −4 M to −60.3 for  ∘ Na + = 1.11 × 10 −3 M. Table 2 and Figure 3 show that the experimental conductivity of sodium chondroitin Λ   2 shows also that the dielectric friction ( d�  > 85%) is the most signi�cant retarding effect by comparison to the electrophoretic effect and the ionic relaxation effect even when taking into account the interference of the local displacement �elds. d�  depends on the interference factor  =   /  .Adjustment between experimental and theoretical conductivities of chondroitin sulfate leads to a group radius equal to   = 2.14 ± 0.2 Å so that  =   /  = 2.71 (a discontinuous linear charge distribution), the maximal value  ∘d�  is, according to (38), equal to  ∘ d�  = (2/3)(  /  ) 3    ≈ 600000 is the average molecular weight in g⋅mol −1 of the used polyelectrolyte.  = −2900 ± 120 is the structural charge number.
= 6.9 ± 0.5 Å is the cylindrical radius of the polyion chain.
Table 3 shows that the degree of dissociation   of Na + from PSS decreases very slightly with dilution from 0.441 to 0.457 in the concentration range: 1.16 × 10 −3 M <  ∘ Na + < 5.14 × 10 −3 M. is behavior seems in contradiction with the Ostwald regime.However this slight variation is not very signi�cant in regard of uncertainties and we can assume that   remains sensibly constant.In fact, theoretical formal calculations according to (1) show that   presents a little minimum (0.413) for  ∘ Na + ≈ 10 −5 M, 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 −5 M <  ∘ Na + < 5.1 × 10 −3 M, in which   remains practically constant, and (b) a formal Ostwald concentration regime:  ∘ Na + ≪ 1 × 10 −5 M, in which   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   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   is in fact different from its Manning's value   =   /|  |  = 0.35.Consequently, the apparent charge number  ap varies with the concentration from −1326 for  ∘ Na + = 5.14 × 10 −3 M to −1279 for  ∘ Na + = 1.16 × 10 −3 M. Table 3 and Figure 4 show that the experimental conductivity Λ exp NaPSS decreases very slowly with concentration in a monotonous way from 35.5 to 34.9 cm 2 ⋅ Ω −1 ⋅ eqv Na + −1 in the indicated concentration range.e mean radius ⟨  ⟩ of the polyion is equal to 521. and   = 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 Λ (41) Now, as   undergoes weak variation in the Manning region, Λ exp NaPSS remains sensibly constant (palier of Λ exp NaPSS ∼ 35 cm 2 ⋅ Ω −1 ⋅ eqv Na + −1 ), in the concentration range: 1 × 10 −3 M <  ∘ Na + < 5 × 10 −3 M. We can however formally expect the emergence of highly diluted regime for  ∘ Na + < 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   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) e 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 ( d�  decreases with concentration from 172% to 142%) in comparison to ionic relaxation effect and electrophoretic effect.
(ii) e adjustment of experimental equivalent conductivities Λ exp NaPSS with the theoretical conductivities Λ  NaPSS leads to a group radius   sensibly constant of about 1.21 Å so that the coherent condition:

F 1 :
Representation of a polyelectrolyte according to the cylindrical model.

exp
NaChondro increases with dilution in a monotonous way from 65.3 to 78.54 cm 2 ⋅ Ω −1 ⋅ eqv Na + −1 in the concentration range: 1.38 × 10 −4 M <  ∘ Na + < 1.11 × 10 −3 M, (i.e., without the appearance of any palier).e hydrodynamic contribution (at in�nite dilution) Λ ∘HD NaChondro to Λ exp NaChondro is obtained in absence of ionic condensation and in absence of other frictional effects from T 2: Variations with counterions concentration  ∘ Na + of the experimental Λ exp NaChondro and theoretical Λ NaChondro , Λ  NaChondro equivalent conductivities, of the degree of Na + condensation (1 −  C ), of the apparent charge number  ap , and of the dielectric frictions in absence ( df ) and in presence ( d� ) of interference, in the case of sodium chondroitin sulfate in water at 25 ∘

exp
NaPSS and   , in the range of dilute solutions for which the stretched conformation is favored, we obtain Henry   +  ir +  df  with   and the relative dielectric friction effect is de�ned by  df  = | df |/| +  ir +  df |, therefore (30) leads to  df  = 2  |  |/3[1   ∞ /  ] Now, in the case of stretched chain of |  | successive charged spheres of charge   =   and of radius   =   ( = 1, |  |), each sphere  moving along direction  with velocity   undergoes from the polarized solvent molecules a dielectric frictional force  df  =    df  , where  df  is the local dielectric relaxation �eld.�ecause of the axial symmetry of the system around the  axis, only the  component Δ df , , and   are the components of the vector    separating a point    of the dielectric medium and the position of the charge   at time    1 .eprincipaldifferencebetween(25)-(29)relatingtoa spherical polyion and (32) and (33), comes primarily from the fact that the expression of the displacement �eld     1 ) created by the stretched chain of charged spheres at   ) is now ) differs from (25) by the sum ∑     1 ) expressing the interference of the displacement �eld       1 ) created at  by   at an anterior time    1 ), with the different       1 ) due to the |  |  1) charges   of the moving chain.Notice however that, if the distance:   =   /|  |, between two successive charged groups, is sufficiently large (  ≫ 2  , i.e.,  =   /  ≫ 2) so that:  ′  >  ′  , then the dielectric friction undergone by the sphere   is essentially due to polarized solvent molecules of its entourage.We could therefore neglect the interference effect and     1 ) is reduced thus to       1 ) =    ′  / ′  ) words, each charged sphere of the moving polyion of radius   and charge   =   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:       .eevaluation of the interference effect in terms of the "interference parameter"  =   /  ≥ 2 is quanti�ed by the ratio between  d�  in presence of interference and  df  in absence of interference: In contrast, it is maximal for a chain of tangent spheres ( = 2) and increases with |  |.It is interesting to note that  d�  increases with dilution as 2 and it reaches its maximal value  ∘d�  at in�nite dilution, that is, when   = 1 (Ostwald) and  ′ AP = ⟨  ⟩: [17] recently, authors of[17]have demonstrated that this last expression remains valid even in the case of an ellipsoidal polyion of minor axis , interfocuses distance , and effective charge  =     but with the proviso of replacing the spherical radius   by an apparent ray  app which is a function of the eccentricity  = /2 so that  app ≈ ⟨ for   1 and  app ≈ /2 for   1.3; in other is equation shows that for a discontinuous charge distribution, that is,   ≫    ≫ 2), the interference effect becomes negligible.