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

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.

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 strength 𝐼^1/2. 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 𝑅_𝐶 radius, 𝐿_𝑆 structural length (𝐿_𝑆=𝐴𝐵) and 𝑍_𝑆𝑒 structural charge. On the other hand, as the linear distribution of the total charge 𝑍_𝑆𝑒 is uniform, then the distance of separation between two successive ionizable groups is 𝑏_𝑆=𝐿_𝑆/|𝑍_𝑆|. Because of the ionic condensation effect, these charged groups are, in absence of additional salt (salt-free), partially neutralized only by counterions “𝑖” of 𝑍_𝑖𝑒 charge and 𝑅_𝑖 effective radius. According to Manning’s rod-like model (𝑅_𝐶/𝐿_𝑆→0) [6–12], the degree of ionic condensation (1−𝛼_𝑀) is independent on the counterions concentration 𝐶_𝑖 so that 𝛼_𝑀=𝑏_𝑆/|𝑍_𝑖|𝐿_𝐵, where 𝐿_𝐵=𝑒²/𝜀_𝑜𝑘_𝐵𝑇 is the Bjerrum length, 𝑘_𝐵 is the Boltzmann constant, and 𝜀_𝑜 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<𝐿_𝑆) [9–11], the corresponding degree of dissociation 𝛼 obeys in general to the Ostwald’s principle of dilution and consequently 𝛼→1 when 𝐶_𝑖→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 𝑑_𝑃 minor axis (see Figure 1). The thickness 𝑑_𝑃 is a function of 𝑅, 𝛼 and 𝐶_𝑖 via the Debye-MSA screen parameter Γ_𝑃MSA [1] and the Debye length 𝜒_𝐷⁻¹. Notice that both first and second layers are equipotential surfaces of total charges equal, respectively, to −(1−𝛼)𝑍_𝑆𝑒 and −𝛼𝑍_𝑆𝑒. In the case of the cylindrical model (𝐴𝐵=𝐿_𝑆, 𝑅≅𝑅_𝐶 and 𝛼≡𝛼_𝐶), calculation of 𝛼_C needs the resolution of the following ensemble of implicit equations: 1−𝛼_𝐶𝛼_𝐶=1−𝑉_𝑃𝐶_𝑃⁻¹𝑉_𝑃𝐶_𝑃×𝑔𝑅_𝐶,𝐿_𝑆𝑔𝑑_𝑃,𝐿_𝑆^{{(𝛼𝐶+|𝑍𝑖/𝑍𝑆|)/𝛼𝑀}},𝑔𝑥,𝐿_𝑆=4𝑥²+𝐿²_𝑆^1/2+𝐿_𝑆4𝑥²+𝐿²_𝑆^1/2−𝐿_𝑆,𝑑_𝑃=𝑅_𝐶+12Γ_𝑃MSA,2Γ_𝑃MSA=−1+1+4𝜒_𝐷𝑅_𝑖^1/22𝑅_𝑖;𝜒_𝐷=4𝜋𝐿_𝐵𝑍²_𝑖𝛼_𝐶𝐶_𝑖^1/2. (1)(2)(3)(4)𝐶_𝑖 and 𝐶_𝑃 are the microscopic concentrations (number of particles/ų) of, respectively, the counterions and the polyions. 𝑉_𝑃=𝜋𝑅²_𝐶𝐿_𝑆 is the structural volume of the cylindrical polyion. 𝑔(𝑅_𝐶,𝐿_𝑆) and 𝑔(𝑑_𝑃,𝐿_𝑆) 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 (𝑅_𝐶/𝐿_𝑆→0) and for dilute solutions (𝑅_𝐶/𝑑_𝑃≪1), 𝛼_𝐶 increases very slowly with dilution so that 𝛼_𝐶 remains approximately constant in a large range of low concentrations. However, 𝛼_𝐶 approaches toward 𝛼_𝑀 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≠𝐿_𝑆, 𝑅≠𝑅_𝐶 and for all 𝑉_𝑃).

2.2. Theoretical Conductivity of Cylindrical Polyelectrolytes in Dilute Solutions

In practice we measure the specific conductance 𝜒 of the polyelectrolyte solution in S·cm⁻¹. 𝜒 is related to the equivalent ionic conductivities 𝜆_𝑃 and 𝜆_𝑖, respectively, of the polyion and the counterion as follows: 𝜒=10⁻³𝛼_𝐶𝑍_𝑆𝜆_𝑃𝐂^∘_𝑃+𝛼_𝐶𝑍_𝑖𝜆_𝑖𝐂^∘_𝑖, (5) where 𝐂^∘_𝑃 is the molar concentration of the polyions and 𝐂^∘_𝑖 is the total molar concentration of the counterions so that 𝛼_𝐶𝐂^∘_𝑖 is the molar concentration of the free counterions, and (𝛼_𝐶|𝑍_𝑆|𝑒) is the apparent (effective) charge |𝑍_ap|𝑒 of the polyion partially neutralized by the condensed counterions. The electroneutrality condition implies that |𝑍_𝑆|𝐂^∘_𝑃=|𝑍_𝑖|𝐂^∘_𝑖. The equivalent conductivity Λ_Poly of the polyelectrolyte is therefore defined by Λ_Poly=1000𝜒𝑍_𝑖𝐂^∘_𝑖=𝛼_𝐶𝜆_𝑃+𝜆_𝑖. (6)𝜆_𝑃 and 𝜆_𝑖 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 Δ𝐗^ir opposed to 𝐄. Quantitatively, these two effects appear in the expression of the ionic conductivity 𝜆_𝑖 of simple ions (in our case, the counterions) via the Δ𝜆^el_𝑖 corrective term and the 𝛽^ir_𝑖 relaxation term as follows: 𝜆_𝑖=𝜆^∘_𝑖−Δ𝜆^el_𝑖1+𝛽^ir_𝑖,Δ𝜆^el_𝑖=𝑍_𝑖𝐹𝑒6𝜋𝜂𝑑_𝑖;𝑑_𝑖=𝑅_𝑖+12Γ_𝑖MSA,Γ²_𝑖MSA≅𝜋𝐿_𝐵𝑍_𝑖²𝛼_𝐶𝐶_𝑖1+2Γ_𝑖MSA𝑅_𝑖²+𝑍²_𝑆𝛼²_𝐶𝐶_𝑃1+2Γ_𝑖MSA𝑅_𝑃²,𝑅_𝑃=𝐿_𝑆Ln𝑔𝑅_𝐶,𝐿_𝑆. (7)(8)(9)(10)𝜆^∘_𝑖 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]). 𝜆^∘_𝑖 is determined experimentally by linear extrapolation at infinite dilution and according to the Debye-Onsager limiting equation [2], of the equivalent conductivity Λ_iX (√𝐼) (with the square root of the ionic strength 𝐼) of any corresponding simple electrolyte (e.g., if 𝑖≡Na⁺, we can choose Λ_iX≡Λ_NaCl; extrapolation of Λ_NaCl in water at 25°C leads to 𝜆^∘_Na⁺=50.1cm²⋅Ω⁻¹⋅eqv_Na+⁻¹). The term 𝐹 is the Faraday, 𝑅_𝑖 is the effective radius of the solvated counterion “𝑖”, 𝑑_𝑖 is the radius of its ionic atmosphere, and Γ_𝑖MSA is its corresponding Debye-MSA screen parameter. Notice that Γ_𝑖MSA differs from the screen parameter Γ_𝑃MSA 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. ⟨𝑅_𝑃⟩ is the mean radius of the polyion (analog to the radius of gyration) which is also equal to the electrostatic capacitance 𝐶_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 𝛽^ir_𝑖 will be examined at the end of this section because of its interdependence with the 𝛽^ir_𝑃 term relating to the polyions.

The expression of the equivalent conductivity 𝜆_𝑃 of the polyion is more complex because its ionic equivalent conductivity at infinite dilution 𝜆^∘_𝑃 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 𝜆^∘_𝑃 by an extrapolation method at infinite dilution. For this reason we decomposed the 𝜆_𝑃 expression as follows [9]: 𝜆_𝑃=𝜆^Henry_𝑃1+𝛽^ir_𝑃+𝛽^df_𝑃=𝜆^∘Hyd_𝑃−Δ𝜆^el_𝑃1+𝛽^ir_𝑃+𝛽^df_𝑃. (11) 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 Δ𝐗^ir and to a dielectric relaxation field Δ𝐗^df slowing down the movement of the polyion. The velocity 𝐯 of the polyion can thus be written in two manners: 𝐯=𝑢_𝑃𝐄=𝑢^Henry_𝑃𝐄+Δ𝐗^ir+Δ𝐗^df≡𝑢^Henry_𝑃𝐄1+𝛽^ir_𝑃+𝛽^df_𝑃. (12)𝑢_𝑃 is the actual electrical mobility of the polyion and 𝑢^Henry_𝑃 is the Henry mobility [9, 21] so that 𝜆_𝑃=𝐹𝑢_𝑃;𝜆^Henry_𝑃=𝐹𝑢^Henry_𝑃. (13)𝛽^ir_𝑃 and 𝛽^df_𝑃 express, respectively, the ionic relaxation effect and the dielectric friction effect acting on the polyion. Equation (12) implies that 𝛽^ir_𝑃=Δ𝐗^ir𝐄+Δ𝐗^ir+Δ𝐗^df,𝛽^df_𝑃=Δ𝐗^df𝐄+Δ𝐗^ir+Δ𝐗^df. (14) The explicit expression of the Henry equivalent conductivity 𝜆^Henry_𝑃 of the polyion is, [9], 𝜆^Henry_𝑃=𝛼_𝐶𝐹𝑒6𝜋𝜂𝑏_𝑆𝑔𝑅Ln^𝐶,𝐿_𝑆𝑔𝑑_𝑃,𝐿_𝑆. (15) Accordingly, we can separate 𝜆^Henry_𝑃 into two contributions: the purely hydrodynamic term 𝜆^∘Hyd_𝑃 due to the viscosity 𝜂 of the solvent, and the so-called electrophoretic term Δ𝜆^el_𝑃, so that 𝜆^∘Hyd_𝑃=𝛼_𝐶𝐹𝑒6𝜋𝜂𝑏_𝑆Ln𝑔𝑅_𝐶,𝐿_𝑆,Δ𝜆^el_𝑃=𝛼_𝐶𝐹𝑒6𝜋𝜂𝑏_𝑆Ln𝑔𝑑_𝑃,𝐿_𝑆. (16) It is remarkable to underline that the expression of 𝜆^∘Hyd_𝑃 coincides with the Hubbard-Douglass general relation [20] expressing the hydrodynamic mobility: 𝑢^∘HD_𝑃≡𝜆^∘HD_𝑃/𝐹 of an arbitrarily shaped unspecified macroion of 𝑍_𝑆𝑒 charge, in terms of its capacitance 𝐶_AP (generalization of Stokes’ law): 𝑢^∘HD_𝑃=𝑍_𝑆𝑒6𝜋𝜂𝐶_AP,𝐶_AP=𝐿_𝑆Ln𝑔𝑅_𝐶,𝐿_𝑆=𝑅_𝑃. (17)(18) We can therefore generalize the Hubbard-Douglass relation given by (17) to the Henry equivalent conductivity 𝜆^Henry_𝑃 as follows [17]: 𝜆^Henry_𝑃=𝛼_𝐶𝑍_𝑆𝐹𝑒6𝜋𝜂𝐶^′_AP. (19)𝐶^′_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 ⟨𝑑_𝑃⟩: 𝐶^′_AP⁻¹=𝑅_𝑃⁻¹−𝑑_𝑃⁻¹,𝑑_𝑃=𝐿_𝑆Ln𝑔𝑑_𝑃,𝐿_𝑆. (20) Notice that the relative importance of the electrophoretic effect can be evaluated by the ratio: Δ𝜆^el_𝑃𝜆^∘Hyd_𝑃=𝑅_𝑃𝑑_𝑃. (21) This last equation implies that the electrophoretic effect vanishes in the range of highly dilute solutions, that is, when ⟨𝑑_𝑃⟩→∞. 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 𝑅_𝐻 was evaluated according to Stokes-Einstein relation: 𝑅_𝐻=𝑘_𝐵𝑇/6𝜋𝜂𝐷^∘_𝑃, where 𝐷^∘_𝑃 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 𝛽^ir_𝑃≅𝛼_𝐶𝑍_𝑖𝑍_𝑆𝐿_𝐵𝑞²_𝑟31+𝑞_𝑟𝑑_𝑃 (22) with: 𝑞_𝑟≅1, if we assume that the ionic atmosphere is free of polyions. However, Manning has demonstrated that for infinite rod-like model (𝐿_𝑆≫𝑑_𝑃), 𝛽^ir_𝑃 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]: 𝛽^ir_𝑃=𝛼_𝐶𝑍_𝑆𝑍_𝑖𝐿_𝐵3𝑑²_𝑃+𝐿²_𝑆/418𝑑²_𝑃+𝐿²_𝑆/4^3/2, (23) so that when 𝐿_𝑆/𝑑_𝑃→0, then 𝛽^ir_𝑃→𝛼_𝐶|𝑍_𝑖𝑍_𝑆|𝐿_𝐵/(6𝑑_𝑃); this limiting expression converges toward the Debye-Onsager relation concerning spherical ions. In contrast, for polyions of large length, 𝐿_𝑆≫𝑑_𝑃, 𝛽^ir_𝑃→1/9 for all 𝐂^∘_𝑖 if 𝛼_𝐶→𝛼_𝑀.

On the other hand, according to linear irreversible thermodynamics (T.I.P) the different 𝛽^ir_𝑘 relaxation terms of all the “𝑘” species (ions or polyions) in solution are interdependent via the general relation [9, 14]: ∑_𝑘𝑍_𝑘𝐶_𝑘𝛽^ir_𝑘=0, with ∑_𝑘𝑍_𝑘𝐶_𝑘=0. This means that in the case of our binary system the two relaxation terms 𝛽^ir_𝑃 and 𝛽^ir_𝑖, respectively, of the polyions and the counterions are equal: 𝛽^ir_𝑃=𝛽^ir_𝑖. (24) 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 𝛽^df_𝑃 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 |𝑍_𝑆| identical charged spheres, each one having a charge 𝑞_𝑛=𝛼_𝐶𝑒 and a radius 𝑅_𝑔 (a linear discontinuous distribution of ionized groups).

The general mechanism of dielectric friction is the following: when a sphere of charge 𝑞 and radius 𝑅_𝑆 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 𝐫_𝑞(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): 𝐃𝐫,𝑡−𝑡₁=𝑞𝐫^′𝑟^′3,𝐫^′=𝐫𝑥,𝑦,𝑧,𝑡−𝐫_𝑞𝑡−𝑡₁. (25) 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]: 𝐏()=1𝐫,𝑡4𝜋^∞₀𝛾𝑡₁𝐃𝐫,𝑡−𝑡₁𝑑𝑡₁ (26) with 𝛾(𝑡)=11−𝜀_∞𝛿(𝑡)+𝜀₀−𝜀_∞𝜏𝜀²_∞⁻¹exp−𝜀₀𝑡𝜀_∞𝜏. (27)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 polarization 𝐏(𝐫,𝑡) exerts back on the charged sphere a resulting dielectric frictional force 𝐅^df=𝑞Δ𝐗^df where Δ𝐗^df is the so called dielectric relaxation field having a direction opposed to the external field 𝐄. The general integral relation between the 𝑧 component Δ𝑋^df_𝑧 of Δ𝐗^df and the 𝐷_𝑥, 𝐷_𝑦, 𝐷_𝑧 components of 𝐃(𝐫,𝑡−𝑡₁) via 𝐏(𝐫,𝑡) and therefore 𝛾(𝑡₁) is, [17], Δ𝑋^df_𝑧=14𝜋^∞₀𝛾𝑡₁𝐼_𝑧𝑡₁𝑑𝑡₁. (28)𝐼_𝑧(𝑡₁) is the so called key integral defined by 𝐼_𝑧𝑡₁=−𝐷_𝑧𝑡−𝑡₁𝑟^′3+3𝑍²𝐷_𝑧𝑡−𝑡₁𝑟^′5+3𝑥𝑍𝐷_𝑥𝑡−𝑡₁𝑟^′5+3𝑦𝑍𝐷_𝑦𝑡−𝑡₁𝑟^′5×𝑑𝑥𝑑𝑦𝑑𝑧, (29) 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, Δ𝑋^df_𝑧→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: Δ𝑋^df_𝑧≠0. Consequently, linearity between causes and response implies that Δ𝐗^df∼(𝐯𝜏). On the other hand, as the dimension of the electrical relaxation force 𝐅^df=𝑞Δ𝐗^df is ∼(𝑞)²/[length]², scaling analysis yields to 𝐅^df∼−(𝑞)²(𝐯𝜏)/𝑅³_𝑆.

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 𝑅_𝑆 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): Δ𝐗^df=±2𝑞/31−𝜀_∞/𝜀_𝑜(𝐯𝜏)𝜀_𝑜𝑅³_𝑆. (30) We can use the above equation to compute the dielectric friction effect on a spherical polyion of effective charge 𝑞=𝑒𝛼_𝑆𝑍_𝑆 and radius 𝑅_𝑆. Indeed, according to (12) and (14), its velocity 𝐯 is given by 𝐯=𝑢^Henry_𝑃(𝐄+Δ𝐗^ir+Δ𝐗^df) with 𝑢^Henry_𝑃=𝜆^Henry_𝑃/𝐹 and the relative dielectric friction effect is defined by 𝛽^df_𝑃=|Δ𝐗^df|/|𝐄+Δ𝐗^ir+Δ𝐗^df|, therefore (30) leads to 𝛽^df_𝑃=(2𝑒𝛼_𝑆|𝑍_𝑆|/3𝐹)[1−𝜀_∞/𝜀_𝑜](𝜆^Henry_𝑃𝜏)/(𝜀_𝑜𝑅³_𝑆). If we explicit the expression of 𝜆^Henry_𝑃 according to (19) and after replacing the relaxation time 𝜏 by its Debye expression in terms of the solvent radius 𝑅_𝑤(𝑅_H2O=1.4Å) and the viscosity 𝜂 so that 𝜏=6𝜋𝜂𝑅³_𝑤/𝑘_𝐵𝑇, we obtain 𝛽^df_𝑃=23𝛼_𝐶𝑍_𝑆²𝑅_𝑤𝑅_𝑆³𝐿_𝐵𝐶^′_AP𝜀1−^∞𝜀_𝑜. (31) 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=𝑒𝛼_𝐸𝑍_𝑆 but with the proviso of replacing the spherical radius 𝑅_𝑆 by an apparent ray 𝑅_app which is a function of the eccentricity 𝛾=𝐿/2𝑅 so that 𝑅_app≈⟨𝑅⟩ for 𝛾<1 and 𝑅_app≈𝐿/2 for 𝛾>1.

Now, in the case of stretched chain of |𝑍_𝑆| successive charged spheres of charge 𝑞_𝑛=−𝑒𝛼_𝐶  and of radius 𝑅_𝑛=𝑅_𝑔 (𝑛=1, |𝑍_𝑆|), each sphere 𝑛 moving along direction 𝑧 with velocity 𝐯_𝑝 undergoes from the polarized solvent molecules a dielectric frictional force 𝐅^df_𝐧=𝑞_𝐧Δ𝐗^df_𝐧, where Δ𝐗^df_𝐧 is the local dielectric relaxation field. Because of the axial symmetry of the system around the 𝑧 axis, only the 𝑧 component Δ𝑋^df_𝑛,𝑧 of Δ𝐗^df_𝐧 is different from zero: Δ𝑋^df_𝑛,𝑧=14𝜋^∞₀𝛾𝑡₁𝐼_𝑛,𝑧𝑡₁𝑑𝑡₁. (32) The corresponding key integral 𝐼_𝑛,𝑧(𝑡₁) is defined by 𝐼_𝑛,𝑧𝑡₁=−𝐷_𝑧𝑡−𝑡₁𝑟^′3_𝑛+3𝑍²_𝑛𝐷_𝑧𝑡−𝑡₁𝑟^′5_𝑛+3𝑥𝑍_𝑛𝐷_𝑥𝑡−𝑡₁𝑟^′5_𝑛+3𝑦𝑍_𝑛𝐷_𝑦𝑡−𝑡₁𝑟^′5_𝑛×𝑑𝑥𝑑𝑦𝑑𝑧. (33)𝑥, 𝑦, and 𝑍_𝑛 are the components of the vector 𝐫^′_𝐧 separating a point 𝑀(𝑥,𝑦,𝑧)  of the dielectric medium and the position of the charge 𝑞_𝐧 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 𝐃𝐫,𝑡−𝑡₁=𝑞_𝐧𝑟^′_𝑛𝑟^′_𝑛³+^{|𝑍𝑆|−1}_𝑖≠𝑛𝐃_𝑖𝐫,𝑡−𝑡₁,𝐃_𝑖𝐫,𝑡−𝑡₁=𝑞_𝐢𝑟^′_𝑖𝑟^′_𝑖³. (34) This equation results from the principle of superposition so that (34) differs from (25) by the sum ∑𝐃_𝑖(𝐫,𝑡−𝑡₁) expressing the interference of the displacement field 𝐃_𝑛(𝐫,𝑡−𝑡₁) created at 𝑀 by 𝑞_𝐧 at an anterior time (𝑡−𝑡₁), with the different 𝐃_𝑖(𝐫,𝑡−𝑡₁) due to the (|𝑍_𝑆|−1) charges 𝑞_𝐢 of the moving chain. Notice however that, if the distance: 𝑏_𝑆=𝐿_𝑆/|𝑍_𝑆|, between two successive charged groups, is sufficiently large (𝑏_𝑆≫2𝑅_𝑔, i.e., 𝑓=𝑏_𝑆/𝑅_𝑔≫2) so that: 𝑟^′_𝑖>𝑟^′_𝑛, then the dielectric friction undergone by the sphere 𝑞_𝐧 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 𝐃_𝑛(𝐫,𝑡−𝑡₁)=𝑞_𝐧𝐫^′_𝐧/(𝑟^′_𝑛)³; in other words, each charged sphere of the moving polyion of radius 𝑅_𝑔 and charge 𝑞_𝑛=−𝑒𝛼_𝐶 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: 𝛽^df_𝑃=23𝛼²_𝐶𝑅_𝑤𝑅_𝑔³𝐿_𝐵𝑏_𝑆𝐿_𝑆𝐶^′_AP𝜀1−^∞𝜀_𝑜. (35) Recent calculations [18] based on (14), (27), and (32)–(34) including the interference effect lead to the following corrected expression 𝛽^dfi_𝑃 in replacement to the above expression of 𝛽^df_𝑃: 𝛽^dfi_𝑃=23𝑓³𝛼²_𝐶𝑅_𝑤𝑅_𝑔³𝐿_𝐵𝑏_𝑆𝐿_𝑆𝐶^′_AP×𝜀1−^∞𝜀_𝑜𝑓³−4.808+6.58𝑍_𝑆,𝑏𝑓=^𝑆𝑅_𝑔. (36) Notice that (35) and (36) are valid only in the case of slowly moving polyions so that 𝑣_𝑃𝜏≤𝑅_𝑔. The evaluation of the interference effect in terms of the “interference parameter” 𝑓=𝑏_𝑆/𝑅_𝑔≥2 is quantified by the ratio between 𝛽^dfi_𝑃 in presence of interference and 𝛽^df_𝑃 in absence of interference: 𝛽^dfi_𝑃𝛽^df_𝑃=1−4.808𝑓³+6.58𝑓³𝑍_𝑆≤1,𝑓≥2. (37) This equation shows that for a discontinuous charge distribution, that is, 𝑏_𝑆≫𝑅_𝑔(𝑓≫2), the interference effect becomes negligible. In contrast, it is maximal for a chain of tangent spheres (𝑓=2) and increases with |𝑍_𝑆|.

It is interesting to note that 𝛽^dfi_𝑃 increases with dilution as 𝛼²_𝐶 and it reaches its maximal value 𝛽^∘dfi_𝑃 at infinite dilution, that is, when 𝛼_𝐶=1 (Ostwald) and 𝐶^′_AP=⟨𝑅_𝑃⟩: 𝛽^∘dfi_𝑃=23𝑅_𝑤𝑏_𝑆³𝐿_𝐵𝑏_𝑆𝐿_𝑆𝑅_𝑃𝜀1−^∞𝜀_𝑜×𝑓³−4.808+6.58𝑍_𝑆. (38) Finally it is important to underline the singular case of Manning’s polyions (𝑅_𝑔≅0 and 𝐿_𝑆→∞, |𝑍_𝑆|→∞). 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=𝑏_𝑆/𝑣_𝑃. As for slowly moving polyions, 𝑣_𝑃𝜏≈𝑏_𝑆, 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 (𝑣_𝑃=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 (Δ𝐗^df=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 Λ^exp_Poly=1000𝜒/𝐂^∘_Na⁺ and theoretical equivalent conductivities Λ^th_Poly=𝛼_𝐶(𝜆_𝑃+𝜆_𝑖)  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 𝐂^∘_Na⁺ of Na⁺ counterions, the experimental equivalent conductivities Λ^exp_Polyincm²⋅Ω⁻¹⋅eqv_Na+⁻¹, the degrees of Na⁺ condensation (1−𝛼_𝐶) on polyions, the apparent charge numbers 𝑍_ap=𝛼_𝐶𝑍_𝑆 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 𝑅_𝑔, the theoretical equivalent conductivities Λ_Poly in absence of interference, the theoretical equivalent conductivities Λ^𝑖_Poly in presence of interference, the % of the dielectric friction effect 𝛽^dfi (36) in presence of interference, and the ratios 𝛽^dfi_𝑃/𝛽^df_𝑃 (37) expressing the relative importance of the interference effect (𝛽^df is the dielectric friction effect in absence of interference).

Table 1 

Variations with counterions concentration 𝐂^∘_Na+ of the experimental Λ^exp_NaHRB and theoretical Λ_NaHRB, Λ^𝑖_NaHRB equivalent conductivities, of the degree of Na⁺ condensation (1−𝛼_C), of the apparent charge number 𝑍_ap, and of the dielectric frictions in absence (𝛽^df) and in presence (𝛽^dfi) of interference, in the case of sodium heparinate RB21055 in water at 25°C.

Table 2 

Variations with counterions concentration 𝐂^∘_Na+ of the experimental Λ^exp_NaChondro and theoretical Λ_NaChondro, Λ^𝑖_NaChondro equivalent conductivities, of the degree of Na⁺ condensation (1−𝛼_C), of the apparent charge number 𝑍_ap, and of the dielectric frictions in absence (𝛽^df) and in presence (𝛽^dfi) of interference, in the case of sodium chondroitin sulfate in water at 25°C.

Table 3 

Variations with counterions concentration 𝐂^∘_Na+ of the experimental Λ^exp_NaPSS and theoretical Λ_NaPSS, Λ^𝑖_NaPSS equivalent conductivities, of the degree of Na⁺ condensation (1−𝛼_C), of the apparent charge number 𝑍_ap, and of the dielectric frictions in absence (𝛽^df) and in presence (𝛽^dfi) of interference, in the case of sodium polystyrene sulphonate in water at 25°C. Distinction between Manning concentration regime and Ostwald concentration regime.

Figure 2 

Comparison between variations with 𝐂∘Na+ of experimental (ΛexpNaHRB, blue ◊ points) and theoretical equivalent conductivities (ΛHy,El,R without dielectric friction, ΛNaHRB with dielectric friction in absence of interference, Λ𝑖NaHRB with dielectric friction in presence of interference) of sodium heparinate RB21055 in water at 25°C.

Figure 3 

Comparison between variations with 𝐂∘Na+ of experimental (ΛexpNaChondro, blue ◊ points) and theoretical equivalent conductivities (ΛHy,El,R without dielectric friction, ΛNaChondro with dielectric friction in absence of interference, Λ𝑖NaChondro with dielectric friction in presence of interference) of sodium chondroitin sulfate in water at 25°C.

Figure 4 

Comparison between variations with 𝐂∘Na+ of experimental (ΛexpNaPSS, blue ◊ points) and theoretical equivalent conductivities (ΛHy,El,R without dielectric friction, ΛNaPSS with dielectric friction in absence of interference, Λ𝑖NaPSS 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 are SO₃⁻ (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: 𝑀_𝑊≈10000 g⋅mol⁻¹ is the average molecular weight of the Sodium Heparinate (RB21055). 𝑍_𝑆=−53±2 at pH≈7 is the structural charge number. 𝐿_𝑆=160±8 Å is the structural length. 𝑏_𝑆=3.2±0.2 Å is the charge-to-charge distance. 𝑅_𝐶=10±1 Å is the cylindrical radius of the polyion chain.

Table 1 shows that the variation of the degree of dissociation 𝛼_𝐶 of Na⁺ from heparin in the concentration range: 5×10⁻⁵M<𝐂^∘_Na⁺<5×10⁻³M is in conformity with the dilution principle so that 𝛼_𝐶 increases with dilution from 0.59 to 0.76 and it differs from its Manning’s value 𝛼_𝑀=𝑏_𝑆/|𝑍_𝑖|𝐿_𝐵=0.44. Consequently, the apparent charge number 𝑍_ap=𝛼_𝐶𝑍_𝑆 varies with the concentration from −37.8 for 𝐂^∘_Na⁺=5.11×10⁻⁵M to −29.5 for 𝐂^∘_Na⁺=5.13×10⁻³M. Notice that this last value seems to be different from the value 𝑍^′_ap=−18 obtained from electrophoretic mobility using Nernst-Einstein relation: 𝑢_𝑃/𝐷_𝑃=|𝑍^′_ap|𝑒/𝑘_𝐵𝑇 with 𝐷_𝑃≈7×10⁻⁷ cm²·s⁻¹ [22]. In fact 𝑍^′_ap differs from 𝑍_ap because it depends at the same time on ionic condensation (𝛼_𝐶) and on ionic friction effects via 𝑢_𝑃. Table 1 and Figure 2 show that the experimental conductivity Λ^exp_NaHRB of sodium heparinate (RB21055) decreases sharply from 82.6 to 61.9cm²⋅Ω⁻¹⋅eqv_Na+⁻¹ in the low concentration range: 5×10⁻⁵M<𝐂^∘_Na⁺<5×10⁻⁴M. Theoretically, the hydrodynamic contribution (at infinite dilution) Λ^∘HD_NaHRB to Λ^exp_NaHRB is obtained in absence of ionic condensation (𝛼_𝐶=1), of ionic frictions (Δ𝜆^el_𝑃=0, 𝛽^ir_𝑃=0) and in absence of dielectric friction (𝛽^df_𝑃=0). According to (6), (10), (16), and (17), Λ^∘HD_NaHRB=(𝜆^∘HD_HRB+𝜆^∘_Na⁺) with 𝜆^∘_Na⁺=50.1, 𝜆^∘HD_HRB=|𝑍_𝑆|𝐹𝑒/6𝜋𝜂⟨𝑅_𝑃⟩. The mean radius ⟨𝑅_𝑃⟩ of heparinate (RB21055) is calculated from (10). We found ⟨𝑅_𝑃⟩=28.8Å, 𝜆^∘HD_HRB=159.84, and Λ^∘HD_NaHRB=210cm²⋅Ω⁻¹⋅eqv_Na+⁻¹. Table 1 shows that the dielectric friction (𝛽^dfi_𝑃>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 𝛽^dfi_𝑃 is proportional to (𝛼_𝐶)² then 𝛽^dfi_𝑃≈(𝛼_𝐶)²𝛽^∘dfi_𝑃, which means that 𝛽^dfi_𝑃 increases with dilution toward its maximal value 𝛽^∘dfi_𝑃 at infinite dilution (𝛼_𝐶=1). According to (38), 𝛽^∘dfi_𝑃 depends on the interference factor 𝑓=𝑏_𝑆/𝑅_𝑔. Adjustment between experimental conductivities Λ^exp_NaHRB and theoretical conductivities Λ^𝑖_NaHRB of heparinate (RB21055) leads to a group radius equal to 𝑅_𝑔=1.65±0.05Å so that 𝑓≈2 (a succession of tangent charged spheres). The maximal value 𝛽^∘dfi_𝑃 is therefore equal to 𝛽^∘dfi_𝑃=(2/3)(𝑅_𝑤/𝑏_𝑆)³(𝐿_𝐵/𝑏_𝑆)(𝐿_𝑆/⟨𝑅_𝑃⟩)[1−𝜀_∞/𝜀_𝑜]|(𝑓³−4.808)+6.58/|𝑍_𝑆||=2.25, for 𝑏_𝑆=3.2Å, 𝑓≈2, 𝐿_𝑆=160Å and 𝑅_𝑤=1.4Å. Consequently, the correct limiting conductivity at infinite dilution Λ^∘_NaHRB must take into account the limiting dielectric friction effect 𝛽^∘dfi_𝑃 in addition to the hydrodynamic friction as follows: Λ^∘_NaHRB=𝜆^∘_Na⁺+𝜆^∘HD_HRB/(1+𝛽^∘dfi_𝑃)=50.1+49.2=99.3cm²⋅Ω⁻¹⋅eqv_Na+⁻¹. This Λ^∘_NaHRB is experimentally inaccessible because of the nonexistence of a universal limiting law allowing a rigorous extrapolation of Λ^exp_HRB 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 Λ^exp_NaHRB and 𝛼_𝐶 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: Λ^exp_NaHRB≈𝛼_𝐶𝜆^∘_Na⁺+𝛼_𝐶𝜆^∘HD_HRB1+𝛼²_𝐶𝛽^∘dfi_𝑃≈𝛼_𝐶50.1+159.84𝛼_𝐶1+2.25𝛼²_𝐶. (39) This expression can be used as an indirect method to evaluate experimentally the degree of ionic condensation (1−𝛼_𝐶) from experimental measurements of the equivalent conductivity of the polyelectrolyte at high dilution. Calculation shows that for 𝐂^∘_Na⁺<10⁻⁴M, 𝛼_𝐶≥0.8 in conformity with theoretical values ((1)–(4)) but incompatible with the Manning’s value 𝛼_𝑀=𝑏_𝑆/𝐿_𝐵≈0.44. It is important to notice at this stage that the constancy of the condensation parameter at high dilution with 𝛼_𝐶=𝛼_𝑀 would imply that Λ^exp_NaHRB increases first with dilution then attains a “palier” (platform) in the range of very low concentrations in which 𝛼_𝐶 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 heparinate Λ^Sph_NaHRB calculated in the case of coiled conformation (quasi-spherical) having a mean radius 𝑅_𝑆≈⟨𝑅_𝑃⟩≈𝑘_𝐵𝑇/6𝜋𝜂𝐷_𝑃≈30Å, to its corresponding experimental equivalent conductivity Λ^exp_NaHRB. According to Zwanzig-Frank notation [19, 25] the expression of the equivalent conductivity 𝜆^Sph_HRB of the spherical polyion at very high dilution is 𝜆^Sph_HRB≈𝐴𝛼_Sph|𝑍_𝑆|/[𝑅_𝑆+(𝛼²_Sph𝑍²_𝑆𝐶/𝑅³_𝑆)], with 𝐴=92, 𝐶=10 at 25°C in water, 𝑅_𝑆=30 Å, and |𝑍_𝑆|=53. The equivalent conductivity of the (spherical) sodium heparinate is therefore Λ^Sph_NaHRB≈𝛼_Sph(𝜆^∘_Na⁺+𝜆^Sph_HRB). Now, according to the ionic association theory of Fuoss [3] the degree of dissociation: 𝛼_Sph>0.68 for 𝐂^∘_Na+<10⁻⁴ M, therefore Λ^Sph_NaHRB>100cm²⋅Ω⁻¹⋅eqv_Na+⁻¹. This expected result is at least twenty percent larger (20%) than the experimental equivalent conductivity Λ^exp_NaHRB  (∼83cm²⋅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.(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 (𝛽^dfi_𝑃 decreases with concentration from 78% to 50%) in comparison to ionic relaxation effect and electrophoretic effect. (ii)The adjustment of the experimental equivalent conductivities Λ^exp_NaHRB with the theoretical conductivities Λ^𝑖_HRB leads to a group radius 𝑅_𝑔 sensibly constant, of about 1.65±0.05 Å, that is, equal to the half of the distance 𝑏_𝑆=3.2±0.2 Å between two successive spherical charged groups, so that the coherent condition: 𝑓=𝑏_𝑆/𝑅_𝑔=constant≥2 is verified in the studied concentration range. (iii)The ionic relaxation effect 𝛽^ir on polyions and counterions becomes important (10%–20%) for 𝐂^∘_Na⁺>10⁻⁴ M, while 𝛽^ir is <6% for highly dilute solutions. This result is not in conformity with the Manning prediction: 𝛽^ir→13%. (iv)The 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)The degree of ionic condensation (1−𝛼_𝐶) 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 sugar 𝑁-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: 𝑀_𝑊≈21430g⋅mol⁻¹ is the average molecular weight of the used polyelectrolyte.  𝑍_𝑆=−75±3 is the structural charge number. 𝐿_𝑆=435±15 Å is the structural length. 𝑏_𝑆=5.8±0.2 Å is the charge-to-charge distance. 𝑅_𝐶=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 separation 𝑏_𝑆, 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−𝛼_𝐶) despite the relative importance of its structural charge number. Table 2 shows that the degree of dissociation 𝛼_𝐶 of Na⁺ from chondroitin increases slightly with dilution from 0.804 to 0.852 in the concentration range: 1.38×10⁻⁴M<𝐂^∘_Na⁺<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 𝛼_𝑀=𝑏_𝑆/|𝑍_𝑖|𝐿_𝐵=0.81. Consequently, the apparent charge number 𝑍_ap=𝛼_𝐶𝑍_𝑆 varies slightly with the concentration from −63.9 for 𝐂^∘_Na⁺=1.38×10⁻⁴M to −60.3 for 𝐂^∘_Na⁺=1.11×10⁻³M.

Table 2 and Figure 3 show that the experimental conductivity of sodium chondroitin Λ^exp_NaChondro increases with dilution in a monotonous way from 65.3 to 78.54 cm²⋅Ω⁻¹⋅eqv_Na+⁻¹ in the concentration range: 1.38×10⁻⁴M<𝐂^∘_Na⁺<1.11×10⁻³M, (i.e., without the appearance of any palier). The hydrodynamic contribution (at infinite dilution) Λ^∘HD_NaChondro to Λ^exp_NaChondro is obtained in absence of ionic condensation and in absence of other frictional effects from (6), (10), (16), and (17), so that Λ^∘HD_NaChondro=(𝜆^∘HD_Chondro+𝜆^∘_Na⁺) with 𝜆^∘_Na⁺=50.1 with 𝜆^∘HD_Chondro=|𝑍_𝑆|Fe/6𝜋𝜂⟨𝑅_𝑃⟩. The mean radius ⟨𝑅_𝑃⟩ of the chondroitin sulfate is equal to 50.8 Å, therefore 𝜆^∘HD_Chondro=136.1 and Λ^∘HD_NaChondro=186.2cm²⋅Ω⁻¹⋅eqv_Na+⁻¹. Table 2 shows also that the dielectric friction (𝛽^dfi_𝑃>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. 𝛽^dfi_𝑃 depends on the interference factor 𝑓=𝑏_𝑆/𝑅_𝑔. Adjustment between experimental and theoretical conductivities of chondroitin sulfate leads to a group radius equal to 𝑅_𝑔=2.14±0.2 Å so that 𝑓=𝑏_𝑆/𝑅_𝑔=2.71 (a discontinuous linear charge distribution), the maximal value 𝛽^∘dfi_𝑃 is, according to (38), equal to 𝛽^∘dfi_𝑃=(2/3)(𝑅_𝑤/𝑏_𝑆)³(𝐿_𝐵/𝑏_𝑆)(𝐿_𝑆/⟨𝑅_𝑃⟩)[1−𝜀_∞/𝜀_𝑜]|(𝑓³−4.808)+6.58/|𝑍_𝑆||=1.47 for 𝑏_𝑆=5.8Å, 𝑓≈2.7, 𝐿_𝑆=435Å and 𝑅_𝑤=1.4Å. Consequently, the new limiting Λ^∘_NaChondro must take into account the limiting dielectric friction effect 𝛽^∘dfi_𝑃 in addition to the hydrodynamic friction as follows: Λ^∘_NaChondro=𝜆^∘_Na⁺+𝜆^∘HD_Chondro/(1+𝛽^∘dfi_𝑃)=50.1+36.7=86.8cm²⋅Ω⁻¹⋅eqv_Na+⁻¹. This Λ^∘_NaChondro 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 Λ^exp_NaChondro and the degree of dissociation 𝛼_𝐶 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 Λ^exp_NaChondro≈𝛼_𝐶𝜆^∘_Na⁺+𝛼_𝐶𝜆^∘HD_Chondro1+𝛼²_𝐶𝛽^∘dfi_𝑃≈𝛼_𝐶50.1+136.13𝛼_𝐶1+1.47𝛼²_𝐶. (40) 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 (𝛽^dfi_𝑃 decreases with concentration from 92% to 86%) in comparison to ionic relaxation effect and electrophoretic effect. (ii)The adjustment of experimental equivalent conductivities Λ^exp_NaChondro with the theoretical conductivities Λ^𝑖_NaChondro leads to a group radius 𝑅_𝑔 sensibly constant of about 2.13±0.2 Å so that the coherent condition: 𝑓=𝑏_𝑆/𝑅_𝑔=2.7≥2 is verified in the studied concentration range. (iii)The ionic relaxation effect 𝛽^ir on chondroitin polyions and Na⁺ counterions remains approximately constant (~15%) in the studied range of relatively low concentrations: 1.38×10⁻⁴M<𝐂^∘_Na⁺<1.11×10⁻³ M in conformity with the Manning prediction.(iv)The electrophoretic effect is relatively weak for the Na⁺ conterions (<4%), while it normally decreases with dilution from 27% to 12% for stretched chondroitin sulfate polyions.(v)The degree of ionic condensation (1−𝛼_𝐶) 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−𝛼_𝑀)=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±300Å 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.

Table 3 shows that the degree of dissociation 𝛼_𝐶 of Na⁺ from PSS decreases very slightly with dilution from 0.441 to 0.457 in the concentration range: 1.16×10⁻³M<𝐂^∘_Na⁺<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 𝛼_𝐶 remains sensibly constant. In fact, theoretical formal calculations according to (1) show that 𝛼_𝐶 presents a little minimum (0.413) for 𝐂^∘_Na⁺≈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<𝐂^∘_Na⁺<5.1×10⁻³M, in which 𝛼_𝐶 remains practically constant, and (b) a formal Ostwald concentration regime: 𝐂^∘_Na⁺≪1×10⁻⁵M, in which 𝛼_𝐶 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 𝑀_𝑤 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 𝛼_𝐶 is in fact different from its Manning’s value 𝛼_𝑀=𝑏_𝑆/|𝑍_𝑖|𝐿_𝐵=0.35. Consequently, the apparent charge number 𝑍_ap varies with the concentration from −1326 for 𝐂^∘_Na⁺=5.14×10⁻³M to −1279 for 𝐂^∘_Na⁺=1.16×10⁻³M.

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 cm²⋅Ω⁻¹⋅eqv_Na+⁻¹ in the indicated concentration range. The mean radius ⟨𝑅_𝑃⟩ of the polyion is equal to 521.2 Å; therefore according to (16)–(18) the hydrodynamic mobility 𝜆^∘HD_PSS of the PSS polyion is equal to 𝜆^∘HD_PSS=|𝑍_𝑆|Fe/6𝜋𝜂⟨𝑅_𝑃⟩=512.8 cm²⋅Ω⁻¹⋅eqv_Na+⁻¹. Adjustment between experimental and theoretical conductivities of NaPSS polyelectrolyte leads to a constant group radius equal to 𝑅_𝑔≅1.21 Å so that 𝑓=𝑏_𝑆/𝑅_𝑔=2.06 (a quasi-continuous linear charge distribution of tangent spheres); the maximal value of 𝛽^∘dfi_𝑃 is therefore 𝛽^∘dfi_𝑃=(2/3)(𝑅_𝑤/𝑏_𝑆)³(𝐿_𝐵/𝑏_𝑆)(𝐿_𝑆/⟨𝑅_𝑃⟩)[1−𝜀_∞/𝜀_𝑜]|(𝑓³−4.808)+6.58/|𝑍_𝑆||=17.9 for 𝑏_𝑆=2.5 Å, 𝑓≈2, 𝐿_𝑆=7250 Å and 𝑅_𝑤=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^exp_NaPSS and 𝛼_𝐶, in the range of dilute solutions for which the stretched conformation is favored, we obtain Λ^exp_NaPSS≈𝛼_𝐶𝜆^∘_Na⁺+𝛼_𝐶𝜆^∘HD_PSS1+𝛼²_𝐶𝛽^∘dfi_𝑃≈𝛼_𝐶50.1+512.8𝛼_𝐶1+17.9𝛼²_𝐶. (41)

Now, as 𝛼_𝐶 undergoes weak variation in the Manning region, Λ^exp_NaPSS remains sensibly constant (palier of Λ^exp_NaPSS∼35 cm²⋅Ω⁻¹⋅eqv_Na+⁻¹), in the concentration range: 1×10⁻³M<𝐂^∘_Na⁺<5×10⁻³M. We can however formally expect the emergence of highly diluted regime for 𝐂^∘_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 𝛼_𝑆 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 (𝛽^dfi_𝑃 decreases with concentration from 172% to 142%) in comparison to ionic relaxation effect and electrophoretic effect. (ii)The adjustment of experimental equivalent conductivities Λ^exp_NaPSS with the theoretical conductivities Λ^𝑖_NaPSS leads to a group radius 𝑅_𝑔 sensibly constant of about 1.21 Å so that the coherent condition: 𝑓=𝑏_𝑆/𝑅_𝑔=2.06≥2 is verified in the studied concentration range.(iii)The ionic relaxation effect 𝛽^ir 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<𝐂^∘_Na⁺<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 𝑑_𝑃. Indeed, 𝑑_𝑃=199 Å for 𝐂^∘_Na⁺=1.16×10⁻³M and 𝑑_𝑃=98 Å for 𝐂^∘_Na⁺=5.14×10⁻³M. Therefore, 𝑑_𝑃/𝐿_𝑆<0.03 and according to (23), 𝛽^ir_𝑃≈(𝛼_𝐶/𝛼_𝑀)(1/9)≈0.14 for all 𝐂^∘_Na⁺. (iv)The electrophoretic effect (Δ𝜆^el_𝑃/𝜆^∘Hyd_𝑃=⟨𝑅_𝑃⟩/⟨𝑑_𝑃⟩) on PSS polyions is more important than 𝛽^ir 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: 𝐂^∘_Na⁺<10⁻⁶M. (v)The observed conductivity variation of NaPSS in the concentration range: 1.16×10⁻³M<𝐂^∘_Na⁺<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 lengths 𝐿_𝑆 and different charge distances 𝑏_𝑆 and without added salt: sodium heparinate (𝐿_𝑆=160Å, 𝑏_𝑆=3.2Å), sodium chondroitin sulfate (𝐿_𝑆=435 Å, 𝑏_𝑆=5.8Å), and sodium polystyrene sulphonate (𝐿_𝑆=7250Å, 𝑏_𝑆=2.5Å). A strong influence on their equivalent conductivities Λ_Poly 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 (Δ𝜆^el_Na+, Δ𝜆^el_𝑃) and to the ionic relaxation effect (𝛽^ir) are less important and become negligible at high dilution. The correlation between Λ_Poly and, respectively, the degree of condensation (1−𝛼_𝐶), the limiting equivalent conductivity 𝜆^∘_Na+, the hydrodynamic equivalent conductivity of the polyion 𝜆^∘HD_𝑃, the limiting dielectric friction coefficient (𝛽^∘df_𝑃), and the ionic friction terms (Δ𝜆^el_Na+, Δ𝜆^el_𝑃, 𝛽^ir) can be expressed by the following general expression: Λ_Poly≈𝛼_𝐶1+𝛽^ir−1×𝜆^∘_Na+−Δ𝜆^el_Na++𝛼_𝐶𝜆^∘HD_𝑃−Δ𝜆^el_𝑃1+𝛼²_𝐶𝛽^∘dfi_𝑃. (42) The limiting dielectric friction coefficient (𝛽^∘df_𝑃) is very sensitive to the conformation of the polyion, to the charge distance 𝑏_𝑆, and to the radius 𝑅_𝑔 of charged groups. Indeed, (𝛽^∘df_𝑃) is weak for coiled conformations and vanishes when 𝑣_𝑃𝜏>𝑅_𝑔, where 𝑣_𝑃 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, (𝛽^∘df_𝑃) is in general larger than 50% and increases with the interference term 𝑓³=(𝑏_𝑆/𝑅_𝑔)³ as follows: 𝛽^∘dfi_𝑃=23𝑅_𝑤𝑏_𝑆³𝐿_𝐵𝑏_𝑆𝐿_𝑆𝑅_𝑃𝜀1−^∞𝜀_𝑜×𝑓³−4.808+6.58𝑍_𝑆𝐿∼Ln^𝑆𝑅_𝐶. (43) 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 𝛼_𝐶 and Λ_Poly increase with dilution, while Δ𝜆^el_Na+, Δ𝜆^el_𝑃, and 𝛽^ir→0; (b) the Manning concentration regime (it is the case of sodium polystyrene sulphonate) for which 𝛼_𝐶, 𝛽^ir, and Λ_Poly remain sensibly constant with dilution, while Δ𝜆^el_Na+ and Δ𝜆^el_𝑃 decrease with concentration. It is important to underline that the ideal Manning behavior (𝐿_𝑆/𝑅_𝐶→∞) implies the restrictive conditions: 𝛼_𝐶=𝛼_𝑀=𝑏_𝑆/𝐿_𝐵, 𝛽^ir=1/9, and 𝛽^∘dfi_𝑃=0 (no dielectric friction: 𝑣_𝑃𝜏>𝑅_𝑔), while less restrictive conditions imply 𝛼_𝐶=constant, 𝛽^ir=(𝛼_𝐶/9𝛼_𝑀), and 𝛽^∘dfi_𝑃≠0. Only the sodium chondroitin sulfate presents an intermediate behavior between Ostwald concentration regime and Manning concentration regime with 𝛼_𝐶≈𝛼_𝑀=0.81 and 𝛽^ir≈0.13. This mixed behavior is characterized by a slowly increasing of its equivalent conductivity with dilution.

Acknowledgment

The authors are grateful to the General Direction of Scientific Research of Tunisia (D.G.R.S.T) for assistance and supporting grants.