About the correlation between the mobility of a polyion and that of its counterions

In this paper, we propose a new approach in order to interpret the variation of the conductibility of the PSS polyion with the nature and the concentration of the alkaline counter-ions Li + , Na + and K + , and the hydrophobic cations Et4N + and Bu4N + . This approach is based on a recent model in which the stretched polyion is represented by a chain of successive charged spheres, partially condensed by the counter-ions. We have found that the moderate variation of the hydrodynamic friction on the polyion with the size RM of condensed counter-ions, cannot completely explain the important decrease (of about 35% from K + to Bu4N + ) of the conductibility (PSS,M ) of the PSS polyion with the nature of the counter-ions. Consequently, we have proposed a supplementary explanation by taking into account of the translational dielectric friction on the moving polyion. Formal analysis of this friction shows that it is very sensitive to the local structure of water surrounding the polyions. As this local structure depends in its turn, on the nature of the condensed counter-ions; we suggested that this specific sensitivity could explain the high dependence of the mobility of the polyion with the nature of its counter-ions.


INTRODUCTION
Contrary to the case of the simple electrolytes, the conductivity of poly electrolytes does not obey to a universal linear limiting law relating their equivalent conductivity at high dilution to the square root of ionic strength: I 1/2 . Consequently, there is no currently available satisfactory theory describing the dynamic behavior of dilute flexible polyelectrolytes in aqueous solution despite some interesting progress toward this objective [1][2][3][4][5][6][7][8][9][10]. This difficulty arises from the complex interdependence between polyion conformation, the processes of condensation of counter-ions M + , the ionic screening effect and frictional effects.
In a series of previous papers [11][12][13], H. Vink proposed a semi-empirical approach for the interpretation of the conductivity of some polyelectrolytes, in order to: a) verify the Manning's theory [3,8] and: b) to study the dependence of the conductibility of some polyions with the nature of their counter-ions. The main results of these studies concerning dilute MPSS polyelectrolyte solutions are: - The degree of condensation (1-α) of counter-ions is practically independent as well as of the concentration and of the nature of studied counter-ions Li + , Na + and K + , and that its value (0.65) is very close to that predicted by the theory of Manning. However, for organic counter-ions (Et4N + and Bu4N + ), α is in general slightly lower than that of the Manning value α Manning . - The ionic friction coefficient on the polyion (βir) can be assumed to be equal to that predicted by the Manning's theory. - The experimental conductibility P,M of the PSS polyion depends strongly on the nature of the counter-ion M + in a range of concentration between 510 -4 M and 3.510 -3 M. Indeed, by combining the experimental equivalent conductivities MP exp with their corresponding experimental "transference numbers", TP,M , the author has been able to extract for each type of counter-ions and for a given concentration, the conductibility P,M of the PSS polyion and he has found that P,M increases with the mobility of the counter-ions in accordance with the following order: However, this last result is qualitatively and quantitatively in complete contradiction with the Manning's theory for which both the degree of condensation (1-α) and the conductibility P,M of the PSS polyion are independent on the nature of counter-ions.
The explanation suggested by the author is that the contribution of the condensed counter-ions to the hydrodynamic friction of the polyion increases with their size.
In fact, as polyanions and positive free counter-ions move along opposed directions, electrostatic coupling gives therefore place to a process of braking rather than to a mutual entrainment. Consequently, P,M will rather decrease with the mobility of the counter-ions (friction effect). In contrast the condensed counter-ions are retained by the polyion and the ensemble forms a stable entity without internal frictional coupling resulting from the difference between motilities.
The present work proposes another approach in order to obtain an adequate analytical expression describing correctly the variation of the global conductibility of the MPSS polyelectrolyte with the nature and the concentration of the counter-ions. This new approach is based on a recent model in which the stretched polyion is represented by a chain of successive charged spheres, partially condensed by the counter-ions, and without introduction of specific parameters other than the structural parameters of the polyion, the valence and the effective radius of the counter-ion. Such approach will make it possible to check if the introduction of these only parameters will allow or not to explain the possible dependence of the conductibility of the polyion with nature of the counter-ions.

THEORETICAL MODEL OF MANNING
According to the Manning's conductance theory of a salt-free polyelectrolyte solution MP, the conductibility P Manning of the corresponding polyion P of LS structural length and ZSe structural charge, is independent on the nature of the counter-ion M and varies with its total concentration C°M (mol.l -1 ) as follows [3,4]: (1)

P
Manning expresses the different friction effects undergone by the polyion i.e.: the hydrodynamic effect, the electrophoretic effect and the ionic relaxation effect via the viscosity η of the solvent (η = 0.8903 10 -2 poises for water at 25 °C), the screen parameter χD and the ionic friction coefficient βir Manning . F is the Faraday so that: Fe  6π = 0.82. On the other hand, P Manning is proportional to the apparent charge of the polyion and therefore to the degree of dissociation α Manning of the counter-ions which is in this case, independent on the concentration and on the nature of the counter-ions: α Manning = bS  |ZM|LB (2) A u g u s t 0 9 , 2 0 1 4 bS = LS  ZS, is the distance of separation between two successive ionizable groups, ZM is the valence of the counter-ion and LB = e 2 (εokBT) is the Bjerrum length, where kB is the Boltzmann constant and εo is the permittivity of the solvent. βir Manning is the Manning ionic friction coefficient on the polyion due to the counter-ions which is also independent on the concentration and on the nature of the counter-ion [3]: βir Manning = 0.13 (3) The screen parameter of Debye χD is given by: NA is the Avogadro number. For water at 25 °C: LB = 7.156A° and C°M (mol.l -1 ) is the total concentration of counter-ions. The practical expression of χD (in A°-1 ) is: Note that χD -1 measure the cylindrical mean radius of the ionic atmosphere around the polyion assumed to be an infinite thread of apparent charge: Zape = (α Manning ZSe). Combination of (Eq.1, Eq. 2, Eq. 3 and Eq. 5) leads to the following practical expression of P Manning (in cm 2  -1 equiv -1 ): It is therefore clear that P Manning is independent of the nature of the counter-ion.
In the other hand, the expression of the ionic conductivity M of the counter-ions M can be decomposed as follows: °M is the ionic equivalent conductivity of the counter-ion at infinite dilution which expresses both the hydrodynamic friction due to the viscosity η of the solvent ( RM is the effective radius of the counter-ion "M" and dM is the radius of its ionic atmosphere, and χD is its corresponding Debye screen parameter. The final expression is therefore: Now, the Manning expression of the conductibility MP Manning of the polyelectrolyte MP is: Combination of (Eq.2, Eq.3, Eq.6, Eq.7, Eq.9 and Eq.10) leads to the following practical expression of MP Manning (in cm 2  -1 equiv -1 ): A u g u s t 0 9 , 2 0 1 4

Principle of Experimental determination of α M and  PSS,M
According to Vink's approach [11][12][13], the experimental equivalent conductivity MP of a polyelectrolyte MP can be written empirically as following: Eq. 12 relates MP to four physical quantities: βirP,M, αM, 'M and 'P,M. As previously, the factor (1 -βirP,M) represents the interionic friction due to the ionic relaxation effect. It is important to note that from irreversible thermodynamics considerations and for a salt-free polyelectrolyte, the friction coefficient βirP,M is the same for polyions and counter-ions [6,[11][12][13]16]. αM is the degree of dissociation of the polyelectrolyte which determines the effective charge Zape of the polyion so that: ZS is the stoichiometric charge number of the polyion.
Note that 'M and 'P,M are different from respectively the equivalent conductivity of the free counter-ions (M) and from the equivalent conductivity of the polyion (P,M).
All these quantities generally depend on the concentration of the polyelectrolyte and on the nature of the counter-ions. This last dependence is indicated by the index M. The experimental determination of βirP,M, αM, 'M and 'P,M needs in addition to Eq.12 of three other relations.
The second relation is obtained from the net charge transported by the polyion which is measured by its electric transport number tP,M: Where the dependence of tP,M with the nature of the counter-ion is indicated by the index M. However, in solutions with ionic association only the corresponding ionic constituent transport number TP,M is experimentally determinable [12]: Tables 2a and 2b, give the variation of TP,M and MP with the concentration for different types of counter-ions.
From equations (Eq.14 and Eq.15) we obtain another form of Eq.14: From Eq.16, P,M appears as well as a pure experimental quantity because it is related to the experimental values: TP,M and MP.
The third relation is obtained by neglecting the electrophoretic friction undergone by the counter-ions M, for dilute solutions, so that: °M is equal to the ionic conductibility of M at infinite dilution.
According to Vink's approach, the fourth relation is obtained by assuming Manning's model for the ionic friction coefficient βir, so that [13]: (1 -βirP,M) = (1 -βir Manning ) = 0.87 (18) It means that βir is independent on the nature and on the concentration of counter-ions. Logically, in turn, if Eq. 18 is valid, then also: αM = α Manning = bS  |ZM|LB = 0.35. (Or at least αM  Cste: Manning's regime [6][7]17]). It is possible to test this assumption by calculating αM after combination of Eqs. (12,15,17): lower (of about respectively 10% and 11.5%). We can conclude that Manning's assumption based on electrostatic condensation seems valid for alkaline ions, but in the case of organic counter-ions, the condensation effect is enhanced by the hydrophobic attraction between polyions and organic cations.
Moreover, in order to justify his assumption expressed by Eq. 18, Vink proposed a new method based on the following equation obtained after rearranging Eq. 19: Indeed, if (in accordance with Manning's theory), both αM and (1-βir) are independent on the counter-ion species, we get (1-βir) as the slope of the line obtained when TP,M is plotted against (°M  MP) (calculated for the different counter-ions; see tables 2a and 2b). However, calculations according to Eqs. (18 and 19), show that for a given concentration C°M + , the αM vary slightly with the nature of the counter-ions. Therefore, we can replace αM in Eq. (20), by its means value <αM> given in table 3, so that: Note that this method is valid even if <αM> depends on the counter-ion concentration.

Vink's interpretation of the dependence of the experimental  PSS,M with the nature of the counter-ions.
As indicated previously, the Manning's model is not able to explain the dependence of the polyion conductibility with the nature of the counter-ions M because the polyion is assumed to be an infinite thread and the counter-ions are modeled as punctual charges. In contrast, Vink suggests a more realistic description which offers an explanation of the dependence of the conductibility of the polyion with the size RM of the condensed counter-ions species. Indeed, according to his model, A u g u s t 0 9 , 2 0 1 4 the hydrodynamic friction on the polyion increases with the minimal distance of approach (RC,M = RC + RM + nRw) between a condensed counter-ion and a charged polyion group. RC is the radius of the polyion, Rw is the radius of water molecule and nRw is a correction due to the hydration effect with: n = 0, 1 or 2.
In the following section we will demonstrate that this interpretation is correct in its principle but it cannot explain the important decrease of PSS,M with the size RM of the condensed counter-ions species (from K + to Bu4N + ) which is of about 35%.

Conductivity of Stretched Polyelectrolyte in Dilute Solutions
The expression of the equivalent conductivity P,M of a polyion is in fact more complex because its ionic equivalent conductivity at infinite dilution P° expresses both hydrodynamic friction effect via P°H yd and dielectric friction effect via βP dfi [6]. For this reason, we can decompose P,M as follows: P,M = (αCMP°H yd -∆λP el )  (1 + βir P + βP dfi ) = P Henry  (1 + βir P + βP dfi ) (23) αCM is the degree of dissociation of the polyelectrolyte for which the polyion is modeled as a chain of charged spheres. P°H yd is the purely hydrodynamic contribution due to the viscosity η of the solvent, and ∆λP el is the so-called electrophoretic effect which expresses the hydrodynamic friction on the ionic atmosphere of the polyion. βir P is the ionic friction coefficient of the polyion.

Hydrodynamic friction and Electrophoretic effect
In previous works [6-8, 16, 17], we have noted that the first term (αCMP°H yd -∆λP el ) is identical to the expression of the Henry equivalent conductivity P Henry given by [6,18]: F is the Faraday, e is the proton charge and C'AP is in fact 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. Eq. 24 is a generalization of the Stokes-Hubbard equation [19].
 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 [18,19].
ZM is the valence of the counter-ion, RM is the effective radius of the solvated counter-ion "M", NA is the Avogadro number and LB is the Bjerrum length. A u g u s t 0 9 , 2 0 1 4

Ionic friction on the polyion
Recall that in addition to the electrophoretic effect, one must take into account the ionic relaxation effect due to the polarization of the ionic atmosphere of the moving polyion by the external field. This local polarization induces a reacting relaxation field in order to moderate the perturbation caused by the external field. The result is a decrease of the mobility of the polyion when the concentration of the counter-ions increases via the ionic friction coefficient βir P as indicated in Eq. 23. The expression of βir P is given by [6][7][8]: Note that for a free salt polyelectrolyte, the ionic friction coefficient βir P of the polyion is equal to the ionic friction coefficient βir M of the counter-ions [6, 13]: Note also that for spherical configuration: LS  dP  0, then: βir P  αCMZM ZSLB (6dP), this limiting expression converges toward the Debye-Onsager relation applicable for spherical ions. In contrast, for polyions of very large length: LS  dP and thus: βir P  1 9  C°M if αCM  α Manning . Consequently, (1 -βir P )  0.889, this limit is in conformity with both Manning and Vink approaches.

Importance of the Dielectric Friction Effect
In this paragraph we will discuss succinctly the dielectric frictional effect on a slowly moving polyion due to dielectric loss in its surrounding medium. Calculations of the dielectric frictional force on a charged sphere were performed successively by R. Zwanzig [15], J. Hubbard and L. Onsager [23] and P. G. Wolynes [21]. However, this dielectric effect is completely ignored by the Manning's model and it is not cited in the Vink's approach.
Indeed, the apparent charge (αCMZSe) of a polyion moving along Oz with a velocity v, induces a polarization of its surrounding dielectric medium which in turn creates after a relaxation time τ (delay) a dielectric frictional force Fz acting on the polyion. In fact, this force depends on the conformation (shape) of the polyion [6-8, [16][17][18]. In the case of a coiled polyion represented by an ellipsoidal conformation, the expression of Fz is [18]: The sign "-" expresses the fact that the direction of the frictional force is opposite to the direction of v. The factor (vτ) expresses the delay effect so that Fz vanishes for immobile polyion: v = 0, or in the case of instantaneous response: τ = 0. o and   are the static and high-frequency dielectric constants of the solvent and the (o -  )o -2 term expresses the dielectric saturation effect. Rapp is the apparent radius of the polyion which depends on its eccentricity [18], and  D is the translational dielectric friction related to the βP dfi coefficient via the Henry mobility uP Henry = P Henry F, according to [18]: For coiled polyions, βP dfi  1, because of the term (Rapp) -3 , and therefore according Eq. 23, the dielectric friction effect remains weak. It is interesting to note that when the moving polyion is assumed to be a thread of infinite length (L ) with a continuous linear charge (Manning's Model), so that the local distribution of its surrounding solvent molecules remains undisturbed, the polyion appears therefore as an equivalent immobile polyion (no dielectric friction).
In the specific case of a stretched polyion modeled as a chain of ZS identical spheres of charge qn = αCMe and of radius Rg,M (a linear discontinuous distribution of ionized groups), each partially charged group undergone a dielectric friction effect proportional to (Rg,M) -3 and not to (Rapp) -3 (with Rg,M  Rapp) and consequently, the conformation transition from coiled state to stretched state will be accompanied by a sharp increase of the dielectric friction on the polyion. The corresponding expression of the dielectric friction coefficient βP dfi is therefore given by [17,22]: ]|(fM 3 -4.808) ZS + 6.58| ; fM = bS  Rg,M With, according to Eqs. (24 -28): uP Henry = (e|ZS|CM  6πη C'AP). Now, if we replace the relaxation time τ by its Debye's expression:  = (6Rw 3  kBT), with LB = e 2  εokBT, the general explicit expression of βP dfi becomes: βP dfi = (2  3)CM 2 (Rw  bS) 3 (LB  C'AP)[1-ε∞ εo] |(fM 3 -4.808) ZS + 6.58 | In fact, the dielectric friction undergone by each spherical group can be decomposed into a self contribution due to the polarization induced by its own charge αCMe and into a crossed contribution due to the polarization caused by the other ZS -1 spherical groups (interference effect indicated by the index "i" in βP dfi ). Notice however that, if the distance: bS = LS  ZS, between two successive charged groups is sufficiently large, fM -1  0 and the interference effect vanishes.
It is interesting to note that βP dfi increases with dilution as CM 2 and it reaches its maximal value β°P dfi at infinite dilution i.e. when CM  1 (Ostwald) and C'AP   RP : Consequently, according to Eqs. (29, 33 and 34), the two friction coefficients: βir P (ionic friction) and βP dfi (dielectric friction), can be written respectively in terms of an ionic parameter B and of a dielectric parameter CM as follows: With: Note that the dependence of αCM and CM with the nature of the counter-ion is indicated by the index M.
In the other hand, the term (αCMP°H yd -∆λP el ) expressing in Eq.23 the hydrodynamic contribution and the electrophoretic contribution can also be written in terms of the hydrodynamic parameter AM defined as follows: (αCMP°H yd -∆λP el ) = αCM ZSFe  6πC'AP  αCMAM The introduction of Eqs. (36 and 38) into Eq. 23 leads to the following "Universal form" of the expression of the conductibility P,M of the stretched polyion: We have used previously this general form in order to distinguish between the Manning's regime for which the degree αCM remains quasi constant, and the Ostwald's regime for which P,M varies more appreciably with the dilution process via αCM [17,22].
In this paper we will use this expression in order to explain the possible specific dependence of the conductibility P,M of the polyion with the nature of the counter-ions. Indeed, According to Eq. 39, this specificity can result from the dependence with the nature of the counter-ion M, of αCM , CM and AM parameters relating to respectively the process of ionic condensation, the dielectric friction (via the specific group radius Rg,M), and the hydrodynamic-electrophoretic friction.
Only the two parameters: αCM and AM are considered in the Vink's approach. Moreover, the comparison for each concentration between the αCM of the different counter-ions given in table 3, show that organic cations Et4N + and Bu4N + present a certain specific association with the charged groups of the PSS polyion due to their hydrophobic character. In contrast, ionic condensation of the alkaline cations Li + , K + and Na + seems to be quasi independent on the nature of the counter-ion and their mean value is equal to the Manning value which is about 0.35. Consequently, according to Vink's interpretation, the important deviations:   (P,M -P,M') observed between two polyion conductibilities of any couple of counter-ions M and M' must be interpreted in terms of hydrodynamic friction parameter AM (via the specific cylindrical radius RCM = RC + RM + nRw). In contrast, it is obvious that this specificity disappears in the case of the Manning's Model. In the following paragraph, we will show that in the case of stretched polyion, the most important factor at the origin of the specificity of P,M to M comes rather from the high sensitivity of the dielectric parameter CM to the local structure parameters characterizing the surrounding of each specific group of apparent radius Rg,M. A u g u s t 0 9 , 2 0 1 4 4.2 Interpretation of the dependence of the conductibility of PSS with the nature of the counter-ion.
In this section we will calculate for each counter-ion M of concentration C°M the different specific parameters: RCM, AM, CM and Rg,M defined previously, in order to interpret the dependence of the conductibility of PSS with the nature of each counter-ion.
The P,M indicates the experimental conductibility of PSS in presence of the counter-ion "M" (see table 4a). The expression of the hydrodynamic-electrophoretic parameter AM is extracted from Eq. (38).
The capacity C'AP is defined by Eqs. (25, 26, 27 and 28). Tables 6 gives the variations with the total ionic concentration C°M of the hydrodynamic-electrophoretic parameters "AM" and the dielectric parameters CM relating to Polystyrene Sulphonate KPSS, NaPSS, LiPSS, Et4NPSS and Bu4NPSS in water at 25 °C.  (see table 5). However, this increase cannot, completely explain the great dependence of the conductibility of the polyion with the nature of its counter-ion. A u g u s t 0 9 , 2 0 1 4 - The dielectric friction is practically the same for Na + and K + , it undergoes a weak increase for Li + and it increases by a factor two for Et4N + and by a factor three for the Bu4N + .
In order to interpret the dependence of CM with the nature of the counter-ions, we will express the parameter CM in terms of the physical parameters influencing the dielectric friction effect (i.e. the dielectric relaxation time τ, the local dielectric permittivities o and   around each group Calculation of Rg,M needs the knowledge of the relaxation time τ of water molecules, the static permittivity εo and the highfrequency permittivity   of water. Experimental values of τ, εo and   for pure water at 298.15 K are: εo = 78.3,   = 4.49 and τ = 8.32 ps [24]. Note that this last value is less than the value estimated according to the Debye's expression:  = (6Rw 3  kBT) = 11.187 ps at 298.15 K. Table 7 gives the variations with the total ionic concentration C°M of the calculated group radii Rg,M in A°, relating to Polystyrene Sulphonate KPSS, NaPSS, LiPSS, Et4NPSS and Bu4NPSS in water at 25 °C. One indicates by < Rg,M >, the mean group radius of the polyion PSS partially condensed by counter-ions M. The Examination of these results shows that the obtained mean radii: < Rg,M >, depend on the nature of the counterions and are in general greater than the structural group radius of the PSS polyion which is equal to Rg = bS  2 = 1.25 A°. Deviation of < Rg,M > from Rg can be interpreted by replacing τ and εo in Eq. 43 by their "local" values τ' and ε'o. Indeed, the relaxation time of water molecules around the polyion PSS and its counter-ions is different from its bulk value τ, so that τ' is in fact a mean value  τ. In same way, ε'o  εo because of the dielectric saturation effect in the surrounding of the polyion and the counter-ions. It is obvious that the local structure of water depend on the nature of the condensed counterions, this could explain the specificity of the dielectric parameter CM to the nature of counter-ions via τ' and ε'o.

CONCLUSION
The Manning's model is not able to explain the dependence of the conductibility of the PSS polyion with the nature of the counter-ions M, because the counter-ions are assumed as punctual charges. However, predictions according to this model of both the degree of condensation and the ionic friction coefficient (  0.35 and βir  0.13) are acceptable in particular for the alkaline cations (Li + , Na + and K + ), but with a small observed negative deviation (  0.31) for the two hydrophobic cations (Et4N + and Bu4N + ).
In contrast, Vink has suggested a more realistic qualitative description in taking into account of the cylindrical radius RC of the polyion and the size RM of the condensed counter-ions, so that the hydrodynamic friction on the polyion increases with the minimal distance of approach (RC,M = RC + RM + nRw) between a specific condensed counter-ion and a charged polyion group.
In this work, we have used a recent model in which the stretched PSS polyion is represented by a chain of successive charged spheres, partially condensed by the counter-ions, in order to quantify the dependence of the hydrodynamic A u g u s t 0 9 , 2 0 1 4 friction on PSS polyion with the size of alkaline cations (Li + , Na + and K + ) and with the size of two hydrophobic cations (Et4N + and Bu4N + ), and we have found that the moderate increasing of this friction with the RCM distance from K + to Bu4N + , cannot completely explain the important decrease (of about 35% from K + to Bu4N + ) of the conductibility (PSS,M ) of the PSS polyion with the size RM of its condensed counter-ion M. Consequently, we have proposed a supplementary explanation to the high dependence of the mobility of the polyion with the nature of its counter-ions, by taking into account of the translational dielectric friction on the moving polyion due to the time dependent polarization of its surrounding water molecules. This supplementary friction which is in the case of this model, the most important friction effect, can be evaluated quantitatively in terms of the specific dielectric parameter CM which is function on the relaxation time τ of water molecules, the static permittivity εo and the high-frequency permittivity   of water. Now, as these physical parameters are sensitive to the local structure of water surrounding the polyions, and in turn, this local structure depends on the nature of the condensed counter-ions, we suggested that the specificity of the parameter CM could explain the high dependence of the mobility of the polyion with the nature of its counter-ions.