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In this review, we focus on the electrical conductivity of aqueous polyelectrolyte solutions in the light of the dynamic scaling laws, recently proposed by Dobrynin and Rubinstein, to take into account the polymer conformations in different concentration regimes, both in good and poor solvent conditions. This approach allows us to separate contributions due to polymer conformation from those due to the ionic character of the chain, and offers the possibility to extend the validity of the Manning conductivity model to dilute and semidilute regimes. The electrical conductivity in the light of the scaling approach compares reasonably well with the observed values for different polyelectrolytes in aqueous solutions, over an extended concentration range, from the dilute to the semidilute regime.

Polyelectrolytes are polymers that bear a large number of ionizable groups distributed along their chain. In aqueous solution and under appropriate conditions, these polymers dissociate, leaving ionized charged groups on the polyion backbone and counterions in the bulk solution [

Polyelectrolytes have attracted much attention in the last decades because of their unusual properties, since the long-range nature of the electrostatic interactions due to the charge distribution along the chain introduces new length and time scales. The delicate balance between attractive hydrophobic interactions and repulsive electrostatic interactions governs the structural and dynamical properties of these systems. One of the most important property of polyelectrolytes is that, because of the dissociation of the ionizable groups along the chain, there is only a partial release of counterions into solution, resulting in an effective (renormalized) charge of the polyion chain significantly lower than its bare structural charge (counterion condensation).

This phenomenon is due to a fine interplay between electrostatic attractions of counterions to the polymer chain and the loss of their conformational entropy due to the localization in the vicinity of the polymer backbone. Counterion condensation strongly influences the transport and the thermodynamic properties of polyelectrolyte solutions.

In order to give a unifying picture of the conductometric properties of these systems, we have proposed a scenario able to take into account different concentration regimes, based on a scaling approach to the dynamic properties of a polyelectrolyte chain.

In this review, we summarize the main results and collect the relevant expressions governing the electrical conductivity of a polyelectrolyte solution in different experimental conditions, on the basis of a series of our own papers, appeared in the last few years [

Electrical conductivity is the key parameter for the understanding of the coupling between chain conformation and counterion condensation. This coupling is of particular importance in biological polyelectrolytes, such as DNA, where the conformation influences the biological functions. Moreover, in the light of this coupling, the knowledge of the electrical conductivity allows the evaluation of the effective charge of the polymer chain in the solution and the scaling approach can be used to calculate this charge in the most general conditions,

The scaling concepts applied to polyelectrolyte solutions have been successfully proposed many years ago, and more recently, Dobrynin, Rubinstein

On the basis of this scaling treatment, we have proposed a modified version of the Manning counterion condensation model [

We will begin our analysis by listing the basic features of the counterion condensation theory originally proposed by Manning [

Consider an aqueous polyelectrolyte solution, made up of polyion chains, each with a degree of polymerization _{p}_{struct} = _{p}_{p}_{p}eN_{1} counterions, each of charge _{1} = _{1}

However, at finite concentration, some counterions, owing to the interplay between the electrostatic interactions and the change in entropy due to their spatial confinement, may condense on the polyion itself, thus, reducing its effective charge. This phenomenon, known as counterion condensation, has been described in detail by Manning [_{B} and the structural charge spacing _{struct} :
_{w} is the permittivity of the aqueous phase and _{B}_{w} is equal to the thermal energy _{B}

If the charge spacing is too small, the electric field becomes so strong that the system can lower its free energy by condensing some of the counterions on the polyion chain. If:

Counterions of valence _{1} become trapped close to the polyion chain (counterion condensation) to reduce its effective charge from the value _{p}_{p}eN

So that the fraction of free (un-condensed) counterions will be:

The charged polyion, together with the condensed counterions, can be considered as a single entity with an effective charge _{p eff} which is considerably lower than the bare structural charge _{p}

Within this model, polyion and condensed counterions, under the influence of an external electric field, move together with the same mobility _{p}

It must be noted that condition (1.4) strictly holds in the case of an infinite isolated linear polyion chain. In the case of finite length rod-like polyions, at a finite concentration, a counterion condensation theory has been, more recently, developed by Netz [

From a physical point of view, counterion condensation is due to a balance between a favorable gain in electrostatic energy as a consequence of the ion collapse and an unfavorable loss of entropy when counterions bind to the polymer chain. This phenomenon plays an important role in biological polyelectrolyte solutions since the effective charge on the polyion backbone affects the morphology of the polyelectrolyte itself and, moreover, its response to external stimuli.

A typical example of counterion condensation occurs in DNA chains, where data obtained from optical tweezers experiments [

During the last decades, a number of studies have focused on the phenomenon of counterion condensation, both experimental [

In order to put things into their proper perspective, we will start by describing the theory of the electrical conductivity of an aqueous solution.

From a phenomenological point of view, the electrical conductivity of an aqueous solution of polyelectrolytes originated by the movement of any charged entity in response to an external applied electric field, depends on three independent contributions, _{i}_{i}e_{i}

Here, the mobility

Equation (2.1) can be written in a more usual way if we express the numeric concentration _{i}_{i}_{i}_{A}_{i}_{A} is the Avogadro number) and the mobility _{i}_{i}_{i}_{i}_{A} is the Faraday constant). We obtain:

In cgs units, the concentrations _{i}^{3}] and the equivalent conductances in [statohm^{−1}·cm^{2}·mol^{−1}] (for the conversion to SI units, 1 statohm ≈ 9 × 10^{11} ohm). Equation (2.5) is the basic equation that governs the whole transport process, depending on the concentration _{i}_{i}

In the case of a polyelectrolyte solution, charge carriers derive from the partial dissociation of the polyion chain and Equation (2.5) can be rewritten as:
_{1}|_{1}λ_{1} + |_{p}_{p}_{p}

Because of the counterion condensation effect, the following relationships hold, _{1} = _{1}_{p}_{1} = _{1}, _{p}_{p}_{p}_{1}|_{1}|λ_{1} + |_{p}_{p}

The equivalent conductance λ_{1} differs from the value

With this substitution, Equation (2.7) becomes:
_{p}_{1}|_{1}| = 0. In the light of this framework, the parameters which define the electrical conductivity of the polyelectrolyte solution are the equivalent conductance λ_{p}_{p}

Before going on, we summarize the main results of the Manning model [_{p}_{p}_{p}_{Etot} according to the expression:

If the electrophoretic coefficient _{E} is calculated according to the general expression given by Kirkwood and Riseman [_{b}_{b}_{b}_{b}_{p}

A sketch of a polyelectrolyte chain in good-solvent conditions, for different (salt free) concentration regimes. The chain is an extended rodlike configuration of electrostatic blobs and a random walk of correlation blobs for dilute (

The second example is taken from reference [

In dilute solution, at very low concentrations, and consequently at a very low ionic strength, the Debye screening length _{D} electrostatic blobs of size _{D}_{e}_{D} = _{p}e f g_{e}_{p}_{D}_{D} ≡ _{p}e f g_{e}N_{D}.

A sketch of a polyelectrolyte chain in good-solvent conditions is shown in

In this case, the derivation of the expression for the equivalent conductance λ_{p}_{b}_{b}_{D}_{b}_{b}_{D}

Consequently, the electrophoretic coefficient _{E} becomes:
_{p}

In this way, the relevant parameters of the model reduce to the number of electrostatic blobs _{D}_{D}_{B} / ^{2/7} ^{4/7}_{D} ~ _{B} / ^{5/7} ^{10/7}

Equation (2.9), together with Equations (3.1.3) and (3.1.4) furnishes the final expression of the electrical conductivity of the polyelectrolyte solution in dilute condition. Note that here and throughout the paper, we drop numerical coefficients and keep our discussion at the scaling level.

In the semidilute solutions, the polyion chain is modeled as random walk of _{ξ0} correlation blobs of size ξ_{0}, each of them containing monomers. The number of correlation blobs is _{ξ0} = _{p}efg_{p}_{ξ0} = _{p}efgN_{ξ0}. In this case, the following substitutions hold:
_{b}_{ξ0}_{b}_{0}_{b}_{ξ}
_{E} becomes:

The friction coefficient ζ_{ξ} can be easily derived taking into account that now we are dealing with a rodlike unit of size ξ_{0} containing _{ξ0} = _{D} / ξ_{0} correlation blobs:

In this concentration regime, the characteristic parameters are the contour length _{ξ0}ξ_{0} of the random walk chain of correlation blobs, the number _{ξ0} of correlation blobs within each polymer chain and the ratio _{e}_{ξ0}ξ_{0} − _{b}^{2/7} ^{4/7}_{ξ0} − ^{3/2}^{1/2}(_{b}^{3/7} ^{6/7}_{e}^{−3/2}^{−1/2}(_{b}^{2/7} ^{4/7}

Here, _{A}, with _{A} the Avogadro number). Analogously to the previous case, Equation (2.9) together with Equations (3.2.3) and (3.2.4) allows the electrical conductivity of the polyelectrolyte solution to be calculated.

Polymer-solvent interactions are of primary importance in determining the spatial configuration of the polymer chain in solution. In the light of the scaling approach, the configuration of a hydrophobic chain in a poor solvent condition has been discussed extensively by Dobrynin and Rubinstein [

Due to the strong hydrophobic interactions between the polymer backbone and the water molecules, in poor solvent conditions, the electrostatic blobs or the correlation blobs, present when good solvent condition applies, will split into a set of smaller charged globules (beads) connected by long and narrow tubules (strings). As pointed out by Dobrynin [

The situation is even more interesting in the case of semidilute regime (_{b}_{b}_{D}

In string-controlled regime, the correlation length is much larger than the size of the beads and the chains assume a bead-necklace structure on length scales smaller than the correlation length. Conversely, in bead-controlled regime, the correlation length is of the order of the size of the beads and the chains assume a bead structure on length scales smaller than the correlation length [

A sketch of the necklace globule model is shown in

A sketch of the necklace globule model for a polyion in poor solvent condition, in dilute (_{b}_{b}_{D}

When the total effective charge of a polyion _{p}_{b}^{1/2} and the Coulomb repulsion becomes comparable to the surface energy, the system tends to reduce its total free energy giving rise to _{b}_{b}_{b}_{s}_{nec} = _{b}l_{s}_{s}_{b}

The quality of the solvent is taken into account by the solvent quality parameter τ defined as τ =

In this case, the friction coefficient _{E} can be written as:

The characteristic quantities scale as:
_{b}_{b}^{−1/3} ^{−2/3}_{s}_{b}^{−1/2} ^{−1}τ^{1/2}

The overlapping concentration _{b}_{b}_{b}_{D}_{b}^{−3} τ^{−1/2}(_{b}^{1/2}

In the string-controlled regime _{b}_{ξ0} = _{ξ} correlation segments of size ξ_{0}, each of them containing _{ξ} monomers.

The electrophoretic friction coefficient is given by:
_{ξ} is given by:

The characteristic quantities scale as:
_{0} − ^{−1/2}^{−1/2} ^{1/4}(_{b}^{−1/4} ^{−1/2}_{ξ} − ^{−3/2}^{−1/2} τ^{3/4}(_{b}^{−3/4} ^{−3/2}

In the bead-controlled regime _{b}_{D}_{0}, containing _{ξ} monomers.

The electrophoretic friction coefficient is given by:
_{ξ} is given by:

The characteristic quantities scale as:
_{0} − ^{1/3} ^{1/3}(_{b}^{−1/3} ^{−2/3}_{ξ} − τ(_{b}^{−1} ^{−2}

To proceed further to the calculation of the polyion equivalent conductance λ_{p}_{1},_{2}) ≠ (0,0) and ξ = 1/|_{1}_{p}_{1} = _{p}

Within the scaling picture, the charge density parameter ξ can be written as:

Consequently, the ratio

In the light of the scaling laws, the two relevant quantities _{e}_{0} /

In poor solvent condition, in dilute regime the following substitution hold:

Typical behavior of the ratio

As can be seen in Equation (4.3.9), in bead controlled regime, there is, contrarily to the other regimes, a dependence on the polyelectrolyte concentration

(Upper panel): the ratio

Bead-controlled regime. Dependence of the ratio

From the above analysis, it turns out that the electrical conductivity of a polyelectrolyte solution in the absence of added salt depends essentially on two parameters, the fraction _{p}

A typical dependence of the equivalent conductance λ_{p}

Behavior of the equivalent conductance λ_{p}

Following an additive rule, in the presence of added salt, Equation (2.7) must be replaced by:

In this case too, the equivalent conductances λ can be writes as:

Substitution of Equation (5.2) into Equation (5.1) yields:

The general expression for the ratio (_{i}

Here, ξ is, as usual, the charge density parameter and _{p}_{s}

To a first approximation, Equations (5.4) and (5.5) can be simplified according to the following expression:
_{p}z_{1}|, Equation (5.3) becomes:

Equation (5.7) reduces to the usual expression in the case of uni-univalent salt and for univalent charges on the polyion chain. In this case, _{p}_{1} = 1; _{1} =1;

In the presence of counterion condensation, _{p}z_{1}|, Equation (5.3) can be written as:

As can be seen, in Equations (5.5) and (5.7), or equivalently, in the Equations (5.6) and (5.8), the key parameter that governs the conductometric behavior of the polyion solution is represented by the polyion equivalent conductance λ_{p}

As previously done, we will start considering the equivalent conductance λ_{p}

Taking into account the “asymmetry field” correction, in the presence of added salt, the general expression for the polyion equivalent conductance λ_{p}_{E} is the electrophoretic coefficient (without the asymmetry field correction), _{p}_{s}_{s}_{p}_{p}_{p}eNf_{p}_{p}eN

The electrophoretic coefficient _{E} in the light of the Manning model is approximated by:

In the presence of added salt, the polyion conformation is characterized by three different concentrations that define different concentration regimes. These concentrations are the concentration _{D}_{e}_{D}_{e}_{e}_{D}. Finally, polymer solution behaves as concentrated solution for _{D}

In analogy with what we have done in the previous section, we will discuss good-solvent and poor-solvent conditions, separately.

In the dilute concentration regime, the polyion chain is represented by a self-avoiding walk of _{rB}_{B} inside which the polyion conformation is extended. Each electrostatic blob contains _{B}_{rB} = _{p}efg_{B}. The polyion bears a charge _{p}_{rB}_{rB}, assuming a flexible conformation with an end-to-end distance given by _{B}(_{B})^{3/5}.

In the semidilute concentration regime, the polyion chain is modeled as a random walk of _{ξ0} = _{0}. Each correlation blob bears an electric charge _{p}efg_{p}_{ξ0}. In this case, the polyion end-to-end length is given by _{0}(^{1/2}.

Contrarily to what happens in the absence of added salt, in this case, electrostatic interactions are reduced by the presence of the added salt and in both the two regimes a random-walk structure dominates.

In dilute concentration regime, when the effective polyion charge becomes larger than a critical value and the Coulombic repulsion becomes comparably with the surface energy, the polyion splits into _{b}_{b}_{b}_{s}_{b}_{b}_{ξ0} = _{0} each of them containing _{0}(^{1/2}. In the bead-controlled regime (_{b}_{D}_{0}(^{1/2}, analogous to the size of the chain in string-controlled regime.

As in the previous cases, in the presence of added salt too, the electrophoretic coefficient _{E} depends on the different concentration regimes, since the elementary unit that contributes to the conductivity, in the light of the Manning theory, differs from a regime to the other.

In good solvent condition and in dilute regime, the elementary unit is the electrostatic blob of size _{b}_{E} is given by:

In good-solvent condition, but in semidilute regime, the elementary unit is the correlation blob of size ξ_{0} and the electrophoretic coefficient _{E} is given by:
_{ξ0} given by:

In the poor solvent condition, when the necklace model applies, the electrophoretic coefficient _{E} in the dilute regime (the elementary unit is the bead of size _{b}_{0}) is given by:

In this latter case, the friction coefficient ζ_{ξ0}, depending on the basic unit of the chain, can be written as:
_{b}_{b}_{D}

In good solvent condition, according to the scaling theory, the characteristic quantities are _{b}_{b}_{0}, _{ξ0} and _{e}_{b}_{B} / ^{−1/3} ^{−2/3}_{b}_{B} / ^{−2}
_{b}_{B} / ^{−1/3} ^{−2/3}_{b}_{B} / ^{−2}

In poor solvent condition, according to the scaling relationships, the characteristic quantities entering Equations (5.2.4) and (5.2.5) are _{b}_{b}_{s}_{0}, _{ξ0} in the semidilute regime. These quantities scale as:
_{b}_{B} / ^{−1/3} ^{−2/3}

Here, the quantity _{s}

For uni-univalent salt and for univalent counterions, the equation reduces to:

On the basis of these scaling relationships, the equivalent conductance λ_{p}_{p}_{p}

The equivalent conductance λ_{p}_{B} = 7 × 10^{−8} cm; ^{−8} cm; η = 1 cP; _{s}

In what follows, we will compare some representative results from the recent literature with the ones predicted by the scaling theory. We confine ourselves to one’s own measurements [

Electrical conductivity measurements of Sodium polyacrylate salts [−CH_{2}CH(CO_{2}Na)−]_{n}, [NaPAA] as a function of polymer concentration to cover the dilute and semidilute regime have been extensively reported in Reference [_{s}_{p}

The equivalent conductance of NaPAA polyions in aqueous solutions in the presence of added salt (

The second example is taken from reference [

We report here some typical results showing the behavior of the equivalent conductance λ_{p} of the polyion as a function of the polyion concentration, derived from the measured electrical conductivity and compared with the values calculated, on the basis of the scaling approach, for semidilute regime in string-controlled conditions. We present two limiting cases,

These results are a strong support for the necklace model for hydrophobic polyions in the light of the dynamic scaling models.

The equivalent conductance of 55% PMVP-Cl (

The equivalent conductance of 17% PMVP-Cl (

The conductometric properties of aqueous polyelectrolyte solutions in the absence and in the presence of added salt have been reviewed in the light of the dynamic scaling models for polyion conformation in different concentration regimes, proposed some years ago by Dobrynin and Rubinstein. Starting from the basic relationship derived by Manning for the equivalent conductance of a polyelectrolyte in high dilution limit, we have extended this approach on the basis of the scaling picture, to more concentrated systems. In the dilute regime, where stretched rod-like polymer conformation prevails, the basic entity that contributes to the electrical conductivity is the electrostatic blob and the polymer chain is represented by a rod-like configuration of electrostatic blobs. In the semidilute regime, where a random walk statistics applies, the polymer chain is modeled as a random walk of correlation blobs. For each of these regimes, it is possible to identify the elementary unit that contributes to the conductive process, together with the characteristic parameters, which model these units. The further step is the introduction of the scaling relationships, which allow the knowledge of the conformal evolution of the polymer chain in the different concentration regimes and in consequence of the solvent quality. This approach furnishes a comprehensive picture of the conductometric behavior of polyelectrolyte solutions in different experimental conditions in good agreement with experimental results.

I acknowledge support from the Department of Physics at the University of Rome “La Sapienza”.

The author declares no conflict of interest.