# Does Electrical Conductivity of Linear Polyelectrolytes in Aqueous Solutions Follow the Dynamic Scaling Laws? A Critical Review and a Summary of the Key Relations

## Abstract

**:**

## 1. Introduction

_{p}. Each monomer, of size b

_{struct}= L/N, bears an ionizable group of valence z

_{p}. In fully ionization condition, each chain will have a charge Q

_{p}= z

_{p}eN and will release in the solution Nv

_{1}counterions, each of charge q

_{1}= z

_{1}e.

_{B}and the structural charge spacing b

_{struct}:

_{w}is the permittivity of the aqueous phase and K

_{B}T the thermal energy. Here, the Bjerrum length is defined as the length scale at which the Coulomb interaction between two elementary charges in a dielectric medium with a dielectric constant ε

_{w}is equal to the thermal energy K

_{B}T.

_{1}become trapped close to the polyion chain (counterion condensation) to reduce its effective charge from the value Q

_{p}= z

_{p}eN (before condensation) to the effective value:

_{p eff}which is considerably lower than the bare structural charge Q

_{p}.

_{p}. However, in the infinite polyion length limit and in the presence of high salt content, simulations have recently shown that this motion decouples and condensed counterions acquire a mobility with respect to the polyion itself [27].

## 2. The Electrical Conductivity. Theoretical Background and Basic Equations

_{i}of the charge carriers of type i, their electrical charge (z

_{i}e) and their mobility u

_{i}, according to the relationship:

_{i}through the molar concentration C

_{i}(n

_{i}= N

_{A}C

_{i}, where N

_{A}is the Avogadro number) and the mobility u

_{i}through the equivalent conductance λ

_{i}(u

_{i}= λ

_{i}/ F, where F = eN

_{A}is the Faraday constant). We obtain:

_{i}are expressed in [mol/cm

^{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 C

_{i}and the equivalent conductance λ

_{i}of each charge carrier.

_{1}|C

_{1}λ

_{1}+ |Z

_{p}|C

_{p}λ

_{p}

_{1}= v

_{1}N f C

_{p}, Z

_{1}= z

_{1}, Z

_{p}= N f z

_{p}, and Equation (2.5) reduces to:

_{p}(v

_{1}|z

_{1}|λ

_{1}+ |z

_{p}|λ

_{p})

_{1}differs from the value of the counterion in the absence of the polyelectrolyte, according to:

_{p}| − v

_{1}|z

_{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}of the polyion chain, the fraction f of free (un-condensed) counterions, besides the degree of polymerization N and the polyion concentration C

_{p}.

#### The Equivalent Conductance in the Manning Model

_{p}is concerned. In this context, the equivalent conductance λ

_{p}can be written as the ratio of the polyion charge Q

_{p}and the total electrophoretic coefficient f

_{Etot}according to the expression:

_{E}is calculated according to the general expression given by Kirkwood and Riseman [32], modeling the polyion as an ensemble of N

_{b}simple spherical units of radius R

_{b}, following the Manning derivation, we have:

_{b}= 6πηR

_{b}is the friction coefficient, with η the viscosity of the aqueous phase. The final expression for the equivalent conductance λ

_{p}of the polyion can be written as:

**Figure 1.**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 (C < C*) and semidilute (C > C*) regimes, respectively.

## 3. Good Solvent Condition

#### 3.1. Dilute Solutions

_{D}electrostatic blobs of size D to form a fully extended chain of length L = N

_{D}D. Each electrostatic blob contains g

_{e}monomers and bears a charge q

_{D}= z

_{p}e f g

_{e}. As usual, f is the fraction of ionized charged groups on the polymer chain and consequently the fraction of free counterions. The total charge of each polyion chain is Q

_{p}= q

_{D}N

_{D}≡ z

_{p}e f g

_{e}N

_{D}.

_{p}of the polyion proceeds analogously to what previously done, with the substitution of the elementary unit of length R

_{b}of the Manning model by the electrostatic blob of size D. This means that the following substitutions hold:

_{b}→ N

_{D}

R

_{b}→ D

ζ

_{b}→ ζ

_{D}

_{E}becomes:

_{p}of the polyion results:

_{D}and their size D. In the light of the scaling approach, these parameters scale as:

_{D}D ~ Nb(l

_{B}/ b)

^{2/7}f

^{4/7}

N

_{D}~ N(l

_{B}/ b)

^{5/7}f

^{10/7}

#### 3.2. Semidilute Solutions

_{ξ0}correlation blobs of size ξ

_{0}, each of them containing monomers. The number of correlation blobs is N

_{ξ0}= N / g. Consequently, each correlation blob bears an electric charge = z

_{p}efg and the charge of the full chain is Q

_{p}= N

_{ξ0}= z

_{p}efgN

_{ξ0}. In this case, the following substitutions hold:

_{b}→ N

_{ξ0}

R

_{b}→ ξ

_{0}

ζ

_{b}→ ζ

_{ξ}

_{E}becomes:

_{ξ}can be easily derived taking into account that now we are dealing with a rodlike unit of size ξ

_{0}containing N

_{ξ0}= N

_{D}/ ξ

_{0}correlation blobs:

_{ξ0}ξ

_{0}of the random walk chain of correlation blobs, the number N

_{ξ0}of correlation blobs within each polymer chain and the ratio g/g

_{e}of monomer inside a correlation blob to the ones inside an electrostatic blob. According to the scaling theory, these quantities scale as:

_{ξ0}ξ

_{0}− Nb(l

_{b}/ b)

^{2/7}f

^{4/7}

N

_{ξ0}− Nb

^{3/2}c

^{1/2}(l

_{b}/ b)

^{3/7}f

^{6/7}

g / g

_{e}− b

^{−3/2}c

^{−1/2}(l

_{b}/ b)

^{2/7}f

^{4/7}

_{A}, with N

_{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.

## 4. Poor Solvent Condition

_{b}) and bead controlled regime (C

_{b}< C < C

_{D}), where a different concentration dependence of the chain size occurs.

**Figure 2.**A sketch of the necklace globule model for a polyion in poor solvent condition, in dilute (C < C*) and semidilute (C > C*) concentrations. The semidilute concentration splits into string controlled regime (C* < C < C

_{b}) and bead controlled regime (C

_{b}< C < C

_{D}).

#### 4.1. Dilute Solutions

_{p}= zefN becomes larger than Q'= ze(Nτb / l

_{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 N

_{b}beads of size D

_{b}containing monomers each and joined (N

_{b}− 1) strings of length l

_{s}. The length of the necklace is given by L

_{nec}= N

_{b}l

_{s}, since the most of the length is stored in the string (l

_{s}>> D

_{b}).

_{E}can be written as:

D

_{b}~ b(l

_{b}/ b)

^{−1/3}f

^{−2/3}

l

_{s}~ b(l

_{b}/ b)

^{−1/2}f

^{−1}τ

^{1/2}

#### 4.2. Semidilute Solutions

_{b}between the string controlled regime (C* < C < C

_{b}) and the bead-controlled regime (C

_{b}< C < C

_{D}) depends, in addition to the monomer size b and the fraction f, on the solvent quality parameter τ and scales according to the relationship:

_{b}− b

^{−3}τ

^{−1/2}(l

_{b}/ b)

^{1/2}f

_{b}, the chain is assumed to be a random walk of N

_{ξ0}= N / g

_{ξ}correlation segments of size ξ

_{0}, each of them containing g

_{ξ}monomers.

_{ξ}is given by:

_{0}− b

^{−1/2}C

^{−1/2}τ

^{1/4}(l

_{b}/ b)

^{−1/4}f

^{−1/2}

g

_{ξ}− b

^{−3/2}C

^{−1/2}τ

^{3/4}(l

_{b}/ b)

^{−3/4}f

^{−3/2}

_{b}* < C < C

_{D}, owing to the screening of the electrostatic interactions between beads, the model predicts only one bead per correlation globule of size ξ

_{0}, containing g

_{ξ}monomers.

_{ξ}is given by:

_{0}− c

^{1/3}τ

^{1/3}(l

_{b}/ b)

^{−1/3}f

^{−2/3}

g

_{ξ}− τ(l

_{b}/ b)

^{−1}f

^{−2}

#### 4.3. The Diffusion Coefficients and in the Light of the Scaling Approach

_{p}and, finally, to the electrical conductivity σ of the polyelectrolyte solution, the ratio / must be appropriately evaluated in the light of the scaling laws. In the presence of counterion condensation (but in the absence of added salt), Manning [25] derived the following expression:

_{1},m

_{2}) ≠ (0,0) and ξ = 1/|z

_{1}z

_{p}|. In the case of uni-univalent polyion (z

_{1}= z

_{p}= 1), for ξ = 1, numerical evaluation of Equation (4.3.1) yields a constant value / ≃ 0.866.

_{e}and ξ

_{0}/ g scale, in good solvent condition, as:

**Figure 3.**(Upper panel): the ratio / for polyelectrolytes solution in dilute regime; full line: good solvent condition; dotted lines: poor solvent condition, with four different values of the parameter τ (τ = 0.2, 0.4, 0.6, 0.8). (Bottom panel): the ratio / for polyelectrolytes solution in semidilute regime. Full line: good solvent condition. Dotted lines: poor solvent condition, in string-controlled regime. Dashed lines: poor solvent condition, in bead-controlled regime with different values of the parameter τ (τ = 0.4, 0.6, 0.8, in the order marked by the arrow).

**Figure 4.**Bead-controlled regime. Dependence of the ratio / on the polyelectrolyte concentration C and on the fraction f for a fixed value of the solvent quality parameter (τ = 0.4).

_{p}in the different concentration regimes (dilute and semidilute concentrations), according to the above stated scaling relationships.

_{p}of the polyion in poor-solvent condition as a function of the polymer concentration, covering both the dilute and semidilute regime, is shown in Figure 5. As can be seen, the dependence is rather complex, reflecting the different conformations assumed by the polyion chain in the different concentration regimes.

**Figure 5.**Behavior of the equivalent conductance λ

_{p}of polyions in poor-solvent condition, as a function of the polyion concentration. According to the scaling picture, three different concentration regimes are clearly shown. In the dilute regime, the conductance is independent of the concentration. In the semidilute regime, a string-controlled regime and a beads-controlled regime are dependent on the concentration.

## 5. Polyelectrolyte Solutions in the Presence of Added Salt

_{i}/ ) (i = 1,2), according to the derivation proposed by Manning [24,25], can be written as:

_{p}/ C

_{s}.

_{p}z

_{1}|, Equation (5.3) becomes:

_{p}= 1; z

_{1}= 1; v

_{1}=1; =1; = 1; = 1; = 1 and the expression for the conducibility becomes:

_{p}z

_{1}|, Equation (5.3) can be written as:

_{p}which contains all the relevant information concerning the concentration regimes and the polyion conformation.

_{p}within the Manning model and how, in the framework of the scaling relationships, this parameter has to be modified as a function of the polyion concentration, the quality of the solvent.

_{p}given by Manning, reads:

_{E}is the electrophoretic coefficient (without the asymmetry field correction), X = Nn

_{p}/ n

_{s}≡ c / n

_{s}takes into account the ratio of the salt and polyion concentration and and are the cation and anion mobilities in the aqueous phase (in the absence of the polyion). Here, the polyion charge Q

_{p}depends on the charge density parameter and stands for Q

_{p}= z

_{p}eNf in the presence of counterion condensation and for Q

_{p}= z

_{p}eN (the structural value) in the absence of counterion condensation.

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

#### 5.1. The Scaling Approach

_{D}where the electrostatic blobs begins to overlap, and the concentration C

_{e}(in between C* and C

_{D}) at which polymer chains begin to entangle. Consequently, the polymer solution behaves as dilute solution for C < C*, as un-entangled semidilute solution for C* < C < C

_{e}and as entangled semidilute solution for C

_{e}< C < C

_{D}. Finally, polymer solution behaves as concentrated solution for C > C

_{D}.

#### 5.1.1. Good-Solvent Condition

_{rB}electrostatic blobs of size r

_{B}inside which the polyion conformation is extended. Each electrostatic blob contains g

_{B}monomers and bears an electric charge q

_{rB}= z

_{p}efg

_{B}. The polyion bears a charge Q

_{p}= N

_{rB}q

_{rB}, assuming a flexible conformation with an end-to-end distance given by R = r

_{B}(N / g

_{B})

^{3/5}.

_{ξ0}= N / g correlation blobs of size ξ

_{0}. Each correlation blob bears an electric charge = z

_{p}efg while the polyion charge is Q

_{p}= N

_{ξ0}. In this case, the polyion end-to-end length is given by R = ξ

_{0}(N / g)

^{1/2}.

#### 5.1.2. Poor-Solvent Condition

_{b}beads of size D

_{b}and joined by (N

_{b}− 1) strings of length l

_{s}. In the semidilute concentration regime, the concentration C

_{b}introduces, in analogy with the case in absence of added salt, two different regimes. In string-controlled regime (C* < C < C

_{b}), the chain is composed of N

_{ξ0}= N / correlation segment of size ξ

_{0}each of them containing monomers. The chain is assumed to be a random walk of size R = ξ

_{0}(N / )

^{1/2}. In the bead-controlled regime (C

_{b}< C < C

_{D}), the screening of the electrostatic interactions produces one bead per correlation globule with a random walk chain conformation of size R = ξ

_{0}(N / )

^{1/2}, analogous to the size of the chain in string-controlled regime.

#### 5.2. The Electrophoretic Coefficient f_{E}

_{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.

_{b}and the electrophoretic coefficient f

_{E}is given by:

_{0}and the electrophoretic coefficient f

_{E}is given by:

_{ξ0}given by:

_{E}in the dilute regime (the elementary unit is the bead of size D

_{b}) is given by:

_{0}) is given by:

_{ξ0}, depending on the basic unit of the chain, can be written as:

_{b}) and as:

_{b}< C < C

_{D}).

_{b}and N

_{b}in dilute regime and ξ

_{0}, N

_{ξ0}and g / g

_{e}in semidilute regime. These quantities scale as:

_{b}~ N(l

_{B}/ b)

^{−1/3}f

^{−2/3}

N

_{b}~ N(l

_{B}/ b) f

^{−2}

_{b}~ N(l

_{B}/ b)

^{−1/3}f

^{−2/3}

N

_{b}~ N(l

_{B}/ b) f

^{−2}

_{b}, D

_{b}and l

_{s}in the dilute regime and ξ

_{0}, N

_{ξ0}in the semidilute regime. These quantities scale as:

D

_{b}~ b(l

_{B}/ b)

^{−1/3}f

^{−2/3}

_{s}, that takes into account the concentration of free ions present in the solution, is defined as:

_{p}can be conveniently evaluated in each concentration regime we are dealing with, taking into account the appropriate quality of the solvent. Once the equivalent conductance λ

_{p}is known, the electrical conductivity σ of the whole polyelectrolyte solution derives directly from the Manning expression. As an example, in Figure 6 we report the equivalent conductance λ

_{p}for polyions in good-solvent condition, in dilute and semidilute regime and in the presence of added salt.

**Figure 6.**The equivalent conductance λ

_{p}of polyions in good solvent regime, for dilute and semidilute conditions, calculated according to Equations (5.2.1) and (5.2.2) in the presence of added salt, for three different values of the parameter X. Dotted lines show the calculated values in the dilute region extrapolated to semidilute region. The parameters are: N = 2250; l

_{B}= 7 × 10

^{−8}cm; b = 2.52 × 10

^{−8}cm; η = 1 cP; = 0.137 cgs units; = 0.235 cgs units. Here, X is defined as X = c/n

_{s}.

## 6. Comparison with Experiments

_{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 [5]. Measurements have been carried out at three different values of the ratio X = c / n

_{s}(X = 0.1, 1, 10), by adding appropriate amount of NaCl electrolyte solution, maintaining constant the ratio between the number of monomers and the number of ions derived from the added salt. In these experiments, good-solvent condition applies. In Figure 7, we show a comparison between the polyion equivalent conductance λ

_{p}directly derived from the experimental values of the electrical conductivity σ and the values calculated on the basis of the scaling approach we have above discussed. As can be seen, both in the dilute regime and in the semidulute regime, the agreement is quite good. In particular, we observe that transition from dilute and semidilute regime occurs exactly at the polyion concentrations predicted by the theory.

**Figure 7.**The equivalent conductance of NaPAA polyions in aqueous solutions in the presence of added salt (X = 0.1, = 1, X = 10) as a function of the polyion concentration. Polymer in good solvent condition. Panel (

**A**) Dilute solution, X = 0.1; (■) experimental values derived from the measured electrical conductivity. Full curve is calculated on the basis of Equation (5.2.1). Panel (

**B**) Dilute and semidilute solution, the transition region being marked by the arrow, X = 1; (●) experimental values derived from the measured electrical conductivity. Upper full curve is calculated on the basis of Equation (5.2.1) in the dilute regime and lower full curve calculated on the basis of Equation (5.2.2) in the semidilute regime. Panel (

**C**) Dilute and semidilute solution, the transition region being marked by the arrow. X = 10; (●) experimental values derived from the measured electrical conductivity. Upper full curve is calculated on the basis of Equation (5.2.1) in the dilute regime and lower full curve calculated on the basis of Equation (5.2.2) in the semidilute regime. Data redrawn from Reference [5].

_{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, i.e., polyions in water (poor-solvent condition) and in ethylene glycol (good-solvent condition) in cases of differently charged poyions (Figure 8, degree of quaternization Q = 55% and Figure 9, degree of quaternization Q = 17%). As can be seen, the agreement with the expected behavior is quite good over the whole concentration range, where the semidilute regime holds. This agreement is further enforced if we compare the values of the equivalent conductance for water solution with the ones calculated for good solvent condition (dotted line in Figure 8 Panel A) or for ethylene glycol with the ones for poor-solvent conditions (dotted line in Figure 8. Panel B).

**Figure 8.**The equivalent conductance of 55% PMVP-Cl (Q = 55%) polyion as a function of concentration C. Panel (

**A**) polyion in water solution (poor-solvent condition). (●): values derived from the measured electrical conductivity. Full line represents the corresponding values calculated according to the necklace model in the semidilute string-controlled condition, assuming = 0.6. The dotted line represents the calculated values in the semidilute good-solvent condition. Panel (

**B**) polyion in ethylene glycol solvent (good-solvent condition). (●): values derived from the measured electrical conductivity. Full line represents the corresponding values calculated according to the semidilute good-solvent condition. The dotted line represents the calculated values in the necklace model in semidilute-string-controlled condition. The arrow marks the concentration c *, indicating the transition between the dilute and the semidilute regime. Data from Reference [12].

**Figure 9.**The equivalent conductance of 17% PMVP-Cl (Q = 17%) polyion as a function of concentration C. Panel (

**A**) polyion in water solution (poor-solvent condition). (●): values derived from the measured electrical conductivity. Full line represents the corresponding values calculated according to the necklace model in the semidilute string-controlled condition, assuming τ = 0.6. The dotted line represents the calculated values in the semidilute good-solvent condition. Panel (

**B**) polyion in ethylene glycol solvent (good-solvent condition). (●): values derived from the measured electrical conductivity. Full line represents the corresponding values calculated according to the semidilute good-solvent condition. The dotted line represents the calculated values in the necklace model in semidilute-string-controlled condition. The arrow marks the concentration c *, indicating the transition between the dilute and the semidilute regime. Data from Reference [12].

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Morawetz, H. Polyelectrolytes; Marcel Dekker: New York, NY, USA, 1993. [Google Scholar]
- Russel, W.B.; Saville, D.A.; Schowalter, W.R. Colloidal Dispersions; Cambridge University Press: Cambridge, UK, 1983. [Google Scholar]
- Mandel, M. Physical Properties of Polyelectrolyte Solutions; Pacini: Florence, Italy, 1999. [Google Scholar]
- Bordi, F.; Cametti, C.; Gili, T. Electrical conductivity of aqueous polyelectrolyte solutions in the presence of counterion condensation: The scaling approach revisited. Phys. Rev. E
**2002**, 66. [Google Scholar] [CrossRef] - Bordi, F.; Cametti, C.; Gili, T. Electrical conductivity of polyelectrolyte solutions in the presence of added salt: The role of the solvent quality factor in the light of the scaling approach. Phys. Rev. E
**2003**, 68. [Google Scholar] [CrossRef] - Truzzolillo, D.; Bordi, F.; Cametti, C.; Sennato, S. Counterion condensation of differently flexible polyelectrolytes in aqueous solution in the dilute and semidilute regime. Phys. Rev. E
**2009**, 79. [Google Scholar] [CrossRef] - Bordi, F.; Cametti, C.; Gili, T.; Colby, R.H. Dielectric relaxations in aqueous polyelectrolyte solutions: A scaling approach and the role of their solvent quality parameter. Langmuir
**2002**, 18, 6404–6409. [Google Scholar] [CrossRef] - Bordi, F.; Colby, R.H.; Cametti, C.; de Lorenzo, L.; Gili, T. Electrical conductivity of polyelectrolyte solutions in the semidilute and concentrated regime: The role of counterion condensation. J. Phys. Chem. B
**2002**, 106, 6887–6893. [Google Scholar] [CrossRef] - Bordi, F.; Cametti, C.; Motta, A. Scaling behavior of the high-frequency dielectric properties of Poly-l-lysine aqueous solutions. Macromolecules
**2000**, 33, 1910–1916. [Google Scholar] [CrossRef] - Bordi, F.; Cametti, C.; Colby, R.H. Dielectric spectroscopy and conductivity of polyelectrolyte solutions. J. Phys. Cond. Matter
**2004**, 16, R1423–R1463. [Google Scholar] [CrossRef] - Bordi, F.; Cametti, C.; Gili, T.; Sennato, S.; Zuzzi, S.; Dou, S.; Colby, R.H. Solvent quality influence on the dielectric properties of polyelectrolyte solutions: A scaling approach. Phys. Rev. E
**2005**, 72. [Google Scholar] [CrossRef] - Bordi, F.; Cametti, C.; Gili, T.; Sennato, S.; Zuzzi, S.; Dou, S.; Colby, R.H. Conductometric properties of linear polyelectrolytes in poor-solvent condition: the necklace model. J. Chem. Phys.
**2005**, 122. [Google Scholar] [CrossRef] - Bordi, F.; Cametti, C.; Sennato, S.; Zuzzi, S.; Dou, S.; Colby, R.H. Dielectric scaling in polyelectrolyte solutions with different solvent quality in the dilute concentration regime. Phys. Chem. Chem. Phys.
**2006**, 8, 3653–3658. [Google Scholar] [CrossRef] - Cametti, C. Dielectric and conductometric properties of highly heterogeneous systems. Riv. Nuovo Cimento
**2009**, 32, 185–260. [Google Scholar] - Truzzolillo, D.; Cametti, C.; Sennato, S. Dielectric properties of differently flexible polyions: A scaling approach. Phys. Chem. Chem. Phys.
**2009**, 11, 1780–1786. [Google Scholar] - Cametti, C.; Zuzzi, S. Radiowave dielectric properties of sodium maleate copolymers in aqueous solutions in light of a scaling approach. J. Phys. Chem. B
**2010**, 114, 7140–7147. [Google Scholar] [CrossRef] - Dobrynin, A.V. Effect of counterion condensation on rigidity of semiflexible polyelectrolytes. Macromolecules
**2006**, 39, 9519–9527. [Google Scholar] [CrossRef] - Dobrynin, A.V.; Rubinstein, M. Theory of polyelectrolyte in solutions and at surfaces. Prog. Polym. Sci.
**2005**, 30, 1049–1118. [Google Scholar] [CrossRef] - Dobrynin, A.V.; Rubinstein, M.; Obukhov, S.P. Cascade of transitions of polyelectrolytes in poor solvent. Macromolecules
**1996**, 29, 2974–2979. [Google Scholar] [CrossRef] - Dobrynin, A.V.; Rubinstein, M. Hydrophobic polyelectrolytes. Macromolecules
**1999**, 32, 915–922. [Google Scholar] [CrossRef] - Manning, G.S. Counterion binding in polyelectrolyte theory. Acc. Chem. Res.
**1979**, 12, 443–449. [Google Scholar] [CrossRef] - Manning, G.S. The critical onset of counterion condensation. A survey of its experimental and theoretical basis. Ber. Bunsenges. Phys. Chem.
**1996**, 100, 909–922. [Google Scholar] [CrossRef] - Manning, G.S. Counterion condensation on charged spheres, cylinders and planes. J. Phys. Chem.
**2007**, 111, 8554–8559. [Google Scholar] [CrossRef] - Manning, G.S. Limiting laws and counterion condensation in polyelectrolyte solutions. I. Colligative properties. J. Chem. Phys.
**1969**, 51, 924–933. [Google Scholar] [CrossRef] - Manning, G.S. Limiting laws and counterion condensation in polyelectrolyte solutions. II. Self diffusion of the small ions. J. Chem. Phys.
**1969**, 51, 934–947. [Google Scholar] [CrossRef] - Manning, G.S. The molecular theory of polyelectrolyte solutions with applications to the electrostatic properties of polynucleotides. Q. Rev. Biophys.
**1978**, 11, 179–246. [Google Scholar] [CrossRef] - Fischer, S.; Naji, A.; Netz, R.R. Salt-induced counterion-mobility anomaly in polyelectriolyte electrophoresis. Phys. Rev. Lett.
**2008**, 101. [Google Scholar] [CrossRef] - Netz, R.R. Polyelectrolytes in electric fields. J. Phys. Chem. B
**2003**, 107, 8208–8217. [Google Scholar] [CrossRef] - Manning, G.S.; Mohanty, U. Counterion condensation on ionic oligomers. Phys. A
**1997**, 247, 196–204. [Google Scholar] [CrossRef] - Keyser, U.F.; Koeleman, B.N.; van Dorp, S.; Krapf, D.; Smeets, R.M.M.; Lemay, S.G.; Dekker, N.H.; Dekker, C. Direct force measurements on DNA in a solid state nanopore. Nat. Phys.
**2006**, 2, 473–477. [Google Scholar] [CrossRef] - Wette, P.; Schope, H.J.; Palberg, T. Comparison of colloidal effective charges from different experiments. J. Chem. Phys.
**2002**, 116, 10981–10988. [Google Scholar] [CrossRef] - Essafi, W.; Lafuma, F.; Williams, C.E. Structural evidence of charge renormalization in semi-dilute solutions of highly charged polyelectrolytes. Eur. Phys. J. B
**1999**, 9, 261–266. [Google Scholar] [CrossRef] - Aubouy, M.; Trizc, E.; Bocquet, L. Effective charge versus bare charge: An analytical estimate for colloids in the infinite dilution limit. J. Phys. A Math. Gen.
**2003**, 36, 5835–5840. [Google Scholar] [CrossRef] - Naji, A.; Netz, R.R. Scaling and universality in the counterion condensation transition at charged cylinders. Phys. Rev. E
**2006**, 96. [Google Scholar] [CrossRef] - Kirkwood, J.K.; Riseman, J. The intrinsic viscosities and diffusion constants of flexible macromolecules in solution. J. Chem. Phys.
**1948**, 16, 565–572. [Google Scholar] [CrossRef] - Dobrynin, A.V.; Rubinstein, M. Counterion condensation and phase separation in solutions of hydrophobic polyelectrolytes. Macromolecules
**2001**, 34, 1964–1972. [Google Scholar] [CrossRef] - Chang, R.; Yethiraj, A. Strongly charged flexible polyelectrolytes in poor solvents: Molecular dynamics simulations with explicit solvent. J. Chem. Phys.
**2003**, 118, 6634–6647. [Google Scholar] [CrossRef] - Rayleight, L. On the equilibrium of liquid conducting masses charged with electricity. Philos. Mag.
**1882**, 14, 184–186. [Google Scholar] [CrossRef] - Jeon, J.; Dobrynin, A.V. Necklace globule and counterion condensation. Macromolecules
**2007**, 40, 7695–7706. [Google Scholar] [CrossRef] - Aseyev, V.O.; Klenin, S.I.; Tenhu, H.; Grillo, I.; Geissler, E. Neutron scattering studies of the structure of a polyelectrolyte globule in a water-acetone mixture. Macromolecules
**2001**, 34, 3706–3709. [Google Scholar] [CrossRef] - Lyulin, A.V.; Dunveg, B.; Borisov, O.V.; Darinskii, A.A. Computer simulation studies of a single polyelectrolyte chain in poor solvent. Macromolecules
**1999**, 32, 3264–3278. [Google Scholar] [CrossRef] - Micka, U.; Kremer, K. Strongly charged flexible polyelectrolytes in poor solvent. From stable spheres to necklace chain. Europhys. Lett.
**2000**, 49, 189–195. [Google Scholar] [CrossRef]

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## Share and Cite

**MDPI and ACS Style**

Cametti, C.
Does Electrical Conductivity of Linear Polyelectrolytes in Aqueous Solutions Follow the Dynamic Scaling Laws? A Critical Review and a Summary of the Key Relations. *Polymers* **2014**, *6*, 1207-1231.
https://doi.org/10.3390/polym6041207

**AMA Style**

Cametti C.
Does Electrical Conductivity of Linear Polyelectrolytes in Aqueous Solutions Follow the Dynamic Scaling Laws? A Critical Review and a Summary of the Key Relations. *Polymers*. 2014; 6(4):1207-1231.
https://doi.org/10.3390/polym6041207

**Chicago/Turabian Style**

Cametti, Cesare.
2014. "Does Electrical Conductivity of Linear Polyelectrolytes in Aqueous Solutions Follow the Dynamic Scaling Laws? A Critical Review and a Summary of the Key Relations" *Polymers* 6, no. 4: 1207-1231.
https://doi.org/10.3390/polym6041207