Polyelectrolytes in Solutions
PArt 2: Flory Theory and Scaling Model

In dilute polymer solutions, intrachain interactions prevail over interchain ones. In this regime, the polyelectrolytes can be treated as single isolated chains that are surrounded by counterions in a large volume with a size equal to the average distance between chains. As has been shown by Rubinstein and others, these isolated polyelectrolyte chains can be envisioned as a sequence of fully extended blobs when dissolved in a salt-free solvent.1

One of the oldest models describing the free energy of isolated polyelectrolyte chains in salt-free dilute solution was developed by Kuhn et al. (1950).2 They considered a flexible polyelectrolyte chain of N Kuhn segments, each of length l and charge q. Neglecting the small counterions in the solution, Kuhn et al. derived following expression for the electrostatic interaction of a polyion with end-to-end distance R:2-4

,

Where f is the degree of ionization, ze is the number of ionizable groups per Kuhn segment, e is the elementary charge, ε is the dielectric constant of the solvent and ξB is the Bjerrum length:

,

Schematic Representation of Elongated Polyelectrolyte Chain

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The conformational (or entropic) contribution to the free energy of a chain was estimated by neglecting the excluded volume interactions between monomers. Then the entropic contribution is simply the energy required to stretch an ideal chain to the end-to-end distance R:

The equilibrium value of R follows from minimizing the total energy f = fe + fc with respect to R:

The authors obtained2,3

This model predicts a proportionality between the end-to-end distance R and the number of segments N, which means that a flexible polyelectrolyte chain in a salt-free solution adopts a rod-like conformation. This model is not very realistic for a weakly charged chains that are not fully stretched but only slightly elongated in one direction. Rubinstein et al. postulated that a weakly charged chain adopts the shape of an ellipsoid with the longitudinal size extended to R and with the transverse size equal to that of an ideal chain lN1/2, that is the size of the polymer coil perpendicular to the long axis of the ellipsoid stays unperturbed. Rubinstein et al. assumed that the charges are uniformly distributed within this volume. The electrostatic energy of such a chain is given by:5-7

Then the total energy reads

Minimizing this expression with respect to R yields

Where u is an interaction parameter, the ratio of the Bjerrum length ξB to the Kuhn segment length l:

According to this model, the chain size grows as N[ln(N)]1/3 which is faster than N as predicted by Kuhn et al.2

The onset of elongation occurs when the electrostatic energy is equal to the thermal energy kT:

For this case, the polyelectrolyte coil has the shape of an ideal coil:

The transition to a stretched chain happens when the number of charged monomers on the chain fzeNe is on the order of u1/2N1/4. At stronger electrostatic forces, the end-to-end distance along the long axis monotonically increases with increasing fraction f of charged repeat units on the polymer backbone.

 

Scaling Model of Dilute Polyelectrolyte Solutions

In a very dilute salt-free polyelectrolyte solution the counterions are more or less evenly distributed throughout the solution volume, because the Debye screening length is much larger than the average distance between the polyelectrolyte molecules. In this regime, the charges on the chains interact through the unscreened Coulomb potential which results in elongated chain conformations. As has been shown above, the degree of extension, i.e., the mean square end-to-end distance R is directly proportional to the number of segments N. The behavior of the charged chains in this regime can be described with the blob model. The smallest correlation length is the electrostatic blob length D (see Figure below). Let Λ be the average number of segments between effective charges, then the total number of charges on a chain is N/Λ. The number of monomers inside each electrostatic blob is g, (g/Λ charges per blob). The statistical behavior of the repeat units inside the blobs is determined by the thermodynamic interaction of uncharged polymers. Then the scaling laws of uncharged polymers can be applied to the electrostatic blobs:

Polyelectrolyte Chain in Dilute Salt-free Solution

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In the good and theta solvent, the electrostatic energy inside the electrostatic blobs is of the order of the thermal energy

Combining these two expressions with the scaling laws for the electrostatic blob, one obtains for the number of monomers inside the electrostatic blob:

and for the size of an electrostatic blob:

On a length scale larger than D, electrostatic forces govern the behavior of the electrostatic blobs which form a ‘necklace’ of fully extended blobs of length L:

Note, the effect of solvent quality only affects the electrostatic blobs, whereas the conformation of the electrostatic blobs always assumes a rodlike configuration (see figure above).

References & Notes
  1. A.V. Dobrynin, R.H. Colby and M. Rubinstein, Macromolecules 28, 1859-1871 (1996)
  2. A. Katchalsky, O. Künzle, W. Kuhn, J. Polym. Sci. 5, 283−300 (1950)
  3. M. Muthukumar, Macromolecules, 50, pp. 9528-9560 (2017)
  4. Kuhn et al. assumed that the total charge of the chain is evenly separated at the two ends by R.
  5. A.V. Dobrynin, R.H. Colby and M. Rubinstein, Prog. Polym. Sci., 30 1049-1118 (2005)
  6. W. Kuhn, O. Kuenzle and A. Katchalsky, Helv. Chim. Acta, 31, 1994 (1948)
  7. T. Swift, L. Swanson, M. Geoghegan, S. Rimmer, Soft Matter, 12, 2542 (2016)
  8. G. Strobl, The Physics of Polymers, 3rd Edition, Heidelberg 2007

August 11, 2021

  • Summary

    Polyelectrolytes

    are polymers with dissociating groups in their repeat units. Like ordinary electrolytes, they dissociate in aqueous solutions and bear one or more charges.

  • The properties of polyelectrolytes in solution are very different from those of uncharged polymers. For example, the viscosity of dilute polyelectrolyte solutions is proportional to the square root of polymer concentration whereas the viscosity of solutions of uncharged polymers in the same regime is proportional to the concentration.

  • Polyelectrolytes carrying ionized groups are surrounded by a cloud of counterions. Their spatial distribution depends on the salt and polymer concentration, as well as on the temperature and pH value.

  • The strength of the electrostatic interaction between charged groups is given by the Bjerrum length and the range is given by the Debye length.

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  • The conformations of charged polyelectrolytes in solution depends on the fraction of charged repeat units, polymer concentration and salt content, and for weakly charged polyelectrolytes also on non-Coulomb interactions.

  • Fully charged polyelectrolytes in a salt-free solution are almost fully extended due to the strong repulsive forces between the charged links along the polymer backbone. Such a rod-like polymer will develop an electric field similar to a thin linear charged wire.

  • For partially charged polyelectrolytes in dilute salt-free solution, the chain can be envisioned as a linear arrangement of electrostatic blobs.

  • The statistical behavior of the repeat units inside the electrostatic blobs is determined by the thermodynamic interaction of uncharged polymers.

  • The conformation of the chain segments inside the electrostatic blobs is practically unperturbed by the electrostatic interaction and can be described with the scaling laws of uncharged chains.

  • For high salt concentrations and low charge density, the electric interactions between chain segments are weak and the polyelectrolytes assume a more coil-like structure.

  • Weakly charged chains adopt the shape of an ellipsoid with the longitudinal size extended and with the transverse size equal to that of an unperturbed chain.