Theory of Star-Shaped Polyelectrolytes

1.2.3. Theory of Star-Shaped Polyelectrolytes

In contrast to linear polyelectrolytes the theoretical description needs to be expanded to a three phase model for star-shaped polyelectrolytes. Besides counterions, which are directly condensed on be backbone, there are also noncondensed counterions to be encountered within the star. Those counterions do not contribute to the free counterions in bulk solution, which are supposed to be responsible for e.g. the osmotic pressure of such systems. Likos et al.⁵⁸ performed molecular dynamics simulations and compared the results with an analytical theory of those polyelectrolyte systems. They varied the degree of polymerisation per arm DParm, the number of arms per star fstar and the fraction of charged monomeric units (α`).

The solution volume V was divided into Nst spherical cells, so called Wigner-Seitz cells. Nst

is the number of stars in volume V. The core of star is placed in the center of this cell, which has a radius R (Figure 1. 8).

l_b

Figure 1. 8: Polyelectrolyte star within an adapted Wigner-Seitz cell

Geometrical considerations give following formula:

st

ρst is the number density of stars. The arm’s distance between core and chain end is given by the variable length R. According to the model, R is dependent on the chains conformation, though the arms are always located within a cylinder, which emanates from the core with radius lB (Bjerrum length). The whole cell is divided into three regions. Volume V3 = 4π(Rw3 -R³)/3 assigns the volume outside the star, which is accessible by N3 free ions. As said, a cylinder with volume V1 and radius lB around the polymer chains is assigned to the number N1

of condensed counterions. The chain’s volume Vσ is not included in V1. To obtain V1 one needs to subtract the monomer’s radius σLJ from lB yielding V1 = fstar.π(lB2-σLJ2)R. Volume V2

with N2 ions is given by the difference between the volume of the star Vstar = 4πR³/3 and V1 + Vσ, since Vstar = V1+V2 +Vσ .

For derivation of the fraction of free ions the free energy F needs to be minimized. The free energy is given by the following formula:

∑

UH is the averaged electrostatic energy of the whole star. It is given by the spatial integral over all pairs of local charge densities ς(r) at spot r. The local charge densities are averaged over time (mean-field approach). Since each pair has been regarded twice, the obtained electrostatical energy needs to be multiplied by a factor ½. This yields:

´ ´

The radial charge density can be derived by a simple consideration. Q^* is the effective charge of the star, which is given by the charge of the sum of all free counterions (N3). For a spherical charge distribution within a radius R and an effective charge Q^* it is given for the integration limits 0 < r < R: extend equation 1. 20. over the whole Wigner-Seitz cell:

3

Equation 1. 19. can be solved analytically by use of equation 1. 21.

UH does not take into account direct interactions between the chains and the counterions.

Those interactions are included in the term for the electrostatical correlation energy, UC. The averaged distance, zm, of the counterions in volume V1 to the charged monomeric units are related to the Bjerrum length, lB, and compared to the thermal energy, kBT. Taking into account the number of interactions, N1, gives:

N1

Figure 1. 9: Averaged geometry of the condensed counterions (black) towards the charged monomer units (grey) (white: uncharged monomers)

zm is approximated by zM = 0.5(lB2+b²)^0.5, since b is the average distance between two charges along the polymer (Figure 1. 9).

Equation 1. 18. has further terms, which specify the entropic contributions of the chains and the counterions. Following equation is given for the elastic properties of the polymeric backbone due to Gaussian statistics:⁵⁹

2

Besides that the free energy caused by the mutual exclusion of two chains needs to be taken into account according to Flory’s formula:⁶⁰

3

v is the Flory-parameter for the excluded volume. Likos et al. have discussed v and they used as approximation v = 30σLJ3.

Finally the entropic contribution of the counterions needs to be taken into account for each region. One can write:

( ) [

]

Λ is the thermal de Broglie wave-length of the counterions. For further considerations the Broglie term is neglected. ρi(r) is the number density of the ions. Within the cylinder around the polymeric backbone and within the bulk volume outside the star a uniform distribution of counterions is expected: ρ1(r) = N1/V1; ρ3(r) = N3/V3. For all N2 confined counterions in V2

one needs to introduce a sphere with the same volume V2, but with radius R´, since not the whole star’s space (4πR³/3) is accessible. Therefore the counterion density is given by:

2

After these considerations each entropic contribution to the free energy can be calculated.

As we are interested in the fraction of free counterions, analytical expressions for N3, N1 and R can be found (N2 is given by N3, N1).

Some examples concerning the counterion distribution can be taken from reference ⁵⁸.

Table 1. 1: Dependence of the counterion distribution for different arm numbers fstar and different fractions of charged monomeric units α`. Arm length DParm = 50, cell radius Rw = 55,83 ^. σLJ (except for fstar

Table 1. 2: Dependence of the counterion distribution for different arm lengths DParm. Arm number fstar = 10, fraction of charged monomeric units α` = 1/3, cell radius R_w = 136,48 ^. σ_LJ (except for N = 50: Rw = 55,83 σLJ); in bracket: results of molecular dynamics (MD) simulations

We can discern that the couterion confinement of star-shaped polyelectrolytes is expected to be of intermediate nature compared to linear (e.g. PAA with α’ ~ 0.25: φ ~ 0.4)⁶¹ and densly grafted brush-like polyelectrolytes.⁶²

A further look at equation 1. 28. reveals that the scaling of the radius R (here: radius of the star`s enclosing sphere) is approximated by following formula for rather highly charged stars (α = 0.33):

0