Handbook of Solvents - George Wypych - ChemTech - Ventech!

732 Ranieri Urbani and Attilio Cesàro
term involving the excess electrostatic free
energy (Figure 12.2.15). The probability
theory guarantees both that the components
(repeating units) of the segment vectors be
distributed in a Gaussian way along the
chain segment, and that high molecular
weight polymers be composed by a statistical
sequence of those segments. Consequence
of the above approximation is that
the distance r between any two points of the
chain (separated by a sufficiently large
number of residues, n) does not depend on
the specific sequence and values of
conformational angles and energies, but
only upon the average potential summed
over the number of residues n.
The calculation of the averaged (electrostatic)
functions is reached in two steps.
At the first, the proper flexibility of the
polymer is evaluated either from
conformational calculation or from suitable
models, then the mean value of each property
is calculated through the averaging
procedure described below.
The computational procedure is the following:
• the conformational energy surface of the uncharged polymer is evaluated by the
standard methods the conformational analysis; 65
• the end-to-end distribution distance W n(r) for the (uncharged) polymer segments is
determined by numerical Monte Carlo methods; 64
• the dependence of the total (conformational) energy G(r) upon chain extension r is
therefore estimated from the distribution of segment lengths; a Boltzmannian
distribution is assumed.
In most cases the distribution function is Gaussian (or approximately so) and the cor-
Figure 12.2.15. Dependence of the probability distribution
function of a model semi-flexible chain, (a) uncharged
and charged (a�), on the end-to-end distance and
the respective total free energies (c and c�). The electrostatic
contribution (b) is also reported.
responding free energy function can be approximated by a simple parabolic equation (Figure
12.2.15). In this case, we assume a Hookean energy (which is correct at least for the
region around the maximum of the distribution curve), so we have:
G
() = − Gr () k( r r )
⎛ ⎞
0
exp ⎜ ⎟ , = −
Wr A
⎝
RT ⎠
[12.2.9]
where:
r 0
average segment length
k a constant which determines the flexibility of the chain
The ionic energy, that results from the process of charging the polymer groups,
changes the probability of the end-to-end distance for the i-th segment, W�(r), to the probability
of the average inter-charge separation distance , W(b), following the definition of
equation [12.2.8] and [12.2.9].