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Polymer Chemistry. The Basic Concepts - Himenz P.C.

Himenz P.C. Polymer Chemistry. The Basic Concepts - Copyright, 1984. - 736 p.
Download (direct link): polymerchemistry1984.djvu
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In the event that the repeat unit and the solvent are not identical in
volume, the chain can be hypothetically divided into n segments which are
equal in volume to the solvent, and this n will be proportional to the
degree of polymerization. The total number of sites in the lattice is Nt
+ nN2 = N.
We assign an index number to each of the polymer molecules and pick up
the analysis of the problem after i polymer molecules have already been
placed on an otherwise empty lattice. Our first question, then, concerns
the number of ways the (i + l)th polymer molecule can be placed in the
lattice. The polymer is to be positioned one repeat unit at a time, so it
is an easy matter to count the number of available positions for the
first segment of the (i + l)th molecule. Since the total lattice consists
of N sites and ni of these are already occupied, the first segment of the
(i + l)th molecule can be placed on any one of the N - ni remaining
sites.
There is nothing unique about the placement of this isolated segment
to distinguish it from the placement of a small molecule on a lattice
filled to the same extent. The polymeric nature of the solute shows up in
the placement of the second segment: This must be positioned in a site
adjacent to the first, since the units are covalently bonded together. No
such limitation exists for independent small molecules. To handle this
development we assume that each site on the lattice has z neighboring
sites and we call z the coordination number of the lattice. It might
appear that the need for this parameter introduces into the model a
quantity which would be difficult to evaluate in any eventual test of the
model. It turns out, however, that the z's cancel out of the final result
for ASm, so we need not worry about this eventuality.
If the molecule under consideration were being placed on an empty
lattice, the second segment could go into any one of the z sites adjacent
to the first. However, ni of the sites are already filled, so there is a
chance that one of the z sites in the coordination sphere of the first
segment is already occupied. To deal with this possibility, we assume
that the fraction of vacant sites on the lattice as a whole also applies
in the immediate vicinity of the segment positioned above. This fraction
is (N - ni)/N, so the number of possible locations for segment 2 of the
(i + l)th molecule is z(N - ni)/N.
The logic that leads us to this last result also limits the
applicability of the ensuing derivation. Applying the fraction of total
lattice sites vacant to the immediate vicinity of the first segment makes
the model descriptive of a relatively concentrated solution. This is
somewhat novel in itself, since theories of solutions more commonly
assume dilute conditions. More to the point, the model is unrealistic for
dilute solutions where the site occupancy within the domain of a
dissolved polymer coil is greater than that for the solution as a whole.
We shall return to a model more appropriate for dilute solutions below.
For now we continue with the case of the more concentrated solution,
realizing
The Flory-Huggins Theory: The Entropy of Mixing
515
that with a significant fraction of the lattice sites already filled, we
can treat the fraction of available sites as constant during the
placement of a specific polymer molecule.
The third segment of molecule i + 1 would have z - 1 available sites
on an otherwise empty lattice: The first segment, after all, already
occupies a site adjacent to segment 2. For a lattice which already
contains ni segments-if we consider those from molecule i + 1 as
insignificant-the number of sites available for segment 3 is (z - 1)(N -
ni)/N. This same number also applies to all subsequent segments.
Therefore the number of ways coi+1 that the (i + l)th molecule can be
placed on a lattice already containing i molecules is given by
coi+1 = (N - ni) z
N- ni N
(z- 1)
f 1-2 XT / N_ ni \
= z(z- l)n N 1- 1
N- ni
~N~
n-2
(8.31)
or for the ith molecule
CO: = z(z - l)1
n-2
N
N- n(i- 1) N
(8.32)
The total number of ways of placing N polymer molecules in the lattice
is given by the product of factors like these, one for each molecule:
CO i CO 2
CO:
"
N,!
1 n2 ----- CO:
N2! i=i 1
(8.33)
In writing Eq. (8.33), we have divided by N2 !, since the polymer
molecules are interchangeable. Equation (8.33) gives the thermodynamic
probabilty for the system according to this model, since there is only
one way to place the solvent molecules once the polymer molecules have
been positioned on the lattice.
Application of the Boltzmann equation to Eq. (8.33) gives the entropy
of the mixture according to this model for concentrated solutions:
S(tm), = kln(^) n "i
Substitution of Eq. (8.32) for coj in the above expression yields the
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