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

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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 1ËÏ-2 XT / N_ ni \Ï

= 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:

ñî"

N,!

1 n2 ----- Ï 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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