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

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inventory of interactions can now be made for the mixture:

1. Each polymer segment (subscript 2) is surrounded, on the average, by

zô2 polymer segments and z0t solvent molecules.

2. The contribution to the energy of this polymer segment interacting

with

its neighbors is zô2w22 + w12 .

3. Since the lattice consists of N sites of which 02N are occupied by

polymer segments, the contribution to the energy of all interactions of

the sort described in item (2) is {lA) z02N[(l - 0i)w22 + 0iW12]. The

factor lA has been introduced, since all pairs are counted twice in

calculating this total.

4. Each solvent molecule (subscript 1) is surrounded, on the average,

by zô2 polymer segments and zôl solvent molecules.

5. The contribution to the energy of this molecule interacting with its

neighbors is zô2wl2 + z0!Wu .

6. In parallel with item (3), the contribution of all solvent

interactions to the total energy is (Vl) z0tN[02w12 + (1 - 02)wn],

7. When either pure component is considered, all w12 terms are zero, as

well as the terms containing the volume fraction of the other component.

Thus, for pure polymer, item (3) becomes (^)z02Nw22, and for the pure

solvent item (6) becomes (1/i)z01Nw11.

The value of AHm can be assembled from the foregoing by adding the

energy contributions of items (3) and (6) and subtracting the

contributions of the pure components given by item (7):

ÄÍò = (1/i)zN(20102W12 - ô! 02 w!! - ôõô2^22)

(8.40)

or, substituting Eq. (8.39),

AHm = (1/i)zN0102 Aw

(8.41)

Since V Aw is the change in interaction energy per 1,2 pair, it can be

expressed as some multiple \ of kT per pair or of RT per mole of pairs.

It is also conventional to consolidate the lattice coordination number

and x into a single parameter x, since z and x' are not measured

separately. With this change of notation Viz Aw is replaced by its

equivalent xRT, and Eq. (8.41) becomes

The Flory-Huggins Theory: The Enthalpy of Mixing

523

AHm = N^xRT = N102 X RT

(8.42)

The quantity x is called the Flory-Huggins interaction parameter: It is

zero for athermal mixtures, positive for endothermic mixing, and negative

for exothermic mixing. These differences in sign originate from Eq.

(8.39) and reaction (8.A).

Until now we have been purposely vague about the quantity Aw. Since

we shall use the notation of Eq. (8.42) from now on, it is convenient to

say a few more things about Aw before we lose sight of it entirely.

Reaction (8.A) is clear enough; what is unspecific are the conditions

under which this "reaction" takes place. Several possibilities come to

mind, and each imparts a slightly different meaning to the "energy" w:

1. Since the solvent molecules, the polymer segments, and the lattice

sites are all assumed to be equal in volume, reaction (8.A) implies

constant volume conditions. Under these conditions, AU is needed and what

we have called Aw might be better viewed as the contribution to the

internal energy of a pairwise interaction AUpair, where the subscript

reminds us that this is the contribution of a single pair formation by

reaction A.

2. Especially for large values of Aw, there could be an additional

entropy effect beyond that calculated in the last section which arises

from the interaction of nearest neighbors. That is, reaction (8.A) might

be characterized by both a AHpair and a ASpair. In this case Aw might be

viewed as the pairwise contribution to a free energy AGpair with

äÑðàè = AHpair-TASpair

(8.43)

3. In writing Eq. (8.41), we have clearly treated Aw as a contribution

to enthalpy. This means we neglect volume changes (AHpair versus AUpair)

and entropy changes beyond the configurational changes discussed in the

last section (AGpair versus AHpair). In a subsequent development it is

convenient to allow an additional entropy effect, and we shall return to

Eq. (8.43) at that point.

The equations we have written until now in this section impose no

restrictions on the species they describe or on the origin of the

interaction energy. Volume and entropy effects associated with reaction

(8.A) will be less if x is not too large. Aside from this consideration,

any of the intermolecular forces listed above could be responsible for

the specific value of x- The relationships for ASm in the last section

are based on a specific model and are subject to whatever limitations

that imposes. There is nothing in the formalism for AHm that we have

developed until now that is obviously inapplicable to certain specific

systems. In the next section we shall introduce another approximation

524

Thermodynamics of Polymer Solutions

which will impose such a limitation, so it is important to recognize the

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