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

Himenz P.C. Polymer Chemistry. The Basic Concepts - Copyright, 1984. - 736 p.
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n = 100 2.25 2.62 2.65 2.53 2.31 1.99 1.66 1.25 0.69
n = 500 3.63 3.87 3.77 3.49 3.11 2.66 2.11 1.52 0.85
Table 8.1 Values of ô Corresponding to the Indicated Mole Fractions for
Values of n = 50, 100, and 500 (These Quantities are Used in Example
8.1.)
x2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Xl 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
For n = 50 02 0.847 0.926 0.955 0.971 0.980 0.987 0.992 0.995 0.998
01 0.153 0.074 0.045 0.029 0.020 0.013 0.008 0.005 0.002
For n = 100 02 0.917 0.962 0.977 0.985 0.990 0.993 0.996 0.998 0.999
01 0.083 0.038 0.023 0.015 0.010 0.007 0.004 0.002 0.001
For n = 500 02 0.982 0.992 0.995 0.997 0.998 0.9987 0.9991 0.9995 0.9998
01 0.018 0.008 0.005 0.003 0.002 0.0013 0.0009 0.0005 0.0002
The Flory-Huggins Theory: The Entropy of Mixing 519
520
Thermodynamics of Polymer Solutions
Figure 8.1 The entropy of mixing (in units of R) as a function of mole
fraction solute for ideal mixing and for the Flory-Huggins lattice model
with n = 50, 100, and 500. Values are calculated in Example 8.1.
A plot of these values is shown in Fig. 8.1. Note the increase in the
entropy of mixing over the ideal value with increasing n value. Also note
that the maximum occurs at decreasing mole fractions of polymer with
increasing degree of polymerization.

Our ultimate goal is to develop expressions for the free energy of
mixing according to the lattice model of the Flory-Huggins theory. The
next step toward realization of this goal is to relax the restriction
which limits the result to athermal solutions. In the next section we
shall develop the theory for the enthalpy of mixing. Then we combine the
equations for entropy and enthalpy to obtain an expression for AGm
according to the Flory-Huggins model.
The Flory-Huggins Theory: The Enthalpy of Mixing
521
8.4 The Flory-Huggins Theory: The Enthalpy of Mixing
In the liquid state molecules are in intimate contact, so the energetics
of molecular interactions generally make a contribution to the overall
picture of the mixing process. There are several aspects of the situation
that we should be aware of before attempting to formulate a theory for
ÄÍò :
1. Our immediate goal is an expression for AHm and we must
remember that
this is the difference in the enthalpies of the solution and the
pure components. We need not worry about the absolute values of
the enthalpies of
the individual states.
2. Enthalpies of mixing have their origin in the forces that operate
between individual molecules. Intermolecular forces drop off rapidly with
increasing distance of separation between molecules. This means that only
nearest neighbors need be considered in the model.
3. Until surface contact, the force between molecules is always one of
attraction, although this attraction has different origins in different
systems. London forces, dipole-dipole attractions, acid-base
interactions, and hydrogen bonds are some of the types of attraction we
have in mind. In the foregoing list, London forces are universal and also
the weakest of the attractions listed. The interactions increase in
strength and also in specificity in the order listed.
4. Since London forces are universal and nonspecific, we shall
eventually emphasize these, realizing that stronger forces may outweigh
the London contribution in some systems. We anticipate the best results
in nonpolar systems where London forces account for the interactions.
The lattice model that served as the basis for calculating ASm in
the last section continues to characterize the Flory-Huggins theory in
the development of an expression for AHm. Specifically, we are concerned
with the change in enthalpy which occurs when one species is replaced by
another in adjacent lattice sites. The situation can be represented in
the notation of a chemical reaction:
(1,1) + (2,2) -*2(1,2)
(8.A)
where 1 and 2 refer to solvent molecules and polymer repeat units,
respectively, and the parentheses enclose the particular pair under
consideration. Suppose we indicate the pairwise interaction energy
between species i and j as w^; then for reaction (8.A),
Aw = 2w12 - wu - w22
(8.39)
522
Thermodynamics of Polymer Solutions
The change in interaction energy per 1,2 pair is thus Vl Aw. Next we
must consider how this scales up for a large array of molecules, and
particularly how to describe the concentration dependence of the result.
Each lattice site is defined to have z nearest neighbors, and ôi and
ô2, respectively, can be used to describe the fraction of sites which are
occupied by solvent molecules and polymer segments. The following
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