Description of the internal interactions of the polymer

In the thermodynamics of gas models, people have often turned to lattice models to solve many statistical mechanics problems. At first sight, a lattice may seem strange because a gas is hardly stable like a crystal of quartz. However, if we were to take many instantaneous snap shots of a gas in motion, and average what we see, we would begin to see a very blurry picture that somewhat resembled a lattice because of the mutual repulsion of the gas both to itself and to the walls of the container. Similarly, liquids are even closer to crystals. Glass does not have long range order like a regular crystal lattice, but the local coordination of the various SiO molecules is hardly random in character. Aside from more extensive defects, the local coordination resembles quartz in many ways. Statistical mechanics is about the long time range characteristics of a system, not the instantaneous activity. Thus, a lattice model should not be seen as all that strange or unphysical.

Flory's theory was derived from these often used lattice-models where he considered the entropy of mixing of a polymer solvent system where the concentration of solvent was small relative to the polymer. The entropy of mixing measures the change in free volume due to replacement of one type of molecule by another.

Let index refers to the solvent and index refers to the polymer. The volume occupied by a polymer is much larger than the volume occupied by a typical solvent; usually by a very large factor. Flory reasoned that one must account for this by considering the number of segments in the polymer; not just the mole fraction of polymer or solvent. He noted that the volume of the polymer-segments may also be different from that of the solvent, but in first approximation, these are mere corrections and a lattice model could account for that in principle. For a binary system of solvent and polymer with mole fraction of solvent ( and polymer (, the total volume of the polymer and solvent should be . The corresponding volume fraction of solvent is and polymer . Hence, .

By considering the volume fraction as opposed to the mole fraction, the
accuracy of the entropy of mixing (Raoult's law) is greatly improved

Now, the enthalpy of solvation is obtained from the van Laar expression for
the heat of mixing in a two component system (here solvent and polymer)

In Flory's model,

Working from the entropy and enthalpy of mixing, partial molar quantities including the osmotic pressure and the chemical potential for solvent and polymer were derived. From these equations, the central concept of Flory's -temperature was established.

From (15) and (17), the free energy (FE) of mixing ( can now be written down as

and the chemical potential of the solvent comes from differentiating (19) with respect to (and noting that and are functions of and yields

where is the number of segments.

This expression is only valid for dilute *solvent* but the main
point of interest is for dilute *polymer* solutions. This was
the issue raised in invoking Raoult's law in the first place. Why is
this important? Equation (20) follows directly from
(19) when the partial derivative is used with constant
and then setting . However, because the definition of
( is known, it is better to use the total
derivative. Doing so, equation (20) results only when the
assumed conditions are the following: ,
and . In short, (20) is valid when . The
solution for the first term should actually contain and
there are other surviving terms in and when the total
derivative is used.

This problem emerges in part because of the way this expression is derived.
From the stand point of the excluded volume for a gas, we have

where is the partial pressure of gas and is the number of species or with and is the total volume. When this expression is expanded into a power series, it has a similar form to (20) but shows more reasonable behavior when the excluded volume is small: the behavior of and .

The argument is now shifted to volume elements from the viewpoint that
in dilute polymer solutions, the polymer-solvent system will consist
of isolated molecules of the polymer surrounded by a large open
regions occupied by the solvent. Within a small volume element
*at the location* of the given polymer, the environment will
resemble the *dilute solvent *condition where (20) is
still valid (in principle)

where emphasizes that we are working with a volume element and not the bulk system.

It is convenient to expand in a power series

One serious objection here is that the presumed conditions are those of a polymer rich solution yet we have invoked a power series approximation of as though it were small and guaranteed convergence. This is not a minor point either because the logarithmic term essentially explodes when .

Nevertheless, proceeding, we simplify (22) to obtain the following

where and with and the enthalpy and entropy of mixing the solvent with the given polymer.

From this formulation, we define a parameter

where the -temperature represents the ideal temperature at which Van't Hoff's law is obeyed for a given solvent-polymer system. Substituting into (23), we have

The excess chemical potential is then

This provides the basis for the equation for the point.

Now that we have shown how to arrive at an expression for the point, we return to (19) to find a way to express the excluded volume of two segments or parts of the same polymer segment. The reason for having side tracked a bit is because the point is so central to Flory's theory and it appears in many places. Without showing where this (25) comes from, it is rather hard to explain how it drops into the other things we need to look at later.

We consider two volume elements and . We introduce the concept of ``segment densities'' for the polymer segments and : and respectively (units: number of segments per unit volume). Let be the volume of such a segment. (I know, haven't we endured enough parameters yet?.... It seems relevant to me because it is hard to grasp the usage of these parameterizations without showing where they come from. Once we understand that point, then it is easier to see the connection between Flory's work and concepts a little more familiar to us.).

The volume fraction in each of the polymer elements is

where the index refers to the ``polymer''. If the two volume elements are brought to a separation distance , the combined concentration will be simply

If we now consider that the volume is mostly occupied by solvent in the regions between these two polymer segments, then we can approximate the total volume by the volume of the solvent (, where is Avogadro's number). The number of solvent molecules in (the region containing both and , will be

hence,

Rewriting (19) for for example, we obtain

and making all the necessary substitutions with equations (27-30), the change in the activity is

and with a little algebra, one arrives at

where the product is where we eventually will arrive at the dependence in the excluded volume term.

To fight our way through this last matter, we need to define and . Flory assumes a radial dependence on the segment density

where is defined as before as the total number of segments in the molecule. According to the general theory for the radius of gyration and . For a Gaussian distribution function as (34), we have

where we find the relationship between and is or .

Now we land on some shaky material again. I don't really find the
method works so well here, but what we want to do is evaluate
in a similar way as
(35). I think the strategy used by Flory here to get the joint
distribution function is a bit questionable, but I will explain what
was done. First, you assume cylindrical symmetry and claim that
and can be expressed as follows

``with the origin midway between the molecules''. The joint probability density for this structure is a bit odd. Anyway, to finish this matter we write

so finally we have

Now!, with just a *few* more definitions, we are
about through the difficult part of this discussion.... Let the volume
of a segment be related to the number of segments as
follows:
where is the molar
specific volume of polymer and is its molecular weight. Further,
let as indicated earlier, where is Avogadro's
number and is the volume fraction of solvent (
.

After doing some more substitution, we arrive at

Here, the constant is

The term is essentially constant for a given polymer because both and have the same dependence on . Hence, for a given solvent:polymer ratio and a given molecular weight of polymer, should be roughly constant.

Now that we have gathered together this large throng of symbols, we are now in the position to develop Flory's famous formula. Parameters associated with the Mayer function and the excluded volume can also be obtained using this approach. The derivations are tedious and major objections can be raised at several steps. Despite these flaws, the interested reader is encouraged to consult the literature to learn how these steps are done for their own edification.

The free energy consists of both an elastic contribution due to the polymer swelling (or contracting), and an internal interaction caused by the interaction of the polymer with itself and with the solvent surrounding it

where, for a GPC,

and likewise

Then, combining (42) and (43) together in (41), the total FE for swelling (or contracting) for the polymer chain of length as measured at the rms end-to-end separation distance is

Finally, we now try to minimize this expression

and, taking the stationary point, we obtain

Considering that , (46) simplifies to

which is Flory's famous solvent swelling expression.

Although we have presented a rough introduction to the original means by which Flory found this property of polymers, in fact, this large cavalcade of parameters does not help bring about significant agreement with experiment for most systems. The important prediction from this theory is the dependence for polymers in good solvent that we will show later.