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,
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
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
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
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
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,
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.