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Next: Van der Waals equation Up: Flory's polymer swelling model Previous: Description of the elastic

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$_2$ 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 $1$ refers to the solvent and index $2$ 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 $x$ 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 ($n_1 )$ and polymer ($n_2
)$, the total volume of the polymer and solvent should be $n_1 + x n_2$. The corresponding volume fraction of solvent is $v_1 = n_1/(n_1 + x n_2)$ and polymer $v_2 = x n_2/(n_1 + x n_2)$. Hence, $v_1 + v_2 \equiv 1$.

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

\Delta S=k_B \left( {n_1 \ln (v_1 )+n_2 \ln (v_2 )} \right)
\end{displaymath} (15)

where $n_1$/$n_2$ corresponds to the mole fraction ( $n_1 +n_2 \equiv 1)$ and $v_1$/$v_2$ corresponds to the partial molar volume (or volume fraction). As shown in Flory's work, the large error in Raoult's law was reduced; particularly when $v_2 > v_1$.

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)

\Delta H_M = z \Delta w_{12} n_1 v_2
\end{displaymath} (16)

where $w_{ij}$ refers to the energy of contact formation between solvent-solvent ($w_{11} )$, polymer-polymer ($w_{22} )$ and solvent-polymer ( $w_{12} =w_{21} )$, and $z$ is the coordination number between the polymer and its environment.

In Flory's model,

\Delta H_M = k_B T \chi_1 n_1 v_2
\end{displaymath} (17)

where for any single segment solvent (a solvent can also be an oligomer)
\chi_1 = \frac{z \Delta w_{12}}{k_B T}.
\end{displaymath} (18)

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 $\Theta$-temperature was established.

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

\Delta F_M
=\Delta H_M -T\Delta S_M
=k_B T\left\{ {n_1 \ln v_1 +n_2 \ln v_2 +\chi _1 n_1 v_2 } \right\}
\end{displaymath} (19)

and the chemical potential of the solvent comes from differentiating (19) with respect to $n_1$ (and noting that $v_1$ and $v_2$ are functions of $n_1$ and $n_2
)$ yields

\mu_1 -\mu_1^o \approx RT\left[ {\ln (1-v_2 )+(1-1/x)v_2 +\chi _1 v_2^2 }
\end{displaymath} (20)

where $x$ 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 $n_2$ constant and then setting $v_1 =1-v_2 $. However, because the definition of $n_2$ ($=1-n_1)$ is known, it is better to use the total derivative. Doing so, equation (20) results only when the assumed conditions are the following: $v_2 \gg v_1 $, $n_2 \gg n_1$ and $x\gg 1$. In short, (20) is valid when $v_2 \sim 1$. The solution for the first term should actually contain $v_1 /v_2$ and there are other surviving terms in $n_1$ and $v_1$ 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

\begin{displaymath}P_j = \frac{n_j k_B T }{(V-\sum_{i} b_i n_i)}\end{displaymath}

where $P_j$ is the partial pressure of gas $j$ and $n_i$ is the number of species $i$ or $j$ with $N=\Sigma n_i$ and $V$ 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: $c.f.$ the behavior of $(1-v_2)/v_2$ and $1/(V-v_2)$.

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)

(\mu_1 -\mu_1^o)_E
\approx RT\left[{\ln (1-v_2 )+(1-1/x)v_2 + \chi_1 v_2^2 } \right]
\end{displaymath} (21)

where $( \cdots )_E$ emphasizes that we are working with a volume element and not the bulk system.

It is convenient to expand $\ln (1 - v_2)$ in a power series

(\mu_1 -\mu_1^o)_E \approx
\left[ {\left( {\frac{1}{2}-\chi_1 } \right) v_2^2 + v_2^3 \cdots } \right]
\end{displaymath} (22)

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 $v_2$ as though it were small and guaranteed convergence. This is not a minor point either because the logarithmic term essentially explodes when $v_2 \to 1$.

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

(\mu_1 - \mu_1^o)_E \approx RT ( \kappa_1 - \psi_1) v_2^2
\end{displaymath} (23)

where $\Delta H_1 = RT \kappa_1 v_2^2$ and $\Delta S_1 = R
\psi_1 v_2^2$ with $\Delta H_1$ and $\Delta S_1$ the enthalpy and entropy of mixing the solvent with the given polymer.

From this formulation, we define a parameter $\Theta$

\Theta = \kappa_1 T / \psi_1
\end{displaymath} (24)

where the $\Theta$-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

\psi _1 -\kappa _1 =\psi _1 \left( {1-\frac{\Theta }{T}} \right).
\end{displaymath} (25)

The excess chemical potential is then

(\mu_1 -\mu_1^o)_E =
-RT\psi _1 \left( {1-\frac{\Theta }{T}} \right)v_2^2
\end{displaymath} (26)

This provides the basis for the equation for the $\Theta$ point.

Now that we have shown how to arrive at an expression for the $\Theta$ 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 $\Theta$ 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 $\delta V_l$ and $\delta V_k$. We introduce the concept of ``segment densities'' for the polymer segments $k$ and $l$: $\rho_k$ and $\rho_l$ respectively (units: number of segments per unit volume). Let $V_s$ 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

v_{2k} =\rho _k V_s ,\;\mbox{and}\;v_{2l} =\rho _l V_s
\end{displaymath} (27)

where the index $2$ refers to the ``polymer''. If the two volume elements are brought to a separation distance $a$, the combined concentration will be simply

v_{2 k l} = (\rho_k + \rho_l) V_s
\end{displaymath} (28)

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 $V_1$ ($=N_A
v_1$, where $N_A$ is Avogadro's number). The number of solvent molecules in $\delta V$ (the region containing both $l$ and $k$, will be

\delta n_{1k} =\frac{\delta V}{V_1 }\left( {1-\rho _k V_s } ...
... n_{1l} =\frac{\delta V}{V_1 }\left( {1-\rho _l
V_s } \right)
\end{displaymath} (29)


\delta n_{1kl} =\frac{\delta V}{V_1 }\left( {1-\rho _k V_s -\rho _l V_s }
\end{displaymath} (30)

Rewriting (19) for $\delta (\Delta F_M)_k$ for example, we obtain

\delta (\Delta F_M )_k =k_B T\left( {\delta n_{1k} ln(1-v_{2k} )+\chi _1
\delta n_{1k} v_{2k} } \right)
\end{displaymath} (31)

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

\delta (\Delta F_a )=\delta \mathop {\left( {\Delta F_M }
...athop {\left( {\Delta F_M }
\right)}\nolimits_{lk} } \right\}
\end{displaymath} (32)

and with a little algebra, one arrives at

\delta (\Delta F_a )=2k_B T\psi _1 \left( {1-\frac{\Theta }{T}} \right)\rho
_k \rho _l \frac{V_s^2 }{V_1 }\delta V
\end{displaymath} (33)

where the product $\rho_k \rho_l$ is where we eventually will arrive at the $N^2$ dependence in the excluded volume term.

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

\rho =x\mathop {\left( {\frac{\beta ^\prime }{\pi ^{1/2}}}
\right)}\nolimits^3 \exp (-\beta ^{\prime 2}s^2)
\end{displaymath} (34)

where $x$ is defined as before as the total number of segments in the molecule. According to the general theory for the radius of gyration $\left\langle {s^2} \right\rangle =\left\langle
{r^2} \right\rangle /6$ and $\left\langle {r^2} \right\rangle
=3/2\beta ^2$. For a Gaussian distribution function as (34), we have

\langle s^2 \rangle
= \int_{0}^{\infty} { (s^2) \frac{\rho}{x} 4 \pi s^2 ds}
= \frac{3}{2 \beta^{\prime 2}}
\end{displaymath} (35)

where we find the relationship between $\beta$ and $\beta^\prime$ is $\beta^{\prime 2} = 6 \beta^2$ or $\beta ^{\prime
2}=9/\left\langle {r^2} \right\rangle $.

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 $\smallint \rho _k \rho _l \delta V$ 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 $s_k$ and $s_l$ can be expressed as follows

s_k^2 =\frac{a^2}{4}+r^2+ar\cos \theta ,\;\mbox{and}\;s_l^2
=\frac{a^2}{4}+r^2-ar\cos \theta

``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

\int {\rho_k \rho_l \delta V} =
\frac{x^2 \beta^{\prime 6}}{...
...i} \right)^{3/2}
\exp \left( - \beta^{\prime 2 } a^2/2 \right)
\end{displaymath} (36)

so finally we have

\Delta F_a =\smallint \delta (\Delta F_a )=2\mathop {\left( ...
...{x^2V_s^2 }{V_1 }\exp \left( {-\beta ^{\prime 2}a^2/2} \right)
\end{displaymath} (37)

Now!, with just a few more definitions, we are about through the difficult part of this discussion.... Let the volume of a segment $V_s$ be related to the number of segments $x$ as follows: $xV_s =M \overline v $ where $\overline v$ is the molar specific volume of polymer and $M$ is its molecular weight. Further, let $V_1
= N_A v_1$ as indicated earlier, where $N_A$ is Avogadro's number and $v_1$ is the volume fraction of solvent ( $v_1 =n_1 /(n_1
+ x n_2))$.

After doing some more substitution, we arrive at

\Delta F_a =2\frac{3^3}{(2\pi )^{3/2}}k_B T\left( {1-\frac{\...
...thop {\overline v }\nolimits^2 }{N_A v_1 }} \right)\exp (-y^2)
\end{displaymath} (38)

where $y = \beta^\prime a / 2^{1/2}$. With this equation, we have enough to solve for the Mayer function (to be discussed in detail in Section 2.3) and obtain the two body term (or the excluded volume). We will not go through the drill here but the methods by which one obtains it are explained in the next section. The form in (38) is close to where we want to go, and it does not seem all that rewarding to wrestle with integrating this function to obtain the two-body interaction term. It turns out that because only $y$ is a function of $a$ and the two body term is obtained by integrating over all $a$ in the Mayer function (Section 2.3), the excluded volume comes out in zeroth order to integrate out $y$ such that $exp(-y^2)\to 1$. Without endeavoring to work through any of the steps here, the result for the FE of chain interaction becomes
\Delta F_{int} (\alpha )=2k_B C_M \left( {1-\frac{\Theta }{T}}
\right)\frac{M^{1/2}}{\alpha ^3}
\end{displaymath} (39)

Here, the constant $C_M$ is
C_M =\mathop {\left( {\frac{3^2}{2\pi }} \right)}\nolimits^{...
...}{\left\langle {r^2} \right\rangle }} \right)}\nolimits^{3/2}
\end{displaymath} (40)

The term $M/\left\langle {r^2} \right\rangle $ is essentially constant for a given polymer because both $r_o^2 =\left\langle {r^2} \right\rangle $ and $M$ have the same dependence on $N$. Hence, for a given solvent:polymer ratio and a given molecular weight of polymer, $C_M$ 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

\Delta F(\alpha) = \Delta F_{int}(\alpha) + \Delta F_{el}(\alpha)
\end{displaymath} (41)

where, for a GPC,

\Delta F_{el} (\alpha )=3k_B T\left\{ {\frac{1}{2}\left( {\alpha ^2-1}
\right)-\ln (\alpha )} \right\}
\end{displaymath} (42)

and likewise

\Delta F_{int} (\alpha )=2k_B TC_M \psi _1 \left\{ {1-\frac{\Theta }{T}} \right\}
\frac{M^{1/2}}{\alpha ^3}
\end{displaymath} (43)

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

\Delta F(\alpha )=
3k_B T\left\{ {\frac{1}{2}\left( {\alpha ...
...eft\{ {1-\frac{\Theta }{T}} \right\}
\frac{M^{1/2}}{\alpha ^3}
\end{displaymath} (44)

Finally, we now try to minimize this expression

\frac{\partial \Delta F(\alpha )}{\partial \alpha }
=3k_B T\...
...eft\{ {1-\frac{\Theta }{T}}
\right\}\frac{M^{1/2}}{\alpha ^4}
\end{displaymath} (45)

and, taking the stationary point, we obtain

\alpha ^5-\alpha ^3=2C_M \psi _1 \left\{ {1-\frac{\Theta }{T}}
\end{displaymath} (46)

Considering that $M \propto N^{1/2}$, (46) simplifies to

\alpha^5 - \alpha^3 = C N^{1/2}
\end{displaymath} (47)

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 $N^{3/5}$ dependence for polymers in good solvent that we will show later.

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Next: Van der Waals equation Up: Flory's polymer swelling model Previous: Description of the elastic
Wayne Dawson 2007-01-10