A model for polymers

Having now worked through the Van der Waals mean field approach and having introduced the approach used in Flory's model with the entropy of mixing, we now seek to place the polymer-solvent equations into perspective.

First, the thermodynamic potentials should be modified to reflect the state variables used in polymers. We seek $V\to r$ and $P\to f$. The equations of state become

$$T dS = dU + f dr$$ (65)

$$F = U - TS = -S dT - f dr$$ (66)

$$f=-\mathop {\left( {\frac{\partial F}{\partial r}} \right)}\nolimits_T$$ (67)

$$S=-\mathop {\left( {\frac{\partial F}{\partial T}} \right)}\nolimits_r$$ (68)

$$H = U + fr$$ (69)

$$dH = TdS + r df$$ (70)

We start by considering the GPC solution as originally found in Flory's model ( $\delta \equiv 2$ and $\gamma \equiv 1)$. In the case of an ideal polymer, the equation of state is found by using the analogous equation for $P = - \partial F/ \partial V$: $i.e.$, $f = - \partial F/\partial r$. For the GPC,

$$S=k_B \left\{ {\ln C+2\ln (r)-\vartheta \mathop {\left( {\frac{r}{b}} \right)}\nolimits^2 } \right\}$$ (71)

and

$$f=-\mathop {\left( {\frac{\partial F}{\partial r}} \right)}\... ...T =2k_B T\left( {\frac{1}{r}-\vartheta \frac{r}{b^2}} \right)$$ (72)

because $(\partial U/ \partial r)_T \sim 0$ just as $(\partial U/ \partial V)_T \sim 0$ for the ideal gas. Equation (72) represents an equation of state for a polymer analogous to $PV = N k_B T$ for the ideal gas.

Now, if there are additional terms such as the excluded volume, we must introduce two-body and three-body density dependent term into an expression for the free energy as a function of swelling. The internal interaction of the chain are expressed as in the Flory case as $\Delta F_{int}(\alpha)$ (39)

$$\Delta F_{int} (\alpha )=k_B T\left\{ {B_2 \frac{N^2}{V}+B_3 \frac{N^3}{V^2}} \right\}$$ (73)

where $B_2$ corresponds to (59) and $B_3$ corresponds to (60). Now we make the approximation that $V\sim \left\langle {r^2} \right\rangle ^{3/2}\equiv r_\alpha ^3$. This is justified because the rms end-to-end separation distance is a function of the actual volume of the polymer. Using the fact that $r_\alpha =\alpha r_o$, we obtain

$$\Delta F_{int} (\alpha )=k_B T\left( {B_2 \frac{N^2}{(\alpha r_o )^3}+B_3 \frac{N^3}{(\alpha r_o )^6}} \right)$$ (74)

For the elastic contribution ( $\Delta F_{el} (\alpha ))$, we again call forth (13). As previously in (44), the total FE from this process is the sum of $\Delta F_{int}(\alpha)$ and $\Delta F_{el} (\alpha ))$

$$\Delta F(\alpha )=k_B T\left\{ {\frac{3}{2}\left( {\alpha ^2... ...2}{(\alpha r_o )^3}+B_3 \frac{N^3}{(\alpha r_o )^6}} \right\}$$ (75)

Then taking the derivative

$$\frac{\partial F}{\partial \alpha }=3k_B T\left\{ {\alpha -\... ...c{N^2}{r_o^3 \alpha ^4}-6k_B TB_3 \frac{N^3}{r_o^6 \alpha ^7}$$ (76)

and solving for the stationary points, we obtain the famous Flory expression again with additional information on the relationship for $B_2$ and $B_3$,

$$\alpha^5 - \alpha^3 = \frac{B_2 N^{1/2}}{b^3} + \frac{2 B_3}{b^6 \alpha^3}$$ (77)

where in good solvent, one can typically ignore the constant contribution of $B_3$. The relationship between $B_2$ and $B_3$ can, in principle, be found either by calculation or by experiment. In the limiting case of large $N$ and in good solvent conditions, one can see that the dominant term is $B_2$ and $\alpha$ as a function of $N$ must be

$$\alpha _{max} \sim \mathop {\left( {\frac{B_2 N^{1/2}}{b^3}} \right)}\nolimits^{1/5} =\frac{B_2^{1/5} N^{1/10}}{b^{3/5}}$$ (78)

where $\alpha \ge 1$.

For a polymer in poor solvent, one finds a situation where $B_2$ is negative and both $B_2$ and $B_3$ are significant. In such cases, we should rewrite (77) as

$$\alpha^8 - \alpha^6 - \frac{B_2 N^{1/2} \alpha^3}{b^3} - \frac{2 B_3}{b^6 } = 0$$ (79)

In the limiting case of large $N$, (79) reduces to

$$\alpha _{min} \sim \mathop {\left( {\frac{2B_3 }{-B_2 N^{1/2... ...mathop {\left( {\frac{2B_3 }{-B_2 }} \right)}\nolimits^{1/3}$$ (80)

Solving for $r$ we in (77) and (79) obtain

$$r_{max} =\alpha _{max} r_o =\mathop {\left( {\frac{B_2 }{b^3}} \right)}\nolimits^{1/5} N^{3/5}b$$ (81)

and

$$r_{min} =\alpha _{min} r_o =\mathop {\left( {\frac{2B_3 }{-B_2 }} \right)}\nolimits^{1/3} N^{1/3}$$ (82)

where $B_2$ has units of volume and $B_3$ volume squared.

Hence, we obtain Flory's original $r \propto N^{3/5}$ dependence and the corresponding globular case where $r \propto N^{1/3}$.

Finally, for completeness, we write the general form for (77) which can be found by substituting (13) into (75) and following the same procedures,

$$\alpha^{\delta+3} - \alpha^3 = \frac{3 B_2 N^{1/2}}{(\delta... ...ma + 1) b^3} + \frac{6 B_3}{(\delta \gamma + 1) b^6 \alpha^3}$$ (83)

One can quickly see that for $\delta \equiv 2$ and $\gamma \equiv 1$, (83) reduces to (77). Moreover, applying the renormalization group solution of $\delta \approx 2.5$ into (83), one obtains $2 \nu = 13/11 \approx 1.18$; almost precisely the renormalization group estimated value for $2 \nu$. Renormalization group theory has not been discussed at all here and is only briefly described in the McKenzie-Moore-Domb-Fisher model, but little need be said about the importance of establishing a sense of unity between different strategies. Indeed, the independent approach of the Flory model appears to have generated some values of the critical exponent that are surprisingly consistent with renormalization group theory.