Description of the elastic free energy

Statistical mechanics is about systems with a very large number of particles. As the number of particles becomes very large, discrete particles (atoms, monomers, molecules, etc.) begin to take on what appears to be a continuous character. Hence, if one were able to count the number of particles possessing a certain state of the system at some specific time, they would find that the number of particles having this state to within some range of tolerance $\delta r$ is roughly equal to the product of the total number of possible configurations times the probability.

The value of $r_{ij}$ in (3) represents a measurable thermodynamic state variable. For the state $k$, there will be $N_k$ polymers with a rms end-to-end separation-distance of $r_k$. For a large ensemble, $N_k \sim N p(r_k) \Delta r$. We let

$$\omega_k = p(r_k) \Delta (r/b)$$ (8)

and we perturb the system such that the parameters in (3) reflect the character of distorted chain ( $r_k^\prime =\alpha r_k$ hence $\vartheta ^\prime =\vartheta /\alpha ^\delta )$. Because of the inverse relationships between $r_k^\prime$ and $\vartheta^\prime$, the overall character of (3) remains unchanged (one can check this by substitution). We approximate the number of molecules having this state as $N_k^\prime \sim Np(r_k /\alpha )$ and use these relationships in the Maxwell-Boltzmann expression

$$\Omega = N ! \prod_{k} { \frac{\omega_k^{N_k^{\prime}}}{N_k^{\prime} ! } }$$ (9)

Now, we take the logarithm of both sides and apply Stirling's approximation at zeroth order yielding

$$\ln \Omega = \sum_{k} { N_k^{\prime} \ln \left( \frac{\omega_k N}{N_k^{\prime}} \right) }$$ (10)

where we have further simplified by using $\ln (N!)\sim N_1 \ln (N)+N_2 \ln (N)\ldots +N_k \ln (N)\ldots$. Using the definition for $p(r_k)$ and $p(r_k/\alpha)$, we have

$$\ln (\omega_k N/N_k^\prime )=\ln \left( {\frac{p(r_k )}{p(r_... ...{1}{\alpha ^\delta }-1} \right)+(\delta \gamma +1)\ln (\alpha)$$ (11)

To find the elastic FE for swelling or contracting the coil, we must integrate over all the microstates of $r_k$. Changing the summation to integration, the Helmholtz FE as a function of $\alpha$ for the elastic contribution to the FE becomes

$$F_{el}(\alpha) = - k_B T C_{ij}^{\delta \gamma} \int_{0}^{\... ...t)^{\delta \gamma} \left( \frac{\Delta r}{\alpha b} \right) }$$ (12)

and the result simplifies to

$$F_{el} (\alpha )=(\delta \gamma +1)k_B T\left\{ {\frac{1}{\delta }(\alpha ^\delta -1)-\ln \alpha } \right\}$$ (13)

If we substitute $\delta \equiv 2$ and $\gamma \equiv 1$ for the GPC solution, we obtain $F_{el}(\alpha) = 3 k_B T [ (\alpha^2-1)/2 - \ln \alpha ]$, which is exactly the Flory solution for the elastic contribution to the FE when the chain is distorted.

Now, differentiating this expression yields the optimal value for $\alpha$

$$\frac{\partial F_{el} (\alpha )}{\partial \alpha }=k_B T(\de... ...mma +1)\left( {\alpha ^{\delta -1}-\frac{1}{\alpha }} \right)$$ (14)

and solving (14) for $\partial F_el/\partial \alpha = 0$, we find that $\alpha = 1$. Surprised? In fact, this is what we should expect, because so far we have done nothing to this system to encourage it to do anything but stay in its current configurational state. To change the current character of the polymer, we need to find properties that render the internal interactions of the polymer non-zero. In short, we must find an expression for $F_{int} (\alpha )\ne 0$ (one such possibility is described in the next section).