Freely Jointed Chain (FJC) model

This is the simplest model for the elasticity of polymers. It assumes that the polymer is made of rigid monomers of length $ b$ (also known as the Kuhn length) connected by junctions that are free to rotate and they do not interact with each other. The contour length of the polymer ($ L_0$) is determined by the number of monomers $ N$ so that $ L_0=Nb$. It is a suitable model for flexible polymers and the elastic response is purely entropic.

The problem can be mapped to a paramagnetic system, where the magnetization is the analogous to the extension and the external magnetic field is the analogous to the force. Differently from the WLC model, the equation of state can be calculated exactly and it is given by,

$\displaystyle x(f)=L_0\cdot \left( \coth \left(\frac{bf}{k_B T}\right) - \frac{k_B T}{bf}\right)$ (F.3)

were $ x$ is the extension, $ f$ is the force applied, $ k_B$ is the Boltzmann constant and $ T$ is the temperature. Strictly speaking, $ L_0=Nb$ where $ b$ is the Kuhn length. However, the resulting FEC does not describe correctly the experimental data. In order to fit the data, this model (Eq. F.3) is used by taking $ L_0=Nd$ where $ d$ is the distance between monomers and $ b$ is the Kuhn length (different from $ d$).

The free energy of the model can also be calculated by integration of Eq. F.3. Note that the equation of state is given in terms of the extension vs. the force, so the calculation of the free energy requires an integration by parts (see Sec. 3.4.2 for further details). Here we focus on the integral of Eq. F.3 which is given by,

$\displaystyle G(f)$ $\displaystyle =$ $\displaystyle \int_{0}^{f} x(f')\,df =$  
  $\displaystyle =$ $\displaystyle \frac{k_BTL_0}{b} \left[ \ln \left(\sinh \frac{bf}{k_BT}\right) - \ln\frac{bf}{k_B T} \right]~.$ (F.4)

For completeness, here we write the free energy of the FJC at controlled position
$\displaystyle G(x)$ $\displaystyle =$ $\displaystyle xf-\int_{0}^{f} x(f')\,df =$  
  $\displaystyle =$ $\displaystyle xf-\frac{k_BTL_0}{b} \left[ \ln \left(\sinh \frac{bf}{k_BT}\right) - \ln\frac{bf}{k_B T} \right]$ (F.5)

where $ x=x(f)$ according to Eq. F.3.

JM Huguet 2014-02-12