User:Tomshalev13/sandbox

Ideal chain under a constant force constraint - calculation
Consider a freely jointed chain of N bonds of length $$l$$ subject to a constant elongational force f applied to its ends along the z axis and an environment temperature $$T$$. An example could be a chain with two opposite charges +q and -q at its ends in a constant electric field $$\vec{E}$$ applied along the $$z$$ axis as sketched in the figure on the right. If the direct Coulomb interaction between the charges is ignored, there is a constant force $$\vec{f}$$ at the two ends.

Different chain conformations are not equally likely, because they correspond to different energy of the chain in the external electric field.

$$U=-q\vec{E}\cdot \vec{R}=-\vec{f}\cdot \vec{R}=-fR_z$$

Thus, different chain conformation have different statistical Boltzmann factors $$exp(-U/k_BT)$$.

The partition function is:

$$Z=\sum_{states}exp(-U/k_BT)=\sum_{states}exp({fR_z \over k_B T})$$

Every monomer connection in the chain is characterize by a vector $$\vec{r_i}$$ of length $$l$$ and angles $$\theta_i, \varphi_i$$ in the spherical coordinate system. The end-to-end vector can be represented as: $$R_z=\sum_{i=1}^N l \text{ } cos\theta_i$$. Therefore:$$Z=\int exp({fl \over k_BT}\sum_{i=1}^N cos\theta_i)\prod_{i=1}^Nsin\theta_id\theta_id\varphi_i =[\int_{0}^{\pi} 2\pi \text{ } sin\theta_i \text{ } exp({fl \over k_BT}cos\theta_i)d\theta_i]^N =[{2\pi \over fl/(k_B T)}[exp({fl \over k_B T})-exp(-{fl \over k_B T})]]^N =[{4\pi \text{ } sinh(fl/(K_B T)) \over fl/(k_B T)}]^N$$

The Gibbs free energy G can be directly calculated from the partition function:

$$G(T,f,N)=-k_B T \text{ } ln \text{ } Z(T,f,N)=-N k_B T [ln(4\pi \text{ } sinh({fl \over k_B T}))-ln({fl \over k_B T})]$$

The average end-to-end distance corresponding to a given force can be obtained as the derivative of the free energy:

$$=-{\partial G \over \partial f}=bN[coth({fl \over k_B T})-{1 \over fl/(k_B T)}]$$

This expression is the Langevin function $$\mathcal{L}$$ ,also mentioned in previous paragraphs:

$$\mathcal{L}(\alpha)=coth(\alpha)-{1 \over \alpha}$$ where,  $$\alpha={fl \over k_B T}$$.

For small relative elongations ($$\left \langle R \right \rangle \ll R_{max} = lN$$) the dependence is approximately linear,

$$\mathcal{L}(\alpha)\cong{\alpha \over 3}$$ for  $$\alpha\ll1 $$

and follows Hooke's law as shown in previous paragraphs:

$$\vec{f}=K_B T {3\left \langle \vec{R} \right \rangle \over Nl^2}$$