User:Tomshalev13/sandbox

Ideal chain under a constant force constraint - calculation

A diagram of an ideal chain constrained by a constant force.

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 $exp(-U/k_{B}T)$.

The partition function is:

$Z=\sum _{states}exp(-U/k_{B}T)=\sum _{states}exp({fR_{z} \over k_{B}T})$

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:$Z=\int exp({fl \over k_{B}T}\sum _{i=1}^{N}cos\theta _{i})\prod _{i=1}^{N}sin\theta _{i}d\theta _{i}d\varphi _{i}=[\int _{0}^{\pi }2\pi {\text{ }}sin\theta _{i}{\text{ }}exp({fl \over k_{B}T}cos\theta _{i})d\theta _{i}]^{N}=[{2\pi \over fl/(k_{B}T)}[exp({fl \over k_{B}T})-exp(-{fl \over k_{B}T})]]^{N}=[{4\pi {\text{ }}sinh(fl/(K_{B}T)) \over fl/(k_{B}T)}]^{N}$

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

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

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

$<R>=-{\partial G \over \partial f}=bN[coth({fl \over k_{B}T})-{1 \over fl/(k_{B}T)}]$

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

The average distance $\left\langle {\vec {R}}\right\rangle$ of the chain as a function of $\alpha$.

${\mathcal {L}}(\alpha )=coth(\alpha )-{1 \over \alpha }$ 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 $\alpha \ll 1$

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

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