Ideal chain

The case of an ideal chain whose two ends are attached to fixed points will be considered in this sub-section. The vector ${\displaystyle {\vec {R}}}$ joining these two points characterizes the macroscopic state (or macro-state) of the ideal chain. Each macro-state corresponds a certain number of micro-states, that we will call ${\displaystyle \Omega ({\vec {R}})}$ (micro-states are defined in the introduction to this section). Since the ideal chain's energy is constant, each of these micro-states is equally likely to occur. The entropy associated to a macro-state is thus equal to:

${\displaystyle S({\vec {R}})=k_{B}\log(\Omega ({\vec {R}}))}$, where ${\displaystyle k_{B}}$ is Boltzmann's constant

The above expression gives the absolute (quantum) entropy of the system. A precise determination of ${\displaystyle \Omega ({\vec {R}})}$ would require a quantum model for the ideal chain, which is beyond the scope of this article. However, we have already calculated the probability density ${\displaystyle P({\vec {R}})}$ associated with the end-to-end vector of the unconstrained ideal chain, above. Since all micro-states of the ideal chain are equally likely to occur, ${\displaystyle P({\vec {R}})}$ is proportional to ${\displaystyle \Omega ({\vec {R}})}$. This leads to the following expression for the classical (relative) entropy of the ideal chain:

${\displaystyle S({\vec {R}})=k_{B}\log(P({\vec {R}}))+C_{st}}$,

where ${\displaystyle C_{st}}$ is a fixed constant. Let us call ${\displaystyle {\vec {F}}}$ the force exerted by the chain on the point to which its end is attached. From the above expression of the entropy, we can deduce an expression of this force. Suppose that, instead of being fixed, the positions of the two ends of the ideal chain are now controlled by an operator. The operator controls the evolution of the end to end vector ${\displaystyle {\vec {R}}}$. If the operator changes ${\displaystyle {\vec {R}}}$ by a tiny amount ${\displaystyle {\vec {dR}}}$, then the variation of internal energy of the chain is zero, since the energy of the chain is constant. This condition can be written as:

${\displaystyle 0=dU=\delta W+\delta Q~}$

${\displaystyle \delta W}$ is defined as the elementary amount of mechanical work transferred by the operator to the ideal chain, and ${\displaystyle \delta Q}$ is defined as the elementary amount of heat transferred by the solvent to the ideal chain. Now, if we assume that the transformation imposed by the operator on the system is quasistatic (i.e., infinitely slow), then the system's transformation will be time-reversible, and we can assume that during its passage from macro-state ${\displaystyle {\vec {R}}}$ to macro-state ${\displaystyle {\vec {R}}+{\vec {dR}}}$, the system passes through a series of thermodynamic equilibrium macro-states. This has two consequences:

• first, the amount of heat received by the system during the transformation can be tied to the variation of its entropy:

${\displaystyle \delta Q=TdS~}$, where T is the temperature of the chain.

• second, in order for the transformation to remain infinitely slow, the mean force exerted by the operator on the end points of the chain must balance the mean force exerted by the chain on its end points. Calling ${\displaystyle {\vec {f}}_{op}}$ the force exerted by the operator and ${\displaystyle {\vec {f}}}$ the force exerted by the chain, we have:

${\displaystyle \delta W=\langle {\vec {f}}_{op}\rangle \cdot {\vec {dR}}=-\langle {\vec {f}}\rangle \cdot {\vec {dR}}~}$

We are thus led to:

${\displaystyle \langle {\vec {f}}\rangle =T{\frac {dS}{\vec {dR}}}={\frac {k_{B}T}{P({\vec {R}})}}{\frac {dP({\vec {R}})}{\vec {dR}}}~}$

${\displaystyle \langle {\vec {f}}\rangle =-k_{B}T{\frac {3{\vec {R}}}{Nl^{2}}}~}$

The above equation is the equation of state of the ideal chain. Since the expression depends on the central limit theorem, it is only exact in the limit of polymers containing a large number of monomers (that is, the thermodynamic limit). It is also only valid for small end-to-end distances, relative to the overall polymer contour length, where the behavior is like a hookean spring. Behavior over larger force ranges can be modeled using a canonical ensemble treatment identical to magnetization of paramagnetic spins. For the arbitrary forces the extension-force dependence will be given by Langevin function ${\displaystyle {\mathcal {L}}}$:

${\displaystyle {\frac {R}{Nl}}=\coth \left({\frac {fl}{k_{B}T}}\right)-{\frac {k_{B}T}{fl}}={\mathcal {L}}\left({\frac {fl}{k_{B}T}}\right),}$

where the extension is ${\displaystyle R=|{\vec {R}}|}$.

For the arbitrary extensions the force-extension dependence can be approximated by:[3]

${\displaystyle {\frac {fl}{k_{B}T}}={\mathcal {L}}^{-1}\left({\frac {R}{Nl}}\right)\approx 3{\frac {R}{Nl}}+{\frac {1}{5}}\left({\frac {R}{Nl}}\right)^{2}\sin \left({\frac {7R}{2Nl}}\right)+{\frac {\left({\frac {R}{Nl}}\right)^{3}}{1-{\frac {R}{Nl}}}}}$,

where ${\displaystyle {\mathcal {L}}^{-1}}$ is the inverse Langevin function, N is the number of bonds[4] in the molecule (therefore if the molecule has N bonds it has N+1 monomers making up the molecule.).

Finally, the model can be extended to even larger force ranges by inclusion of a stretch modulus along the polymer contour length. That is, by allowing the length of each unit of the chain to respond elastically to the applied force.[5]

Throughout this sub-section, as in the previous one, the two ends of the polymer are attached to a micro-manipulation device. This time, however, the device does not maintain the two ends of the ideal chain in a fixed position, but rather it maintains a constant pulling force ${\displaystyle {\vec {f}}_{op}}$ on the ideal chain. In this case the two ends of the polymer fluctuate around a mean position ${\displaystyle \langle {\vec {R}}\rangle }$. The ideal chain reacts with a constant opposite force ${\displaystyle {\vec {f}}=-{\vec {f}}_{op}}$

For an ideal chain exchanging length with a reservoir, a macro-state of the system is characterized by the vector ${\displaystyle {\vec {f}}}$.

The change between an ideal chain of fixed length and an ideal chain in contact with a length reservoir is very much akin to the change between the micro-canonical ensemble and the canonical ensemble (see the Statistical mechanics article about this).[6] The change is from a state where a fixed value is imposed on a certain parameter, to a state where the system is left free to exchange this parameter with the outside. The parameter in question is energy for the microcanonical and canonical descriptions, whereas in the case of the ideal chain the parameter is the length of the ideal chain.

As in the micro-canonical and canonical ensembles, the two descriptions of the ideal chain differ only in the way they treat the system's fluctuations. They are thus equivalent at the thermodynamic limit. The equation of state of the ideal chain remains the same, except that ${\displaystyle {\vec {R}}}$ is now subject to fluctuations:

${\displaystyle {\vec {f}}=-k_{B}T{\frac {3\langle {\vec {R}}\rangle }{Nl^{2}}}~}$.

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

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

${\displaystyle U=-q{\vec {E}}\cdot {\vec {R}}=-{\vec {f}}\cdot {\vec {R}}=-fR_{z}}$

Thus, different chain conformation have different statistical Boltzmann factors ${\displaystyle \exp(-U/k_{B}T)}$.[4]

The partition function is:

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

Every monomer connection in the chain is characterize by a vector ${\displaystyle {\vec {r_{i}}}}$ of length ${\displaystyle l}$ and angles ${\displaystyle \theta _{i},\varphi _{i}}$ in the spherical coordinate system. The end-to-end vector can be represented as: ${\displaystyle R_{z}=\sum _{i=1}^{N}l{\text{ }}\cos \theta _{i}}$. Therefore:

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

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

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

The Gibbs free energy is used here because the ensemble of chains corresponds to constant temperature ${\displaystyle T}$ and constant force ${\displaystyle f}$ (analogous to the isothermal-isobaric ensemble, which has constant temperature and pressure).

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

${\displaystyle \langle R\rangle =-{\partial G \over \partial f}=Nl\left[\coth \left({fl \over k_{B}T}\right)-{k_{B}T \over fl}\right]}$

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

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

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

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

${\displaystyle {\mathcal {L}}(\alpha )\cong {\alpha \over 3}}$ for ${\displaystyle \alpha \ll 1}$

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

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