EFJC model thermodynamics (isotensional)

class EFJC(number_of_links, link_length, hinge_mass, link_stiffness)

The extensible freely-jointed chain (EFJC) model thermodynamics in the isotensional ensemble.

The number of links in the chain.

The length of each link in the chain in units of nm.

hinge_mass

The mass of each hinge in the chain in units of kg/mol.

The stiffness of each link in the chain in units of J/(mol⋅nm^2).

asymptotic

The thermodynamic functions of the model in the isotensional ensemble approximated using an asymptotic approach.

legendre

The thermodynamic functions of the model in the isotensional ensemble approximated using a Legendre transformation.

end_to_end_length(force, temperature)

The expected end-to-end length as a function of the applied force and temperature,

\[\xi(f, T) = -\frac{\partial\varphi}{\partial f}.\]
Parameters:

  • force (numpy.ndarray) – The force \(f\).

  • temperature (float) – The temperature \(T\).

Returns:

The end-to-end length \(\xi\).

Return type:

numpy.ndarray

The expected end-to-end length per link as a function of the applied force and temperature.

Parameters:

  • force (numpy.ndarray) – The force \(f\).

  • temperature (float) – The temperature \(T\).

Returns:

The end-to-end length per link \(\xi/N_b=\ell_b\gamma\).

Return type:

numpy.ndarray

nondimensional_end_to_end_length(nondimensional_force)

The expected nondimensional end-to-end length as a function of the applied nondimensional force.

Parameters:

  • nondimensional_force (numpy.ndarray) – The nondimensional force \(\eta\equiv\beta f\ell_b\).

  • temperature (float) – The temperature \(T\).

Returns:

The nondimensional end-to-end length \(N_b\gamma=\xi/\ell_b\).

Return type:

numpy.ndarray

The expected nondimensional end-to-end length per link as a function of the applied nondimensional force, calculated from Balabaev and Khazanovich[1], Buche et al.[2] as

\[\gamma(\eta) = -\frac{\partial\varrho}{\partial\eta} = \mathcal{L}(\eta) + \frac{\eta}{\kappa}\left[1 + \frac{1 - \mathcal{L}(\eta)\coth(\eta)}{1 + (\eta/\kappa)\coth(\eta)}\right] + \frac{\partial}{\partial\eta}\,\ln\left[1+g(\eta)\right],\]

where \(\mathcal{L}(x)=\coth(x)-1/x\) is the Langevin function, and \(g(\eta)\) is defined as

\[g(\eta) \equiv \frac{e^{\eta}\left(\frac{\eta}{\kappa} + 1\right) \,\mathrm{erf}\left(\frac{\eta+\kappa}{\sqrt{2\kappa}}\right) - e^{-\eta}\left(\frac{\eta}{\kappa} - 1\right) \,\mathrm{erf}\left(\frac{\eta-\kappa}{\sqrt{2\kappa}}\right)}{4\sinh(\eta)\left[1 + (\eta/\kappa)\coth(\eta)\right]} - \frac{1}{2}.\]
Parameters:

  • nondimensional_force (numpy.ndarray) – The nondimensional force \(\eta\equiv\beta f\ell_b\).

  • temperature (float) – The temperature \(T\).

Returns:

The nondimensional end-to-end length per link \(\gamma\equiv \xi/N_b\ell_b\).

Return type:

numpy.ndarray

gibbs_free_energy(force, temperature)

The Gibbs free energy as a function of the applied force and temperature,

\[\varphi(f, T) = -kT\ln Z(f, T).\]
Parameters:

  • force (numpy.ndarray) – The force \(f\).

  • temperature (float) – The temperature \(T\).

Returns:

The Gibbs free energy \(\varphi\).

Return type:

numpy.ndarray

The Gibbs free energy per link as a function of the applied force and temperature.

Parameters:

  • force (numpy.ndarray) – The force \(f\).

  • temperature (float) – The temperature \(T\).

Returns:

The Gibbs free energy per link \(\varphi/N_b\).

Return type:

numpy.ndarray

relative_gibbs_free_energy(force, temperature)

The relative Gibbs free energy as a function of the applied force and temperature.

Parameters:

  • force (numpy.ndarray) – The force \(f\).

  • temperature (float) – The temperature \(T\).

Returns:

The relative Gibbs free energy \(\Delta\varphi\equiv\varphi(f,T)-\varphi(0,T)\).

Return type:

numpy.ndarray

The relative Gibbs free energy per link as a function of the applied force and temperature.

Parameters:

  • force (numpy.ndarray) – The force \(f\).

  • temperature (float) – The temperature \(T\).

Returns:

The relative Gibbs free energy per link \(\Delta\varphi/N_b\).

Return type:

numpy.ndarray

nondimensional_gibbs_free_energy(nondimensional_force, temperature)

The nondimensional Gibbs free energy as a function of the applied nondimensional force and temperature.

Parameters:

  • nondimensional_force (numpy.ndarray) – The nondimensional force \(\eta\equiv\beta f\ell_b\).

  • temperature (float) – The temperature \(T\).

Returns:

The nondimensional Gibbs free energy \(\beta\varphi=N_b\varrho\).

Return type:

numpy.ndarray

The nondimensional Gibbs free energy per link as a function of the applied nondimensional force and temperature.

Parameters:

  • nondimensional_force (numpy.ndarray) – The nondimensional force \(\eta\equiv\beta f\ell_b\).

  • temperature (float) – The temperature \(T\).

Returns:

The nondimensional Gibbs free energy per link \(\varrho\equiv\beta\varphi/N_b\).

Return type:

numpy.ndarray

nondimensional_relative_gibbs_free_energy(nondimensional_force)

The nondimensional relative Gibbs free energy as a function of the applied nondimensional force.

Parameters:

  • nondimensional_force (numpy.ndarray) – The nondimensional force \(\eta\equiv\beta f\ell_b\).

  • temperature (float) – The temperature \(T\).

Returns:

The nondimensional relative Gibbs free energy \(\beta\Delta\varphi=N_b\Delta\varrho\).

Return type:

numpy.ndarray

The nondimensional relative Gibbs free energy per link as a function of the applied nondimensional force, given by Buche et al.[2] as

\[\Delta\varrho(\eta) = -\frac{\eta^2}{2\kappa} - \ln\left[w^+(\eta) + w^-(\eta)\right],\]

where the functions \(w^+(\eta)\) and \(w^-(\eta)\) are defined as as

\[w^\pm(\eta) \equiv e^{\pm\eta}\left(\frac{1}{\kappa} \pm \frac{1}{\eta}\right) \left[1 \pm \mathrm{erf}\left(\frac{\eta\pm\kappa}{\sqrt{2\kappa}}\right)\right].\]
Parameters:

  • nondimensional_force (numpy.ndarray) – The nondimensional force \(\eta\equiv\beta f\ell_b\).

  • temperature (float) – The temperature \(T\).

Returns:

The nondimensional relative Gibbs free energy per link \(\Delta\varrho\equiv\beta\Delta\varphi/N_b\).

Return type:

numpy.ndarray

References