FJC model thermodynamics (isotensional)

class FJC(number_of_links, link_length, hinge_mass)

The freely-jointed chain (FJC) 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.

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, given by Rubinstein and Colby[1] as

\[\xi(f, T) = -\frac{\partial\varphi}{\partial f} = N_b \ell_b \mathcal{L}(\beta f\ell_b),\]

where \(\mathcal{L}(x)=\coth(x)-1/x\) is the Langevin function.

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 \(\eta\equiv\beta f\ell_b\).

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, given by as

\[\gamma(\eta) = -\frac{\partial\varrho}{\partial\eta} = \mathcal{L}(\eta),\]

where \(\mathcal{L}(x)=\coth(x)-1/x\) is the Langevin function.

Parameters:

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

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,

\[\Delta\varphi(f, T) = N_bkT\ln\left[\frac{\beta f\ell_b}{\sinh(\beta f\ell_b)}\right].\]
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 \(N_b\varrho=\beta\varphi\).

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 \(\eta\equiv\beta f\ell_b\).

Returns:

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

Return type:

numpy.ndarray

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

\[\Delta\varrho(\eta) = \ln\left[\frac{\eta}{\sinh(\eta)}\right].\]
Parameters:

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

Returns:

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

Return type:

numpy.ndarray

References