def pot_chem_dp_excess(self, c, ion_size, x_pair, a1_d=1, a2_d=1):
"""
Excess chemical potential of the dipole function
:param c: concentration of ions (both + and -) mols/litre (M)
:param ion_size: ionic size (in Angstrom)
:param x_pair: fraction of Bjerrum pairs
:param a1_d: dipole distance, in units of ion_size
:param a2_d: diameter of enclosing dipole, in units of ion_size
:return: excess chemical potential (ndarray)
"""
l_dh = dh.DebyeHuckel(self.bjerrum_object).debye_length(
(1 - x_pair) * c)
rho = un.mol_lit_2_mol_angstrom(c) * ion_size**3
rho1 = 0.5 * rho * (1 - x_pair)
rho2 = 0.5 * rho * x_pair
z = ion_size / l_dh
t_star = self.bjerrum_object.temp_star(ion_size)
mu_0 = -z**2 * a1_d**2 * self.omega(z * a2_d) / (t_star * a2_d)
val1 = -z**2 * a1_d**2 * rho2 * self.omega(
z * a2_d) / (t_star * a2_d**3)
val2 = self.der_x2omega(z * a2_d) * z * a2_d
mu_p = val1 * val2 * 0.5 * rho1
return np.array([mu_p, mu_p, mu_0])
def pressure_hc_excess(self, c, ion_size):
"""
Excess pressure for the hard core
:param c: concentration of the different species mols/litre (M)
:param ion_size: ionic size (in Angstrom)
:return: excess pressure (float)
"""
cc_molecular = un.mol_lit_2_mol_angstrom(c)
arg_vol = np.dot(self.b_fac, cc_molecular) * ion_size**3
pressure_hc_excess = np.sum(cc_molecular, axis=0) / (1 - arg_vol)
return pressure_hc_excess
def debye_length(self, c):
"""
Debye length according to :cite:`Levin1996`
.. math::
:label: debye_length
\\lambda_D = \\frac{1}{\\sqrt{4\\pi l_B(T)n_1}}
:param c: Concentration of ions (both + and -) mols/litre (M)
"""
c_m = un.mol_lit_2_mol_angstrom(c)
debye_length = 1.0 / np.sqrt(
4 * np.pi * c_m * self.bjerrum_object.bjerrum_length)
return debye_length
def free_energy_hc_excess(self, c, ion_size):
"""
Excess free energy of the hard core
:param c: concentration of the different species mols/litre (M)
:param ion_size: ionic size (in Angstrom)
:return: excess free energy (ndarray)
"""
cc_molecular = un.mol_lit_2_mol_angstrom(c)
arg_vol = np.dot(self.b_fac, cc_molecular) * ion_size**3
free_energy_hc_excess = np.sum(cc_molecular,
axis=0) * np.log(1 - arg_vol)
return free_energy_hc_excess
def pot_chem_hc_excess(self, c, ion_size):
"""
Excess chemical potential of the hard core
:param c: concentration of the different species mols/litre (M)
:param ion_size: ionic size (in Angstrom)
:return: excess chemical potential (float)
"""
cc_molecular = un.mol_lit_2_mol_angstrom(c)
arg_vol = np.dot(self.b_fac, cc_molecular) * ion_size**3
t1 = -np.log(1 - arg_vol)
t2 = ion_size**3 * np.sum(cc_molecular, axis=0) / (1 - arg_vol)
pot_chem_hc_excess = np.outer(self.b_fac, t2) + t1
return pot_chem_hc_excess
def free_energy_dp_excess(self, c, ion_size, x_pair, a1_d=1, a2_d=1):
"""
Excess free energy of the dipole function
:param c: concentration of ions (both + and -) mols/litre (M)
:param ion_size: ionic size (in Angstrom)
:param x_pair: fraction of Bjerrum pairs
:param a1_d: dipole distance, in units of ion_size
:param a2_d: diameter of enclosing dipole, in units of ion_size
:return: excess free energy in units of :math:`ion\\_size^3` (float)
"""
l_dh = dh.DebyeHuckel(self.bjerrum_object).debye_length(
(1 - x_pair) * c)
rho2 = un.mol_lit_2_mol_angstrom(c) * x_pair * 0.5 * ion_size**3
z = ion_size / l_dh
t_star = self.bjerrum_object.temp_star(ion_size)
f_dip = -z**2 * a1_d**2 * rho2 * self.omega(z * a2_d) / (t_star * a2_d)
return f_dip
def test_mol_lit_conversion(self):
# 1 Mol/litre = to Molecule /A^3
self.assertEqual(un.mol_lit_2_mol_angstrom(1),
un.avogadro() / (1e-3 * un.m_2_angstrom(1)**3))