*4.1. Modified Langevin Poisson–Boltzmann model*

In the following, we shall describe the theoretical consideration of the electrostatic interaction (adhesion) between lipid head groups of the proximal leaflet of the lipid bilayer and charged solid surface where the orientational degree of freedom of lipid headgroups is taken into account. Among others, we shall derive, within the modified Langevin Poisson– Boltzmann [38,125,126] model, an analytical expression for the osmotic pressure between two charged surfaces, which can then also be used for the calculation of osmotic pressure between the planar lipid bilayer and charged planar surface.

We shall start with a short description of the modified Langevin Poisson–Boltzmann (LPB) model of the electric double layer [38,125,126], which presents the generalization of classic Poisson–Boltzmann (PB) theory for point-like ions by taking into account the orientational ordering of water molecules in an EDL. In the modified LPB model, the orientational ordering of water dipoles is also considered close to the saturation regime or in the saturation regime, which leads to the prediction that the relative permittivity close to the charged surface is considerably reduced. The modified LPB model also accounts for the electronic polarization of the water [38,126]. The space dependency of the relative permittivity within the modified LPB model is [38,43,126]:

$$
\varepsilon\_r(r) = n^2 + \frac{n\_w p\_0}{\varepsilon\_0} \left(\frac{2 + n^2}{3}\right) \left(\frac{\mathcal{L}(\gamma p\_0 E(r) \mathcal{\beta})}{E(r)}\right), \tag{1}
$$

where *n* is the refractive index of water, *n<sup>w</sup>* is the bulk number density of water, *p*<sup>0</sup> is the magnitude of the dipole moment of a water molecule, L(*u*) = cot h(*u*) − 1/*u* is the Langevin function, *γ* = 2 + *n* 2 /2, *E*(*r*) is the magnitude (absolute value) of the electric field strength, *β* = 1/*kT*, and *kT* is the thermal energy. The above expression for the space dependency of the relative permittivity (Equation (1)) then appears in the modified LPB equation for electric potential *φ* [38,43,126]:

$$\nabla \cdot \left[ \varepsilon\_0 \varepsilon\_r(r) \nabla \right] = 2e\_0 n\_0 \sin \mathbf{h} (e\_0 \phi(r) \beta) \,, \tag{2}$$

where we take into account the macroscopic (net) volume charge density of the electrolyte solution written in the form:

$$
\rho(r) = e\_0 \ n\_+(r) - e\_0 \ n\_-(r) = -2e\_0 n\_0 \sinh(e\_0 \phi(r)\beta) \tag{3}
$$

and Boltzmann distribution functions for the number densities of monovalent cations and anions:

$$n\_+(r) = n\_0 \exp(-e\_0 \phi(r)\beta), \ n\_-(r) = n\_0(e\_0 \phi(r)\beta) \tag{4}$$

where *n*<sup>0</sup> is the bulk number density of ions. In the limit of vanishing electric field strength, the above expression for the relative permittivity (Equation (1)) yields the Onsager limit expression for bulk relative permittivity [38,43,82]:

$$
\varepsilon\_{r,b} = n^2 + \left(\frac{2+n^2}{3}\right)^2 \frac{n\_w p\_0^2 \beta}{2\varepsilon\_0} \tag{5}
$$

At room temperature *<sup>T</sup>* <sup>=</sup> <sup>298</sup> K, *<sup>p</sup>*<sup>0</sup> <sup>=</sup> 3.1 Debye (the Debye is 3.336 <sup>×</sup> <sup>10</sup>−<sup>30</sup> C/m), and *nw*/*N<sup>A</sup>* = 55 mol/L, Equation (5) gives *εr*,*<sup>b</sup>* = 78.5 for bulk solution. The value *p*<sup>0</sup> = 3.1 Debye is smaller than the corresponding value in previous similar models of electric double layers, also considering orientational ordering of the water dipole (*p*<sup>0</sup> = 4.86 Debye) (see, for example, [86,127]), which did not take into account the cavity field and electronic polarizability of water molecules. In addition, the model [127] predicts the increase in the relative permittivity in the direction toward the charged surface contrary to the prediction of the modified LPB model, which predicts the decrease in relative permittivity in the electrolyte solution near the charged surface [38,43,82] in agreement with experimental results. The predicted substantial increase in relative permittivity near the charged surface in [127], therefore, opposes the experimental results and defies the common principles in physics [87,123,125,126].
