*4.3. Osmotic Pressure between Dipolar Zwitterionic Lipid Bilayer and Charged Rigid Surface*

 <sup>ଵ</sup> = 0 <sup>ଶ</sup> = In the model, the zwitterionic dipolar lipid headgroup is composed of the lipids with a positively charged trimethylammonium group and a negatively charged carboxyl group, theoretically described by two charges at fixed distance, *D* (see Figure 9) [38,43,128]. The negative charges of the phosphate groups of dipolar (zwitterionic) lipids are described by negative surface charge density, *σ*<sup>1</sup> at *x* = 0, while the opposite charged surface with surface charge density *σ*<sup>2</sup> is located at *x* = *H*. The corresponding Poisson equation in a planar geometry then reads [38,43,128]:

$$\frac{d}{d\mathbf{x}}\left[\varepsilon\_0 \varepsilon\_r(\mathbf{x})\frac{d\boldsymbol{\phi}}{d\mathbf{x}}\right] = 2\varepsilon\_0 n\_0 \sin\mathbf{h} (e\_0 \boldsymbol{\phi}(\mathbf{x})\boldsymbol{\beta}) - \rho\_{\text{ZW}}(\mathbf{x})\_\prime \tag{13}$$

ௐ() = |ఙభ<sup>|</sup> (௫) where *ρZW*(*x*) is the volume charge density due to the positively charged trimethylammonium group (Figure 9):

$$\rho\_{ZW}(\mathbf{x}) = \frac{|\sigma\_1| \mathbf{P}(\mathbf{x})}{D} \text{ and } \rho\_{ZW}(\mathbf{x} > D) = 0,\tag{14}$$

P( ) = Λ ఈ ୣ୶୮(ିబథ(௫)ఉ) ఈ ୣ୶୮(ିబథ(௫)ఉ)ାଵ 0<≤ and P(*x*) the probability density function [38,43,128]:

ௐ()

P( )

.

$$P(\mathbf{x}) = \Lambda \frac{\mathfrak{a} \exp(-e\_0 \phi(\mathbf{x}) \beta)}{\mathfrak{a} \exp(-e\_0 \phi(\mathbf{x}) \beta) + 1}, \ 0 < \mathbf{x} \le D \tag{15}$$

The normalization constant is determined from the condition:

$$\frac{1}{D} \int\_{0}^{D} \mathbb{P}(\mathbf{x}) d\mathbf{x} = 1. \tag{16}$$

P(*x*) describes the probability that the positive charge of a dipolar lipid headgroup is located at the distance *x* from the negatively charged surface at *x* = 0. The parameter *α* is equal to the ratio between the average volume of the positively charged parts of dipolar P()

(zwitterionic) headgroups and the average volume of the salt solution in the headgroup region, meaning that the finite size of the positively charged part of the zwitterionic lipid headgroup is taken into account. The corresponding boundary conditions at *x* = 0 and *x* = *H* should be taken into account [38]. The predictions of the model agree well with the results of MD simulations, as shown in [38,129]. =

1  න P( 

=0

) = 1.

=0

<sup>ଶ</sup> <

()

To calculate the osmotic pressure between the zwitterionic headgroup region and positively charged surface (Figure 9), we can use Equation (8) with the input *φ*(*x*) and *E*(*x*) determined from Equations (13)–(16) at appropriate boundary conditions, where the values *φ*(*x*) and *E*(*x*) inserted in Equation (8) can be calculated for any *D* ≤ *x* ≤ *H* because, in thermodynamic equilibrium, the osmotic pressure is equal everywhere between the two charged surfaces. However, as we are using expression Equation (8), which neglects *ρZW*(*x*) (Equation (14)), we can calculate the osmotic pressure between the dipolar zwitterionic lipid bilayer and charged rigid surface (Figure 10) by using Equation (8) only in the region *D* ≤ *x* ≤ *H*, where *ρZW*(*x*) is different from zero. () () () ≤≤ ௐ() ≤≤ ௐ()

<sup>ଵ</sup> =0 ௐ() 0<≤ **Figure 9.** Schematic figure of the headgroup region composed of zwitterionic lipids with a positively charged trimethylammonium group and a negatively charged carboxyl group. The negative charges of the phosphate groups of the dipolar (zwitterionic) lipids are described by a negative surface charge density *σ*<sup>1</sup> at *x* = 0, while the electric charge due to the positively charged trimethylammonium group is described by the spatially dependent volume charge density *ρZW*(*x*), defined in the region <sup>0</sup> <sup>&</sup>lt; *<sup>x</sup>* <sup>≤</sup> *<sup>D</sup>* (see Equations (14)–(16)). An example of a zwitterionic lipid is SOPC.

0 ⟨⟩ When the zwitterionic lipid layer approaches the negatively charged surface (*σ*<sup>2</sup> < 0), the average orientation of the lipid headgroup orientation angle (h*ω*i) decreases with decreasing *H* due to the electrostatic attraction between the positively charged parts of the lipid headgroups and the negatively charged surface, as schematically shown in Figure 11, based on the results presented in [38,43]. Accordingly, the osmotic pressure between the headgroups and the negatively charged surface decreases with decreasing *H*, as calculated in [38,43].

<sup>ଵ</sup> = <sup>ଶ</sup> = = / = ௪/ = **Figure 10.** Calculated osmotic pressure between the dipolar headgroups and planar negatively charged surface as a function of the distance between the plane of the lipid phosphate groups and the charged surface (H) for alpha = 5. The values of model parameters are: T = 298 K, *σ*<sup>1</sup> = –0.30 As/m<sup>2</sup> , *σ*<sup>2</sup> = 0.30 As/m<sup>2</sup> , dipole moment of water *p*<sup>0</sup> = 3.1 Debye, bulk concentration of salt *n*0/*N<sup>A</sup>* = 0.01 mol/L, and concentration of water *nw*/*N<sup>A</sup>* = 55 mol/L. Reprinted from [38] with permission from Elsevier. <sup>ଵ</sup> = <sup>ଶ</sup> = = / = ௪/ =

**Figure 11.** Schematic of the average orientation of the zwitterionic head-group at two different distances from the negatively charged surface. The figure, based on the results of theoretical modeling and MD simulations [38,128], shows that at smaller distances from the charged surface, the average orientation of the zwitterionic head-groups is more perpendicular to the charged surface.
