*4.2. Osmotic Pressure between Two Charged Surfaces within Modified Langevin Poisson–Boltzmann Model*

In the following, we shall derive, within the modified LPB theory, the expression for osmotic pressure between two charged planar surfaces (see Figure 7). First, we shall rearrange the modified LPB equation (Equation (2)) into a planar geometry in the form [38,43,125]:

$$-\frac{d}{d\mathbf{x}}\left[\varepsilon\_0 n^2 \frac{d\phi}{d\mathbf{x}}\right] - n\_{0w} p\_0 \left(\frac{2+n^2}{3}\right) \frac{d}{d\mathbf{x}} \mathcal{L}(\gamma p\_0 \mathcal{E}(\mathbf{x})\beta) + 2\varepsilon\_0 n\_0 \sin\mathbf{h}(e\_0 \phi \beta) = 0,\tag{6}$$

where we take into account Equation (1) for relative permittivity. Equation (6) is first multiplied by *φ* ′ = *dφ*/*dx* and then integrated to obtain [38,43]

$$\begin{aligned} \left[-\frac{1}{2}\varepsilon\_0 n^2 E(\mathbf{x})\right]^2 + 2n\_0 kT \cos\mathbf{h} (-e\_0 \boldsymbol{\phi} \boldsymbol{\mathcal{A}}) - n\_w p\_0 \left(\frac{2+n^2}{3}\right) E(\mathbf{x}) \mathbf{L} (\gamma p\_0 E(\mathbf{x}) \boldsymbol{\mathcal{A}}) + \\\ \left(\frac{2+n^2}{3}\right) \frac{n\_w}{\gamma \boldsymbol{\mathcal{A}}} \ln\left[\frac{\sin\mathbf{h} (\gamma p\_0 E(\mathbf{x}) \boldsymbol{\mathcal{A}})}{\gamma p\_0 E(\mathbf{x}) \boldsymbol{\mathcal{A}}}\right] = \mathbf{K}, \end{aligned} \tag{7}$$

where the constant *K* in Equation (7) is the local pressure between the charged surfaces. Equation (7) is equivalent to the contact theorem. In the second step, we subtract the bulk values (outside the space between the charged surfaces) from the local pressure between the charged surfaces to obtain the expression for the osmotic pressure difference <sup>Π</sup> = <sup>Π</sup>*inner* − <sup>Π</sup>*bulk* in the form [38,43]:

$$\begin{array}{c} \Pi = -\frac{1}{2}\varepsilon\_{0}n^{2}E(\mathbf{x})^{2} + 2n\_{0}kT(\cos\mathbf{h}(-e\_{0}\boldsymbol{\phi}(\mathbf{x})\boldsymbol{\beta}) - 1) -\\ n\_{w}p\_{0}\left(\frac{2+n^{2}}{3}\right)E(\mathbf{x})\mathcal{L}(\gamma p\_{0}E(\mathbf{x})\boldsymbol{\beta}) + \left(\frac{2+n^{2}}{3}\right)\frac{n\_{W}}{\gamma\beta}\ln\left[\frac{\sin\mathbf{h}(\gamma p\_{0}E(\mathbf{x})\boldsymbol{\beta})}{\gamma p\_{0}E(\mathbf{x})\boldsymbol{\beta}}\right]. \end{array} \tag{8}$$

− ௨

ଶ

=− <sup>ଵ</sup> ଶ 

ଶା<sup>మ</sup> ଷ

−௪ ቀ

By taking into account Equation (4), we can rewrite Equation (8) in the form: =− <sup>ଵ</sup> <sup>ଶ</sup>()<sup>ଶ</sup> + (ା() + ି() − 2 ) − ௪ ቀ ଶା<sup>మ</sup> ቁ ()L(()) +

ቁ ()L(()) + ቀଶା<sup>మ</sup>

$$\begin{array}{c} \Pi = -\frac{1}{2}\varepsilon\_{0}n^{2}E(\mathbf{x})^{2} + kT(n\_{+}(\mathbf{x}) + n\_{-}(\mathbf{x}) - 2n\_{0}) - \\ n\_{w}p\_{0}\left(\frac{2+n^{2}}{3}\right)E(\mathbf{x})\mathbf{L}(\gamma p\_{0}E(\mathbf{x})\boldsymbol{\beta}) + \left(\frac{2+n^{2}}{3}\right)\frac{n\_{W}}{\gamma\boldsymbol{\beta}}\ln\left[\frac{\sinh\left(\gamma p\_{0}E(\mathbf{x})\boldsymbol{\beta}\right)}{\gamma p\_{0}E(\mathbf{x})\boldsymbol{\beta}}\right]. \end{array} \tag{9}$$

<sup>ଶ</sup>()<sup>ଶ</sup> + 2(cosh(−()) − 1) −

ଷ ቁ ೢ

ఊఉ ln ቂୱ୧୬୦(ఊబா(௫)ఉ) ఊబா(௫)ఉ <sup>ቃ</sup>

The osmotic pressure is constant everywhere in the solution between the charged plates (Figure 7). <sup>ଵ</sup> = <sup>ଶ</sup>

If both surfaces have equal surface charge density (*σ*<sup>1</sup> = *σ*2), the electric field strength in the middle (*x* = *H*/2 in Figure 7) is zero; therefore, Equation (8) simplifies to the form [38]: = /2

$$\Pi = 2\eta\_0 kT(\cos \mathbf{h}(-e\_0 \phi(\mathbf{x} = H/2)\beta) - 1). \tag{10}$$

=

<sup>ଵ</sup> < 0 <sup>ଶ</sup> > 0 **Figure 7.** Schematic figure of an electrolyte solution between two charged surfaces at the distance *H*, where the surface charge densities *σ*<sup>1</sup> < 0 and *σ*<sup>2</sup> > 0.

() For small values of *γp*0*E*(*x*)*β* everywhere in the solution between the two charged surfaces, we expand the third and fourth term in the above Equation (9) into series to obtain:

$$\begin{split} \Pi \approx & -\frac{1}{2}\varepsilon\_0 \left( n^2 + \left( \frac{2 + n^2}{3} \right)^2 \frac{n\_w p\_0^2 \beta}{2\varepsilon\_0} \right) E(\mathbf{x})^2 + kT(n\_+(\mathbf{x}) + n\_-(\mathbf{x}) - 2n\_0) \\ &= -\frac{1}{2}\varepsilon\_0 \varepsilon\_{r,b} E(\mathbf{x})^2 + kT(n\_+(\mathbf{x}) + n\_-(\mathbf{x}) - 2n\_0) \\ &= -\frac{1}{2}\varepsilon\_0 \varepsilon\_{r,b} E(\mathbf{x})^2 + 2n\_0 kT(\cos \mathbf{h}(-e\_0 \phi \beta) - 1) \end{split} \tag{11}$$

= − 2 ,()<sup>ଶ</sup> + (ା() + ି() − 2 ) = = − ଵ ଶ ,()<sup>ଶ</sup> + 2(cosh(−) − 1) , where *εr*,*<sup>b</sup>* is the Onsager expression for relative permittivity, defined by Equation (5). As, in thermodynamic equilibrium, the osmotic pressure is equal everywhere between the two charged surfaces (Figure 7), we can calculate the value of the magnitude of the electric field strength in Equation (10) also at the right charged surface (Figure 7) from the corresponding boundary condition, so Equation (11) then reads

$$\Pi \approx -\frac{\sigma\_2^2}{2\varepsilon\_0 \varepsilon\_{r,b}} + 2n\_0 kT(\cos \mathbf{h}(-e\_0 \phi(\mathbf{x} = H)\beta) - 1),\tag{12}$$

where *H* is the distance between the two charged surfaces.

Figure 8 presents the osmotic pressure between negatively and positively charged flat surfaces as a function of the decreasing distance (*H*) between them, calculated within the modified LPB model.

≈− <sup>ఙ</sup><sup>మ</sup>

మ ଶఌబఌ,್

<sup>ଵ</sup> = <sup>ଶ</sup> = −<sup>ଵ</sup> = ௪/ = = **Figure 8.** The calculated osmotic pressure between negatively and positively charged flat surfaces as a function of the distance between the two surfaces (H) (see Figure 7), calculated within the modified LPB model (Equations (1) and (2)) for two values of the bulk salt concentration. Other model parameters are: *σ*<sup>1</sup> = 0.2 As/m<sup>2</sup> , *σ*<sup>2</sup> = −*σ*<sup>1</sup> , *T* = 298 K, concentration of water *nw*/*N<sup>A</sup>* = 55 mol/L, and dipole moment of water *p*<sup>0</sup> = 3.1 Debye, where *N<sup>A</sup>* is the Avogadro number.

+ 2(cosh(−( = )) − 1)
