Approximate analytic expressions for the electrostatic interaction energy between two colloidal particles based on the modified Poisson-Boltzmann equation

Abstract

Simple analytic expressions are derived for the electrostatic interaction energy between two charged colloidal particles in an electrolyte solution. The obtained expressions are based on an approximate form of the modified Poisson-Boltzmann equation taking into account the ion size effects through Carnahan-Starling activity coefficients of electrolyte ions. We derive the electrostatic interaction energy between two parallel plates on the basis of the linear superposition approximation. We further employ Derjaguin’s approximation to derive the corresponding expressions for the electrostatic interaction energy between two spheres, two parallel cylinders, or two crossed cylinders.

Appendix

Analytic expressions for the interaction between two parallel similar plates at separation h can be derived on the basis of the modified Poisson-Boltzmann equation (19). The interaction force P _pl(h) per unit area between the two plates is given by

$$ {P}_{\mathrm{pl}}(h)=4nkT\frac{1}{g\left({\phi}_{\mathrm{B}}\right)} \ln \left[1+g\left({\phi}_{\mathrm{B}}\right){ \sinh}^2\left(\frac{y_{\mathrm{m}}}{2}\right)\right] $$

(52)

with

$$ g\left({\phi}_{\mathrm{B}}\right)=\frac{16{\phi}_{\mathrm{B}}}{1+8{\phi}_{\mathrm{B}}} $$

(53)

where y _m is the scaled potential at the midpoint x = h/2 between the two plates. Eq. (52) can be derived from Eq. (35) by choosing h/2 as x′. Note that Eq. (52) can be applied for both of the constant surface potential and surface charge density models. For the constant surface potential model, y _m is related to the scaled surface potential y _o  = zeψ _o /kT, viz.,

$$ \kappa h=\sqrt{g\left({\phi}_{\mathrm{B}}\right)}{\mathit{\int}}_{\left|{y}_{\mathrm{m}}\right|}^{\left|{y}_{\mathrm{o}}\right|}\frac{dy}{\sqrt{ \ln \left[\frac{1+g\left({\phi}_{\mathrm{B}}\right){ \sinh}^2\left(y/2\right)}{1+g\left({\phi}_{\mathrm{B}}\right){ \sinh}^2\left({y}_{\mathrm{m}}/2\right)}\right]}} $$

(54)

which is derived by applying Eq. (20) for the system of two parallel plates. For the constant surface charge density model, on the other hand, we have

$$ \kappa h=\sqrt{g\left({\phi}_{\mathrm{B}}\right)}{\mathit{\int}}_{\left|{y}_{\mathrm{m}}\right|}^{\left|y(0)\right|}\frac{dy}{\sqrt{ \ln \left[\frac{1+g\left({\phi}_{\mathrm{B}}\right){ \sinh}^2\left(y/2\right)}{1+g\left({\phi}_{\mathrm{B}}\right){ \sinh}^2\left({y}_{\mathrm{m}}/2\right)}\right]}} $$

(55)

Note that the scaled surface potential y (0), which is a function of the plate separation h, is related to the scaled unperturbed surface potential y _o  = zeψ _o /kT in the absence of interaction (h = ∞) by

$$ {\sigma}^2={\left(\frac{2{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}\kappa kT}{ze}\right)}^2\frac{1}{g\left({\phi}_{\mathrm{B}}\right)} \ln \left[\frac{1+g\left({\phi}_{\mathrm{B}}\right){ \sinh}^2\left(y(0)/2\right)}{1+g\left({\phi}_{\mathrm{B}}\right){ \sinh}^2\left({y}_{\mathrm{m}}/2\right)}\right] $$

(56)

which can be derived by integrating Eq. (19) once and applying the boundary condition: dψ/dx| _x=0 ⁺ =  − σ/ε _r ε _o and dψ/dx| _x = h/2 = 0 (from symmetry of the system). The surface charge density σ of the plates is related to the scaled unperturbed surface potential y _o  = zeψ _o /kT (in the absence of interaction) by

$$ \sigma =\mathrm{s}\mathrm{g}\mathrm{n}\left({\psi}_{\mathrm{o}}\right)\frac{2{\varepsilon}_{\mathrm{r}}{\varepsilon}_{\mathrm{o}}\kappa kT}{ze}\sqrt{\frac{1}{g\left({\phi}_{\mathrm{B}}\right)}\cdot \ln \left[1+g\left({\phi}_{\mathrm{B}}\right){ \sinh}^2\left(\frac{y_{\mathrm{o}}}{2}\right)\right]} $$

(57)

By combining Eqs. (56) and (57), we obtain

$$ 1+g\left({\phi}_{\mathrm{B}}\right){ \sinh}^2\left({y}_{\mathrm{o}}/2\right)=\frac{1+g\left({\phi}_{\mathrm{B}}\right){ \sinh}^2\left(y(0)/2\right)}{1+g\left({\phi}_{\mathrm{B}}\right){ \sinh}^2\left({y}_{\mathrm{m}}/2\right)} $$

(58)

Equations (55) and (58) form coupled equations for y _m for the given values of y _o and κh.

Once the value of y _m is obtained from Eq. (54) for the constant potential model (in which y(0) is always equal to the unperturbed surface potential y _o) or the coupled Eqs. (55) and (58) for the constant surface charge density model, one can calculate the interaction force P _pl(h) via Eq. (52).