XDLVO theory

According to the XDLVO theory, the foulant-membrane interaction energy can be calculated as the sum of the LW, electrostatic double layer (EL), and Lewis acid–base interactions (AB)²⁶:

The LW, AB, and EL interaction energy components between a spherical foulant and an infinite planar surface can be presented as follows^27,28:

where f, w, and m indicate the foulant, water, and membrane, respectively; a denotes the foulant radius; h represents the separation distance between the membrane and foulant;h0 represents the minimum equilibrium separation distance (usually assigned a value of 0.158 ± 0.009 nm); ε_rε₀ represents the dielectric permittivity of the suspending fluid (F/m)²²; κ is the inverse Debye screening length; and λ (≅ 0.6 nm) is the characteristic decay length of the LW interaction. ζ_f and ζ_m are the surface potentials of the foulant and membrane, respectively.

When the separation distance between the two surfaces is close to the minimum equilibrium separation distance, the AB, LW and EL adhesion free energies per unit area can be obtained by the following²⁹

In addition, the surface tension parameters (γs-, γs+, and γsLW) of the membrane and foulants were expressed as follows^29–32:

where ΔGh0LW, ΔGh0AB, andΔGh0EL are the Lifshitz-van der Waals, acid–base, and electrostatic double-layer free energy components at a separation distance of h₀; the subscript (s) represents the solid surface of the membrane or foulant; (l) represents the liquid used in each determination; θ refers to the contact angle; γTOT, γLW, γ_, and γ+ denote the total surface tension, the LW component, the electron-donor parameter, and the electron-acceptor parameter, respectively.