Electrical potentials of two identical particles with fixed surface charge density in a salt-free medium

Analytical solutions of nonlinear Poisson–Boltzmann equation for colloidal particles immersed in a general electrolyte solution by homotopy perturbation technique

“…(2) The valid application ranges of the present solutions are complementary with those of previous expressions available in literatures [32,[36][37][38][39][40][41], and accordingly, global valid expressions for I/y 0 relationship and electrical potential distribution will certainly result from a combination of these complementary expressions. Using appropriate mathematical methods and techniques, the integrated expressions will be with desirable continuous and smooth features; therefore, one does not worry about possible undesirable discontinuity and non-smoothness originating from the unlike mathematical structures and error sizes of the constituent expressions involved.…”

“…As a result, we present analytic expressions for the electrical potential distribution and the surface charge density/surface potential relationship (I/y 0 ) for spherical colloidal particle, and electrical potential distribution expression for planar colloidal particle. Two striking features of our analytical solutions are summarized as follows: (1) they readily apply irrespective of the types of electrolyte considered or whether single electrolyte or mixed electrolytes are being considered, and this contrasts sharply with previous analytical solutions [32,[36][37][38][39][40][41], most of which only applies for particular electrolyte types or applies only for saltfree solution.…”

“…Moreover, many of these investigations are only concerned with specific electrolyte type [32,[39][40][41], for example, symmetric electrolytes, single electrolyte type, or salt-free water medium. Aim of this study is to overcome the limitations of existing approximate solutions of the nonlinear PB equation by deriving out analytical σ/y 0 expression, which applies for general electrolyte solutions.…”

Approximate Analytic Expression of Surface Charge Density/Surface Potential Relationship for a Spherical Colloidal Particle Immersed in a General Electrolyte Solution

“…On the other hand, we propose to approximate the highly nonlinear term of the PBE by a polynomial of proper order, and this makes analytically possible application of the variational iteration method to any complex differential equations; specific to the PBE of the present interest, advantage of the polynomial approximation of the nonlinear term resides in its universality, and this makes possible dealing with all kinds of electrolyte types in a unified way, and consequently surpasses well beyond the previously proposed approximate solutions of the PBE. [15,[20][21][22][23][24] Finally, the valid application range of the variational iteration solution changes as different electrolyte types are considered. Considering that the solution property plays its role only through changing the non-linear term Q(y) of the PBE, the sensitivity of the valid application range on the electrolyte type considered indicates that the effectiveness of the variational iteration method crucially depends on the mathematical structure of the non-linear term, and this is well understood as the linear term of the target equation is taken into account by the use of the linear solution as the initial iteration solution and the non-linear term effect is only considered by subsequent iterations.…”

“…In the last 100 years, although a great deal of research had been done in analytically and approximately solving the PBE, the issue is not yet dealt with satisfactorily, and the acquired approximate solutions are exclusively valid only for high value of reduced colloid radius ja (Debye-Hü ckel screening parameter j being defined below, and a representing radius of the spherical particle), a relevant parameter of the PBE. [15][16][17][18][19] Moreover, many of these investigations are only concerned with specific electrolyte types, [15,[20][21][22][23][24] for example, symmetric electrolyte, single electrolyte, or salt-free water medium. It is noted that the non-linear term of the PBE is determined by the electrolyte types considered.…”