RAPIDLY CONVERGENT SERIES FOR THE COMPUTATION OF THE INTERACTION BETWEEN DISSIMILAR PLANE PARALLEL DOUBLE-LAYER (y0 > 0 > yd ≥ -y0)

2005 ◽  
Vol 12 (02) ◽  
pp. 145-153 ◽  
Author(s):  
SHIMIN ZHANG
Keyword(s):  
Power Series ◽  
Interaction Energy ◽  
Double Layer ◽  
Surface Potential ◽  
Convergent Series ◽  
Integration Constant ◽  
Double Layers ◽  
Absolute Value ◽  
Plane Parallel ◽  
Significant Figures

Several rapidly-convergent series for the computation of the interaction energy between dissimilar plane parallel double layers (y0 > 0 > yd ≥ -y0) are derived by expanding the interaction energy in the power series of ω0[ω0 < tanh (y0/64)]. The series terms required to obtain the interaction energy with six significant figures do not exceed 1 when the dimensionless surface potential of colloid particles y0 is less than or equal to 2.00000 × 10. The results of Devereux and de Bruyn are discovered to be incorrect when the integration constant C and absolute value of the surface potential is larger or the integration constant C is smaller.

APPROXIMATE EXPRESSIONS FOR INTERACTION OF TWO SIMILAR PLANE PARALLEL DOUBLE LAYERS AT ARBITRARY POTENTIAL

2004 ◽  
Vol 11 (03) ◽  
pp. 311-320 ◽  
Author(s):  
SHI-MIN ZHANG
Keyword(s):  
Power Series ◽  
Interaction Energy ◽  
Surface Potential ◽  
High Potential ◽  
Double Layers ◽  
Plane Parallel ◽  
Colloid Particles ◽  
Arbitrary Potential ◽  
Significant Figures ◽  
Approximate Expressions

Two approximate expressions used to compute the interaction energy between two similar plane parallel double layers are derived by expanding the interaction energy in the power series of [Formula: see text] at low dimensionless surface potential y0 or the power series of [Formula: see text] at high dimensionless surface potential y0, respectively. Two series converge very fast. In the range of dimensionless surface potential of colloid particles y0 from 1.00000×10-1 to 3.00000×10, when C<-4.30000, the high potential expression is usable; when C≥-4.30000, the low potential expression is usable. The required series terms will not exceed four to obtain the interaction energy with six significant figures if a low potential expression and a high potential expression are combined.

INTERACTION OF DISSIMILAR PLANE PARALLEL DOUBLE LAYER AT ARBITRARY POTENTIAL

2005 ◽  
Vol 12 (04) ◽  
pp. 523-537 ◽  
Author(s):  
SHIMIN ZHANG
Keyword(s):  
Interaction Energy ◽  
Double Layer ◽  
Surface Potential ◽  
Convergent Series ◽  
Elliptic Integrals ◽  
Parallel Plates ◽  
Arbitrary Potential ◽  
Significant Figures ◽  
Approximate Expressions ◽  
Very High

Several elliptic integrals related to the interaction energy between two dissimilar parallel plates (the potentials on the two plates are of the same sign) are expanded in several fast convergent series for lower and higher surface potentials, respectively. The number of series terms required to obtain the interaction energy with six significant figures is not more than four for the dimensionless surface potential from 0 to 20 if the series fit for the lower potential is combined with the series fit for the higher potential. The approximate expressions with different precisions can be obtained by retaining different series terms. The results of Devereux and de Bruyn are discovered to be incorrect when the surface potential is very low or very high.

INTERACTION OF TWO SIMILAR PLANE DOUBLE LAYERS FOR CaCl2-TYPE ASYMMETRIC ELECTROLYTES AT POSITIVE SURFACE POTENTIAL

2006 ◽  
Vol 13 (01) ◽  
pp. 17-26 ◽  
Author(s):  
SHIMIN ZHANG
Keyword(s):  
Power Series ◽  
Surface Potential ◽  
Numerical Results ◽  
Double Layers ◽  
Interaction Energies ◽  
Absolute Value ◽  
Integral Constant ◽  
Negative Surface ◽  
Approximate Expressions

The interaction energies between two similar plane double layers for CaCl 2-type asymmetric electrolytes at positive surface potential are expanded in the power series at lower potential and smaller absolute value of integral constant as well as higher potential and larger absolute value of integral constant, respectively. When dimensionless surface potential y0 ≤ 20, the number of the series terms required to obtain the interaction energies with six significant digits are not more than eight if higher potential expressions are combined with lower potential expressions. The accurate numerical results are given and they can be used to check up the validity of approximate expressions that will be obtained. The present results are also fit for Na 2 SO 4-type asymmetric electrolytes at negative surface potential.

A NOVEL METHOD OF CALCULATING THE INTERACTION ENERGIES OF DOUBLE LAYERS

2006 ◽  
Vol 13 (01) ◽  
pp. 127-133
Author(s):  
SHIMIN ZHANG
Keyword(s):  
Power Series ◽  
Surface Potential ◽  
Efficient Method ◽  
Double Layers ◽  
Parameter Method ◽  
Interaction Energies ◽  
Plane Parallel ◽  
Elliptical Integral ◽  
Novel Method

A novel, simple, and efficient method to calculate the interaction energies of double layers, λ parameter method, is presented by introducing a parameter λ in the elliptical integral. The interaction energies between two similar plane parallel double layers are expanded in the power series with such a method. The series converge very fast. When the dimensionless surface potential of double layers is less than or equal to 30, the number of the series terms required does not exceed 3 to obtain the interaction energies with six significant digits.

scholarly journals On the diffuse double layer

1933 ◽  
Vol 139 (839) ◽  
pp. 596-603 ◽  
Keyword(s):  
Double Layer ◽  
Colloidal Particles ◽  
Mutual Influence ◽  
Streaming Potential ◽  
Specific Adsorption ◽  
Double Layers ◽  
Diffuse Double Layer ◽  
Plane Parallel ◽  
Parallel Surfaces

1. The formulæ derived by Gouy for the diffuse double layer hold only in the case of a single surface in an infinite amount of medium. In practice they are not very suitable, for very frequently we have to deal either with capillaries, as in streaming potential measurements, or with colloidal particles which may be near enough to influence one another. In these cases it is too difficult to calculate the effect of the mutual influence of two double layers, though in less complicated systems it is possible. Suppose we have two plane parallel surfaces separated by a distance 2 h and charged both to the same potential, in a solution of an electrolyte. If the dimensions of the surfaces are large compared with the distance h there will be no drop of potential between the two. We therefore need to make the assumption (which was superfluous in Gouy’s case) of a specific adsorption by the surface of one or more of the ions in the solution, otherwise there will be no double layer at all; the same effect will also be obtained if ions of the surface dissolve.

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