Cole–Cole equation

The Cole–Cole equation is a relaxation model that is often used to describe dielectric relaxation in polymers.

It is given by the equation

where ε ∗ is the complex dielectric constant, ε s and ε ∞ are the "static" and "infinite frequency" dielectric constants, ω is the angular frequency and τ is a time constant.

The exponent parameter α , which takes a value between 0 and 1, allows to describe different spectral shapes. When α = 0 , the Cole-Cole model reduces to the Debye model. When α > 0 , the relaxation is stretched, i.e. it extends over a wider range on a logarithmic ω scale than Debye relaxation.

The separation of the complex dielectric constant ε (ω) was reported in the original paper by Cole and Cole as follows:

ε ′ = ε ∞ + ( ε s − ε ∞ ) 1 + ( ω τ ) 1 − α s i n α π / 2 1 + 2 ( ω τ ) 1 − α s i n α π / 2 + ( ω τ ) 2 ( 1 − α )

ε ″ = ( ε s − ε ∞ ) ( ω τ ) 1 − α c o s α π / 2 1 + 2 ( ω τ ) 1 − α s i n α π / 2 + ( ω τ ) 2 ( 1 − α )

Upon introduction of hyperbolic functions, the above expressions reduce to:

ε ′ − ε ∞ = 1 2 ( ε 0 − ε ∞ ) [ 1 − s i n h ( 1 − α ) x c o s h ( 1 − α ) x + c o s α π / 2 ]

ε ″ = 1 2 ( ε 0 − ε ∞ ) c o s α π / 2 c o s h ( 1 − α ) x + s i n α π / 2

Here x = l n ( ω τ ) .

These equations reduce to the Debye expression when α = 0 .

Cole-Cole relaxation constitutes a special case of Havriliak-Negami relaxation when the symmetry parameter (β) is equal to 1 - that is, when the relaxation peaks are symmetric. Another special case of Havriliak-Negami relaxation (β<1, α=1) is known as Cole-Davidson relaxation, for an abridged and updated review of anomalous dielectric relaxation in dissored systems see Kalmykov.