A rounded stretched exponential function is introduced, C(t)=exp{(τ_0 /τ_E)~β[1-(1+(t /τ_0)~2)~(β/2)]},where t is time, and τ_0 and τ_E are two relaxation times. This expression can be used to represent therelaxation function of many real dynamical processes, as at long times, tτ_0, the functionconverges to a stretched exponential with normalizing relaxation time, τ_E, yet its expansion is evenor symmetric in time, which is a statistical mechanical requirement. This expression fits well theshear stress relaxation function for model soft soft-sphere fluids near coexistence, with τ_Eτ_O. Thefunction gives the correct limits at low and high frequency in Cole–Cole plots for dielectric andshear stress relaxation (both the modulus and viscosity forms). It is shown that both the dielectricspectra and dynamic shear modulus imaginary parts approach the real axis with a slope equal to 0at high frequency, whereas the dynamic viscosity has an infinite slope in the same limit. Thisindicates that inertial effects at high frequency are best discerned in the modulus rather than theviscosity Cole–Cole plot. As a consequence of the even expansion in time of the shear stressrelaxation function, the value of the storage modulus derived from it at very high frequency exceedsthat in the infinite frequency limit (i.e., G_∞).
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