To calculate the effective dielectric constant of a complex solute from computer simulation, it is necessary to map the properties of the solute observed in the simulation onto a simpler geometry, one amenable to analytical treatment. Specifically, the solute is approximated as a spherical cavity of volume V and dielectric constant ɛ embedded in a uniform dielectric continuum with dielectric constant ɛ_RF. The charge distribution of the solute is represented as a point charge and point dipole placed at the center of the spherical cavity. From the fluctuations of the solute dipole moment, M, observed during a computer simulation, the absolute temperature, T, the volume, V, of the solute cavity, and the external dielectric, ɛ_RF, it is then possible to determine the dielectric constant ɛ inside the solute cavity: (3) ɛ   =   1 + [ ( < M 2 >   −   < M > 2 ) / ( 3 ɛ 0 V k B T ) ] • 2 ɛ R F / ( 2 ɛ R F + 1 ) 1 − [ ( < M 2 >   −   < M > 2 ) / ( 3 ɛ 0 V k B T ) ] / ( 2 ɛ R F + 1 )where M is the dipole moment of the computational box, < > denotes Boltzmann ensemble averaging, ɛ₀ is the dielectric constant of vacuum, k_B is Boltzmann’s constant, and all of the above parameters are used SI units.

Figure 1 describes the dielectric constant ɛ as a function of the composition. The error bars were estimated using the blocking method, considering that the intermolecular potentials used here lack of any polarizability effects beyond the usual dipole moment enhancement and the fact that the SPC/E underestimates water’s dielectric constant by about 10%, the MD values of ɛ for the mixtures are small than those from Ref. [2] and [27].

Thermodynamic and statistical perturbation theory originally developed by Zwanzig [28] can be used to calculate relative and absolute free energies for molecularly detailed systems [29]. The change of the Gibbs free energy of a system with N particles at a given temperature, T, and pressure, P, is given by [30]: (4) Δ G   =   − k T ln   < e − Δ V / k T > 0Where ΔV is a perturbation of the Hamiltonian.

The composition dependence of the calculated values of ΔG is given in Figure 2. From this Figure, it can be seen that the free energy of activation increases with the concentration of the nonelectrolyte in water, reaching a maximum value of 3.5 at the concentration fraction of 0.35, whereafter it decreases. This difference can be attributed to the fact that the components are capable of specific interactions with one another.

Eyring et al. have applied the theory of rate processes to molecular dipole relaxation with the following equation [31]: (5) τ =   ( h / k B T ) exp ( Δ G / R T )where ΔG is the free energy change of activation for the dipole relaxation, k_B is Boltzmann’s constant, h, R are Planck’s constant and the molar gas constant, respectively.

The mixture’s overall dielectric relaxation time τ is in good agreement with the experimental, except for compositions between 0.3 and 0.42, where the simulations predict significantly higher relaxation times seen from Figure 3.

In Figure 3, the open symbols show mole fraction dependence of the relaxation time is calculated by Equation 5.

The complex permittivities are fitted by the nonlinear least-squares fit method to the Havriliak-Negami expression to obtain various dielectric parameters: (6) ɛ * ( ω )   =   ɛ ∞   +   ( ɛ 0   +   ɛ ∞ ) / [ 1 + ( j ω τ ) 1 − α ] βWhere ɛ* (ω) is the complex permittivity at an angular frequency ω, ɛ_∞ is the permittivity at high frequency, ɛ _∞ is the static dielectric constant, α is the shape parameter representing symmetrical distribution of relaxation time and β̣ is the shape parameter of an asymmetric relaxation curve. Equation 4 includes Cole-Cole (β̣ =1,0 < α <1) [32], Davidson-Cole (α̣ =0, 0 < β < 1) [33] and Debye (α=0, β̣=1) [34] relaxation models. In our system, it is observed that under experimental error the dielectric relaxation can be described by the Debye-like (β ≈ 1[35]). Therefore, Equation 6 can be rewritten to Equation 7: (7) ɛ * ( ω )   =   ɛ ∞ + ( ɛ 0   +   ɛ ∞ ) / ( 1   +   j ω τ )where the complex permittivity ɛ* (ω) is (8) ɛ * ( ω )   =   ɛ ′ ( ω )   −   i ɛ ″ ( ω )

For the Debye Equation, the real ɛ′ and imaginary ɛ″ contributions to the dielectric behavior can be expressed by: (9) ɛ ′ ( ω )   =   ɛ ∞   +   ( ɛ   −   ɛ ∞ ) / ( 1   +   ω 2 τ 2 ) ɛ ″ ( ω )   =   ( ɛ   −   ɛ ∞ ) ω τ / ( 1   +   ω 2 τ 2 ) ɛ ∞   ≈   1.1 n D 2

Permittivity and losses for DMSO-water mixtures at 298 K are shown in Figure 4, where a single relaxation peak is observed for the entire concentration range of all the DMSO-water mixtures in the range of 1~40 GHz. Figure 4(a) obviously indicates that the higher the frequency the lower of the real ɛ’. Nevertheless, the real ɛ’ decreases with the concentration of DMSO increasing, however, a completely opposite result were obtained when x_D is more than 0.35. It can be addressed that the real ɛ’ of the mixtures strongly depend on the DMSO concentration. It is expected that this phenomenon is due to the cooperative motion of DMSO-water molecules through hydrogen bonds.

From Figure 4(b), the imaginary ɛ″ of dielectric constant increases with the frequency till it reaches the summit, after which the imaginary part reduce gradually with the frequency. Generally, the primary process observed for various associated liquids and those water mixtures exhibit an asymmetric loss peak. In a theoretical study, it has been suggested that the motional units in the correlated domains cooperatively move and the heterogeneity of the size distribution of the domain results in the asymmetric shape of the loss peak. The concentration dependence of DMSO-water mixture solution had been calculated by Equation 9, the Cloe-Cloe diagrams can be drawn as shown in Figure 5. It can be concluded that the relation between ɛ″ and ɛ″ are circle, but those centre aren’t the cross axle ɛ″.