The effective complex conductivity of real and imaginary part [14] of ac conductivity is presented as:

σ ∗ ( ω ) = σ ′ ( ω ) + j σ ″ ( ω ) (1)

The real part of the measured conductivity σ', at a given frequency range, is separable into dc and ac components and can be given by Jonscher’s power law [14]:

σ ′ ( ω ) = σ d c + A ω S (2)

where; σ_dc represents the direct current resulting from drift mobility of Na⁺ as the main charge carriers and in most cases is temperature-dependent. σ_ac = Aω^S is the “true a.c.” conductivity which may be obtained through subtraction of the measured dc conductivity (σ_dc) from the total conductivity ( σ ′ ( ω ) ) measured at the frequency ω. The pre-factor constant A can be assumed as

A = π N 2 e 2 / 6 K B T ( 2 α ) (3)

where; α is the polarizability of a pair of sites.

n is the number of sites per unit volume at which the hopping process takes place

sis the exponent of frequency and differs between 0.5 and 1 that characterize the interaction between charged species to be determined as the slope of curves at variable temperatures.

ω is the angular frequency of the applied electric field,

ω = 2 π f (4)

Figure 2 presented the frequency dependence of ac conductivity at different concentrations of CuO in the ternary CuO-Na₂O-B₂O₃ glass system. The sodium borate glasses containing different CuO concentrations have shown nearly like behavior with minor changes. Electrical conductivity was calculated as a function of frequency within the range (0.1 Hz - 20 MHz) at room temperature. It was observed that the values of ac conductivity increase with monotonic manner with increasing frequency especially in the high-frequency region. The rate of change is lower in the lower frequency region (≤2 KHz) which can be attributed to the effect of space charges [13] [14]. The effect of space charge was disappeared at higher frequencies, since the conductivity is increased with a higher rate than that detected in lower frequency region. It can also be shown from Figure 2 that increasing CuO leads to an obvious decrease in ac conductivity.

The decrease of conductivity with increasing CuO contents, Figure 3 leads one to suggest that the main portion from CuO can share in the constructing of the glass network and as a consequence CuO plays the role of a network forming

species (CuO₄). The formation of the latter structural units causes a reduction in the concentration of Na₂O which required for transformation of BO₃ to BO₄ groups. Then part of Na₂O can modify CuO to form CuO₄ tetrahedral units. Then more CuO₄ and BO₃ units are formed with more addition of CuO at the expense of tetrahedral BO₄ groups. As a result more bridging oxygen atoms are formed and consumed to bridge Cu and B forming Cu₄-O-B₃ species. Therefore, some of the well formed BO₄ units are replaced with CuO₄ ones.

The above arguments are in the same lines suggested upon analysis based on NMR spectroscopy of ¹¹B data (Figure 4). The total result extracted from Figure 3 is that the main role played by CuO is forming CuO₄ through presenting more bridging units in the glass structure and their abilities to shield and coordinate with BO₃ units. Therefore, increasing of CuO results in increasing of Cu-O-Cu and B₃-O-Cu bonds at the expense of BO₄ units. The glass structure is then

mainly consisting of Cu-O bond in CuO₄ groups. In such a case, the linkage between CuO₄ and BO₃ groups is being the most abundant within the glass network. The possibility for the cross linking of these CuO₄ units with BO₃ groups to form Cu-O-B bonds in the glass network obviously reduce the mobility and the number of the charge carriers which leads to decrease in the values of ac conductivity of the study samples with increasing CuO contents as presented by Figure 3. The conductivity is decreased by more than 3 orders of magnitudes upon increasing CuO from 0 to 22.5 mol%.

Change of dynamic ionic conductivity of Na₂O containing CuO borate glasses, xCuO-(25-x)Na₂O-75B₂O₃ versus frequency can be represented by three distinct regions in the conductivity spectrum see Figure 2. Lower frequency region in which the change in conductivity is relatively small with changing frequency (low frequency plateau). This limited change is clearly observed in glass free from CuO and in a glass contains 7.5 mol% CuO. The plateau length is decreased with increasing CuO content. On the other hand, glasses containing lower concentration of Na₂O (high CuO) did not show the low frequency plateau, where steeply increase in the conductivity is observed. In the second region, the conductivity increases with higher rate with increasing frequency (jump relaxation) [15] - [23]. Finally, at certain value of frequency (log f = 6.5), the conductivity tends to be fixed around a specific maximum single value for all composition (higher frequency limit). Comparing the experimental data of conductivity of sodium rich glass with the dispersion of the conductivity according to the jump relaxation model reflects a good resemblance [23] [24] [25] [26].

Actually, several models [24] [25] [26] have been proposed claiming that some glass systems contain small clusters or micro domains. These clusters are formed during the cooling process and are surrounded by a residual phase or connective tissue. In these clusters the ions can move more freely and more easily depending only on the effective frequency value which required for charge carriers to gain the highest mobility. The ion mobility is therefore higher than that of hopping process in connective tissue between the well formed clusters. Then the highest conductivity is found to depend only on the specific value of frequency frequency regardless changing composition.

Dielectric materials are characterized by their ability to store energy when they exposed to an external electric field. In all cases there is an induced polarization due the interaction between the external electric field and the field produced by the investigated sample. The behavior of dielectric materials generally described by the complex permittivity that formulated with two parts [23]:

ε ∗ = ε ′ − j ε ″ (6)

where ε' and ε'' are real and imaginary components of the complex permittivity respectively, and can be calculated using the following equations:

ε ′ = C d A ε 0 (7)

ε ″ = ε ′ tan δ (8)

where (C) is the sample capacitance measured in parallel mode in farads, d is the thickness of the glass samples in meters, A represents the cross-sectional area of the sample in square meters and ε₀ (ε₀ = 8.85 × 10⁻¹² F/m) is the permittivity of free space, ε' represents the amount of electrical energy stored in the material in response to an external electric field. The imaginary part of permittivity ε'' represents energy dissipated in the heat form. ε'' is usually much smaller than ε' (0 < ε'' < ε').

When the alternating voltage is applied to the dielectric materials, the material will absorb an electrical energy and dissipated in the form of heat which may be represented by the term (tan δ) which includes the influence of both dielectric loss and conductivity. Based on the values of ε' & ε'', the term tan δ gives the phase difference due to the loss of energy within the structure. This means that tan δ is a dissipation factor (tangent loss) which represents the electrical loss. Dissipation factor, tan δ is equal to the ratio between the electrical energy dissipated in heat to the total electrical energy stored per cycle in the dielectric materials, which is given by the following relation:

tan δ = ε ″ ε ′ (9)

Figure 5 and Figure 6 show dependence of the real (ε') and an imaginary (ε'') parts of the dielectric permittivity on the frequency respectively. All samples of CuO-Na₂O-B₂O₃ showed the same trend except the composition free from CuO.

Obviously, the dielectric constant and losses both are decreased with increasing the CuO concentrations (Figure 7 and Figure 8). The decreasing trends may be related to the hopping process of the charge carriers’ species. The change in CuO concentrations leads to increasing Cu-O-B₃ which makes the charge carrier unable to follow such change in the electric field. This leads to a decrease in the dielectric constant and losses (ε' & ε'').

The electric modulus approach was used to assist in understanding the response of bulk materials to the electrode polarization effect. The latter is negligible and enables the relaxation processes to be assessed at the lower frequency range of dielectric spectra [25] [26]. The complex electric modulus M* is related to the complex dielectric constant as described in the following expressions:

M ∗ = ( ε ∗ ) − 1 (10)

M ∗ = M ′ + i M ″ = ε ′ ε ′ 2 + ε ″ 2 + ε ″ ε ′ 2 + ε ″ 2 (11)

where M' and M'' are the real and imaginary parts of dielectric modulus respectively.

All investigated samples have showed the same behavior (Figure 9 and Figure 10) except the sample which free from CuO. It can be observed from Figure 9 that the values of M' at low frequencies are small (M' ≈ 0) for CuO = 0 which is attributed to the negligible contribution of electrode polarization. With increasing frequency, M' increases gradually and tends to approach maximum constant values M_∞ due to the relaxation process. This is attributed to the reduced space charge effects. The frequencies dispersion may be attributed to the conductivity relaxation processes. The graph also clearly shows that the M' increases with a rise of CuO contents which indicates that the conduction in the investigated samples can be attributed to the long-range mobility of charge carriers. It is observed that the normalized plot does not overlap on a single curve, which indicates that the conductivity relaxation depends on the glass composition.