**3.1. Dielectric relaxation.**

The earliest model of relaxation behavior is originally derived from Debye relaxation model [12] In this model, real and imaginary part of dielectric constant can be represented as in Figure 5

**Figure 5.** Debye dielectric dispersion curve.

where

10 Dielectric Material

following Figure 4

environment of polarisable groups.

**3.1. Dielectric relaxation.** 

Figure 5

**3. Structure-properties relationship.** 

**Figure 5.** Debye dielectric dispersion curve.

The dielectric loss will show maxima at respective relaxation mechanisms as the temperature is increased. The loss in dielectric can be schematically represented as in the

The γ relaxation occur at lower temperature as it involved small entities of phenyl rings and C-H units whose motion are readily perturbed at low thermal energy. This is followed by β relaxation and finally α relaxation corresponding to the longer scale segmental motion. The broadness for each peaks signify dispersion in relaxation time as the result of different local

The earliest model of relaxation behavior is originally derived from Debye relaxation model [12] In this model, real and imaginary part of dielectric constant can be represented as in

**Figure 4.** Schematic dielectric loss curve for polymer as temperature is increased.

$$
\varepsilon' = \mathcal{E}\_{\phi} + \frac{\varepsilon\_{\phi - \varepsilon\_{\varepsilon}}}{1 + o^2 \tau^2}
$$

$$
\varepsilon'' = \frac{\varepsilon\_{\phi - \varepsilon\_{\varepsilon}}}{1 + o^2 \tau^2} o \sigma
$$

This model relates the dielectric properties with the relaxation time. The relationship between ε'and ε'' can be formulated by eliminating the parameter ωτ to give Equation (6):

$$
\varepsilon \left( \varepsilon^{\cdot} - \frac{\varepsilon\_s + \varepsilon\_\alpha}{2} \right)^2 + \varepsilon^{\cdot^\*2} = \left( \frac{\varepsilon\_s - \varepsilon\_\alpha}{2} \right)^2 \tag{6}
$$

This is a form of a semispherical plot which is popularly known as Cole-Cole plot. See Figure 6.

**Figure 6.** Cole-Cole Plot showing the relationship between dielectric constant and dielectric loss.

The plot shows that at dielectric constant of infinite frequency, ε∞ and static dielectric constant, εs there will be no loss. Maximum loss occur at the midpoint between the two dielectric values. The larger the different between the static and infinite dielectric constant, the higher will be the loss. This model fit very well with polar small molecular liquids. However, polymeric materials are bigger in size, higher viscosity with entanglement between chains. This contribute to visco-elastic properties which requires some modifications to the original model. It can be noted that the above relationship involved only one specific relaxation time. This is contrary in polymeric system whose relaxation time is dependent on mobility of dipoles which behave differently in varying local environments. This result in distribution in relaxtion time. Modification include Cole and Cole semiemperical equation [13] Davidson and Cole [14] Williams and Watt [15] and Navriliak and Nagami [16]. The last modification lead to the new equation (7):

$$\mathcal{E}' = \mathcal{E}\_{\boldsymbol{\phi}} + \frac{\mathfrak{E}\_{\boldsymbol{\alpha}-\mathfrak{E}\_{\boldsymbol{w}}}}{(1 + \left(o^{2}\,\,\tau^{2}\right)^{\alpha})\,^{\beta}} \tag{7}$$

#### 12 Dielectric Material

where α and β is in the range 0 and 1. No physical meaning as yet is assignable to these parameters.[17] This modification result in a broader peak and smaller loss value with asymmetrical in features. The behaviour of dielectric constant and loss at variable frequencies and temperatures is exemplified in the following Figure 7 for polyvinylchloride.

Polymeric Dielectric Materials 13

1 MHz 3 GHz

1 MHz 3 GHz

100 MHz 3 GHz

100 Hz 3 GHz

Material Dielectric constant (ε') Loss tangent (tan δ) Frequency (Hz)

0.001 0.0009

0.038 0.34

0.00033

0.00028

Polymers are often cross-linked to improve their properties. The cross linking or curing process can be conveniently monitored based on relaxation time changes with the progress of reaction. This is exemplified during curing of diglycidylether bisphenol A (DGEBA) with diethyltetraamine (DETA)[19]. During the cross-linking process, the chains are covalently

This rigidity is proportional to cross-link density henceforth affecting the change in

Uncross-link chains Cross-link network

ABS (plastic) 2.0 – 3.5 0.005 – 0.0190

HDPE 1.0 – 5.0 0.00004 – 0.001

Polycarbonate 2.8 – 3.4 0.00066 – 0.01

Polystyrene 2.5 – 2.6 0.0001

Teflon (PTFE) 2.0 – 2.1 0.0005

**3.2. Effect of cross-link between chains** 

bonded to each other which induce chain rigidity.

**Scheme 1.** Affect of crosslink network on rigidity of polymer chains

relaxation time. This can be illustrated in the following Figure 8:

**Table 2.** Dielectric parameters for some polymers at various frequencies.

2.35

3.9 2.9

4.0

Butyl rubber 2.35

Gutta percha 2.6

Neoprene rubber 6.26

Nylon 3.2 - 5 Polyamide 2.5 – 2.6

Polypropylene 2.2

PVC 3 Silicone (RTV) 3.6

Kapton (Type 100) (Type 200)

**Figure 7.** Plot of dielectric constant (a) and dielectric loss with the change in frequency and temperature for polyvinylchloride. (From Ref 18)

Figure 7a shows the variation of ε' and ε" at the region of glass transition (85 0C) of polyvinylchloride. At the onset of glass transition the PVC showed a relatively low dielectric constant of 4.1 to 3.2 within the measured frequency range. With the increased in temperature, chain mobility begin to increase thus reducing the relaxation time. The dipole polarization of the polymer chain is better able to align in phase with the changing frequency and this account for the increase in dielectric constant as the temperature is increased. However this alignment with the applied oscillating field gradually failed as the frequency is increased. The optimum rate of decreased of dielectric constant occur at higher frequency as the temperature is increased. This correspond to the maximum dielectric loss in Fig 7b. Based on Cole-Cole plot, when there is a big difference in static and infinite dielectric constant, as under high thermal treatment, then the dielectric loss will be correspondingly large. It can be noted that at temperature 128 oC, there is a large dielectric loss occurring at higher frequency compared to those of lower temperature. Glass transition of polymer is a vital consideration that need to take into account during use of polymers as this affect the dielectric properties substantially. Substitution of fluorine into polyimide, for example, only affect the electronic polarization since PI is mostly used at temperature lower then its Tg (<260 oC). At this temperature, no effective polar orientation occurr which reduce any possibility of intrusion effect from this mechanism into the dielectric properties. The following Table 2 present the dielectric constant and loss of commercially used polymers.


**Table 2.** Dielectric parameters for some polymers at various frequencies.
