**1.1. Interaction of microwaves with ceramic materials: theoretical aspects**

Microwaves are electromagnetic waves with the electromagnetic radiations in the frequency band of 300 MHz to 300 GHz, and their corresponding wavelength between 1 m and 1 mm can be used successfully to heat many ceramic materials. Since most of the microwave band is used for communication purpose, the Federal Communications Commission has allocated only very few specific frequencies for industrial, scientific, and medical applications. A major portion of these microwave used in the communication sector and only certain frequencies, viz., 0.915, 2.45, 5.85, and 21.2 GHz, are chosen for medical and industrial applications. Among these allowed frequencies, 2.45 GHz is the most common microwave frequency used for industrial and scientific applications. The interaction and heating generation of ceramics under microwave field depends on the dielectric, magnetic, and conductive loss of the material and temperature dependent parameters.

The ability of a material to be heated in microwave field depends on its dielectric properties, characterized by the dielectric complex constant *ε*\*:

#### 4 Advanced Ceramic Processing

$$
\mathbf{e}^\* = \mathbf{e}^\cdot - j\mathbf{e}^\cdot \tag{1}
$$

Dielectric permittivity *ε*′ represents the material capacity to store electromagnetic energy and loss factor *ε*″ to dissipate it.

The dielectric constant of a material varies with its temperature, frequency, and composition. The power P absorbed in the material is proportional to the loss factor *ε*″, the frequency *f* [Hz], and the electric field intensity *E* [V/m]:

$$P = 2\Pi f \varepsilon^0 \varepsilon^\circ E^2 \tag{2}$$

(or)

$$P = 55.63 \times 10^{-2} \, f \, \dot{\varepsilon} \, \tan \delta E^2 \tag{3}$$

where the dielectric loss angle (loss tangent) is

$$\tan \delta = \frac{\mathcal{E}}{\mathcal{E}} \, \text{\AA} \tag{4}$$

Eq. (3) shows that for a fixed value of electric field (E), the power in microwave absorbed in the material mass is proportional to the frequency (*f*) (which is practically 2.45 GHz), the dielectric permittivity (*ε*′), and loss factor (*ε*″) (through the loss tangent tan *δ*), which vary with the materials temperature and humidity in their turn.

The diffusion of electromagnetic power into the absorber is characterized by skin depth (*D*) and expressed as

$$D = \left(\pi f \,\mu \sigma\right)^{-1/2} \tag{5}$$

where *μ* and *σ* are magnetic permeability and electrical conductivity and *f* is the frequency, respectively.

The effective penetration depth decreases with increase in frequency which in turn causes less heating. Hence, a suitable combination of parameters in Eqs. (2) and (4) is required for achieving optimum coupling. It can be inferred from this discussion that low dielectric loss materials take longer time and high dielectric loss materials take shorter duration in the microwave sintering.

On a microscopic scale, the phenomenon of dielectric heating is the effect of impurity dipolar relaxation in the microwave frequency region. When the vacancy jumps around the impurity ion to align its dipole moment with the electric field the internal friction of the rapidly oscillating dipole cause a homogeneous (volumetric) heating. Where the maximum absorption of microwave energy at the frequency or temperature at which the loss factor (tan *δ*) attains its maximum. This is equivalent to an elastic relaxation resulting in damping of mechanical vibrations in solids.

The efficiency of the microwave dielectric heating is dependent on the ability of a specific material (powder, solvent, or reagent or anything else) to absorb microwave energy and convert it into heat. The heat is generated by the electric component of the electromagnetic field through two main mechanisms, i.e., dipolar polarization and ionic contribution [19]. According to the electromagnetism, the effect of a material upon heat transfer rates is often expressed as

$$\frac{\Delta T}{t} = \frac{0.56 \times 10^{-10} \,\mathrm{s}^{\mathrm{-}}\_{\mathrm{eff}} f \mathrm{E}^2}{\rho \mathrm{C}\_p} \tag{6}$$

where *εeff* '' is the effective relative dielectric loss factor, *f* is the frequency of microwave, *E* is the magnetic fields of microwave within the material, *ρ* is the mass density of the sample, and *C*<sup>P</sup> is the isotonic specific heat capacity [19]. In this case, the energy efficiency can easily reach 80– 90% utilization and higher than the conventional heating methods [20, 21]. However, the essential nature of the interaction between microwaves and reactant molecules during the preparation of materials is fairly uncertain and speculative.

### **1.2. Benefits of microwave sintering comparison to conventional sintering method**

In recent years, microwave sintering has shown significant advantages against conventional sintering for the synthesis of ceramic materials. Microwave sintering has attained worldwide attention due to its major advantages against conventional sintering methods, especially in ceramic materials.

Microwave sintering can significantly shorten the sintering time leading to consume much lower energy than conventional sintering.

There are major potential and real advantages using microwave energy for material processing over conventional heating. These include the following:

**•** Time and energy savings

\* ' '' ee

loss factor *ε*″ to dissipate it.

4 Advanced Ceramic Processing

(or)

and expressed as

respectively.

microwave sintering.

and the electric field intensity *E* [V/m]:

where the dielectric loss angle (loss tangent) is

with the materials temperature and humidity in their turn.

 e

Dielectric permittivity *ε*′ represents the material capacity to store electromagnetic energy and

The dielectric constant of a material varies with its temperature, frequency, and composition. The power P absorbed in the material is proportional to the loss factor *ε*″, the frequency *f* [Hz],

> 0 '' 2 *P fE* = P2 e e

2' 2 *P fE* 55.63 10 tan e

''

' tan , e d

Eq. (3) shows that for a fixed value of electric field (E), the power in microwave absorbed in the material mass is proportional to the frequency (*f*) (which is practically 2.45 GHz), the dielectric permittivity (*ε*′), and loss factor (*ε*″) (through the loss tangent tan *δ*), which vary

The diffusion of electromagnetic power into the absorber is characterized by skin depth (*D*)

( ) 1/2

where *μ* and *σ* are magnetic permeability and electrical conductivity and *f* is the frequency,

The effective penetration depth decreases with increase in frequency which in turn causes less heating. Hence, a suitable combination of parameters in Eqs. (2) and (4) is required for achieving optimum coupling. It can be inferred from this discussion that low dielectric loss materials take longer time and high dielectric loss materials take shorter duration in the

On a microscopic scale, the phenomenon of dielectric heating is the effect of impurity dipolar relaxation in the microwave frequency region. When the vacancy jumps around the impurity

*D f* p ms-

e

 d

= - *j* (1)

(2)

<sup>=</sup> (4)

= (5)



**•** Controllable electric field distribution
