Polarization of Materials

1.1 Polarization
Polarization is a property of waves that describes the orientation of their oscillations. For transverse waves such as many electromagnetic waves, it describes the orientation of the oscillations in the plane perpendicular to the wave's direction of travel. The oscillations may be oriented in a single direction (linear polarization), or the oscillation direction may rotate as the wave travels (circular or elliptical polarization). Circularly polarized waves can rotate rightward or leftward in the direction of travel, and which of those two rotations is present in a wave is called the wave's chirality. For longitudinal waves such as sound waves in fluids the direction of oscillation is by definition along the direction of travel (i.e. polarization is not possible). The polarization is described by specifying the direction of the wave's electric field. According to the Maxwell equations, the direction of the magnetic field is uniquely determined for a specific electric field distribution and polarization. 1

1.2 Polarizability
Polarizability is the relative tendency of a charge distribution, like the electron cloud of an atom or molecule, to be distorted from its normal shape by an external electric field, which may be caused by the presence of a nearby ion or dipole.
The electronic polarizability is defined as the ratio of the induced dipole moment p of an atom to the electric field E that produces this dipole moment. 6

### p=αE

Polarizability has the SI units of C•m2•V-1=A2•s4•kg-1 but is more often expressed as polarizability volume with units of cm3 or in Å3 = 10-24 cm3.

### α (cm3)=α106/4πε0(C•m2•V-1)

where ε0 is the vacuum permittivity. The polarizability of individual particles is related to the average electric susceptibility of the medium by the Clausius-Mossotti relation. The sources of polarizability have been reported in ching-chang's paper.

1.3 Dipolar polarization
Dipolar polarization is a polarization that is particular to polar molecules. This polarization results from permanent dipoles, which retain polarization in the absence of an external electric field. The assembly of these dipoles forms a macroscopic polarization.
When an external electric field is applied, the distance between charges, which is related to chemical bonding, remains constant in the polarization; however, the polarization itself rotates. Because this rotation completes not instantaeously but in the delay time τ, which depends on the torque and the surrounding local viscosity of the molecules, dipolar polarizations lose the response to electric fields at the lowest frequency in polarizations. 5 T he delay of the response to the change of the electric field causes friction and heat.

For exmaple, a molecule in a polar liquid such as water, there are intermolecular forces which give any motion of the molecule some inertia. Under a very high frequency electric field, the polar molecule will attempt to follow the field, but intermolecular inertia stops any significant motion before the field has reversed, and no net motion results. If the frequency of field oscillation is very low, then the molecules will be polarized uniformly, and no random motion results. In the intermediate case, the frequency of the field will be such that the molecules will be almost, but not quite, able to keep in phase with the field polarity. In this case, the random motion resulting as molecules jostle to attempt in vain to follow the field is the heating observed in the sample.

Fig. 1 Molecular rotations of hydrogen-bonded system. 5

It is interesting to note that whist the efficiency of microwave absorbance varies markedly with frequency for any liquid, the frequency of a domestic microwave oven (2.45GHz) is not selected so that it is at the maximum absorbency for water (something like 10GHz). 5 Note, though, that this is a very simplistic model of microwave heating in hydrogen-bonded systems, which actually involves a complex mechanism that does not simply result from molecular rotations.

1.31 Conduction Mechanisms
Where the irradiated sample is an electrical conductor, the charge carriers (electrons, ions, etc.) are moved through the material under the influence of the electric field, E, resulting in a polarisation, P. These induced currents will cause heating in the sample due to any electrical resistance. For a very good conductor, complete polarisation may be achieved in approximately 10-18 seconds, indicating that under the influence of a 2.45GHz microwave, the conducting electrons move precisely in phase with the field.

Fig. 2 Polarization of an electric conductor. 5

If the sample is too conducting, such as a metal, most of the microwave energy does not penetrate the surface of the material, but is reflected. However, the colossal surface voltages which may still be induced are responsible for the arcing that is observed from metals under microwave radiation.
Thus, if one takes pure water and heats it in a microwave oven, where a variant of the polarization mechanism dominates, we find that the heating rate is significantly less than when one takes the same volume of water and add salt. In the latter case, both mechanisms occur, and contribute to the heating effect.

1.32 Interfacial Polarisation
This mechanism is important for systems comprised of conducting inclusions in a second, non-conducting material. An example would be a dispersion of metal particles in sulphur. Sulphur is microwave transparent and metals reflect microwaves yet, curiously, the combination forms an extremely good microwave absorbing material (in fact, that interfacial polarisation effects are reputed to be the basis of 'Stealth' radar absorbent materials).
Interfacial polarization is an effect which is very difficult to treat in a simple manner, and is most easily viewed as a combination of the conduction and dipolar polarization effects.
For a (non-superconducting) metal, there will always be a very thin surface layer in which some of the incident microwaves are attenuated, and in which induced currents will give rise to heating. 5 For a bulk metal this heating effect is so small as to be irrelevant, but in powders this surface layer makes up a large proportion of the material. However, the polarisation induced in the metal is also subject to the properties of the surrounding medium - in simple terms, it induces a 'drag' on the polarization of the metallic inclusions - making it less effective than it might otherwise be. Under these circumstances, the polarization of the metallic particles does not take place instantaneously, but lags behind the induced field, as for the polar molecule in the dipolar polarization mechanism. Hence, the frequency dependence of the sample's heating properties is similar to that of the dipolar polarization mechanism, despite being due to a conduction mechanism.

1.4 Application
1.41 Solid Oxide Fuel Cell
A solid oxide fuel cell (SOFC) is an electrochemical conversion device that produces electricity directly from oxidizing a fuel. Fuel cells are characterized by their electrolyte material and, as the name implies, the SOFC has a solid oxide, or ceramic, electrolyte. Advantages of this class of fuel cells include high efficiencies, long term stability, fuel flexibility, low emissions, and cost. The largest disadvantage is the high operating temperature which results in longer start up times and mechanical/chemical compatibility issues.

Fig. 3 Scheme of a solid-oxide fuel cell 4

La1-xSrxMnO3 (LSM) oxide is generally used as the cathode of solid oxide fuel cells (SOFCs). LSM cathodes have low levels of chemical reactivity and good thermal expansion compatibility with yttria-stabilized-zirconia (YSZ). However, they have low ionic conductivity, which results in poor oxygen reduction reactions. Both the oxygen diffusivity and oxygen surface exchange kinetics of LSM oxides are low compared to those of cobalt-containing perovskite oxides. 3 Composite cathodes composed of LSM and YSZ have been investigated extensively with the intention of enhancing the coefficients of the oxygen diffusion and surface exchange. LSM and YSZ are assumed to be a pure electronic conductor and a pure ionic conductor, respectively, at typical SOFC operational temperatures and pressure ranges. For this reason, the ionic conductivity of composite cathodes is typically estimated by taking into account the connectivity and ionic conductivity of the ionic conductor. 2 However, an oxygen reduction reaction must occur when measuring the ionic conductivity of a certain material (Fig. 4), which implies that it is one of the most important factors for oxygen mass transport in bulk material. If the oxygen reduction reaction is not generated at the material, measuring the ionic conductivity becomes impossible.

Fig. 4. Schematic of a general direct current method for measuring the ionic conductivity of a material. 2

It is accepted that there are two principle pathways for the performance of the cathodic reaction: 3 (i) for metals and oxides with low ionic conductivity as LSM the reaction should take place at the triple-phase boundary oxygen (air)/cathode/electrolyte (Fig. 5a) and thus the increase of the triple-phase boundary length would influence strongly the electrode properties (Fig. 5b); (ii) in mixed conductors with higher ionic conductivity (as LSC and LSCF), the reaction pathway should be through the bulk electrode (Fig. 5c).

Fig. 5. Possible mechanisms of cathode reaction in SOFC: (a) reaction at the triple-phase boundary for single-phase electrode; (b) reaction at the triple-phase boundary for composite electrode; (c) reaction involving ion transport through the bulk of the electrode.3

1.42 Sources of Polarization in SOFC
Polarizations are losses in voltage due to imperfections in materials, microstructure, and design of the fuel cell. Polarizations result from ohmic resistance of oxygen ions conducting through the electrolyte (iRΩ), electrochemical activation barriers at the anode and cathode, and finally concentration polarizations due to inability of gases to diffuse at high rates through the porous anode and cathode (shown as ηA for the anode and ηC for cathode)

### V = E0 − iRω − ηcathode − ηanode

In SOFCs, it is typically most important to focus on the ohmic and concentration polarizations since high operating temperatures experience little activation polarization. 4However, as the lower limit of SOFC operating temperature is approached (~600°C), these polarizations do become important.
a) Ohmic Polarization
Ohmic losses in an SOFC result from ionic conductivity through the electrolyte. This is inherently a materials property of the crystal structure and atoms involved. However, to maximize the ionic conductivity, several methods can be done. Firstly, operating at higher temperatures can significantly decrease these ohmic losses. Also, substitutional doping methods to further finetune the crystal structure and control defect concentrations can also play a significant role in increasing the conductivity. Another way to decrease ohmic resistance is to decrease the thickness of the electrolyte layer.
b) Concentration Polarization
The concentration polarization is the result of the finite gas diffusion processes that govern movement of the gases into and out of the electrochemical reaction. The rate of mass transport of gases is described by Fick’s first law. Therefore the maximum rate of gas diffusion (which is directly related to the maximum current density that can be obtained) is found when the concentration of fuel at the electrochemically active area is assumed to be zero. The potential difference between operation where current is
flowing and not flowing is the concentration polarization and is equal to: 4

### ηconc=(RT/nF)×ln[1-(i/il)]

where:
• R = gas constant
• T0 = operating temperature
• n = number of electrons exchanged in electrochemical reaction
• i = operating current
• il = max current
The concentration polarization is highly dependent on the gases used as well as the distance that they must diffuse through. Pore volume percentage as well as diffusion length can be varied to optimize these properties. For similar geometries, cathode concentrations are much larger than anode concentrations due to the lower diffusivities of O2/N2 in the cathode than H2/H2O in the anode.
c) Activation Polarization
The activation polarization is the result of the kinetics involved with the electrochemical reactions. Each reaction has a certain activation barrier that must be overcome in order to proceed and this barrier leads to the polarization. The activation barrier is the result of many complex electrochemical reaction steps where typically the rate limiting step is responsible for the polarization. The polarization equation shown below is found by solving the Butler-Volmer equation in the high current density regime (where the cell typically operates) 4

### ηact=(RT/βzF)×ln(i/i0)

where:
• R = gas constant
• T0 = operating temperature
• β = electron transfer coefficient
• z = electrons associated with the electrochemical reaction
• i = operating current
• i0 = exchange current density
The polarization can be modified most easily by microstructural optimization. The Triple Phase Boundary (TPB) length, which is the length where porous, ionic and electronically conducting pathways all meet, directly relates to the electrochemically active length in the cell. The larger the length, the more reactions can occur and thus the less the activation polarization. 4 Optimization of TPB length can be done by processing conditions to affect mircrostructure or by materials selection to use a mixed ionic/electronic conductor to further increase TPB length.

References
1. http://en.wikipedia.org/wiki/Polarization
2. C.B. Lee, J. Bae. Effect of effective areas on ionic conductivity in dense composite material composed
of ionic and electronic conductors for solid oxide fuel cells. Solid State Ionics 179 (2008) 2031-2036
3. D.E. Vladikova, Z.B. Stoynov, A. Barbucci, M. Viviani, P. Carpanese, J.A. Kilner, S.J. Skinner, R. Rudkin. Impedance studies of cathode/electrolyte behaviour in SOFC. Electrochim. Acta 53 (2008) 7491–7499
4. http://en.wikipedia.org/wiki/Solid_oxide_fuel_cell
5. http://www.tan-delta.com/basics.html
6. http://en.wikipedia.org/wiki/Polarizability
7. M. E. Lynch, D. S. Mebane, Y. J. Liu, M. Liua. Triple-Phase Boundary and Surface Transport in Mixed Conducting Patterned Electrodes. J. Electrochem. Soc. 155 [6] (2008) B635-B643