Orientation polarization

Water molecule as a permanent electric dipole
red: negative partial charge
blue: positive partial charge
green: directed dipole

The permittivity of water (20 ° C) is frequency dependent.
The imaginary part shown in red is decisive for the energy absorption by flipping the dipoles.

As orientation polarization signifies the polarization caused by the alignment (orientation) of permanent electric dipoles , z. B. water, is effected in an electric field . Their thermal movement counteracts this alignment of the dipoles . The orientation polarization therefore depends on the temperature (the higher the temperature, the lower the orientation polarization), which is described by the Debye equation .

Permanent dipole moments are generally much larger (about a factor of 10 ³ ) than induced dipole moments that are first generated by the electric field ( displacement polarization ).

If the direction of the electric field is reversed , the dipole molecules have to reorient or realign themselves ( relaxation process ). Due to their relatively great inertia , they require a certain amount of time (typical rotation time of a molecule in liquid 10 ⁻⁹ ... 10 ⁻¹¹  s), which is why the absorption maximum is around 20  GHz (corresponds to a period T = 0.5 · 10 ⁻¹⁰ s, see 2nd fig.). Orientation polarization can no longer be observed at even higher frequencies , but only displacement polarization, and the Debye equation changes into the Clausius-Mossotti equation .

Derivation of the temperature dependence

The interaction energy W of a permanent electric dipole with an external electric field is:

${\displaystyle W=-{\vec {p}}\cdot {\vec {E}}=-pE\cos \vartheta }$

The complete alignment in the electric field is opposed to the thermal energy , which strives for an equal distribution of all directions. If the dipoles can rotate freely and are in thermodynamic equilibrium at the temperature , the probability of encountering a dipole with the energy or the angle is proportional to the Boltzmann factor : ${\displaystyle W\propto kT}$ ${\displaystyle T}$ ${\displaystyle W}$ ${\displaystyle \vartheta }$

${\displaystyle \exp \left(-{\frac {W}{kT}}\right)=\exp \left({\frac {pE\cos \vartheta }{kT}}\right)}$

For a constant electric field in the z-direction the mean dipole moment in the z-direction is equal to: ${\displaystyle {\vec {E}}=E{\hat {e}}_{z}}$

${\displaystyle \left\langle p_{z}\right\rangle =p\left\langle \cos \vartheta \right\rangle =p\,{\frac {\int _{0}^{\pi }{\cos \vartheta \;\operatorname {e} ^{pE\cos \vartheta /kT}\sin \vartheta \;\mathrm {d} \vartheta }}{\int _{0}^{\pi }{\operatorname {e} ^{pE\cos \vartheta /kT}\sin \vartheta \;\mathrm {d} \vartheta }}}=p\left[\coth \left({\frac {pE}{kT}}\right)-{\frac {kT}{pE}}\right]}$

The sum of all mean dipole moments per volume gives the macroscopic polarization (N is a density, namely dipoles per volume):

${\displaystyle P=N\left\langle p_{z}\right\rangle =Np\left[\coth \left({\frac {pE}{kT}}\right)-{\frac {kT}{pE}}\right]}$

The expression in square brackets is the Langevin function . The Langevin function can be developed for high temperatures or small field strengths:

${\displaystyle L(x)=\coth(x)-{\frac {1}{x}}\ {\overset {x\ll 1}{\mathop {=} }}\ {\frac {x}{3}}-{\frac {x^{3}}{45}}+{\mathcal {O}}(x^{5})}$     With     ${\displaystyle x={\frac {pE}{kT}}}$

Thus it follows for the macroscopic polarization with a first approximation: ${\displaystyle pE\ll kT}$

${\displaystyle P={\frac {Np^{2}}{3kT}}E}$

At room temperature is about 1/40  eV = 0.025 eV and the orientation energy of the dipoles with dipole moment is about 10 ⁻³⁰  A · s · m at a field strength of 10 ⁷  V / m is about 0.00062 eV. The above assumption is thus fulfilled . ${\displaystyle kT}$ ${\displaystyle pE/kT=1/40}$ ${\displaystyle \ll 1}$

For weak electric field strengths, the polarization is a linear function of the electric field

${\displaystyle {\vec {P}}=\varepsilon _{0}\chi {\vec {E}}}$

The previous equation gives a temperature-dependent electrical susceptibility

${\displaystyle \chi ={\frac {Np^{2}}{3\varepsilon _{0}kT}}}$

The orientation polarization is therefore proportional to the reciprocal temperature ( Curie law ). Note that this result only applies to dipoles that can rotate freely. This is generally not the case with a solid.

See also

• dielectric

literature

• Gerhard H. Findegg, Thomas Hellweg: Statistical Thermodynamics . 2nd Edition. Springer, Berlin / Heidelberg 2015, ISBN 978-3-642-37871-3 , Chapter 6: Ideal gases , doi : 10.1007 / 978-3-642-37872-0_6 .