Clausius–Mossotti relation

The Clausius–Mossotti relation is named after the Italian physicist Ottaviano-Fabrizio Mossotti, whose 1850 book[1] analyzed the relationship between the dielectric constants of two different media, and the German physicist Rudolf Clausius, who gave the formula explicitly in his 1879 book[2] in the context not of dielectric constants but of indices of refraction. The same formula also arises in the context of conductivity, in which it is known as Maxwell's formula[citation needed]. It arises yet again in the context of refractivity, in which it is known as the Lorentz–Lorenz equation.

The Clausius–Mossotti law applies to the dielectric constant of a dielectric that is perfect, homogeneous and isotropic. It is the second of the following three equalities:[3][4]

\( \frac{\epsilon - \epsilon_0}{\epsilon + 2\epsilon_0} \cdot \frac{M}{d} = \frac{\epsilon_\mathrm{r} - 1}{\epsilon_\mathrm{r} + 2} \cdot \frac{M}{d} = \frac{4\pi N_A \alpha'}{3} = \frac{N_A \alpha}{3\epsilon_0} \)

where

\( \epsilon = \epsilon_r\epsilon_0 \) is the dielectric constant of a substance
\( \epsilon_0 \)is the permittivity of a vacuum
M is the molar mass of the substance
d is its density
\( N_A \) is Avogadro's number,
\( \alpha \) is the molecular polarizability in SI-units (C·m²/V) and
\( \alpha' = \alpha/(4\pi\varepsilon_0 \)) is the molecular polarizability volume (m₃ in SI) or the polarizability in the CGS system of units.[4]

Clausius–Mossotti factor

The Clausius–Mossotti factor can be expressed in terms of complex permittivities:[5][6][7]

\( K(\omega) = \frac{\epsilon^*_p - \epsilon^*_m}{\epsilon^*_p + 2\epsilon^*_m} \)

\( \epsilon^* = \epsilon + \frac{\sigma}{i\omega} = \epsilon - \frac{i\sigma}{\omega} \)

where

\( \epsilon \) is the permittivity (the subscript p refers to a lossless dielectric sphere suspended in a medium m)
\( \sigma \) is the conductivity
\( \omega \) is the angular frequency of the applied electric field
i is the imaginary unit, the square root of -1

In the context of electrokinetic manipulation, the real part of the Clausius–Mossotti factor is a determining factor for the dielectrophoretic force on a particle, whereas the imaginary part is a determining factor for the electrorotational torque on the particle. Other factors are, of course, the geometries of the particle to be manipulated and the electric field. Whereas \( Re(K(\omega)) \) can be directly measured by application of different AC potentials directly on electrodes,[8] \( Im(K(\omega)) \) can be measured by electro-rotation measurements thanks to optical trapping methods.

Derivation

Assume a simple cubic lattice of polarisable points with polarisability \alpha. Application of an external field \( E _{ext} \) will induce a dipole at each site \( R_i \). Due to symmetry, the local (microscopic) field inside lattice is identical at each lattice point: \( E_{local} = E(R_i) \).

Away from the lattice points, the electric field is given by:

\( E = E_{ext} + \sum_i E(p_i,r-R_i) \)

where the dipole electric field \( E(p_i,r-R_i) = \frac{3(p\cdot \hat{r})\hat{r} - p}{r^3} \)

Richard Feynman on the Clausius–Mossotti equation

In his Lectures on Physics (Vol.2, Ch32), Richard Feynman has a background discussion deriving the Clausius–Mossotti Equation, in reference to the index of refraction for dense materials. He starts with the derivation of an equation for the index of refraction for gases, and then shows how this must be modified for dense materials, modifying it, because in dense materials, there are also electric fields produced by other nearby atoms, creating local fields. In essence, Feynman is saying that for dense materials the polarization of a material is proportional to its electric field, but that it has a different constant of proportionality than that for a gas. When this constant is corrected for a dense material, by taking into account the local fields of nearby atoms, one ends up with the Clausius–Mossotti Equation.[9] Feynman states the Clausius–Mossotti equation as follows:

\( \mathcal N \alpha = 3\, \frac{n^2 - 1}{n^2 + 2} , \)

where

\( \mathcal N \) is the number of particles per unit volume of the capacitor,
\( \ \alpha \)is the atomic polarizability,
\( \ n \)is the refractive index.

Feynman discusses "atomic polarizability" and explains it in these terms: When there is a sinusoidal electric field acting on a material, there is an induced dipole moment per unit volume which is proportional to the electric field - with a proportionality constant \alpha that depends on the frequency. This constant is a complex number, meaning that the polarization does not exactly follow the electric field, but may be shifted in phase to some extent. At any rate, there is a polarization per unit volume whose magnitude is proportional to the strength of the electric field.
Dielectric constant and polarizability

The polarizability \( \alpha \), of an atom is defined in terms of the local electric field at the atom by

\( \ \alpha = \ \rho / E _{local} \)

where

\( \ \rho \) is the dipole moment,
\( \ E _{local} \) is the local electric field at the orbital[clarification needed]

The polarizability is an atomic property, but the dielectric constant will depend on the manner in which the atoms are assembled to form a crystal. For a non-spherical atom, \( \alpha \) will be a tensor.[10]

The polarization of a crystal may be expressed approximately as the product of the polarizabilities of the atoms times the local electric field:

Now, to relate the dielectric constant to the polarizability, which is what the Clausius–Mossotti equation (or relation) is all about,[10] one must consider that the results will depend on the relation that holds between the macroscopic electric field and the local electric field:

\( P = \sum_{j} N_j\ \rho_j = \sum_{j} N_j \alpha_j E _{local}(j) \)

where

\( \ N_j \) is the concentration,
\( \ \alpha_j \) is the polarizability of atoms j,
\( \ E _{local}(j) \) Local Electrical Field at atom sites \( \ j \).

References

C.J.F. Böttcher, Theory of electric polarization, Elsevier Publishing Company, 1952

Mossotti, O. F. (1850). Mem. di mathem. e fisica in Modena. 24 11. p. 49.
Clausius, R. (1879). Die mechanische U’grmetheorie. 2. p. 62.
Rysselberghe, P. V. (January 1932). "Remarks concerning the Clausius–Mossotti Law". J. Phys. Chem. 36 (4): 1152–1155. doi:10.1021/j150334a007.
Atkins, Peter; de Paula, Julio (2010). "Chapter 17". Atkins' Physical Chemistry. Oxford University Press. p. 622-629. ISBN 978-0-19-954337-3.
Hughes, Michael Pycraft (2000). "AC electrokinetics: applications for nanotechnology". Nanotechnology 11 (2): 124–132. Bibcode:2000Nanot..11..124P. doi:10.1088/0957-4484/11/2/314.
Markov, Konstantin Z. (2000). "Elementary Micromechanics of Heterogeneous Media". In Konstantin Z. Markov and Luigi Preziosi. 'Heterogeneous Media: Modelling and Simulation' (PDF). Boston: Birkhauser. pp. 1–162. ISBN 978-0-8176-4083-5.
Gimsa, J. (2001). "Characterization of particles and biological cells by AC-electrokinetics". In A.V. Delgado. Interfacial Electrokinetics and Electrophoresis. New York: Marcel Dekker Inc. pp. 369–400. ISBN 0-8247-0603-X.
T. Honegger, K. Berton, E. Picard et D. Peyrade. Determination of Clausius–Mossotti factors and surface capacitances for colloidal particles. Appl. Phys. Lett., vol. 98, no. 18, page 181906, 2011.
Feynman, R. P., Leighton, R. B.; Sands, M (1989). Feynman Lectures on Physics. Vol. 2, chap. 32 (Refractive Index of Dense Materials), sec. 3: Addison Wesley. ISBN 0-201-50064-7.
Kittel, Charles (1995). Introduction to Solid State Physics (8th ed.). Wiley. ISBN 0-471-41526-X.

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