The theory of dielectric polarization

One main goal of studies of dielectric polarization is to relate macroscopic properties such as the dielectric constant to microscopic properties such as the polarizability.

Non-polar molecules in the gas phase

This is done quite simply for non-polar molecules in the gas phase where intermolecular interactions can be igored. The polarization can be immediately expressed in terms of both electric susceptibility (macroscopic) and polarizability (microscopic).

We can see that

and since e_r = 1 + c_e

Furthermore since e_r = n² we have

The last step is due to a Taylor's series expansion. Experimentally, we see that the index of refraction of a gas is a linear function of the density (N/V) provided that the density is not too high.

Non-polar molecules in the condensed phase

Interactions between non-polar molecules cannot be neglected in condensed phases. The treatment considers a local field F inside the dielectric and its relation to an applied field E. The Lorentz local field considers a spherical region inside a dielectric that is large compared to the size of a molecule. The field inside this uniformly polarized sphere behaves as if it were due to a dipole given by:

Since P is the polarization per unit volume and 4pa³/3 is the volume of the sphere we see that m is the induced dipole moment/polarization (these are equivalent). The local field is the macroscopic field E minus the contribution of the due to the matter in the sphere:

Since

the Lorentz local field is

Since e_r = 1 for vacuum and e_r > 1 for all dielectric media it is apparent that the local field is always larger than the applied field. This simple consequence of the theory of dielectric polarization causes confusion. We usually think of the dielectric constant as providing a screening of the applied field. So therefore we might be inclined to think of a local field as smaller than the applied field. However, this naïve view ignores the role of the polarization of the dielectric itself. Inside the sphere we have carved out of the dielectric we observe the macroscopic (applied) field plus the field due to the polarization of the medium. The sum of these two contributions leads to a field that is always larger than the applied electric field.

The polarization is the number density times the polarizability times the local field.

We eliminate E to obtain the Clausius-Mossotti equation.

This equation connects the macroscopic dielectric constant e_r to the microscopic polarizability. Since e_r = n² we can replace these to obtain the Lorentz-Lorentz equation:

Again here the equation connects the index of refraction (macroscopic property) to the polarizability (microscopic property). The number density N/V can be replaced by the bulk density r (gm/cm³) through

where N_A is Avagadro's number and M is the molar mass.

Polar molecules in condensed phases

The polarization we have discussed up to now is the electronic polarization. If a collection of non-polar molecules is subjected to an applied electric field the polarization is induced only in their electron distribution. However, if molecules in the collection possess a permanent ground state dipole moment, these molecules will tend to reorient in the applied field. The alignment of the dipoles will be disrupted by thermal motion that tends to randomize the orientation of the dipoles. The nuclear polarization will then be an equilibrium (or ensemble) average of dipoles aligned in the field.

The angle brackets indicate the equilibrium average. If the permanent dipole moment is m₀, then the interaction with the field is W = - m₀×F = - m₀Fcosq where q is the angle between the dipole and the field direction. Thus, the average dipole moment is

The average indicated is an average over a Boltzmann distribution.

Here

Substituting in for the interaction energy W we find

We make the substitutions

The integral is

The function coth(u) - 1/u is known as the Langevin function. It approaches u/3 for u << 1 and 1 when u is large. The limit for large u is easy to see. The limit for small u requires carrying out a Taylor's series expansion of the function to many higher order terms.

For typical fields employed m₀F/kT << 1. You can convince yourself of this using the following handy conversion factors

m₀F = 1.68 x 10^-5 cm^-1/(DV/cm)

k = 0.697 cm^-1/K

For example, at 300 K, thermal energy is 209 cm^-1. For liquid water (m₀»2.4 D) in a 10,000 V/cm field we have W = 0.4 cm^-1. Here u = m₀F/kT is of the order of 1/1000.

Thus, we can express the orientational polarization as

The total polarization is the sum of the electronic and orientational polarization terms

Following the same protocol used above to derive the Clausius-Mossotti equation, we obtain the Debye equation for the molar polarization

This equation works reasonable well for some organics, however, it fails for water. The reason for the failure of the Debye model is that the Lorentz local field correction begins with a cavity large compared to molecular dimension and thus ignores local interactions of solvent dipoles.

The local field problem

The local field problem is one of the most vexing problems of condensed phase electrostatics. Following Lorentz there are two models, the Onsager model and the Kirkwood model that attempt to account for the local interactions of solvent molecules in an applied electric field. The approaches discussed here are all continuum approaches in that there is a cavity and outside that cavity the medium is treated as a continuum dielectric with dielectric constant e_r. The models differ in how they define the cavity. As stated above, Lorentz model assumes a large cavity (a is much larger than the molecule size). The Onsager model focuses on the creation of a cavity around a single molecule of interest (a is equal to the molecule size). The Kirkwood model includes a cluster around the molecule to account for local structure.

The Onsager model

The Debye model assumes that the dipole m₀ is not affected by the solvation shell. Yet consider water which has a gas phase dipole moment of 1.86 D and in condensed phase has a dipole moment in the range 2.3 - 2.4 D. The neighboring water molecules have a large effect inducing a dipole moment more than 25% larger than the gas phase dipole moment. The dipole moment m is the sum of the permanent and induced parts

The local field F has two contributions, the cavity field G and the reaction field R.

The cavity field is given the spherical cavity approximation in terms of the applied field

Notice that the cavity field is always greater than one. This is exactly analogous to the Lorentz local field. However, the Lorentz local field increases without bound as e_r increases. The Onsager cavity field increases from 1 to 1.5 as e_r approaches ¥.

The reaction field is proportional to the dipole moment of the molecule in the cavity:

The reaction field is always parallel to the permanent dipole moment. Only the cavity field can exert a torque on the dipole and cause it to align in the applied field. By separating these two effects the Onsager model improves upon the Debye equation. The Onsager reaction field is also an important relation for understanding the effect of solvents on the absorption and emission spectra of polar and polarizable molecules. Solvatochromism is the measurement of the effect of the solvent on the maximum position of the absorption band. Relaxation dynamics are also measured by determining the change in fluorescence maximum in fluorescent dyes in order to obtain an estimate of the reorientational dynamics of solvents.