EIT and Van Vleck–Weisskopf–Froehlich dielectric polarization

In order to sharpen the thermodynamic perspective as suggested by one of the reviewers we show how the EIT can be used to extend the range of validity of the Debye theory to higher frequencies restricting ourselves to a single polarization process.

EIT assumes an equation of state for the entropy per unit volume to depend additionally on the non-equilibrium flux which is denoted as p⃗˙ below with the internal energy per unit volume u and polarization per unit volume p⃗:

sˉ is essentially the inversion of eq. (8) extended by the inclusion of the vector variable p⃗˙ called polarization rate which is chosen as the temporal rate of change of p⃗. The Gibbs-relation then takes the form:

The first law remains unchanged from eq. (6) for q⃗=0⃗:

Defining

and assuming (isotropic) polarization:

where ω−1(u) is a time scalar, and the entropy balance according to EIT reads:

So eq. (13) for a single polarization process yields:

and we finally arrive at the Van Vleck–Weisskopf–Froehlich equation [14]

A small perturbation version takes the form:

which is the linearized Van Vleck–Weisskopf–Froehlich equation. For ω0−2=0 it reduces to the Debye equation (eq. (4)) for a single polarization process. The first term in eq. (108) takes account of the rotary inertia of polar molecules within the Maxwell–Lorentz picture [4].

Kluitenberg–Debye theory

The Gibbs relation for processes in a rigid isotropic dielectric non-magnetic insulator, using a vectorial internal variable ξ⃗ (Kluitenberg), reads:

The introduction of the vectorial internal variable ξ⃗ and its corresponding vectorial affinity A⃗ is formal at this point; its physical meaning is specified below. Electrical field strength E⃗, vector affinity A⃗, conjugate to ξ⃗, and temperature T are derived from the canonical form of the internal energy (per unit volume) function u=uˉ(s,p⃗,ξ⃗) according to

The Maxwell relations imply the second-order tensor relation, in index and symbolic notation, resp.:

where ∇(ξ⃗,p⃗) denotes the gradient operator w.r.t the components of ξ⃗,p⃗. For isotropic material a linearized form of the state equations is

with α,γ,εrf denoting scalar constants with ε0 the vacuum dielectric constant. The state eqs. (112), (113) satisfy the Maxwell relations (111). Within the CIT framework the entropy production rate (per unit volume – assuming absence of heat and electrical current conduction – is:

A linear flux–force relationship between the internal variable and its affinity

with L representing a positive scalar kinetic coefficient,

ensures fulfillment of the second law. To give an interpretation to the internal variable ξ⃗ we observe that in equilibrium the Gibbs relation takes the form

stipulating that the affinity A⃗ vanishes in equilibrium. The difference between eq. (109) and the last equation implies that

for the choice ξ⃗=p⃗. Thermodynamic stability of the rest state, i.e. convexity of the state function uˉ(s,p⃗,ξ⃗) implies, with ε0 denoting the (positive) vacuum dielectric constant,

Elimination of A⃗,ξ⃗ from eqs. (112), (113), (115) yields a single evolution equation connecting E⃗,p⃗, addressed as Kluitenberg–Debye equation (KDE) below:

Low- and high-frequency aspects of the KDE are given by

resp., where an equilibrium susceptibility, εre, is introduced according to

τ is a positive relaxation time as implied by the salient inequalities according to eqs. (8), (9); for the same reason:

The spectroscopic consequence of the last inequality is that the real part of the susceptibility in the low frequency regime, i.e. near thermodynamic equilibrium, is larger than the real part of the dielectric susceptibility in the high-frequency regime, i.e. under nearly frozen conditions.

The Kluitenberg version of eqs.(40)–(42)

We start from eqs. (41), (42), which are repeated here for reference aiming at putting them into the Kluitenberg form previously derived, see eq. (120). The steps are self-explanatory.

The last equation, renaming τ2↦τ,p⃗2⋆↦p⃗ is rewritten in the form of the KDE given above,

where, finally, the identifications

are obvious. The same conclusion would, of course, be implied by using p1 instead of p2 in the preceding argument. Thus using eqs. (128) and (40) in place of eqs. (40)–(42) is equivalent.

The appendix thus shows the close connections between the Kluitenberg and Debye approaches within CIT. The EIT approach leading to the Van-Vleck–Weisskopf–Fröhlich material goes beyond the KDE approach and provides an extension to higher frequencies.