Thermodynamics and forces in charged solid insulators

A thermodynamic framework is a powerful method to synthesise our knowledge on the material properties. This is done by introducing fields and related properties of the vacuum. The thermodynamic description of the medium is achieved by adding two new variables to the classical description (temperature T, deformation E, composition γ∞). These variables are the polarisation P for the matter and the Maxwell (or mean) electric field E for the vacuum if the free energy is used as thermodynamic potential to characterize the system. Then, the expression of the matter free energy depends on the polarisation P and the vacuum related term that only depends on E must be added: f_vol(T, ε, γ_α, P, E)=f_vol(T, ε, γ_α, P)+½ε₀E². From the thermodynamic potential the physical quantities (entropy, energy, chemical potentials, stresses) are obtained as usual using partial derivatives. The mechanical forces that a dielectric can support are deduced after having added the Maxwell electric tensor: σ_el=ε₀(E⊗D-½E.D δ) where δ is the unit tensor, D=ε₀E+P and ⊗ the tensor product. The equation giving the evolution toward the equilibrium is also obtained. It is governed by the associated thermodynamic force: A_P=δf_vol/δP-E. Then partial equilibrium between the matter and the electric field is considered. We show that at the polarisation equilibrium: δf_vol/δP=E and a new internal variable D=ε ₀E+P (displacement field) must be substituted in replacement of E and P while the value of the thermodynamic potential remains unchanged f_eq(-, D)=f_vol(-, P, E)