Thermodynamics of Dielectric Relaxations in Complex Systems

1. Thermodynamics of Dielectric Relaxations in Complex Systems TUTORIAL 3

2. Static dipoles • It is necessary to found the Relation between microscopic polarizability and macroscopic permittivity. • From the phenomenological point of view, it is necessary to know the kinetic of the Polarization. • From molecular one it’s required the knowledge of the effective Electric field at which the dipole is subjected. • 4 different ways are proposed to evaluate the molecular: • Claussius – Mossotti • Debye • Onsager • Fouss – Kirkwood

3. Lorentz local field • The basic idea is to consider a a spherical zone containing the dipole under study, immersed in the dielectric. • The sphere is small in comparison with the dimension of the condenser, but large compared with the molecular dimensions. • We treat the properties of the sphere at the microscopic level as containing many molecules, but the material outside of the sphere is considered a continuum. • The field acting at the center of the sphere where the dipole is placed arises from the field due to • (1) the charges on the condenser plates • (2) the polarization charges on the spherical surface, and • (3) the molecular dipoles in the spherical region.

4. d - - - - Eo - - + + + + + + Lorentz local field  Lorentz local field

5. Claussius – Mossotti equation valid for nonpolar gases at low pressure. This expression is also valid for high frequency limit. The remaining problem to be solved is the calculation of the dipolar contribution to the polarizability.

6. Debye, extended the Claussius – Mossotti equation adding a new term in the polarization (orientational polarization). • By this way the dipolar contribution it’s taking into account • Debye equation for the static permittivity

7. Onsager, generalize the Debye equation taking into account the effect of the if the permanent dipole moment of a molecule by the polarization of the environment. • 1 – The cavity field, G, (the field produced in the empty cavity by the external field.) • 2 - The reaction field, R (the field produced in the cavity by the polarization induced by the surrounding dipoles).

8. Onsager treatment of the cavity differs from Lorentz’s because the cavity is assumed to be filled with a dielectric material having a macroscopic dielectric permittivity. • Also Onsager studies the dipolar reorientation polarizability on statistical grounds as Debye does. • The remaining problem is to take into account the interaction between dipoles • Kirkwood and Fröhlich develop a fully statistical argument to determine the short – range dipole – dipole interaction.

9. gwill be different from1when there is correlation between the orientations of neighboring molecules. • When the molecules tend to direct themselves withparallel dipole moments, will be positive and g>1. • When the molecules prefer an ordering withanti-parallel dipoles, g<1. • g =1 in the case of no dipolar correlation between neighboring molecules, or equivalently a dipole does not influence the position and orientations of the neighboring ones. • gdepends on the structure of the material, and for this reason it is a parameter that fives information about the forces of local type.

10. From Kremer – Schonhals book

12. Claussius – Mossotti: Only valid for non polar gases, at low pressure • Debye: Include the distortional polarization. • Onsager: Include the orientational polarization, but neglected the interaction between dipoles. describe the dielectric behavior on non-interacting dipolar fluids • Kirkwood: include correlation factor (interaction dipole-dipole) • Fröhlich – Kirkwood – Onsager

13. Dynamic theory E(t) s Orientational polarization ()  Induced polarization

14. 1,14 Debye equation • First order kinetic: • Decay function: • In frequency domain

15. Debye equation doesn’t represent in a good way the experimental data. • Some modifications in the decay function was proposed by Williams – Watt, ussing a previously Kolraush equation. • Advantages: represent better the experimental data. • Disadvantages: the  parameter it’s an artificial parameter and no molecular relation for this parameter have been yet found

18. DISPERSION RELATIONS • The real and imaginary part of the complex permittivity are, respectively, the cosine and sine Fourier transforms of the same function, that is, (). As a consequence, ’ and " are no independent. Kramer-Kronigs relationships

20. Thermodynamics • Thermodynamics appear in the XIX century because of the necessity of describe the thermal machines. • It is based in postulates, without mathematical demonstration, and as the mechanics and electromagnetic postulates, establish the basic physic laws.

21. FUNCTIONS Enthalpy (H) Entropy (S) Internal Energy (U) Free Energy (G) VARIABLES Temperature Density or volume Pressure Thermodynamic Characteristic properties of materials Calorific capacity Expansion coefficient Electric Permittivity

22. Thermodynamic postulates • Thermodynamic are based in 4 fundamentals laws: • U= Q-W (First law) (Energy balance) • Siso≥ 0 (Second law) • Thermal equilibrium (Zero law): 2 systems in equilibrium with one 3rd are in equilibrium between them. • Perfect Crystals at 0 K, define 0 entropy. (Third law)

23. Thermodynamic point of view • Thermodynamics relates the properties of macroscopic systems. • The macroscopic properties are originated in the statistical average properties of microscopic properties. Thermodynamic property Microscopic property Statistical average

24. Thermodynamics is usually concerned with very specific systems at equilibrium. • In nature, the processes are mainly irreversible. • Their description requires going beyond equilibrium.

25. THERMODYNAMICS OF IRREVERSIBLEPROCESSES • The four main postulates of the theory are: • 1 - The local and instantaneous relations between thermal and mechanical properties of a physical system are the same as for a uniform system at equilibrium. This is the so-called local equilibrium hypothesis. • 2 - The internal entropy arising from irreversible phenomena inside a volume element is always a non-negative quantity. This is a local formulation of the second law of thermodynamics • 3 - The internal entropy has a very simple character. It is a sum of terms, each being the product of a flux and a thermodynamic force • 4 - The phenomenological description relating irreversible fluxes to thermodynamic forces are assumed to be linear.

26. MAXWELL EQUATIONS • where c is the velocity of light. The total charge density, , and the total current density, J are taken as the sources of the field • If the magnetization is assumed to be zero, for a polarizable fluid, the Current will correspond to free charges and the polarization rate. • And the charge density correspond to the free and polarization density • J total current, Jf electric current of free charges, f density of free charges, p density of polarization charges, and P/t polarization current

27. Taking into account the relations between the electric displacement and the electric field, we can obtain a different version of the Maxwell equation can be written as:

28. Conservation equations • Mass: In the absence of chemical reactions, the rate of change in mass within a volume V can be written as the flux through the surface dS according to: , and v are, respectively, the mass density and velocity. • Charge: free charges Polarization charges

29. Conservation equations Linear momentum: The equation indicates that the force exerted by the electromagnetic field on the material within the volume V is equal to the rate of decrease in electromagnetic momentum within V plus the rate at which electromagnetic momentum is transferred into V across the surface V . =ED + HB - ½ (E·D + H·B)I is the Maxwell stress tensor; can be interpreted as the moment flux density. is the momentum density is the density of the Lorentz force identified with the body forces

30. Conservation equations • Energy: • The Poynting vector determines the density and direction of this flux at each point of the surface. • work per time unit spent in production of conduction currents. • time rate of change in the field energy within the region Poynting vector • electromagnetic energy flux through the surface.

31. Internal Energy Equation • The total energy can be expressed as Electromagnetic energy Kinetic energy Potential energy

32. ENTROPY EQUATION • For a single component system • the corresponding Gibbs equation is polarization charge density Internal energy

33. RELAXATION EQUATION • Using the entropy and energy balance equation, it is possible to express the relationship between the polarization rate and the thermodynamics functions • if To is constant and q= dJ/dt=0, Eq (3.6.1) reduces to the well-known Debye equation: • Debye equation predict instantaneous propagation of the perturbation

34. The Debye equations derived do not adequately represent the experimental behavior of polymers. • Instead of a symmetric semicircular arc, an asymmetric and skewed arc is observed. • To represent in a more accurate way the actual behavior, some modifications to the former theory must be made. • A more general relationship between forces and fluxes as follows where the operator D1represents the fractional derivatives of order  (0<<1).

35. Fractional derivatives were introduced in the theory of viscoelastic relaxations to give account of the deviations of the experimental data from those predicted by classical linear models, such as Maxwell and Kelvin-Voigt models, which are combinations of springs and dashpots • In terms of decay equation, the fractional derivatives it’s equivalent to stretch the decay function instead of the exponent. • Or which is the same, to chose a kinetic order different to 1(0<<1).

36. Under some considerations, the Laplace transform of the fractional derivative equation leads to the Havriliak Negammi empirical equation • This equation, contrary to the Debye equation, adequately predicts the shape of the actual dielectric data in the relaxation zones

38. Advantages: It’s possible to fit experimental data with HN equation. Disadvantages: It’s based on phenomenological point of view. It’s not possible to relate the exponents with the molecular structure. (no physical meaning)

39. Summary • From thermodynamic point of view it is also possible to obtain the relaxation equations. • The use of thermodynamics in dielectric materials relate the macroscopic properties with microscopic ones.