A dielectric is a nonconducting substance, i.e. an insulator. The term was coined by William Whewell in response to a request from Michael Faraday.Although dielectric and insulator are generally considered synonymous, the term dielectric is more often used to describe materials where the dielectric polarization is important, such as the insulating material between the metallic plates of a capacitor, while insulator is more often used when the material is being used to prevent a current flow across it.
Dielectrics is the study of dielectric materials and involves physical models to describe how an electric field behaves inside a material. It is characterized by how an electric field interacts with an atom and is therefore possible to approach from either a classical interpretation or a quantum one.
Many phenomena in electronics, solid state and optical physics can be described using the underlying assumptions of the dielectric model. This can mean that the same mathematical model can be used to describe different physical phenomena.
1.1 Definition:
Diectric field interaction with an atom under the classical dielectric model.Arthur R. von Hippel, in his seminal work, Dielectric Materials and Applications, stated:
Dielectrics... are not a narrow class of so-called insulators, but the broad expanse of nonmetals considered from the standpoint of their interaction with electric, magnetic, or electromagnetic fields. Thus we are concerned with gases as well as with liquids and solids, and with the storage of electric and magnetic energy as well as its dissipation.
1.2Classical:
In the classical approach to the dielectric model, a material is made up of atoms. Each atom consists of a cloud of negative charge bound to and surrounding a positive point charge at its centre. Because of the comparatively huge distance between them, none of the atoms in the dielectric material interact with one another[citation needed]. Note: Remember that the model is not attempting to say anything about the structure of matter. It is only trying to describe the interaction between an electric field and matter.
In the presence of an electric field the charge cloud is distorted, as shown in the top right of the figure.
This can be reduced to a simple dipole using the superposition principle. A dipole is characterized by its dipole moment, a vector quantity shown in the figure as the blue arrow labeled M. It is the relationship between the electric field and the dipole moment that gives rise to the behavior of the dielectric. Note: The dipole moment is shown to be pointing in the same direction as the electric field. This isn't always correct, and it is a major simplification, but it is suitable for many materials.[citation needed]
When the electric field is removed the atom returns to its original state. The time required to do so is the so-called relaxation time; an exponential decay.
1.3 Behavior:
This is the essence of the model in physics. The behavior of the dielectric now depends on the situation. The more complicated the situation the richer the model has to be in order to accurately describe the behavior. Important questions are:
The relationship between the electric field E and the dipole moment M gives rise to the behavior of the dielectric, which, for a given material, can be characterized by the function F defined by the equation:
When both the type of electric field and the type of material have been defined, one then chooses the simplest function F that correctly predicts the phenomena of interest. Examples of possible phenomena:
1.4 Dielectric model applied to vacuum:
From the definition it might seem strange to apply the dielectric model to a vacuum, however, it is both the simplest and the most accurate example of a dielectric.
Recall that the property which defines how a dielectric behaves is the relationship between the applied electric field and the induced dipole moment. For a vacuum the relationship is a real constant number. This constant is called the permittivity of free space, ε0.
1.5 Dielectric dispersion:
In physics, dielectric dispersion is the dependence of the permittivity of a dielectric material on the frequency of an applied electric field. Because there is always a lag between changes in polarization and changes in an electric field, the permittivity of the dielectric is a complicated, complex-valued function of frequency of the electric field. It is very important for the application of dielectric materials and the analysis of polarization systems.
This is one instance of a general phenomenon known as material dispersion: a frequency-dependent response of a medium for wave propagation.
it becomes impossible for dipolar polarization to follow the electric field in the microwave region around 1010 Hz;
in the infrared or far-infrared region around 1013 Hz, ionic polarization loses the response to the electric field;
electronic polarization loses its response in the ultraviolet region around 1015 Hz.
In the wavelength region below ultraviolet, permittivity approaches the constant ε0 in every substance, where ε0 is the permittivity of the free space. Because permittivity indicates the strength of the relation between an electric field and polarization, if a polarization process loses its response, permittivity decreases.
1.6 Dielectric relaxation
Dielectric relaxation is the momentary delay (or lag) in the dielectric constant of a material. This is usually caused by the delay in molecular polarization with respect to a changing electric field in a dielectric medium (e.g. inside capacitors or between two large conducting surfaces). Dielectric relaxation in changing electric fields could be considered analogous to hysteresis in changing magnetic fields (for inductors or transformers). Relaxation in general is a delay or lag in the response of a linear system, and therefore dielectric relaxation is measured relative to the expected linear steady state (equilibrium) dielectric values. The time lag between electrical field and polarization implies an irreversible degradation of free energy(G).
In physics, dielectric relaxation refers to the relaxation response of a dielectric medium to an external electric field of microwave frequencies. This relaxation is often described in terms of permittivity as a function of frequency, which can, for ideal systems, be described by the Debye equation. On the other hand, the distortion related to ionic and electronic polarization shows behavior of the resonance or oscillator type. The character of the distortion process depends on the structure, composition, and surroundings of the sample.
Chapter -2
In this chapter we introduce the concept of polarization in dielectric media. We will treat
the induced polarization as a dynamical response of the system to an externally applied
electric field. In this thesis we will only consider nonmetallic systems that do not possess
a static polarization. This excludes for instance ferroelectrica, which do possess a finite
polarization in the ground state due to a symmetry-breaking lattice deformation.
2.1 Electric Polarization
When a solid is placed in an externally applied electric field, the medium will adapt to this
perturbation by dynamically changing the positions of the nuclei and the electrons. We
will only consider time-varying fields of optical frequencies. At such high frequencies the
motion of the nuclei is not only effectively independent of the motion of the electrons, but
also far from resonance. Therefore we can assume the lattice to be rigid.
The reaction of the system to the external field consists of electric currents flowing through
the system. These currents generate electromagnetic fields by themselves, and thus the
motion of all constituent particles in the system is coupled. The response of the system
should therefore be considered as a collective phenomenon. The electrical currents now
determine in what way the externally applied electric field is screened. The induced field
resulting from these induced currents tends to oppose the externally applied field, effectively
reducing the perturbing field inside the solid. In metals the moving electrons are
able to flow over very large distances, so they are able to completely screen any static
(externally applied) electric field to which the system is exposed. For fields varying in
time, however, this screening can only be partial due to the inertia of the electrons. In
insulators this screening is also restricted since in these materials the electronic charge is
bound to the nuclei and can not flow over such large distances. The charge density then
merely changes by polarization of the dielectricum.
2.2The 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 er = 1 + ce
Furthermore since er = 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 er = 1 for vacuum and er > 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 er to the microscopic polarizability. Since er = 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 NA is Avagadro's number and M is the molar mass.
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 er. 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 er increases. The Onsager cavity field increases from 1 to 1.5 as er 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.
