Nikiya Anton Bettey 2021: 09/25/21

Saturday, September 25, 2021

09-25-2021-1755 - 4052/3-4,5

09-25-2021-1754 - electric displacement field (denoted by D) or electric induction

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In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in the related concept of displacement current in dielectrics. In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding of Gauss's law. In the International System of Units(SI), it is expressed in units of coulomb per meter square (C⋅m⁻²).

In linear, homogeneous, isotropic media, ε is a constant. However, in linear anisotropic media it is a tensor, and in nonhomogeneous media it is a function of position inside the medium. It may also depend upon the electric field (nonlinear materials) and have a time dependent response. Explicit time dependence can arise if the materials are physically moving or changing in time (e.g. reflections off a moving interface give rise to Doppler shifts). A different form of time dependence can arise in a time-invariant medium, as there can be a time delay between the imposition of the electric field and the resulting polarization of the material. In this case, P is a convolution of the impulse response susceptibility χ and the electric field E. Such a convolution takes on a simpler form in the frequency domain: by Fourier transforming the relationship and applying the convolution theorem, one obtains the following relation for a linear time-invariant medium:

\mathbf{D(\omega)} = \varepsilon (\omega) \mathbf{E}(\omega) ,

where $\omega$ is the frequency of the applied field. The constraint of causality leads to the Kramers–Kronig relations, which place limitations upon the form of the frequency dependence. The phenomenon of a frequency-dependent permittivity is an example of material dispersion. In fact, all physical materials have some material dispersion because they cannot respond instantaneously to applied fields, but for many problems (those concerned with a narrow enough bandwidth) the frequency-dependence of ε can be neglected.

At a boundary, $(\mathbf{D_1} - \mathbf{D_2})\cdot \hat{\mathbf{n}} = D_{1,\perp} - D_{2,\perp} = \sigma_\text{f}$ , where σ_f is the free charge density and the unit normal $\mathbf {\hat {n}}$ points in the direction from medium 2 to medium 1.^[1]

https://en.wikipedia.org/wiki/Electric_displacement_field

09-25-2021-1754 - time-invariant (TIV)

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A time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is a function of the time-dependent input function. If this function depends only indirectly on the time-domain (via the input function, for example), then that is a system that would be considered time-invariant. Conversely, any direct dependence on the time-domain of the system function could be considered as a "time-varying system".

Mathematically speaking, "time-invariance" of a system is the following property:^[4]^{: p. 50}

Given a system with a time-dependent output function $y(t),$ and a time-dependent input function $x(t);$ the system will be considered time-invariant if a time-delay on the input $x(t+\delta )$ directly equates to a time-delay of the output $y(t+\delta )$ function. For example, if time $t$ is "elapsed time", then "time-invariance" implies that the relationship between the input function $x(t)$ and the output function $y(t)$ is constant with respect to time $t$ :

y(t)=f(x(t),t)=f(x(t)).

In the language of signal processing, this property can be satisfied if the transfer function of the system is not a direct function of time except as expressed by the input and output.

In the context of a system schematic, this property can also be stated as follows, as shown in the figure to the right:

If a system is time-invariant then the system block commutes with an arbitrary delay.

If a time-invariant system is also linear, it is the subject of linear time-invariant theory (linear time-invariant) with direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. Nonlinear time-invariant systems lack a comprehensive, governing theory. Discrete time-invariant systems are known as shift-invariant systems. Systems which lack the time-invariant property are studied as time-variant systems.

Block diagram illustrating the time invariance for a deterministic continuous-time SISO system. The system is time-invariant if and only if

y_{2}(t)=y_{1}(t-t_{0})

for all time

t

, for all real constant

t_{0}

and for all input

x_{1}(t)

.^[1]^[2]^[3] Click image to expand it.

https://en.wikipedia.org/wiki/Time-invariant_system

09-25-2021-1752 - polarization density (or electric polarization, or simply polarization)

In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.^[1]^[2]

Polarization density also describes how a material responds to an applied electric field as well as the way the material changes the electric field, and can be used to calculate the forces that result from those interactions. It can be compared to magnetization, which is the measure of the corresponding response of a material to a magnetic field in magnetism. The SI unit of measure is coulombs per square meter, and polarization density is represented by a vector P.^[2]

Relationship between the fields of P and E[edit]

Homogeneous, isotropic dielectrics[edit]

Field lines of the D-field in a dielectric sphere with greater susceptibility than its surroundings, placed in a previously-uniform field.^[6] The field lines of the E-field are not shown: These point in the same directions, but many field lines start and end on the surface of the sphere, where there is bound charge. As a result, the density of E-field lines is lower inside the sphere than outside, which corresponds to the fact that the E-field is weaker inside the sphere than outside.

In a homogeneous, linear, non-dispersive and isotropic dielectric medium, the polarization is aligned with and proportional to the electric field E:^[7]

\mathbf {P} =\chi \varepsilon _{0}\mathbf {E} ,

where ε₀ is the electric constant, and χ is the electric susceptibility of the medium. Note that in this case χ simplifies to a scalar, although more generally it is a tensor. This is a particular case due to the isotropy of the dielectric.

Taking into account this relation between P and E, equation (3) becomes:^[3]

-Q_{b}=\chi \varepsilon _{0}\

$\oiint$

\scriptstyle {S}

\mathbf{E} \cdot \mathrm{d}\mathbf{A}

The expression in the integral is Gauss's law for the field E which yields the total charge, both free $(Q_{f})$ and bound $(Q_{b})$ , in the volume V enclosed by S.^[3] Therefore,

{\begin{aligned}-Q_{b}&=\chi Q_{\text{total}}\\&=\chi \left(Q_{f}+Q_{b}\right)\\[3pt]\Rightarrow Q_{b}&=-{\frac {\chi }{1+\chi }}Q_{f},\end{aligned}}

which can be written in terms of free charge and bound charge densities (by considering the relationship between the charges, their volume charge densities and the given volume):

\rho _{b}=-{\frac {\chi }{1+\chi }}\rho _{f}

Since within a homogeneous dielectric there can be no free charges $(\rho _{f}=0)$ , by the last equation it follows that there is no bulk bound charge in the material $(\rho _{b}=0)$ . And since free charges can get as close to the dielectric as to its topmost surface, it follows that polarization only gives rise to surface bound charge density (denoted $\sigma_b$ to avoid ambiguity with the volume bound charge density $\rho_b$ ).^[3]

$\sigma_b$ may be related to P by the following equation:^[8]

\sigma _{b}=\mathbf {\hat {n}} _{\text{out}}\cdot \mathbf {P}

where $\mathbf {\hat {n}} _{\text{out}}$ is the normal vector to the surface S pointing outwards. (see charge density for the rigorous proof)

Anisotropic dielectrics[edit]

The class of dielectrics where the polarization density and the electric field are not in the same direction are known as anisotropic materials.

In such materials, the ith component of the polarization is related to the jth component of the electric field according to:^[7]

P_{i}=\sum _{j}\epsilon _{0}\chi _{ij}E_{j},

This relation shows, for example, that a material can polarize in the x direction by applying a field in the z direction, and so on. The case of an anisotropic dielectric medium is described by the field of crystal optics.

As in most electromagnetism, this relation deals with macroscopic averages of the fields and dipole density, so that one has a continuum approximation of the dielectric materials that neglects atomic-scale behaviors. The polarizability of individual particles in the medium can be related to the average susceptibility and polarization density by the Clausius–Mossotti relation.

In general, the susceptibility is a function of the frequency ω of the applied field. When the field is an arbitrary function of time t, the polarization is a convolution of the Fourier transform of χ(ω) with the E(t). This reflects the fact that the dipoles in the material cannot respond instantaneously to the applied field, and causality considerations lead to the Kramers–Kronig relations.

If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear and is described by the field of nonlinear optics. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is usually given by a Taylor series in E whose coefficients are the nonlinear susceptibilities:

{\frac {P_{i}}{\epsilon _{0}}}=\sum _{j}\chi _{ij}^{(1)}E_{j}+\sum _{jk}\chi _{ijk}^{(2)}E_{j}E_{k}+\sum _{jk\ell }\chi _{ijk\ell }^{(3)}E_{j}E_{k}E_{\ell }+\cdots

where $\chi ^{(1)}$ is the linear susceptibility, $\chi ^{(2)}$ is the second-order susceptibility (describing phenomena such as the Pockels effect, optical rectification and second-harmonic generation), and $\chi ^{(3)}$ is the third-order susceptibility (describing third-order effects such as the Kerr effect and electric field-induced optical rectification).

In ferroelectric materials, there is no one-to-one correspondence between P and E at all because of hysteresis.

https://en.wikipedia.org/wiki/Polarization_density

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09-25-2021-1752 - Antiferroelectricity

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Antiferroelectricity is a physical property of certain materials. It is closely related to ferroelectricity; the relation between antiferroelectricity and ferroelectricity is analogous to the relation between antiferromagnetism and ferromagnetism.

An antiferroelectric material consists of an ordered (crystalline) array of electric dipoles (from the ions and electrons in the material), but with adjacent dipoles oriented in opposite (antiparallel) directions (the dipoles of each orientation form interpenetrating sublattices, loosely analogous to a checkerboardpattern).^[1]^[2] This can be contrasted with a ferroelectric, in which the dipoles all point in the same direction.

In an antiferroelectric, unlike a ferroelectric, the total, macroscopic spontaneous polarization is zero, since the adjacent dipoles cancel each other out.

Antiferroelectricity is a property of a material, and it can appear or disappear (more generally, strengthen or weaken) depending on temperature, pressure, external electric field, growth method, and other parameters. In particular, at a high enough temperature, antiferroelectricity disappears; this temperature is known as the Néel point or Curie point.

References[edit]

^ Compendium of chemical terminology - Gold Book (PDF). International Union of Pure and Applied Chemistry. 2014. Archived from the original (PDF) on 2016-09-13. Retrieved 2012-10-01.
^ Charles Kittel (1951). "Theory of Antiferroelectric Crystals". Phys. Rev. 82 (5): 729–732. Bibcode:1951PhRv...82..729K. doi:10.1103/PhysRev.82.729.

ategories:

https://en.wikipedia.org/wiki/Antiferroelectricity

09-25-2021-1751 - mesogen

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A mesogen is a compound that displays liquid crystal properties.^[1]^[2] Mesogens can be described as disordered solids or ordered liquids because they arise from a unique state of matter that exhibits both solid- and liquid-like properties called the liquid crystalline state.^[1] This liquid crystalline state (LC) is called the mesophase and occurs between the crystalline solid (Cr) state and the isotropic liquid (Iso) state at distinct temperature ranges.^[2]

The liquid crystal properties arise because mesogenic compounds are composed of rigid and flexible parts, which help characterize the order and mobility of its structure.^[2] The rigid components align mesogen moieties in one direction and have distinctive shapes that are typically found in the form of rod or disk shapes.^[2] The flexible segments provide mesogens with mobility because they are usually made up of alkyl chains, which hinder crystallization to a certain degree.^[2] The combination of rigid and flexible chains induce structural alignment and fluidity between liquid crystal moieties.^[2]

In doing so, varying degrees of order and mobility within mesogens results in different types of liquid crystal phases, Figure 1. The nematic phase (N) is the least ordered and most fluid liquid crystalline state or mesophase that is based on the rigid core of mesogen moieties.^[1]^[2] The nematic phase leads to long range orientational order and short range positional order of mesogens.^[1]^[2] The smectic (Sm) and columnar (CoI) phases are more ordered and less fluid than their nematic phases and demonstrate long range orientational order of rod-shaped and disk-shaped rigid cores, respectively.^[1]^[2]

Mesophase.

Figure 1 – Organization of rod-like and disk-like rigid cores in liquid crystal phases of mesogens, where Iso is the isotropic liquid state; N is the nematic phase of the liquid crystal state; SmA is the smectic A phase; SmC is the smectic C phase; and CoI is the columnar phase.^[2]

Thermotropic mesogens are liquid crystals that are induced by temperature^[1] and there are two classical types, which include discotic mesogens and calamitic mesogens.^[3]

Other Calamitic and Discotic Mesogens.

Discotic mesogens contain a disk-shaped rigid core and tend to organize in columns, forming columnar liquid crystal phases (CoI) of long range positional order, Figure 2.^[1]^[2]

Figure 2 – Two-dimensional lattices of mesogens with columnar mesophases, where CoIhex, CoIsqu, CoIrec and CoIob are hexagonal, square, rectangular and oblique lattice types, respectively ^[2]

An example of a discotic mesogen type rigid core is a triphenylene based disk molecule, where the hexagonal columnar liquid crystal phase exists between the 66 °C (crystalline solid phase) and 122 °C (isotropic liquid phase) temperature range, Figure 3.^[2]

Figure 3 – Example of triphenylene based disk-shaped rigid core (Temperature in °C)^[2]

Calamitic mesogens contain a rod-shaped rigid core and tend to organize in distinctive layers, forming lamellar or smectic liquid crystal phases (Sm) of long range positional order.^[1]^[2] Low order smectic phases, Figure 4, include smectic A (SmA) and smectic C (SmC) phases, while higher ordered smectic phases include smectic B, I, F, G and H (SmB/I/F/G/H) phases.^[3]

Figure 4 – Lamellar (layer) organization of low order calamitic mesogens, including the smectic A phase and smectic C phase (tilted)^[2]

An example of a calamitic mesogen type rigid core is a benzyl cyanide based rod molecule, where the smectic A liquid crystal phase exists between the 60 °C (crystalline solid phase) and 62 °C (isotropic liquid phase) temperature range, Figure 5.^[2]

Figure 5 - Example of benzyl cyanide based rod-shaped rigid core (Temperature in °C)^[2]

Bent-rod mesogens are special calamitic mesogens that contain a nonlinear rod-shaped or bent- rod shaped rigid core and organize to form ‘banana-phases.’^[3] The rigid units of these phases pack in a way so that the highest density and polar order are achieved, typically with the apex of the bent rod pointing in one direction.^[2] When a layer of bent-rods points in the same polar direction as its adjacent layers the lamellar organization is known as the smectic PF (SmPF) phase, where the F subscript indicates ferroelectric switching, Figure 6.^[2] Smectic PA (SmPA) is the term given to a layer of bent-rods that points in the opposite polar direction as its neighbouring layers, where A stands for antiferroelectic switching, Figure 6.^[2]

Figure 6 - Lamellar (layer) organization of bent-rod calamitic mesogens, including the smectic PF and smectic PA phases^[2]

Other variations of bent-rod liquid crystal phases include: antiferroelectric/ferroelectric smectic C (SmCPA/SmCPF) phases and antiferroelectric/ferroelectric smectic A (SmAPA/SmAPF) phases, which have distinctive tilt and orthogonal modes of lamellar organization.^[2]

The figure below illustrates an example of a nonlinear rod-shaped rigid core that gives rise to a bent-rod (calamitic) mesogen, where the antiferroelectric smectic C (Sm CPA) liquid crystal phase exists between the 145 °C (crystalline solid phase) and 162 °C (isotropic liquid phase) temperature range, Figure 7.^[2]^[3]

Figure 7 - Example of nonlinear rod-shaped rigid core of bent-rod mesogen, where Cr 145 Sm CPA* 162 Iso (Temperature in °C)^[3]

https://en.wikipedia.org/wiki/Mesogen

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