Deformations external fields

Now many physical properties depend mainly on the behaviour of the electron in the outer part of its orbit. As an example we may mention the mole refraction or polarizability of an atom, which arises from deformation of the orbit in an external field. This deformation is greatest where the ratio of external field strength to atomic field strength is greatest that is, in the outer part of the orbit. Let us consider such a property which for hydrogen-like atoms is found to vary with nrZ t. Then a screening constant for this property would be such that... 

By assuming that the external field - deformed by the presence of the colloidal particle-and the field of the double layer are additive, D. C. Henry derived the following expression for mobility ... 

To confirm the above conjectures we have performed a numerical simulation of equation (29) on the Brusselator model chemical reaction.46 The results are shown in Fig. 7. We start with an initial condition corresponding to a clockwise wave. Under the effect of the counterclockwise field this wave is deformed and eventually its sense of rotation is reversed. In other words, the system shows a clear-cut preference for one chirality. As a matter of fact we are witnessing an entrainment phenomenon of a new kind, whereby not only the frequency but also the sense of rotation of the system are adjusted to those of the external field. More complex situations, including chaotic behavior, are likely to arise when the resonance condition w = fl, is not satisfied, but we do not address ourselves to this problem here. 

Fig. 7. Numerical simulation of equation (29) for the Brusselator model on a ring. At t = 0 a clockwise wave propagates along the ring at t = 4.49 the effect of the counterclockwise external field deforms this wave appreciably after a while the sense of rotation is reversed and for t > 22.49 one obtains a stable wave solution in the counterclockwise direction. The period of both the external field and of the linearized solution of the unperturbed system is 4.28 time units.

A precondition for the investigation of deformation of a I.c. in an external field is a uniform alignment of the I.c. with respect to the measuring cell, in order to get quantitative informations. Normally the I.c. is aligned by surface effects in the measuring cell, which usually consists of two glass plates separated by a distance of about 10 pm. We will consider three principal modes of alignment of the I.c. (Fig. 16) ... 

The title of the book, Optical Rheometry of Complex Fluids, refers to the strong connection of the experimental methods that are presented to the field of rheology. Rheology refers to the study of deformation and orientation as a result of fluid flow, and one principal aim of this discipline is the development of constitutive equations that relate the macroscopic stress and velocity gradient tensors. A successful constitutive equation, however, will recognize the particular microstructure of a complex fluid, and it is here that optical methods have proven to be very important. The emphasis in this book is on the use of in situ measurements where the dynamics and structure are measured in the presence of an external field. In this manner, the connection between the microstructural response and macroscopic observables, such as stress and fluid motion can be effectively established. Although many of the examples used in the book involve the application of flow, the use of these techniques is appropriate whenever an external field is applied. For that reason, examples are also included for the case of electric and magnetic fields. 

Composites with filler concentrations close to the percolation threshold exhibit conductivity which is sensitive to compressive deformation, since this brings the metal particles into contact, thereby forming percolation pathways. This sensitivity has been exploited especially in anisotropic composites. These are made by prealigning the metal particles with either electric or magnetic fields. This alignment is identical with that produced by external fields in electro- and magneto-rheological fluids where at a critical field continuous threads of... 

The birefringence in external electric and magnetic fields (the Kerr and Cotton-Mouton effects) can be explained by the anisotropy of the properties of the medium that is due to either the orientation of anisotropic molecules in the external field (the Langevin-Bom mechanism) or the deformation of the electric or magnetic susceptibilities by this field, i.e., to hyperpolarizabilities (Voight mechanism). The former mechanism is effective for molecules that are anisotropic in the absence of the field and... 

Cross-linked liquid crystalline polymers with the optical axis being macroscopically and uniformly aligned are called liquid single crystalline elastomers (LSCE). Without an external field cross-linking of linear liquid crystalline polymers result in macroscopically non-ordered polydomain samples with an isotropic director orientation. The networks behave like crystal powder with respect to their optical properties. Applying a uniaxial strain to the polydomain network causes a reorientation process and the director of liquid crystalline elastomers becomes macroscopically aligned by the mechanical deformation. The samples become optically transparent (Figure 9.7). This process, however, does not lead to a permanent orientation of the director. 

Kirkwood31 has also considered, and treated by statistical mechanics, the orientation polarization but not the deformation effects in a polar liquid. He considers a sphere in vacuocontaining a set of nondeformable molecules characterized by an internal moment p (and not fi). The total potential energy U is divided in two parts Ulf due to London-Van der Waals and dipole forces, is practically independent of the field U2 is due to electrostatic interactions of the dipoles with the external field. In Kirkwood s model, one finds ... 

In particular, we note that ( ) = y represents the deformability of a system. Similarly, svff) represents the ease with which the molecular electron density is deformed by an external field we refer to as the polarizability... 

The liquid crystals can be deformed by applying external fields. Even a small electric or magnetic field, shear force, surface anchoring, etc., is able to make significant distortion or deformation to liquid crystals. Thus, n is actually a function of position r. According to the symmetry of liquid crystals there exist three kinds of deformations in liquid crystals splay, twist and bend deformations, shown in Figure 1.17. The short bars in the figure represent the projections of the local directors. 

In this work, we performed electric field-assisted ion exchange on commercial borosilicate glass tubes, thus avoiding the problems associated with the deformation of the samples. The aim was to analyze the possibility of using an external field for speeding up the ion exchange process and obtaining improved mechanical properties. 

G. A. Arteca and P. G. Mezey, Chem. Phys., 161, 1 (1992). Deformation of Electron Densities in Static External Fields Shape Group Analysis for Small Molecules. 

Takatrai SC, Brady JF (2014) Swim stress, motion, and deformation of active matter effect of an external field. Soft Matter 10 9433-9445... 

Due to the effect of external fields, the order can vary in space and gradient terms have to be added to the Landau expansion (8.9). Usually, only the terms up to the quadratic order are considered. There are many symmetry allowed invariants related to gradients of the tensorial order parameter [29]. However, in the vicinity of the phase transition, one is not interested in elastic deformations of the nematic director but rather in spatial variations of the degree of nematic order. Therefore, the pretransitional nematic system is described adequately within the usual one-elastic-constant approximation. 

Since the flexoelectric effect is associated with curvature distortions of the director field it seems natural to expect that the splay and bend elastic constants themselves may have contributions from flexoelectricity. The shape polarity of the molecules invoked by Meyer will have a direct mechanical influence independently of flexoelectricity and can be expected to lower the relevant elastic constants.The flexoelectric polarization will generate an electrostatic self-energy and hence make an independent contribution to the elastic constants. In the absence of any external field, the electric displacement D = 0 and the flexoelectric polarization generates an internal field E = —P/eo, where eq is the vacuum dielectric constant. Considering only a director deformation confined to a plane, and described by a polar angle 9 z), and in the absence of ionic screening, the energy density due to a splay-bend deformation reads as ... 

Conventional low molar mass LCs as well as linear LC-polymers can be macroscopically ordered by external electric or magnetic fields, which is widely applied in optoelectronics in the case of low molar mass LCs. For LC-elastomers it is very important to know whether a macroscopic mechanical deformation of the polymer network influences the liquid crystalline side groups and whether a mechanical stress or strain produces similar effects as observed for conventional LCs by external fields. 

All above means that the ferroics can be regarded as a general notation for the materials, where at T < (so-called low-temperature phase) some reorientable physical quantities (order parameters) spontaneously appear. Latter order parameters can be of vector (spontaneous electric polarization, spontaneous magnetization) or tensor (second order tensor like spontaneous deformation or higher order tensors like elastic moduli and piezoelectric coefficients) nature. In low-temperature phase ferroics can usually split into domains, their switching being possible by the external fields. 

Polymer solutions are isotropic at equilibrium. If there is a velocity gradient, the statistical distribution of the polymer is deformed from the isotropic state, and the optical property of the solution becomes anisotropic. This phenomena is called flow birefringence (or the Maxwell effect). Other external fields such as electric or magnetic fields also cause birefringence, which is called electric bire ingence (or Kerr effect) and magnetic birefiingence (Cotton-Mouton effect), respectively. 

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