Shear planes stabilities

Stabilization of Shear Planes.—The development of our ideas on the factors controlling shear-plane stability is aided by a simple schematic illustration of the relationship... 

Non-stoichiometric oxides with high levels of disorder may adopt two modes of stabilization aggregation or elimination of point defects. Point defect aggregates forming clusters are examples of the former and extended defect structures like crystallographic shear-plane structures are examples of the latter. 

Until recently very little was understood as to the factors which determine whether point or extended defects are formed in a non-stoicheiometric phase, although interesting empirical correlations between shear-plane formation and both dielectric and lattice dynamical properties of the defective solid had been noted. Theoretical techniques have, however, provided valuable insight into this problem and into the related one of the relative stabilities of extended and point defect structures. The role of these techniques is emphasized in this article. 

These alternative descriptions of shear-plane formation will be valuable in our discussion of mechanism in Section 4. In the account now presented of the relative stabilities of shear plane and point defect structures we will assume that vacancies are the predominant point defects. However, the arguments we present could be adapted for metal interstitial defect structures. 

Thus to summarize, the extent of cation relaxation around a shear plane has emerged from our analysis as the most decisive factor in stabilizing shear planes with respect to point defect structures. Our discussion now continues with an account of the behaviour of the crystals at low deviations from stoicheiometry where an equilibrium may exist between point and extended defect structures. 

It seems therefore that little or no stability is to be expected for the point defect aggregates which provide the necessary shear-plane precursors in the homogeneous shear-plane formation mechanisms. These homogeneous nucleation mechanisms are therefore unlikely to operate, and we turn our attention now to a heterogeneous mechanism, in which point defects aggregate at pre-existing planar-defect sites. 

The so-called potential can be taken as a first estimate for the surface potential. The potential is the electrostatic potential at the hydrodynamic shear plane close to the particle surface. It can be determined from electrophoretic mobility measurements of the particles in an electric field (see for example Ref. [23]). The potential is zero when the charge within the shear plane is zero. This is the case as the surface charge plus the charge due to adsorbed ions other than hydrogen (for example AIOH2 in the case of alumina suspensions) is zero. This point is the iso-electro-point (i.e.p.) of the material in the dispersion medium. The suspension pH with respect to the i.e.p. is an important criterion for a first judgement of possible electrostatic stability. 

Starting with the basic concept of the electrokinetic potential of colloidal particles, the so-called zeta potential, i.e., the electrokinetic potential at the shear plane, the most important well-established methods of zeta potential determination are discussed separately. Taking into account the peculiarities of kaolin particles, the relevance of these methods for characterizing kaolin particles in the absence and the presence of polyelectrolytes are outlined here. Thereby a mixed stabilization by oppositely charged polyelectrolytes is discussed in more detail. 

At some distance from the particle surface (usually identified as the beginning of the diffuse layer in Fig. 3), a hydrodynamic shear plane exists that is characterized by the potential. The magnitude of is directly related to dispersion stability [71]. For oxides, hydroxides, and related materials, is strongly influenced by solution pH and electrolyte concentration and may be modified by surface-active species, such as oxyanions and polyelectrolytes. The key parameter characterizing a powder surface is the isoelectric point pHiep. Under pristine conditions (i.e., no surface contamination), pHiep defines the solution pH at which C = 0 and the particles exhibit a net surface charge of zero. 

Some nematic liquid crystals show positive 3 values in the neighbourhood of a nematic/smectic transition. For positive Oj the sign of the torque component does not depend on 0. This leads to a continuous rotation if the director is orientated in the shear plane (0=90°). The sign of the torque component Fq depends on 0, that is, the director is stabilized in the shear plane for two quarters of a revolution and destabilized for the other two quarters. Because of the additional influence of surface alignment and elastic torques the real movement of the director is difficult to predict [31]. 

The conditions for the first two viscosity coefficient ratios correspond to Eq. (100). A discussion of the stability of the various solutions is presented in the paper of Saupe [59]. Brand and Pleiner [61] as well as Leslie [62] discuss the flow alignment without the restriction that one director is perpendicular to the shear plane. 

The Ni octahedra derive their stability from the interactions of s, p, and d electron orbitals to form octahedral sp3d2 hybrids. When these are sheared by dislocation motion this strong bonding is destroyed, and the octahedral symmetry is lost. Therefore, the overall (0°K) energy barrier to dislocation motion is about COCi/47r where = octahedral shear stiffness = [3C44 (Cu - Ci2)]/ [4C44 + (Cu - C12)] = 50.8 GPa (Prikhodko et al., 1998), and the barrier = 4.04 GPa. The octahedral shear stiffness is small compared with the primary stiffnesses C44 = 118 GPa, and (Cn - C12)/2 = 79 GPa. Thus elastic as well as plastic shear is easier on this plane than on either the (100), or the (110) planes. 

Linear stability theory results match quite well with controlled laboratory experiment for thermal and centrifugal instabilities. But, instabilities dictated by shear force do not match so well, e.g. linear stability theory applied to plane Poiseuille flow gives a critical Reynolds number of 5772, while experimentally such flows have been observed to become turbulent even at Re = 1000- as shown in Davies and White (1928). Couette and pipe flows are also found to be linearly stable for all Reynolds numbers, the former was found to suffer transition in a computational exercise at Re = 350 (Lundbladh Johansson, 1991) and the latter found to be unstable in experiments for Re > 1950. Interestingly, according to Trefethen et al. (1993) the other example for which linear analysis fails include to a lesser degree, Blasius boundary layer flow. This is the flow which many cite as the success story of linear stability theory. 

The Bromwich contour for point A was chosen in the a -plane on a line extending from -20 to 4-20 that is below and parallel to the areal axis at a distance of 0.009 and in the w-plane it extended from -1 to - -1, above and parallel to the u>reai axis at a distance of 0.02. For the other points, the Bromwich contour in the a- plane is located at a distance of 0.001 below the Ureal axis. The choice of the Bromwich contour in the a- plane was such that all the downstream propagating eigenvalues lie above it. Orr-Sommerfeld equation was solved along these contours with 8192 equidistant points in the a- plane and 512 points in the w-plane. Orr-Sommerfeld equation was solved taking equidistant 2400 points across the shear layer in the range 0 < 2/ < 6.97. Spatial stability analysis produced waves for the four points of Fig. 4.2 with the properties shown in Table 4.1. 

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