Measurement of Surface Elastic Constants

R286 D. G. Cory, R. Laflamme, E. Knill, L. Viola, T. F. Havel, N, Boulant, G. Boutis, E. Fortunato, S. Lloyd, R. Martinez, C. Negrevergne, M. Pravia, Y. Sharf, G. Teklemariam, Y. S. Weinstein and W. H. Zurek, NMR Based Quantum Information Processing Achievements and Prospects , Forschr. Phys., 2000, 48, 875 R287 G. P. Crawford, Measurement of Surface Elastic Constants , EMIS Datarev. Ser., 2000,25, 230... 

The effect of surface elastic constants on the nematic director configurations is of basic interest for the elastic theory of liquid crystals and plays a critical role in those device applications where the nematic is confined to a curved geometry. The saddle-splay surface elastic constant, K24, and the splay-bend surface elastic constant, K13, defied measurement for more than sixty years, since the pioneering work of Oseen, who made the first steps toward the elastic theory of liquid crystals. 

As the current pulse is largely dominated by the stress differences, a short duration current pulse is observed upon loading with a quiescent period during the time at constant stress. With release of pressure upon arrival of the unloading wave from the stress-free surface behind the impactor, a current pulse of opposite polarity is observed. The amplitude of the release wave current pulse provides a sensitive measure of the elastic nonlinearity of the target material at the peak pressure in question. 

Fig. 5 Dependency of the measured shell spring constant on speed the speed of the AFM probe deforming the surface is varied over 3 orders of magnitude which results, within the accuracy of the measurement, in no change of the elastic constants...

The simplest method of measuring the three elastic constants of a nematic liquid crystal is by studying the deformations due to an external magnetic field (Freedericksz and Tsvetkov, Zocher ). The geometry has to be so chosen that the orienting effect of the field conflicts with the orientations imposed by the surfaces with which the liquid crystal is in contact. To develop a static theory of such deformations we apply the equation of... 

Membrane shear modulus A measure of the elastic resistance of the membrane to surface shear deformation that is, changes in the shape of the surface at constant surface area (Equation 60.8). (Units 1 mN/m = 1 dyn/cm)... 

The simplest theory for the analysis of surface elasticity based on AFM force-distance curve measurement is Hertzian contact mechanics [Landau and Lifshitz, 1967], As shown schematically in Figure 3.1, a force-distance curve is a plot of the displacement, z, of the piezoelectric scanner normal to the specimen s surface as the horizontal axis and the cantilever deflection, A, as the vertical axis. Hertzian contact mechanics cannot treat adhesive force in principle. We need to make some effort to minimize the adhesive force in a practical experiment. Measurement in aqueous conditions is effective for polymeric materials with low water absorbability. A cantilever with a large spring constant also hides weak van der Waals forces. Figure 3.1a shows the... 

Equation (102) shows that MAQO can provide important information about the electronic parameters (extremal Fermi surface cross-sectional area, effective masses, electronic relaxation times) and about the electron-phonon interaction (strain derivatives of the cross-sectional area for different symmetry strains). With the help of this technique, combined with de Haas-van Alphen susceptibility measurements, one can put the deformation potential interaction and the temperature dependence of the elastic constants, discussed above in sect. 3.2, on a solid basis. In the following we discuss some compounds. 

All other methods start with a director deformation the relaxation of which takes place in a plane perpendicular to the boundaries. In this case standard methods for the determination of the director orientation, which are also applied to the determination of elastic constants, can be used (e.g. light intensity measurements behind crossed polarizers or measurement of the dielectric constant). Usually a cell with planar surface orientation is used. By means of a magnetic [65] or electric [66] field perpendicular to the plates and somewhat larger than the critical value, a small splay deformation of the director is produced. The relaxation to the planar equilibrium orientation after switching off the field is followed by the methods discussed above. In contrast to the twist deformation, inhomogeneous rotation of the director causes a shear flow in the cell, which is called backflow. The observed relaxation time for a small deformation will be shorter than the analogue to Eq. (40)... 

Nevertheless an independent measurement of the eompressional elastic constant would be extremely important. The low value of 7 is due to the perfluorinated tail. Subsequently, a direet measurement of surface tension confirmed this value of 7 [19]. In comparison with terminal aliphatic compounds, the reduction of 7and enhancement in B leads to v =jl BK) )<. Thus surface damping of layer fluctuations is weaker. 

As an optical technique, Brillouin scattering allows the determination of the elastic constants and hence of the bulk moduli through the interaction of light with thermal excitation in a material, in particular acoustic phonons in a crystal. In this technique, the elastic constants Cn and Cgg can be directly obtained from measurement of the phase velocity of the longitudinal mode and of the shear horizontal mode traveling parallel to the crystal surface. The remaining constants,... 

Measurements of the saddle-splay surface elastic constant, K24, and the splay-bend surface elastic constant, K13, were first introduced by Oseen [1] in 1933 from a phenomenological viewpoint, and later by Nehring and Saupe [2] from a molecular standpoint. These constants tend to be neglected in conventional elastic continuum treatments for fixed boundary conditions because they do not ent the Euler-Lagrange equation for bulk equilibrium. Experimental determination of tiie two surface elastic constants is undoubtedly a difficult task, since their effects are hard to discriminate from those of ordinary sur ce anchoring [3]. 

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