Surface elastic constants

Our intention is to give a brief survey of advanced theoretical methods used to detennine the electronic and geometric stmcture of solids and surfaces. The electronic stmcture encompasses the energies and wavefunctions (and other properties derived from them) of the electronic states in solids, while the geometric stmcture refers to the equilibrium atomic positions. Quantities that can be derived from the electronic stmcture calculations include the electronic (electron energies, charge densities), vibrational (phonon spectra), stmctiiral (lattice constants, equilibrium stmctiires), mechanical (bulk moduli, elastic constants) and optical (absorption, transmission) properties of crystals. We will also report on teclmiques used to study solid surfaces, with particular examples drawn from chemisorption on transition metal surfaces. 

Mehl M J and Papaconstantopoulos D A 1996 Applications of a tight-binding total-energy method for transition and noble metals Elastic constants, vacancies and surfaces of monatomic metals Phys. Rev. B 54 4519... 

As computer power continues to increase over the next few years, there can be real hope that atomistic simulations will have major uses in the prediction of phases, phase transition temperatures, and key material properties such as diffusion coefficients, elastic constants, viscosities and the details of surface adsorption. 

By measuring V z), which includes examining the reflectance function of solid material, measuring the phase velocity and attenuation of leaky surface acoustic waves at the liquid-specimen boundary, the SAM can be used indetermining the elastic constants of the material. 

The first term in both Equations 17 and 18 is the constant surface-tension contribution and the second term gives the first-order contribution resulting from the presence of a soluble surfactant with finite sorption kinetics. A linear dependence on the surfactant elasticity number arises because only the first-order term in the regular perturbation expansion has been evaluated. The thin film thickness deviates negatively by only one percent from the constant-tension solution when E = 1, whereas the pressure drop across the bubble is significantly greater than the constant-tension value when E - 1. 

R y z Density [kg/m3] Specific surface area [m2/g] Elasticity constant cu [MPa]... 

As is known, if one blows air bubbles in pure water, no foam is formed. On the other hand, if a detergent or protein (amphiphile) is present in the system, adsorbed surfactant molecules at the interface produce foam or soap bubble. Foam can be characterized as a coarse dispersion of a gas in a liquid, where the gas is the major phase volume. The foam, or the lamina of liquid, will tend to contract due to its surface tension, and a low surface tension would thus be expected to be a necessary requirement for good foam-forming property. Furthermore, in order to be able to stabilize the lamina, it should be able to maintain slight differences of tension in its different regions. Therefore, it is also clear that a pure liquid, which has constant surface tension, cannot meet this requirement. The stability of such foams or bubbles has been related to monomolecular film structures and stability. For instance, foam stability has been shown to be related to surface elasticity or surface viscosity, qs, besides other interfacial forces. 

For a nematic LC, the preferred orientation is one in which the director is parallel everywhere. Other orientations have a free-energy distribution that depends on the elastic constants, K /. The orientational elastic constants K, K22 and K33 determine respectively splay, twist and bend deformations. Values of elastic constants in LCs are around 10 N so that free-energy difference between different orientations is of the order of 5 x 10 J m the same order of magnitude as surface energy. A thin layer of LC sandwiched between two aligned surfaces therefore adopts an orientation determined by the surfaces. This fact forms the basis of most electrooptical effects in LCs. Display devices based on LCs are discussed in Chapter 7. 

In (6.49) and (6.50) the x-dependence of 0 and tf/y is the same (this is one of the boundary conditions, equivalent to Snell s law), and the z-dependence is simply unity at the surface. The explicit dependence on the elastic constants has been eliminated, so two simultaneous equations remain. From (6.49),... 

Surface acoustic-wave (SAW) elements Plate-mode oscillators Interface impedance elements Fiber optic elements sensitive to elastic constants... 

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

Successful assembly requires matching features of the template, meaning that wavelength and height have to be of the same dimensions. Additionally, adhesion of particles and surface must be avoided by weak repulsive forces. In this context, polyelectrolyte multilayer-wrinkles are particularly useful, as the wettability of the multilayer is determined by the part of the layer adjacent to the film/solution or film/air interface respectively, while the elastic properties are determined by the total film [84], Thus, elastic constants can be adjusted largely independent from wettability properties. 

The dilational rheology behavior of polymer monolayers is a very interesting aspect. If a polymer film is viewed as a macroscopy continuum medium, several types of motion are possible [96], As it has been explained by Monroy et al. [59], it is possible to distinguish two main types capillary (or out of plane) and dilational (or in plane) [59,60,97], The first one is a shear deformation, while for the second one there are both a compression - dilatation motion and a shear motion. Since dissipative effects do exist within the film, each of the motions consists of elastic and viscous components. The elastic constant for the capillary motion is the surface tension y, while for the second it is the dilatation elasticity e. The latter modulus depends upon the stress applied to the monolayer. For a uniaxial stress (as it is the case for capillary waves or for compression in a single barrier Langmuir trough) the dilatational modulus is the sum of the compression and shear moduli [98]... 

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...

Surface stress — The surface area A of a solid electrode can be varied in two ways In a plastic deformation, such as cleavage, the number of surface atoms is changed, while in an elastic deformation, such as stretching, the number of surface atoms is constant. Therefore, the differential dUs of the internal surface energy, at constant entropy and composition, is given by dUs = ydAp + A m g m denm, where y is the interfacial tension, dAp is the change in area due to a plastic deformation, gnm is the surface stress, and enm the surface strain caused by an elastic deformation. Surface stress and strain are tensors, and the indices denote the directions of space. From this follows the generalized Lippmann equation for a solid electrode ... 

However, this is not always true. Complications arise, for example, if the adsorbent undergoes some form of elastic deformation or if the pore structure is modified as a result of the adsorption process. We adopt this convention in order to simplify the thermodynamic treatment. Similarly, we assume that the area of the Gibbs dividing surface is equal to the constant surface area of the adsorbent. We must not forget that we have made these simplifying assumptions when we come to interpret experimental data - especially if there is any indication of low pressure hysteresis. 

The oscillating bubble method proves to be very convenient and precise for the evaluation of the non-equilibrium elasticity of surfaces in a wide range of frequencies of external disturbances and surface coverage (adsorption of surfactant) [103-105]. It is based on registration of the sinusoidal variation of bubble volume. The bubble is situated in a capillary containing surfactant solution in which oscillations of different frequencies and amplitudes are created. The treatment of the U = f(ft)) curves (where U is the tension needed to initiate oscillations of constant amplitude) allows the determination of Marangoni elasticities [105]. 

The spatial and temporal response of a nematic phase to a distorting force, such as an electric (or magnetic) field is determined in part by three elastic constants, kii, k22 and associated with splay, twist and bend deformations, respectively, see Figure 2.9. The elastic constants describe the restoring forces on a molecule within the nematic phase on removal of some external force which had distorted the nematic medium from its equilibrium, i.e. lowest energy conformation. The configuration of the nematic director within an LCD in the absence of an applied field is determined by the interaction of very thin layers of molecules with an orientation layer coating the surface of the substrates above the electrodes. The direction imposed on the director at the surface is then... 

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