Film elasticity theories

Theories concerning the mechanisms of operation of this film elasticity depend on two observations concerning the surface tension of aqueous solutions of surface-active solutes (1) its increase in value with decrease in concentration of the surface-active solute, at concentrations of the latter below the CMC, and (2) the time required for it to obtain its equilibrium value (the fact that the initial value of these surface tension at a new surface is always greater than the equilibrium... 

The elementary surface excited states of electrons in crystals are called surface excitons. Their existence is due solely to the presence of crystal boundaries. Surface excitons, in this sense, are quite analogous to Rayleigh surface waves in elasticity theory and to Tamm states of electrons in a bounded crystal. Increasing interest in surface excitons is provided by the new methods for the experimental investigation of excited states of the surfaces of metals, semiconductors and dielectrics, of thin films on substrates and other laminated media, and by the extensive potentialities of surface physics in scientific instrument making and technology. 

Hydrogen incorporation in thin films can produce an extremely high out-of-plane expansion due to the clamping of the thin film to the substrate [20]. Within the sohd solution phase the expansion can be predicted by linear elastic theory [21]. However, deviations from linear elastic theory were reported for high H concentrations due to the onset of plastic deformation in the film [22, 23]. 

Tandon GP, Weng GJ (1984) The effect of aspect ratio of inclusion on elastic properties of unidirectionally aligned composites. Polym Compos 5 327-333 Tanner RI (1960) Full-film lubrication theory for a Maxwell liquid. Int J Mech Sci 1 206-215 Tanner RI (1975) Stresses in dilute solutions of bead-nonlinear-spring macromolecules, ii. Unsteady flows and approximate constitutive relations, constitutive relations. J Rheol 19 37-65... 

Fibrin can also be prepared in the form of films and fibers which undergo supercontraction on heating. Preliminary attempts to apply the elasticity theory of Flory (1956b), discussed in Chapter IV, to such systems thus far indicate that fibrin may not show rubberlike elastiiaty above the transition temperature (I oeb, 1960). It may be that this transition is a polymorphic one between two crystalline forms rather than between crystalline and amorphous phases, as discussed in Chapter IV. At any rate, such studies may shed further light on the nature of fibrin and the mechanism of its formation. 

The polymer-surfactant complex has high surfeice viscosity and elasticity (i.e. surfeice viscoelasticity), both will enhance the foam stability (see below). The amphoteric surfactants such as betaines and the phospholipid surfeictants when used in conjunction with alkyl sulfeites or alkyl ether sulfeites can also enhance the foam stability. All these molecules strengthen the film of surfactant at the air/water interface, thus modifying the lather from a loose lacy structure to a rich, dense, small bubble size, luxurious foam. Several foam boosters have been suggested and these include fatty acid alkanolamide, amine oxides. Fatty alcohol and fatty acids can also act as foam boosters when used at levels of 0.25-0.5 %. Several approaches have been considered to explain foam stability (a) Surface viscosity and elasticity theory The adsorbed surfeictant film is assumed to control the mechanical-dynamical properties of the surface layers by virtue of its surface viscosity and elasticity. This may be true for thick films (> 100 nm) whereby intermolecular forces are less dominant. Some correlations... 

Theoretically, the elasticity theory of continuous media may be used to study the long-wavelength modes. To determine the microscopic modes, numerical approaches are necessary. Most of them have used Born s model to estimate the inter-atomic forces. The semi-infinite crystals are modelled by thin films, whose thickness must be larger than the attenuation length of the surface modes. The complete MgO(OOl) phonon spectrum has been calculated, neglecting (Chen et al, 1977 Barnett and Bass, 1979) or taking into account (Lakshmi and de Wette, 1980) the surface relaxation. The same has been done for SrTiO3(001) (Prade et al, 1993). 

For a liquid to form a foam, it must be able to form a membrane around the gas bubble possessing a form of elasticity that opposes the thinning of the lamellae as a result of drainage. Foaming does not occur in pure liquids because no such mechanism for the retardation of lamellae drainage or interfacial stabilization exists. When amphiphilic materials or polymers are present, however, their adsorption at the gas-liquid interface serves to retard the loss of liquid from the lamellae and, in some instances, to produce a more mechanically stable system. Theories related to such film formation and persistence, especially film elasticity, derive from a number of experimental observations about the surface tension of liquids ... 

Before discussing how the dislocations are introduced for strain relief, let us return to consideration of when strain energies will favor introduction of dislocations in more detail. For a detailed discussion of the following model, see Hirth and Lothe or any similar book for background on elasticity theory and Matthews or other sources for the specific application of elasticity theory to lattice-mismatched structures. A complete analysis shows that the elastic strain energy, Ed, in a film of thickness h on a much thicker substrate is given by ... 

Films of high misfit such as might lead to dislocation loop formation do not grow as uniform thin layers, but rather form islands on the substrates. The critical thickness analysis does not hold well for such a system, as strain relief can occur both by dislocation formation and by relaxation of the island without dislocations. Relaxation without dislocation formation has been extensively modeled using linear elasticity theory by Johnson and Freimd [10], as in Figure 7.23. The model shows that for moderate misfits and moderate film thicknesses, strain can be maintained in small islands well beyond the critical thickness calculated with Equation 7.30. 

It has been also shown that when a thin polymer film is directly coated onto a substrate with a low modulus ( < 10 MPa), if the contact radius to layer thickness ratio is large (afh> 20), the surface layer will make a negligible contribution to the stiffness of the system and the layered solid system acts as a homogeneous half-space of substrate material while the surface and interfacial properties are governed by those of the layer [32,33]. The extension of the JKR theory to such layered bodies has two important implications. Firstly, hard and opaque materials can be coated on soft and clear substrates which deform more readily by small surface forces. Secondly, viscoelastic materials can be coated on soft elastic substrates, thereby reducing their time-dependent effects. 

This type of lubrication provides the answer to why many mechanisms operate under conditions that are beyond the limits forecast by theory. It was previously thought that increasing pressure reduced oil film thickness until the aspirates broke through, causing metal-to-metal contact. Research has shown, however, that the effect on mineral oil of high contact pressure is a large increase in the viscosity of the lubricant. This viscosity increase combined with the elasticity of the metal causes the oil film to act like a thin solid film, thus preventing metal-to-metal contact. 

Flow of trains of surfactant-laden gas bubbles through capillaries is an important ingredient of foam transport in porous media. To understand the role of surfactants in bubble flow, we present a regular perturbation expansion in large adsorption rates within the low capillary-number, singular perturbation hydrodynamic theory of Bretherton. Upon addition of soluble surfactant to the continuous liquid phase, the pressure drop across the bubble increases with the elasticity number while the deposited thin film thickness decreases slightly with the elasticity number. Both pressure drop and thin film thickness retain their 2/3 power dependence on the capillary number found by Bretherton for surfactant-free bubbles. Comparison of the proposed theory to available and new experimental... 

Frankel and Acrivos11 have obtained models with well-defined hydrodynamics for very high concentrations of rigid and elastic particles. Here the solvent forms thin films and we enter the region of lubrication theory. The expressions describing the flow do bear some similarities to the semi-empirical expressions developed at lower concentrations. For example Frankel and Acrivos give... 

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