The elasticity modulus

The reorientation of molecules in the adsorption layer should have a strong effect on the surface elasticity modules (cf paragraph 4.5 of Chapter 4). The Gibbs elasticity modulus (, =-(dy/dlnT), =(dy/dln A)p can be calculated from the equation of state (2.84) together with the relationships (2.85)-(2.88). Therefore, this value should reflect the processes involved in the equilibrium transition between the adsorption states.  

The viscoelasticity is a complex number determined by the dilatational elasticity and viscosity [19, 94, 95]. The viscoelasticity modulus (or surface dilatational modulus) incorporates a real and imaginary constituent, elasticity and viscosity, respectively. 

The diffusion relaxation frequency, in turn, is determined by the diffusion coefficient, adsorption and bulk concentration of the surfactant 

As example we will calculate the elasticity module for CjoEOg, using the following parameters (compare with Chapter 3, Table 3.13) water/air interface - t02 = 4.010 mVmol, 1 = 1.2-10 mVmol, a = 3.0, water/hexane interface - 2 = 4.M0 mVmol, 

It seems instructive to discuss the nature of the minima in the plots of Fig. 2.10. For CioEOg at the water/hexane interface the elasticity modulus attains a minimum at n = 25 mN/m. This can be explained by the influence of the surface pressure on the composition of the surface layer comprised of surfactant molecules capable to adsorb in two states, see Fig. 2.11. Moreover, at n 11 mN/m for the water/hexane interface, and H 7 mN/m for the water/air interface, the adsorption in state 1 starts to decrease significantly. At the same time, adsorption in state 2 increases continuously as usual - with increasing pressure the adsorption becomes higher. As the changes of adsorption in states 1 and 2 are opposite to each other, the resulting changes in the total adsorption and surface tension are small. 

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In AFM, the relative approach of sample and tip is nonnally stopped after contact is reached. Flowever, the instrument may also be used as a nanoindenter, measuring the penetration deptli of the tip as it is pressed into the surface of the material under test. Infomiation such as the elastic modulus at a given point on the surface may be obtained in tliis way [114], altliough producing enough points to synthesize an elastic modulus image is very time consuming. 

Secondly, the ultimate properties of polymers are of continuous interest. Ultimate properties are the properties of ideal, defect free, structures. So far, for polymer crystals the ultimate elastic modulus and the ultimate tensile strength have not been calculated at an appropriate level. In particular, convergence as a function of basis set size has not been demonstrated, and most calculations have been applied to a single isolated chain rather than a three-dimensional polymer crystal. Using the Car-Parrinello method, we have been able to achieve basis set convergence for the elastic modulus of a three-dimensional infinite polyethylene crystal. These results will also be fliscussed. 

Moduli and Poisson s Ratio. The Young s modulus of vitreous sihca at 25°C is 73 GPa (<1.06 x 10 psi), the shear modulus is 31 GPa (<4.5 X 10 psi), and the Poisson s ratio is 0.17. Minor differences in values can arise owing to density variations. The elastic modulus decreases with increasing density and Poisson s ratio increases (26). 

Elasticity is another manifestation of non-Newtonian behavior. Elastic Hquids resist stress and deform reversibly provided that the strain is not too large. The elastic modulus is the ratio of the stress to the strain. Elasticity can be characterized usiag transient measurements such as recoil when a spinning bob stops rotating, or by steady-state measurements such as normal stress ia rotating plates. 

Here c[-], which will be called the elastic modulus tensor, is a fourth-order linear mapping of its second-order tensor argument, while b[-], which will be called the inelastic modulus tensor, is a linear mapping of k whose form will depend on the specific properties assigned to k. They depend, in general, on and k. For example, if k consists of a single second-order tensor, then in component form... 

While c in (5.112) is a linear function of d, it may be an arbitrary function of s. Truesdell considered cases where c is a polynomial in s, terming (5.112) a hypoelastic equation of grade n, where n is the power of the highest-order term in the polynomial. For a hypoelastic equation of grade zero, the elastic modulus c is independent of s and linear in dand therefore has the representation (A.89). It is convenient to nondimensionalize the stress by defining s = sjljx. Since the stress rate must vanish when d is zero, Cq = 0 and the result is... 

Orowan (1949) suggested a method for estimating the theoretical tensile fracture strength based on a simple model for the intermolecular potential of a solid. These calculations indicate that the theoretical tensile strength of solids is an appreciable fraction of the elastic modulus of the material. Following these ideas, a theoretical spall strength of Bq/ti, where Bq is the bulk modulus of the material, is derived through an application of the Orowan approach based on a sinusoidal representation of the cohesive force (Lawn and Wilshaw, 1975). 

To conclude, the concept of bond stiffness, based on the energy/distance curves for the various bond types, goes a long way towards explaining the origin of the elastic modulus. But we need to find out how individual atom bonds build up to form whole pieces of material before we can fully explain experimental data for the modulus. The... 

Note When the elastic modulus is used, the elastic limit or proportional limit should be used with it in the formula. When the plastic modulus or Secant Modulus is used, it should be used with the corresponding yield strength. 

The elastic modulus and strength are related by a Griffith theory type relationship. 

Via an ad hoc extension of the viscoelastic Hertzian contact problem, Falsafi et al. [38] incorporated linear viscoelastic effects into the JKR formalism by replacing the elastic modulus with a viscoelastic memory function accounting for time and deformation, K t) ... 

Thus the Quesnel model predicts a range of power law values from 2/3 to 0.771 depending on the ratio of the elastic modulus of the particle and the substrate. The 2/3 behavior is for a compliant particle and the 0.771 is for a compliant substrate. The transition should be smooth with changes in modulus, but does not depend on the size of the particle. 

The question of the actual form of the relation between toughness and crack speed is still rather unclear. It is tempting to relate 4> V) to the imaginary part of the elastic modulus, but the size parameter required to relate V to a time or... 

Tackifying resins enhance the adhesion of non-polar elastomers by improving wettability, increasing polarity and altering the viscoelastic properties. Dahlquist [31 ] established the first evidence of the modification of the viscoelastic properties of an elastomer by adding resins, and demonstrated that the performance of pressure-sensitive adhesives was related to the creep compliance. Later, Aubrey and Sherriff [32] demonstrated that a relationship between peel strength and viscoelasticity in natural rubber-low molecular resins blends existed. Class and Chu [33] used the dynamic mechanical measurements to demonstrate that compatible resins with an elastomer produced a decrease in the elastic modulus at room temperature and an increase in the tan <5 peak (which indicated the glass transition temperature of the resin-elastomer blend). Resins which are incompatible with an elastomer caused an increase in the elastic modulus at room temperature and showed two distinct maxima in the tan <5 curve. 

Material added to a plastic to increase its workability and flexibility. Plasticizers tend to lower the melt viscosity, the glass transition temperature and/or the elastic modulus. 

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