Polymers elastic constants

Fig. 2. Range of measured polymer elastic constants in the chain direction (left-hand side) compared with moduli based on fundamental deformation mechanisms (right-hand side). (Reproduced by kind permission of the Internationa Union of Pure and...

Using these general notions, the authors of Ref. [6, 7] offered the fractal models for polymers elastic constants description. The quasiequilibrium state of polymers structure is characterized by the criterion D = 3 [8, 9], where is dimension of excess energy localization domains. A loosely packed matrix is totality of such domains. The value can be determined within the frameworks of free volume fractal theory according to the equation [9] ... 

Since according to the indicated above reasons two order parameters are required, as a minimum, for solid-phase polymers elastic constants description, then variable percolation threshold should be introduced in the Eq. (3. 1), that is,p should be replaced on A. Besides, it has been shown earlier, that for polymers structure Vp 1 (see Table 1.1) [10] and therefore, T[=d - 1 can be assumed in the Eq. (3.2) as the first approximation. Then the Eq. (3.1) assumes the following form [6, 7] ... 

Novikov, V. U., Kozlov, G. V. (2000). Fractal Analysis of Polymers Elastic Constants. Materialovedenie, 1,2-12. 

Table 8 presents a survey of the basic elastic constants of a series of polymer fibres and the relation with the various kinds of interchain bonds. As shown by this table, the interchain forces not only determine the elastic shear modulus gy but also the creep rate of the fibre. 

Table 8 The basic elastic constants g and ec, the highest filament values of the modulus ( ) and the strength (q,), together with the average values of the creep compliance (/(f)) at 20 °C (ratio of creep rate and load stress) and the interchain bond for a variety of organic polymer fibres... 

DNA, on a length scale beyond some tens of base pairs, can be described as a stiff polymer chain with an electrostatic charge. In the most basic view, four parameters are sufficient to describe this chain with sufficient precision the diameter, the elastic constants for bending and torsion and the electrostatic potential. 

Another class of thermodynamic barrier theories focuses on the large increases in the elastic constants that accompany glass formation. (These theoretical approaches seem especially appropriate to polymer fluids below the crossover temperature Fj.) In particular, the barrier height governing particle displacement in the shoving model [57] is taken to be on the order of the elastic energy GqoVo required to displace a particle on a scale comparable to the interparticle distance,... 

Figure 5.120 Elastic constants for unidirectional (a) glass fiber reinforced epoxy and (b) graphite fiber-reinforced epoxy laminae. Reprinled, by permission, from P. C. Powell and A. J. I. Housz, Engineering with Polymers, pp. 222, 223. Copyrighl 1998 by Stanley Thorned Publishers.

Some fermentation broths are non-Newtonian due to the presence of microbial mycelia or fermentation products, such as polysaccharides. In some cases, a small amount of water-soluble polymer may be added to the broth to reduce stirrer power requirements, or to protect the microbes against excessive shear forces. These additives may develop non-Newtonian viscosity or even viscoelasticity of the broth, which in turn will affect the aeration characteristics of the fermentor. Viscoelastic liquids exhibit elasticity superimposed on viscosity. The elastic constant, an index of elasticity, is defined as the ratio of stress (Pa) to strain (—), while viscosity is shear stress divided by shear rate (Equation 2.4). The relaxation time (s) is viscosity (Pa s) divided by the elastic constant (Pa). 

The ratio of elastic constants Ku, calculated for the S-effect according to the equation (4) appeared to be (Kn (polymer XIV)/Kn (polymer XIII)) x 1 100 and (Ku (polymer XVI)/Kn (polymer XV)) x 1 36. Yet, as we have just indicated, taking into account molecular masses of the LC polymers and reducing k, values for various polymers to equal values of DP one may come to substantially different values for ratios of constants presented. It is necessary to note that up to date no quantitative data on the determination of elastic constants of LC polymers has been published (excluding the preliminary results on Leslie viscosity coefficients for LC comb-like polymer127)). Thus, one of the important tasks today is the investigation of elastic and visco-elastic properties of LC polymers and their quantitative description. 

For anisotropic materials torsion is discussed in the books by Love, Lekhnitskii175 and Hearmon185. The torque M now depends not upon one elastic constant only, as in the isotropic case, but upon two. This makes the determination of shear modulus by a torsion test a difficult task and requires careful experimentation. Early work on this for polymers was done by Raumann195, by Ladizesky and Ward205 and by Arridge and Folkes165. 

The sets of equations are solved by the assumption of periodic waves and, by expansion in powers of the wave number, a relation is found for the limiting case of long waves so that the elements of the dynamical matrix elastic constants of the continuum. It is also possible to derive the Raman frequencies from the lattice dynamics analysis but this does not seem to have been done for polymer crystals, though they have been derived for example, for NaCl and for diamond. 

The high viscosity of the PMMA damps out orientational contributions so that the yqeo that is measured is thought to be =60-90% electronic. This has been ascertained by measuring the electric field induced second harmonic generation (EFISH) below the Tg of the polymer. From this can be obtained the microscopic elastic constant, which can in turn be used to estimate the magnitude of the two orientational contributions to yQEO- Details are provided else where (13,16). 

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]... 

This replica-trick method was used in the Refs. [60,61] for the polymer network free energy calculation. For averaging of Z"-value over the ensemble of realizations the probability distribution should be chosen. The authors of Refs. [60,61] have used the condition of the thermodynamical equilibrium for the Gibbs probability distribution corresponding to the conditions of network preparation. To our mind, it is a beautiful idea but it should be considered more deeply because not all the types of networks can be described in such a way - many networks cannot be prepared under any equilibrium conditions. Using some additional tube-like approximations, the authors have obtained rather simple results for network elastic constants and for some other parameters. 

For amorphous polymers the constant n may vary between 0.5 and 1.0 n, a dimensionless number, is a measure of the relative importance of elastic and viscous contributions to stress relaxation it is closely related to tan 8 ... 

The big difference between normal isotropic liquids and nematic liquids is the effect of anisotropy on the viscous and elastic properties of the material. Liquid crystals of low molecular weight can be Newtonian anisotropic fluids, whereas liquid crystalline polymers can be rate and strain dependent anisotropic non-Newtonian fluids. The anisotropy gives rise to 5 viscosities and 3 elastic constants. In addition, the effective flow properties are determined by the flow dependent and history dependent texture. This all makes the rheology of LCPs extremely complicated. 

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