Thermal equilibrium linear polymers

PET contains about 2-3 % of short chain oligomers, which cause problems in the processing of the polymer. Oligomers can occur as linear or cyclic molecules and can be extracted by suitable solvents. Different compounds have been identified depending on the solvent and the analysis technique used [49-52], After their extraction from the polymer, oligomers will reform by thermal treatment of the extracted sample [49], and a dynamic equilibrium between polymer and oligomers has been proposed. 

The dynamics of a generic linear, ideal Gaussian chain - as described in the Rouse model [38] - is the starting point and standard description for the Brownian dynamics in polymer melts. In this model the conformational entropy of a chain acts as a resource for restoring forces for chain conformations deviating from thermal equilibrium. First, we attempt to exemphfy the mathematical treatment of chain dynamics problems. Therefore, we have detailed the description such that it may be followed in all steps. In the discussion of further models we have given references to the relevant literature. 

CycUc polyamides were reported to be isolated from Nylon 6 polymers in 1956 [18,19]. Thermal polycondensation of co-amino acid (carbon number > 6) gave a cycUc and linear polymer [82]. Moreover, upon heating polyamide in the presence of a transamidation catalyst, the cyclization equilibrium is eventually reached, and both Unear and cyclic constituents are present [83]. The proportion of the latter depends on the concentration, and cycUc compounds predominate in high dilute solutions. 

ABS is another polymer in which the associated thermal effects can become quite large even at modest stress levels and frequencies. In this polymer, in tests made at 27.6 MPa, it has been noted that AT increases linearly with frequency, in accord with Eq. (5), from a value of about 2.5 °C at 2 Hz to a value above 25 °C at 21 Hz. The influence of stress magnitude on the temperatiare rise, for a constant frequency of 21 Hz, is shown for two different polymers, PSAN and ABS, in Fig. 5. In PSAN over the whole stress range investigated, and in ABS in the range where thermal equilibrium is achieved, AT varies approximately as the square of the stress, as predicted by the preceding equations. It may also be noted from the figure that the associated thermal effects are much more severe for the rubber-modified polymer than for the unmodified PSAN. 

The term supramolecular polymer applies to any type of polymer-tike assembly that spontaneously forms by the reversible linear aggregation of one or more type of molecule in solution or in the melt. The crucial factor discriminating supramolecular from conventional or so-called dead polymers, is that for the former the monomeric and the polymeric states are in thermal equilibrium with each other, while for the latter this is not so (on the relevant experimental timescale). Examples of supramolecular polymers include the so-called giant surfactant micelles [1], peptide )3-sheet ribbons [2], self-assembled stacks of discotic molecules [3], protein fibers such as those formed by sickle cell hemoglobin [4], and so on. Chains of colloidal particles found in quite diverse contexts [5-8] and living polymers of chemically reactive species [9] also belong to the class of supramolecular polymers, if only in principle. 

The photomicrographic measurements refer directly to polymer motion under the influence of an external force. However, measurements of migration velocity v as a function of applied electrical field E show that some of these electrophoretic measurements were made in a low-field linear regime, in which the electrophoretic mobility jx is independent of E. Linear response theory and the fluctuation-dissipation theorem are then applicable they provide that the modes of motion used by a polymer undergoing electrophoresis in the linear regime, and the modes of motion used by the same polymer as it diffuses, must be the same. This requirement on the equality of drag coefficients for driven and diffusive motion was first seen in Einstein s derivation of the Stokes-Einstein equation(16), namely thermal equilibrium requires that the drag coefficients / that determine the sedimentation rate v = mg/f and the diffusion coefficient D = kBT/f must be the same. 

Logarithmic dependences of shear stress on shear rate are seen in many bulk materials such as amorphous polymers. The logarithm follows naturally from a simple model proposed by Eyring [55] that assumes shear is thermally activated and that the activation barrier decreases linearly with the applied stress. If the equilibrium barrier U is much larger than kuT, one can ignore backward motion and write that the velocity v = Doexp(—(LA — xV )/kBT ) where the coefficient V has units of volume and is called the activation volume. Solving for r, one finds... 

Recently DuPre et al reported that, S increases linearly with temperature rise. Qualitatively, their results are consistent with ours. However, the time required to reach the equilibrium pitch, varied with temperature, the concentration of polymer, and also the thermal history. For PBLG solution in dichloro-ethane (EDC), which concentration is 0.12 vol/vol, the variation of pitch with time was measured at a constant temperature by T-jump method. Fig. 1 shows the time dependence of cholesteric pitch by the T-jump method from -2°C to +30°C, 40°C and 50°C respectively. It is clear that the time required to arrive at the equilibrium pitch is shorter at higher temperature but is still over several hours. Therefore, the equilibrium pitch must be measured after prolonged aging at each measuring temperature. It was found that the... 

Conformational relaxation of polymers at temperatures below their glass transition temperature is retarded by lack of segmental motions. The conformation and free volume at the glass transition temperature continues at lower temperatures since equilibrium cannot be attained over typical experimental times. Cooperative relaxation towards conformational equilibrium depends upon temperatures, relaxation time spectrum, and the disparity between the actual and equilibrium states. The approach of the vitrified polymer to equilibrium is called thermal aging. Aging is both non-linear and nonexponential and several descriptions and models have been proposed. One model is based on a concept of temporary networks where the viscoelastic... 

Several transport properties can be evaluated from equilibrium simulations with use of linear response theory, which relates correlation fimctions of spontaneously fluctuating molecular properties to phenomenological transport coefficients. These relations can be used to evaluate diffusion coefficients, thermal conductivities, viscosities, IR spectra, and so on. However, most of these properties are evaluated more directly using appropriately devised techniques of nonequilibrium molecular dynamics. Particularly challenging for polymers is the direct... 

In most cases, the flow properties of polymers in solution or in a molten state are Newtonian, pseudoplastic, or a combination of both. In the case of liquid crystal polymer solutions, the flow behavior is more complex. The profound difference in the rheological behavior of ordinary and liquid crystalline polymers is due to the fact that, for the flrst ones, the molecular orientation is entirely determined by the flow process. The second ones are anisotropic materials already at equilibrium (Acierno and Brostow 1996). The spontaneous molecular orientation is already in existence before the flow and is switched on, varying in space, over distances of several microns or less (polydomain). If one ignores the latter, one can discuss the linear case (slow flow) as long as the rate of deformation due to flow (the magnitude of the symmetric part of the velocity gradient) is lower than the rate at which molecules rearrange their orientational spread by thermal motions. 

---