Thermal Analysis of Polymeric Materials Part 10 docx

5 Structure and Properties of Materials ___________________________________________________________________ 526 Fig. 5.100 Fig. 5.99 By inspection, one can see that the kink is the result of the collision of a transverse (A), torsional (B), and longitudinal vibration (C) between 0.5 and 1.1 ps (see also Fig. 1.47). After formation, this particular kink defect had a life time of about 2 ps. The mechanism of twisting of a single chain is illustrated in Fig. 5.102. One can see a very gradual twist starts at about 1.5 ps, and is completed after 2.3 ps. After this 5.3 Defects in Polymer Crystals ___________________________________________________________________ 527 Fig. 5.101 Fig. 5.102 twist, the whole chain has been rotated by about 90 o . It remains with minor fluctuations in this position until 7 ps when the rotation reverses. The twist is so gradual that no gauche conformations are necessary for its occurrence. The sequence of Figs. 5.103 105 illustrates the details of the diffusion of a chain through the crystal. The figures refer to a chain that was driven into the direction of 5 Structure and Properties of Materials ___________________________________________________________________ 528 Fig. 5.103 Fig. 5.104 the indicated gradient by application of an external force. Figure 5.103 illustrates the gauche conformations that appear during the diffusion (compare to Fig. 5.98). There is no direct involvement of the gauche defects in the motion. At later time, the increase in gauche bonds is connected with the chain having moved out of the crystal in the positions above 85. 5.3 Defects in Polymer Crystals ___________________________________________________________________ 529 Fig. 5.105 Figure 5.104 shows more details how the chain moves. The almost horizontal, dotted lines represent the non-moving chains which surround the chain being moved out of the crystal by the force gradient. One should note the initial pulling of the lower chain-end into the crystal at 0 2 ps. This is followed by an expulsion of the upper chain-end out of the crystal between 2 and 4 ps. The mechanism of the diffusion of the chain is shown even better in the enlarged plot of the end-to-end distance of the moving chain in Fig. 5.105. The longitudinal acoustic mode of vibration (LAM) is clearly visible. From the LAM frequency one can extract a sound velocity in the chain direction which agrees with experiments. On this LAM vibration a series of spikes is superimposed which occur on motion of the chain into or out-of the crystal as seen in Fig. 5.104. The sliding diffusion of the chain through the crystal is, thus, coupled strongly with the skeletal vibrations and does not involve diffusion of the point defects themselves, but rather seems to be connected with the twisting motion of the chain which is illustrated in the movie of the chain dynamics of a single chain in Fig. 5.102. The change of the rate of diffusion with temperature is illustrated in Fig. 5.106. Although point defects are not directly involved in the motion, they help in the motion of the chain. The motion decreases towards zero in the temperature range where the concentration of gauche bonds reaches zero in Fig. 5.100, i.e., with gauche defects present, the diffusion is more facile. Instantaneous projections of segments of C 50 H 100 within a crystal at different temperatures are shown in Fig. 5.107. The MD simulation represents the hexagonal polymorph. One can see that the crystal is divided in nanometer-size domains and has increasingly averaged chain cross-sections at higher temperatures. Both yield on macroscopic X-ray analysis hexagonal, average symmetry. At higher temperature, chains start to leave the surface of the crystal, indicating the first stages of fusion. 5 Structure and Properties of Materials ___________________________________________________________________ 530 Fig. 5.107 Fig. 5.106 5.3.5 Deformation of Polymers Turning to the macroscopic deformation of polyethylene, as experienced on drawing of fibers, one observes frequently the formation of a neck at the yield point, as illustrated in Fig. 5.108. The stress-strain curve in Fig. 5.108 illustrates a typical 5.3 Defects in Polymer Crystals ___________________________________________________________________ 531 Fig. 5.108 Fig. 5.109 drawing of quenched polyethylene. During plastic deformation in the neck, the cross- section is drastically reduced, so that the true stress and strain are actually increasing instead of decreasing. Figure 5.109 illustrates a calculation of the true draw ratio by following the changes of the cross-section at various positions along the fiber, starting at the point of initial necking. In Fig. 5.110 the true stress-strain curves are plotted as calculated 5 Structure and Properties of Materials ___________________________________________________________________ 532 Fig. 5.110 from the actual cross-section at the moment of stress measurement for experiments on drawing at different temperatures. Figure 5.110 shows three regions of different behaviors. Region (1) is the Hookean region. It indicates elastic deformation with a constant Young’s modulus as defined in Fig. 4.143. In the scale of the stress in the figure the slope appears almost vertical, i.e., Young’s modulus is rather large. This region of elastic deformation decreases with increasing temperature parallel to the increase of gauche defects seen in Fig. 5.100. In the region of the neck of Fig. 5.108, a catastrophic change occurs, the polymer yields as marked by (2). Rather than breaking, the modulus increases in parallel with the production of a stronger fiber morphology in a process called strain hardening (3). At the ultimate break, most molecules are arranged along the fiber axis in fibrillar crystals and inter-fibrillar, oriented intermediate-phase material. The oriented, noncrystalline material can often be described as a mesophase. Its structure was derived in Sect. 5.2 on the example of poly(ethylene terephthalate) in Figs. 5.68–72. A more detailed deformation mechanism is schematically given in Fig. 5.111. It was suggested already in 1967 by Peterlin and coworkers (see also [34]). The simulations discussed in the previous Section begin to explain some of the details of the drawing process. In particular, they show the enormous speed that is possible on a molecular scale for the sliding of the chain within the crystal. The ratio between macroscopic time scales of about 1.0 s and the molecular time scale of 1.0 ps is 10 12 ! The simulations indicated that the initiation of the diffusion seems to come from a sessile gauche defect or kink. On application of an external force, this can lead to a moving of the chains, catastrophically deforming the crystal morphology with the possible creation of a metastable mesophase as an intermediate, as indicated schematically in Fig. 5.111 and shown sometimes to be able to remain as a third phase in Fig. 5.69–72. On annealing, part or all of the initial structure may be regained. 5.3 Defects in Polymer Crystals ___________________________________________________________________ 533 Fig. 5.111 5.3.6 Ultimate Strength of Polymers To use flexible linear macromolecules as high-strength materials, themolecular chains should be arranged parallel to the applied stress and be as much extended as possible. The resulting modulus is then dependent on the properties of the isolated chain and the packing of the chains into the macroscopic cross-section. The properties of the chain can be derived from the same force constants as are used for the molecular dynamics calculation in Sect. 5.3.4 with Figs. 1.43 and 1.44. The ultimate strength, also called tenacity or tensile strength, in turn, depends on the defects in form of chain ends and other defects that produce the slipping mechanism and reduce the ultimate number of chains available at break to carry the load. The moduli, thus, are close-to structure insensitive, while the ultimate strengths are structure sensitive as was suggested in the summary discussion of Fig. 5.80 of Sect. 5.3.1. Figure 5.112 compares the specific tensile strengths and moduli of a number of strong materials. The specific quantities are defined as shown in the figure as the modulus divided by the density. This quantity is chosen to emphasize the materials of high strength at low weight. For these properties most desirable are the materials found in the upper right corner of the plot. Many rigid macromolecules as defined in Fig. 1.6 collect along the drawn diagonal with a rather small ratio of strength-to- modulus because of the above-mentioned deformation mechanism followedultimately by a crack development that breaks the crystals by separating a few crystal layers at a time needing much less stress. In flexible polymers, the covalent backbone bonds are stronger than metallic or ionic bonds (see Sect. 1.1). High-strength fibers result when chains of a proper mesophase gradually orient parallel to the stress and the molecules become fully extended in large numbers over a small deformation limit. 5 Structure and Properties of Materials ___________________________________________________________________ 534 Fig. 5.112 Fig. 5.113 Figures 5.113–115 illustrates the need to parallelize the chains of poly(ethylene terephthalate) applying the three-phase structure, identified in Figs. 5.69–72 [25]. Figure 4.113 shows the model for the combination of the three phases [35,36]. The orientation in the mesophasematrix, measuredbyitsaverageorientationfunction (OF) obtained from the X-ray diffraction pattern [24], is of most importance for the modulus and, surprisingly, also for the ultimatestrength,asindicatedbythe left curves 5.3 Defects in Polymer Crystals ___________________________________________________________________ 535 Fig. 5.114 Fig. 5.115 of Figs. 4.114 and 4.116. The right curves are computed assuming the fibers consist only of two phases and, using the total orientation, were summed over all chains. Only the left curves extrapolate within a factor of two to the known ultimate modulus of the crystal of poly(ethylene terephthalate). The extrapolated tenacity is, as expected, smaller than the extrapolated modulus. [...]... for their description all tools of thermal analysis of polymeric materials In this section, the upper temperature limit of the crystalline state is explored on the basis of experimental data on the thermodynamics of melting, extrapolated to equilibrium The more common nonequilibrium melting will see its final discussion in Sects 6.2 and 7.2 The other condensed states of macromolecules, the mesophases,... analyzed [41] The majority of these salts follow the same rule of constant entropy of fusion per mole of ions, irrespective of the 5.4 Transitions and Prediction of Melting 541 _ Fig 5.120 charge of the ions There are, however, a reasonable number of exceptions Smaller as well as larger entropies of fusion have been reported Lower entropies of fusion can be observed,... with coordination number, CN, the changing number of electrons in the conduction band, as well as changes of the character of the bonds from the metallic CN (usually 12) to the covalent CN (often 4, see Fig 1.5) The rule of constant entropy of melting was first observed by Richards [40], and should be compared to the rule of constant entropy of evaporation of 90 J K 1 mol 1 by simple liquids as proposed... understanding of the differences of transition temperatures for different materials A general application of Richards’s rule requires, however, 540 5 Structure and Properties of Materials _ modification whenever other than positional disorder is produced on melting Before exploring the melting of such different molecules, Fig 5.118 shows also some entropies of melting of larger,... usually larger volume of the melt (2) The disordering of the crystalline arrangement to the liquid which shows short-range order only (3) The introduction of additional defects into the liquid structure which, at least for spherical motifs, is often a quasi-crystalline structure and can have defects of the type described for crystals in Sect 5.3 Of the three types of disorder of Fig 2 .102 which can be introduced... presents, next, the entropy of melting of 20 alkali halides All of these crystals have the Fm3m space group of the NaCl structure which is depicted in Fig 5.2 The average entropy of fusion of these 20 salts is 24.43 ±1.7 J K 1 mol 1 Since their formula mass refers to two ions, the average positional entropy of fusion is 12.2 J K 1 mol 1, in good accord with Richards’s rule A total of 76 other salts, with... entropy of fusion are approximately proportional to the introduced disorder, one can include all within the rather wide, empirically derived, limits of the positional entropy of fusion Similar proportionalities are assumed for the other types of disorder included in Fig 2 .103 The metals in Fig 5.117 expand the table of entropies of fusion Perhaps it is not surprising that the metals have an entropy of melting... about 10 to 50 J K 1 mol 1 This uncertainty of the entropy change is larger than for Spositional, but the rotational degrees of freedom have a larger variation of rotational motion The rotation may require different amounts of volume, be coupled between neighboring molecules, and, depending on the symmetry of the molecules, may have a reduced entropy contribution 542 5 Structure and Properties of Materials. .. known only with errors of ±3% The often heard argument that equilibrium information is not available for polymers is not valid in this range of precision, and it will be shown in this section that valuable information can be deduced from a discussion of the equilibrium entropy of fusion The discussion of Tmo must involve four independent variables, namely the enthalpies and entropies of both the melt and... isotropization For the discussion of the entropies of fusion these results indicate that the size of the motif does not significantly influence the entropy of fusion All data treated up to now, thus, can be summarized by stating that under the given conditions the entropy of fusion, or better isotropization, is made up of mainly positional disordering, and that the entropy of disordering varies little among . all tools of thermal analysis of polymeric materials. In this section, the upper temperature limit of the crystalline state is explored on the basis of experimental data on the thermodynamics of melting,. of Figs. 5 .103 105 illustrates the details of the diffusion of a chain through the crystal. The figures refer to a chain that was driven into the direction of 5 Structure and Properties of Materials ___________________________________________________________________ 528 Fig is often a quasi-crystalline structure and can have defects of the type described for crystals in Sect. 5.3. Of the three types of disorder of Fig. 2 .102 which can be introduced on melting of