Fold surfaces

Use Eq. (4.14), the results in Fig. 4.5, and the data in Table 4.1 to estimate a value for 7 for polyethylene. Figure 4.10 shows the unit cell of polyethylene Fig. 4.10b shows the equivalent of two chains emerging from an area 0.740 by 0.493 nm. On the basis of the calculated value of 7 and the characteristics of the unit cell, estimate the free energy of the fold surface per mole of repeat units. 

A weld bead included in a test-piece is, to some extent, peculiar to itself and may not necessarily be representative of nominally similar welds to be made by other welders under other circumstances. To this extent, results of tests on welds must be subject to some qualification in interpretation, having in mind that what will be disclosed principally will be the overall ability of the composition of the weld metal to resist the corrosive environment. In some cases, entrapped flux, craters, fissures, folds, surface oxides etc. may introduce localised corrosion that may or may not occur with all welds of the type studied (see Section 9.5). 

Lamellar thickness Minimum stable thickness Thickness deviation l — lmin Surface area of the fold surface Width of a stem Thickness of a stem Fold surface free energy Lateral surface free energy... 

Fig. 2.1. The crystalline model a single crystal in which molecules traverse the lamella perpendicular to the fold surface. Cilia are formed at the end of the molecules outside the crystalline core. The folds are predominantly adjacent and the loop sizes may vary...

Hoffman assumes that aj, has the same interpretation as for infinite chain length, that is the surface tension of the fold surface. However, as pointed out above, effects of a non-folded surface are already included in AH[ Tm(0, p)] and AS[ 7, (0, p)], and at best a e could be regarded as the contribution to the surface tension from just the folds, but more realistically as a parameter which is related to the surface tension but which also varies with the thickness of the lamella, that is as the proportion of the number of folds to free ends in the surface changes. 

Furthermore, different fold surface free energies for the different sectors and a maximum crystallization temperature are predicted. For details the reader is referred to the original paper. 

The elementary process of growth is treated as the attaching or detaching of one repeating unit on the surface. There are two possible ways in which a unit may add to a nucleus, which are shown in Fig. 3.20 (from Ref. [146]). A unit may diffuse from the liquid to the side of the nucleus with a small activation energy compared with kT. However, it is very difficult for a new unit from the liquid to add directly onto the fold surface, and the thickening of the nucleus is due to the... 

The structure of the fold surface has long been a most controversial topic (ever since the finding of the chain folded lamellae). What kind of fold structure will the direct MD simulation predict Figure 29 shows (a) the crystalline domains and (b) the fold loops at 330 K after a sufficiently long time period of 38.4 ns the crystallinity reaches about 52%. We noticed that most of the fold loops near the substrate are rather short. The presence of looser and longer loops and abundant cilium in the middle of the MD cell is obviously... 

Fig. 11 Role of loop entropy on the fold surface free energy...

This result is to be contrasted with the standard model [53] of Fig. 13, where the fold surfaces are simply treated as planar interfaces with fold surface free energy oy per unit area. In the latter case, the free energy of the nucleus is given by... 

Keller, A., E. Martuscelli, D. J. Priest, and Y. Udagawa. Fold surface of polyethylene single crystals as assessed by selective d radation studies. III. Application of the improved techniques to single crystak. J. Polymer Sci. Part A-2 9 1807-1837, 1971. 

Note A lamellar crystal is usually of a thickness in the 5-50 nm range, and it may be found individually or in aggregates. The parallel-chain stems intersect the lamellar plane at an angle between 45° and 90°. The lamellae often have pyramidal shape owing to differences in the fold domains, as a result, one can deduce different fold planes and fold surfaces from the lamellar morphology. 

The small ratio of lamellar thickness to the contour length of a polymer molecule clearly implies that chains must fold back and forth into stems with chain direction essentially perpendicular to the lamellar surface, as originally declared by Storks. The large surfaces of the lamellae containing the chain folds are called fold surfaces, and the thin surfaces are called lateral surfaces. 

The estimated value of the free energy of the fold surface (q3 = oy) is 90 mJ/m for polyethylene, whereas that of the lateral surface (ai = a/) is 15 mJ/m. Therefore we expect the lamellar thickness to be 6 times larger than the lateral dimension, specifically, a cylinder shape, instead of a disklike shape. This is in stark contrast to the facts described in Section II. The thermodynamic estimate of lamellar thickness is about two orders of magnitude larger than the observed values for polyethylene and other polymers. In view of this discrepancy, we are led to the conclusion that lamellae are not in equilibrium. 

The free energy of a lamella, where the fold surface area A(= L1L2) is taken to be much larger than lateral surface area (L3L1 + L2L3), follows from Eq. (1.20) as... 

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