Lamellar stacking model

The statistics by which the phases are related depend on the theoretical model used. In general, three theoretical models are considered and these are the homogeneous lamellar model, the network model, and the lamellar stacking model. In the following paragraphs, the particular distribution relationships are presented depending on the index of the distance. 

Lamellar Stacking Model (LSM) In this case, the function g[(r) results from the independent variation of the quantities aj and a2 of the distances r and r2, respectively (equivalent to the thickness Zj and I2 of the lamellae of phases 1 and 2), within a given stacking. The change of a, is given by... 

Figure 4.35 Calculated structure for an HT 3-hexylthiophene tetramer obtained by using molecular mechanics modeling, where the globally minimized tetramers have been docked in an idealized manner to X-ray structural parameters, (a) Intermolecular p-stacking between the thiophene rings as inferred from a (90 °C) X-ray pattern of the film, (b) Lamellar stacking as inferred from X-ray scans of intensity versus 20 data. Reprinted from R.D. McCullough, S. Tristram-Nagle, S.P. Williams, R.D. Lowe and M. Jayaraman, ]. Am. Chem. Soc., 115,4910 (1993). Copyright (1993) American Chemical Society...

Highly ordered lamellar gel microstructures are formed by certain surfactants and mixtures of a surfactant and long-chain fatty alcohols in water. Using small angle X-ray scattering (SAXS), an ordered lamellar stack lattice model was proposed for the gel formed by 10% w/w cetostearyl alcohol containing 0.5% cetri-mide surfactant. In contrast, the microstructure of a Brij 96 gel depends on the surfactants concentration. A hexagonal liquid-crystalline gel structure was... 

Fig. 5a-c. Diffraction patterns of lamellar stacks with perfect stacking order (a), with disorder of the first kind (b), and disorder of the second kind (c). For details of the model parameters, see Ref. 4. (From Ref. 4, with permission)... 

In SAXS studies of a real sample, all the lamellar layers or stacks are oriented at random with respect to the incident beam therefore the intensity function /(g)obs which corresponds to the measured intensity shows spherical symmetry [20]. However, when using theoretical one-dimensional models (where the lateral width of the lamellae is much greater than the periodicity), the dispersion intensity is calculated assuming that the lamellar stacks are correctly oriented (perpendicular) with respect to the incident beam [21]. This implies that the observed intensity (with spherical symmetry) should be corrected to a perpendicular intensity to the lamellar stacks. Because of this, the Lorentz factor that is described for lamellar systems is also used. 

In most spherulitic polymers, touching spherulites occupy whole of the space. Their microstructure is too complex to be completely modelled, especially if there is twisting of lamellar stacks about spherulite radii. Consequently, models simplify the structure, and use composite micromechanics concepts. A stack of parallel lamellar crystals with interleaved amorphous layers (Fig. 3.20) has a similar geometry to a laminated rubber/metal spring (Fig. 4.1). The crystals have different Young s moduli E, Eb and E (Section 3.4.3), and different shear moduli when the... 

Fig. 4.20a shows a variety of deformation mechanisms at the equator of the spherulite, due to differing orientations of the lamellar stacks relative to the tensile stress axis. Computer models are needed to consider the variety of lamellar stack orientations, and calculate the macroscopic stresses. Using an axisymmetric model of a spherulite (in a regular array), the tensile yield stress was predicted to be a nearly linear function of the crystallinity (Fig. 4.20b), and in the same range as experimental data. 

There is evidence from neutron-scattering experiments that lamellar crystallisation from the melt can occur without significant change of overall molecular dimensions, such as the radius of gyration. A model that fits this evidence and incorporates the ideas described above is shown in fig. 5.7. Lamellar stacks are discussed further in section 5.4.3. 

Similar to polyethylenes the morphology of these polymers is also described as a lamellar stack of crystalline and non-crystalline layers. This so-called two phase model is applied for the interpretation of X-ray diffraction data as well as for heat of fusion or density measurements. However, it is well known that several mechanical properties, as well as the relaxation strength at the glass transition temperature, cannot be described by such a simplistic two-phase approach, as discussed by Gupta [59]. Prom standard DSC measurements [60], dielectric spectroscopy, shear spectroscopy [61], NMR [62], and other techniques probing molecular dynamics at the glass transition (a-relaxation) temperature, the measured relaxation strength is always smaller than expected... 

SWD for other polymer, namely poly (butylene isophthalate)(PBI) [49], and the corresponding (Z e) orm data are also included in Fig. 21.13 for comparison. One may propose the idea that in the second regime, which we can associate with the secondary crystallization, crystalUzation takes place essentially in the inter lamellar stacks amorphous phase. These secondary crystals should be arranged either as independent lamellae or as very defective stacks. This mechanism should not produce significant amounts of RAP because, as previously discussed, the RAP can be assigned to an intra lamellar stacks amorphous phase. Additional support for this model on the basis of structural experiments has been discussed extensively for PBI [49]. A similar view has been recently proposed to explain secondary crystalUzation in poly(ethylene isophthalate-co-terephtalate) copolymers crystallized from the melt [50]. 

However, this simple model is not generally valid, particnlarly for lower crystallinity materials. In the latter, the (linear) crystallinity of lamellar stacks can be relatively high, but the bulk crystallinity low. In this case (pc = Vs x Wc (where Vs is the volnme fraction of lamellar stacks) and Ys < 1 and Wc > [Pg.1988]

Figure 26 Schematic illustration of the two possible models of the lamellar stack of semirigid chain polymers. Left, stacks with thin crystals and thicker interlamellar amorphous regions. Right stacks with thicker crystals and thinner interlamellar amorphous regions. The ambiguity of the microstructural model Is due to the Babinet principle, which makes the scattering from the two structures indistinguishable.

Important morphological parameters such as the long period (I), crystal thickness (Ic), and amorphous layer thickness (la) of semipolymer melts and blends can be determined using SAXS via two different approaches. In the first approach, standard models such as the Hosemarm-Tsvankin [23] and the Vonk-Kortleve [24,25] for lamellar stacks are fitted to data obtained for the SAXS profile. The second approach is based on performing a Fourier transform for the SAXS profile to produce a one-dimensional correlation function, y(z) (which is Fourier transform of the measured I(q) in SAXS) or an interphase distribution function (IDF) in real space. 

Small-angle x-ray scattering (SAXS) probes structural features at the mesoscopic length-scale (Section 1.9.2). It can be used to measure the radius of gyration of polymers in solution, via Eq. (1.27), which can be up to hundreds of nanometres. Semicrystalline polymers form lamellae, which are layered systems with a layer period of the order of 10 nm. Such lamellar stacks diffract x-rays at small angles, indeed they are a one-dimensional lattice, and Bragg s law can be used to determine the layer period. Furthermore, the relative intensities of the peaks depend on the distribution of amorphous and crystalline material, and this can be analysed to provide a model of the electron density profile normal to the layers. 

Figure 7.31 Simple model of lamellar stack showing two adjacent crystal lamellae and the two types of chain present random tie chain and tight, (regular) fold.

Modeling the Interface Distribution Function for a ID Lamellar Stack As demonstrated in the last section, the nonideality of a real semicrystalline polymer can lead to a broadening and overlapping of the peaks in K (z), which makes it difficult to extract the correct structure parameters simply from the peak positions. The one-dimensional paracrystalline stack has been suggested as an analytical model for the semicrystalline structure [2,13,16], We here present a procedure that allows simulating and modeling the measured IDF based on this model. A simulated IDF... 

Figure 5.15 (a) Electron density distribution p(z) of the stacked lamellar structural model... 

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