Crystallography - Wikipedia
John Hoffman was the first to describe this relationship although his derivation of a crystallite thickness law borrowed heavily on asymmetric growth models form low molecular weight, particularly ceramic an metallurgical systems. Hoffman's law is given in equation 4. Hoffman's law can be obtained very quickly for a free energy balance following the "Gibbs-Thomson Approach" Strobl pp. The deeper the quench, T [infinity] - T , the thinner the crystal and for a crystal crystallized at T [infinity] , the crystallite is of infinite thickness.
Crystallization does not occur at T [infinity].
Nature of the Chain Fold Surface: In addition to determination of T [infinity] , the specific nature of the lamellar interface in terms of molecular conformation is of critical importance to the Hoffman analysis. The synoptic or comprehensive model involves interconnection between neighboring lamellae through a combination of adjacent and Switchboard models. The interzonal model involves non-adjacent reentry but considers a region at the interface where the chains are not randomly arranged, effectively creating a three phase system, crystalline, amorphous and interzonal.
Several distinguishing features of the lamellar interfaces are characteristic of each of these models. The Hoffman equation states that the lamellar thickness is proportional to the interfacial energy so we can say that Adjacent reentry favors thicker lamellae since adjacent reentry has the highest interfacial energy and the more random interfacial regions should display thinner lamellae.
Colloidal Scale Structure in Semi-Crystalline Polymers: Lamellae crystallized in dilute solution by precipitation can form pyramid shaped crystallites which are essentially single lamellar crystals figure 4. Pyramids form due to chain tilt in the lamellae which leads to a strained crystal if growth proceeds in 2 dimensions only.
In some cases these lamellae which have an aspect ratio similar to a sheet of paper can stack although this is usually a weak feature in solution crystallized polymers. Lamellae crystallized from a melt show a dramatically different colloidal morphology as shown in figure 4.
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In these micrographs the lamellae tend to stack into fibrillar structures. The stacking period is usually extremely regular and this period is called the long period of the crystallites. The long period is so regular that diffraction occurs from regularly spaced lamellae at very small angles using x-rays. Small-angle x-ray scattering is a primary technique to describe the colloidal scale structure of such stacked lamellae.
Figure 4. The peak occurs at about 0. In some cases the x-ray data has been Fourier transformed to obtain a correlation function for the lamellae which indicate an average lamellar profile as shown in figure 4. The degree of stacking of lamellae would appear to be a direct function of the density of crystallization, i. You can think of lamellar stacking as resulting from a reeling in of the lamellae as chains which bridge different lamellae further crystallize as well as a consequence of spatial constraints in densely crystallized systems.
In melt crystallized systems, many lamellar stacks tend to nucleate from a single nucleation site and grow radially out until they impinge on other lamellar stacks growing from other nucleation sites. The lamellar stacks have a dominant direction of growth, that is, they are laterally constrained in extent, so that they form ribbon like fibers. The lateral constraint in melt crystallized polymers is primarily a consequence of exclusion of impurities from the growing crystallites. Some of these "Impurities" will crystallize at a lower temperature so it is possible to have secondary crystallization occur in the interfibrillar region.
Despite the complexity of the "impurities" it can be postulated that the impurities display an average diffusion constant, D. The Fibrillar growth front displays a linear growth rate, G. This rule implies that faster growth rate will lead to narrower fibrils. Also, the inclusion of high molecular weight impurities, which have a high diffusion constant, D, leads to wider fibrils. There is extensive, albeit qualitative, data supporting the Keith and Padden del parameter approach to describe the coarseness of spherulitic growth in this respect.
Branching of Fibrils: Dendrites versus Spherulites. Low molecular weight materials such as water can grow in dendrite crystalline habits which in some ways resemble polymer spherulites collections of fibrillar crystallites which emerge from a nucleation site. One major qualitative difference is that dendritic crystalline habits are very loose structures while spherulitic structures, such as shown in Strobl, fill space in dense branching.
At first this difference might seem to be qualitative. In low-molecular weight materials such as snowflakes or ice crystallites branching always occurs along low index crystallographic planes low Miller indices. In spherulitic growth there is no relationship between the crystallographic planes and the direction of branching. It has been proposed that this may be related to twinning phenomena or to epitaxial nucleation of a new lamellar crystallite on the surface of an existing lamellae.
A definitive reason for non-crystallographic branching in polymer spherulites has not been determined but it remains a distinguishing feature between spherulites and dendrites. Incidentally, the growth of dendrites can occur due to similar impurity transport issues as the growth of fibrillar habits in polymers.
In some cases a similar mechanism has been proposed where rather than impurity diffusion, the asymmetric growth is caused by thermal transport as heat is built up following the arrows in the diagram on the previous page. Non-crystallographic branching leads to the extremely dense fibrillar growth seen in figures 4.
In the absence of non-crystallographic branching, many of the mechanical properties of semi-crystalline polymers would not be possible.
The formation of polymer spherulites requires two essential features as detailed by Keith and Padden in from a wide range of micrographic studies: 1 Fibrillar growth habits. Polymer Spherulites. The micrographs in figure 4. The Maltese cross is an indication of radial symmetry to the lamellae in the spherulite, supporting fibrillar growth, low angle branching and nucleation at the center of the spherulite.
In some systems, especially blends of non-crystallizable and crystallizable polymers, extremely repetitive banding is observed in spherulites as a strong feature, figure 4. It has been proposed by Keith that banding is related to regular twisting of lamellar bundles in the spherulite circa Keith has proposed that this twisting is induced by surface tension in the fold surface caused by chain tilt in the lamellae circa Since most spherulites crystallize in an extremely dense manner it has been difficult to support Keith's hypothesis with experimental data.
Regular banding has, apparently, no consequences for the mechanical properties of semi-crystalline polymers so has been essentially ignored in recent literature.
The main difference is that polymer crystals can not be formed in perfect crystals, so single crystal or Laue patterns are not possible. Also, polymer crystals tend to be of low symmetry, orthorhombic or lower symmetry, due to the asymmetry in bonding of the crystalline lattice, i. Additionally, the unit cell form factor tends to be fairly complicated in polymer crystals. Line widths are broad for polymer diffraction and a substantial amorphous peak is usually present.
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C67 C66 Unknown. More options. Find it at other libraries via WorldCat Limited preview. Contributor Ortiz, Oscar L. Bibliography Includes bibliographical references and index. Topics discussed include hybrid vanadates and metal organic frameworks; structure and magnetic properties of mono- and poly-nuclear complexes containing Re IV l; metal organic framework applications in the fields of hydrogen storage and catalysis; MOF-Based mixed-matrix-membranes for industrial applications; coordination polymers in heterogeneous catalysis; high pressure gas storage on porous solids; metal organic frameworks for CO2 capture and halogen bonding in the assembly of high-dimensional supramolecular coordination polymers.
Subject Coordination polymers. Bibliographic information.
go to link Publication date Series Chemical engineering methods and technology Materials science and technologies ISBN hardcover hardcover. Browse related items Start at call number: QD C67 C66