Interpenetration dimensions

A continuous lipidic cubic phase is obtained by mixing a long-chain lipid such as monoolein with a small amount of water. The result is a highly viscous state where the lipids are packed in curved continuous bilayers extending in three dimensions and which are interpenetrated by communicating aqueous channels. Crystallization of incorporated proteins starts inside the lipid phase and growth is achieved by lateral diffusion of the protein molecules to the nucleation sites. This system has recently been used to obtain three-dimensional crystals 20 x 20 x 8 pm in size of the membrane protein bacteriorhodopsin, which diffracted to 2 A resolution using a microfocus beam at the European Synchrotron Radiation Facility. 

Since the power 7 is easier to detect in two than in three dimensions, the first MC study [62] sampled a two-dimensional MWD in a range of temperatures (that is, of (L)), so that a change in the degree of interpenetration should trigger a crossover from dilute to semi-dilute regime at some density 0. Evidently, indeed, from Fig. 4, the MWD follows the form of Eq. (16). At 0 one observes a power 7eff 1.300 0.005 which comes closely to the expected one. Above 0 one finds 7eff —> 1, and the distribution (11) becomes relevant. 

The toughest challenge and the greatest opportunity in chemical engineering for high-performance materials lie in the development of wholly new designs for composite solids. Such materials are typified by composites reinforced by three-dimensional networks and trass-works—microstractures that are multiply cormected and that interpenetrate the multiply cormected matrix in which they are embedded. In such materials, both reinforcement and matrix are continuous in three dimensions the composite is bicontinuous. Geometric prototypes of... 

Table 17.1 also displays the mass-fractal dimension, df, given by the power-law value, P, for the second level (aggregate). The df= 1.63 for A1 is less than that, and is less than 2.37 for A2 and 2.1 for A3. The lower value of df could indicate a more open structure of the A1 sample, " but it could also arise from less interpenetration of the silica aggregates occurring during drying. ... 

A polymer coil does not only possess a structure on the atomistic scale of a few A, corresponding to the length of covalent bonds and interatomic distances characteristic of macromolecules are coils that more or less, obey Gaussian statistics and have a diameter of the order of hundreds of A (Fig. 1.2) [17]. Structures of intermediate length scales also occur e. g., characterized by the persistence length. For a simulation of a polymer melt, one should consider a box that contains many such chains that interpenetrate each other, i. e., a box with a linear dimension of several hundred A or more, in order to ensure that no artefacts occur attributable to the finite size of the simulation box or the periodic boundary conditions at the surfaces of the box. This ne-... 

Insulation is effectively achieved if one particular set of interactions is very much stronger than the rest. This is the situation that arises in many coordination polymers (S. R. Batten, R. Robson, Interpenetrating nets ordered, periodic entanglement , Angew. Chem Int. Ed. Engl, 1998, 37, 1469-1494 Angew. Chem, 1998, 110, 1558-1595). But coordination polymers are not molecular in all three dimensions, and as such are not within the scope of this essay. 

It may now be additionally suggested that Noircarme also made his own peculiar contribution in 1918 to the complicated metaphysical framework of Duchamp s Tu m... (fig. 13). In this instance, the particular hypothesis is that Duchamp s pointe de fuite, a vanishing point emphatically signalled by a rodlike and n-dimensionally interpenetrative bottle brush stuck into an illusionistic rip, most likely represents what Noircarme calls a no-dimensional point, a point of departure leading to the Absolute, and thereby that it announces the presence of the cycle of dimensions. More to the point is Noircarme s very concrete description of a certain Measure of the Fourth Dimension, which turns out to be a rod, and this rod (tige), as 1 believe, initially represented the very specific role reenacted by Duchamp s Bottle Brush in 1918. 

Ca and Cg) for simplicity. In this case two types of vernier structure are considered as shown in Fig. 2.32. Structure [a] is composed of A- and B-planes, which are stacked alternatively along the h-axis as. .. ABAB. .. Due to the difference between the dimensions of the unit cells of each plane, the unit length of the supercell structure along the c-axis is expressed by eqn (2.7). In structure [b], the structures A and B are mutually interpenetrated. The dimension of the super structure along the c-axis is also expressed by eqn (2.7). In each case, the relation between structures A and B is very similar to that between the main scale and vernier scale in vernier calipers. 

The class of cubane clusters (4w) is large and includes complexes of metals distributed throughout the Periodic Table, with diverse bridging ligands X and with 0-4 terminal ligands L. There is substantial variability of shape, symmetry and dimensions of the central core apart from dissymmetry due to partial substitution of M, X or L, the concentric interpenetrating M4 and X4... 

The proportionality constant Nf in Eq. (21) is a generalized Flory-Number of order one (Np=l) that considers a possible interpenetrating of neighboring clusters [22]. For an estimation of cluster size in dependence of filler concentration we take into account that the solid fraction of fractal CCA-clusters fulfils a scaling law similar to Eq. (14). It follow directly from the definition of the mass fractal dimension df given by NA=( /d)df, which implies... 

Figure 2. Concentration profiles of poly(2-chloroethyl acrylate) in poly-(methyl methacrylate) along the thickness (L0) dimension of the samples. (GRAD) Gradient polymer, (IPN) interpenetrating networks.

If the power of R is negative, the probability of intersection decreases as the size of the fractal, R, increases. For this reason, two fractals with the same fractal dimension will have a low intersection probability ifZ>i = D2 > 1.5. As a result, fractals with fractal dimensions greater than 1.5 act as individual spheres in dilute solution. In concentrated solution, fractals with a fractal dimension less than 1.5 can be forced to interpenetrate strongly, unlike hard spheres, during settling and dewatering of fractal sol-gel ceramic powders. 

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