Theory of Critical Distances

The theory of critical distances summarizes several approaches for predicting the effects of notches and other stress concentration features with regard to fracture and fatigue under a wide variety of conditions (6). The basic framework of this theory is suitable for 

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The problem of distorted flow and voids between the membranes in the stack does not exist in cartridges that accommodate only a single thicker disk of porous material. In addition, Belenkii s theory of critical distance in adsorption chromatography leads to a conclusion that even a very short layer of a separation medium such as the disk can be sufficient for an efficient separation of large molecules [85], These provisions led to development of rigid disks for rapid chromatography of biopolymers by Tennikova and Svec in the late 1980s [86]. 

D. Taylor, The theory of critical distances. Engineering Fracture Mechanics, 2008. In Press, Corrected Proof at 2008-01-27. 

D. Taylor, "Poljoners Brittle fracture in polymeric materials," in The Theory of Critical Distances, pp. 93-118. Elsevier Science Ltd, Oxford, 2007. 

Close to the boundaries Sj and S2 in the three phase domain, the interfacial tensions were found to be very low. In that case, the theoretical model presented above is no longer valid, first of all because the middle phase microemulsion structure is not simply a droplet dispersion. Furthermore the interaction term F becomes evidently dominant and is difficult to evaluate since the nature of the forces is not perfectly known. However, such low interfacial tensions are characteristic of critical consolute points. It was then tempting to check that the behavior of the interfacial tensions was compatible with the universal scaling laws obtained in the theory of critical phenomena. In these theories the relevant parameter is the distance e to the critical point defined by ... 

In numerous applications of polymeric materials multilayers of films are used. This practice is found in microelectronic, aeronautical, and biomedical applications to name a few. Developing good adhesion between these layers requires interdiffusion of the molecules at the interfaces between the layers over size scales comparable to the molecular diameter (tens of nm). In addition, these interfaces are buried within the specimen. Aside from this practical aspect, interdififlision over short distances holds the key for critically evaluating current theories of polymer difllision. Theories of polymer interdiffusion predict specific shapes for the concentration profile of segments across the interface as a function of time. Interdiffiision studies on bilayered specimen comprised of a layer of polystyrene (PS) on a layer of perdeuterated (PS) d-PS, can be used as a model system that will capture the fundamental physics of the problem. Initially, the bilayer will have a sharp interface, which upon annealing will broaden with time. 

J. Bjerrum (1926) first developed the theory of ion association. He introduced the concept of a certain critical distance between the cation and the anion at which the electrostatic attractive force is balanced by the mean force corresponding to thermal motion. The energy of the ion is at a minimum at this distance. The method of calculation is analogous to that of Debye and Hiickel in the theory of activity coefficients (see Section 1.3.1). The probability Pt dr has to be found for the ith ion species to be present in a volume element in the shape of a spherical shell with thickness dr at a sufficiently small distance r from the central ion (index k). 

A somewhat different interpretation has been given by Johansson who applied the Mott-Hubbard theory of localized versus itinerant electron behaviour also to compounds. This interpretation differs from the above one mainly in that it assumes complete localization for magnetic compounds, and that at a certain critical inter-atomic distance we have to switch our description from a metallic state to an insulating one for the 5 f electrons (see Eq. (42)). In Eq. (42), an is substituted by a convenient measure of the spatial extension of the 5 f orbital, the expectation values (analogous to (of Fig. 10) and Xmoh is calculated from the R j radii of actinide metals (Fig. 3). The result is given in Table 6. 

Three common methods of measuring crosslinking (swelling, elastic modulus, and gel point measurements) have recently been critically appraised by Dole (14). A fourth method using a plot of sol + sol against the reciprocal dose has also been used extensively. However, Lyons (23) has pointed out that this relation, even for polyethylenes of closely random distribution, does not have the rectilinear form required by the statistical theory of crosslinking. Flory (19) pointed out many years ago that the extensibility of a crosslinked elastomer should vary as the square root of the distance between crosslinks. More recently Case (4, 5) has calculated that the extensibility of an elastomer is given by ... 

There have been many theories of metal-ammonia solutions that differ from that presented here. Cohen and Thomson (1968) and more recently Cohen and Jortner (1973) supposed that large fluctuations in concentration occur over large distances, sufficient to invalidate the pseudogap model and substitute a semiclassical percolation theory. For the reasons given above, we think that this is only likely to be so within a few degrees of the critical point. 

The theory has been examined by measuring the ratio K,c/Klcs as a function of j/g, as shown in Fig. 17. The theoretical lines have been fitted to the experimental points by choosing suitable values of the critical distance, c, which is the only fitting parameter. The agreement between theory and experiment has been found to be equally good for many different epoxy polymers cured with many different hardeners, both unmodified 44- 45,51), rubber-modified 45) and containing glass particles 22) and even, under certain circumstances, for structural adhesive joints S3). Values of critical stress, ct,c, and distance, c, for various epoxy materials, obtained from bulk and... 

Two methods appear to be very powerful for the study of critical phenomena field theory as a description of many-body systems, and cell methods grouping together sets of neighboring sites and describing them by an effective Hamiltonian. Both methods are based on the old idea that the relevant scale of critical phenomena is much larger than the interatomic distance and this leads to the notion of scale invariance and to the statistical applications of the renormalization group technique.93... 

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