By Herbert Jr. Oertel, M. Böhle, J. Delfs, D. Hafermann, H. Holthoff
Dieses Buch wendet sich an Studenten der Ingenieurwissenschaften und Ingenieure der Raumfahrtindustrie und der Energieverfahrenstechnik. Es verkn?pft die klassischen Gebiete der Aerodynamik mit der Nichtgleichgewichts-Thermodynamik hei?er Gase. Am Beispiel des Wiedereintritts einer Raumkapsel in die Erdatmosph?re werden die aerothermodynamischen Grundlagen und numerischen Methoden zur Berechnung des Str?mungsfeldes der Raumkapsel im gaskinetischen und kontinuumsmechanischen Bereich der Wiedereintrittstrajektorie behandelt. Am Beispiel von Raumfahrtprojekten werden die Methoden entwickelt. Die Autoren sind anerkannte Spezialisten f?r dieses Fachgebiet.
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26 IDEAL CHAIN MODELS the solvent quality for the polymer is lowered, usually by decreasing the temperature, attractive interactions among polymer segments are transmitted by the solvent medium. 1) is recovered. A similar phenomenon occurs in a homopolymer melt (de Gennes, 1979). 5 nm) in space, repel each other strongly. However, this net repulsion is cancelled (“screened”) in the asymptotic limit of N → ∞ by the many other chains (of number ∼N 1/2 ) that protrude into the random coil of the chain of interest.
Correlated in orientation since the short-ranged interferences have died oﬀ between two segments separated by such a large distance along a chain. Through the random coiling of the polymer, however, those same two segments could ﬁnd themselves separated by a small distance in space, as with particles 6 and 12 in the coarse-grained chain of Fig. 1. The typically strong interactions between two such monomers lead to positional correlations that are referred to as long-ranged interferences. One manifestation of long-ranged interferences is the so-called excluded volume eﬀect, which is the constraint that any two segments in a coiled polymer cannot occupy the same location in space (Flory, 1953; de Gennes, 1979).
Of particular interest are moments of the end-to-end vector R = rN − r0 . This object can be conveniently expressed as N N R = i=1 bi = b i=1 ni . 10) i=1 j=1 where we use Greek subscripts to denote Cartesian components of vectors and tensors. The terms in the double sum with i = j evidently vanish because of the independence of the respective unit vectors. 1) is exactly unity for the freely jointed chain. Higher moments of the end-to-end vector are similarly calculated. 13) From this result, it is apparent that the probability distribution of the end-to-end vector R for the freely jointed chain is not a simple Gaussian distribution, since for such a distribution (see Appendix B) (R·R)2 = ( R·R )2 +2 RR : RR , which leads to (R · R)2 0 = (5/3)b4 N 2 .