Free random

Barabasi, A-L., Albert Reka, and Jeong Hawoong, Mean-Field Theory for Scale-Free Random Networks, Physics A, 272, (1999) 173-187. 

Unfortunately, for the investigation of random walk statistics in the regular 3D lattice of obstacles the approach based on the idea of conformal transformations cannot be applied. Nevertheless, due to the analogy established in the 2D-case it is naturally to suppose that between random paths statistics in the 3D lattice of uncrossable strings and the free random walk in Lobachevsky space the similar analogy remains. Let us present below some arguments confirming that idea. 

A number of different mechanisms have been proposed to account for the rubberlike properties of materials. In classical rubber theory these properties are attributed to a decrease in conformational entropy on deforming a network of kinetically free random polymer molecules. Stress orders the polymer chains and decreases their entropy by limiting their conformational freedom, thus providing the restoring force to the relaxed state. Such a theory was developed for elastin by Hoeve and Flory... 

A first step towards the consideration of free, randomly oriented molecules (or microcrystals) is the superposition of S(A) with S(—A) ... 

S. Zhang and H. Krakauer (2003) Qnantnm Monte Carlo Method using Phase-Free Random Walks with Slater Determinants. Phys. Rev. Lett. 90, p. 136401... 

The new results on experimental and theoretical investigations of the linear chain-like structure presented below confirm the proposed model of its atomic and crystal structure. The major question still remains is sp -hybridized chain-like carbon stable or unstable with respect to the formation of cross-linkages among carbon chains The viewpoint that sp -hybridized chain-like carbon is unstable is based on studies of free, randomly oriented sp carbon chains, which can simply be cross-linked. In the case when the chain ends are strongly fixed on the surface of substrate, the substrate plays an important role of a command surface [8]. This means that the surface controls the orientation of the chains and keeps them at a distance apart from cross-linking. However, the substrate can... 

A first step towards the consideration of free, randomly oriented molecules (or microcrystals) is the superposition of S(h) on S(—h). Only the cosine terms in Eq. (12) are conserved. Introducing the wavelength dependence of f and f" ex()licitly, we obtain by integration of S(h) over the solid angle... 

Two papers have been presented on the photochemistry of 5-methylphena-zinium salts in aqueous solution. Fluorescence, optical flash photolysis, and electron paramagnetic resonance (e.p.r.) techniques have been used to elucidate various aspects of product formation and quantum yield. Two products have been identified, namely the 5-methyl-10-hydrophenazinium cation radical (MPH ) and the pyocyanine (l-hydroxy-5-methyl-phenozinium) cation (PyH ) in a stoicheiometric ratio of 2 1. The quantum yield of formation of (MPH ) was found to be 0.29 0.03 at pH 7.0 and 1.1 0.1 at pH 3.0. The triplet state of MP (Ti) has also been detected by triplet-triplet absorption and is found to have a lifetime of 0.5 ns. Flash photolysis and e.p.r. have also been used to study a geminate triplet radical pair obtained from hydrogen abstraction by excited triplet acetone from propan-2-ol. The authors demonstrate that the geminate pairs contribute most of the polarization in photochemically-induced dynamic electron polarization (CIDEP) as compared with free random-phase pairs. 

Figure 13.3 Gelation, of polymer solutions (a) polymer chains in free random motion, (b) polymer chains forming helices which interact to form a gel structure, (c) gel formation in polyelectrolytes, often enhanced by the addition of e.g., Ca2+ ions.

Gases have no fixed shape or volume. They expand spontaneously to fill the container and are easily compressed. The molecules have almost free random motion. 

The structural picture of the xerogel is that of a dense network structure consisting of tension-free randomly kinked chains, here and there interconnected by permanent junction points. 

Table 7 Effect of increased acrylate level in TEMPO-mediated stable free random copol5anerization with styrene.

Naturally the rate of electron transport through the semiconductor to the anode, in direct competition with recombination and interception, is pertinent in the overall performance of a DSSC. Photo-injected electrons are transported by diffusion in a free random walk [15] as a result of an electron concentration gradient. The diffusion time will depend on the distance to the anode (the semiconductor film thickness) and the diffusion coefficient of electrons. But electrons also seem to become trapped within the semiconductor particles for some time (picoseconds to nanoseconds). This trapping and thermal release (detrapping) mechanism requires energy, and is important in retarding charge recombination in DSSCs. The process... 

Figure 6.10. Sketch of force response of free random coil and adsorbed chain in dependence on the coil strain related to the maximum elongation, 100% 0.81 Nh. Whilst free coil exhibits Gaussian response under given low strain approximately up to 20%, adsorbed loops and bridges, represented by A = 10 and 100, are highly Langevineau with incremental stiffness considerably increased. Strain induced perturbation of interphase accompanied by desorption of trains causes significant local stress and strain relief and, consequently, a lower stiffness...

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