Molecular jump frequency

Go is the molecular jump frequency and is given by bak T/h where and hp are the Boltzman and Planck constants, respectively, bo is a crystal dimension (see Fig. 5.21), values of which are given in Table 5.16 for PP and in Appendix C for a number of other polymers. The quantity —Eq/RT) represents the diffusive transport of the molecules in the melt while IxF is the free energy of a nucleus with n-dimensional... 

Lauger, P., and Apell, H. J. (1982). Jumping frequencies in membrane channels. Comparison between stochastic molecular dynamics simulation and rate theory. Biophys. Chem. 16, 209-221. 

For diffusion, the net jump frequency k was related to molecular quantities by viewing the ionic jumps as a rate process (Equation 4.111). In this view, for an ion to jump, it must possess a certain free energy of activation to surmount the free-energy barrier. It was shown that the net jump frequency is given by... 

Smets and Evens assumed that the pre-exponential term of the Arrhenius equation of the rate constant is proportional to the jump frequency of a molecular segment from one position to another, i.e. proportional to the reciprocal of the internal viscosity of the bulk polymer at a given temperature. 

In the process of identification of condis crystals it was observed that conformational mobility alone is not sufficient to prove the presence of a condis phase. Large amplitude molecular jump motion may be possible already in crystals without disorder if the symmetry is identical before and after the jump. The frequencies of these jumps can be surprisingly large and the moving parts of the molecules substantial. In the condis phase quick reptation can lead to extension of folded chain crystals, and is possibly also involved in rearrangements on mechanical deformation and membrane functions. 

In Sect. 3, the Noyes approach to analysing reaction rates based on the molecular pair approach is discussed [5]. Both this and the diffusion equation analysis are identical under conditions where the diffusion equation is valid and when the appropriate recombination reaction rate for a molecular pair is based on the diffusion equation. Some comments by Naqvi et al. [38] and Stevens [455] have obscured this identity. The diffusion equation is a valid approximation to molecular motion when the details of motion in a cage are no longer of importance. This time is typically a few picoseconds in a mobile liquid. When extrapolating the diffusion equation back to such times, it should be recalled that the diffusion is a continuum form of random walk [271]. While random walks can be described with both a distribution of jump frequencies and distances, nevertheless, the diffusion equation would not describe a random walk satisfactorily over times less than about five jump periods (typically 10 ps in mobile liquids). Even with a distribution of jump distances and frequencies, the random walk model of molecular motion does not represent such motion adequately well as these times (nor will the telegrapher s or Fokker-Planck equation be much better). It is therefore inappropriate to compare either the diffusion equation or random walk analysis with that of the molecular pair over such times. Finally, because of the inherent complexity of molecular motion, it is doubtful whether it can be described adequately in terms of average jump distances and frequencies. These jump characteristics are only operational terms for very complex quantities which derive from the detailed molecular motion of the liquid. For this very reason, the identification of the diffusion coefficient with a specific jump formula (e.g. D = has been avoided. 

Figure 3. Example of a jumping pentacene molecule in p-terphenyl (class II, see text), for which the frequency undergoes irregular jumps. Top successive excitation scans of the laser bottom Molecular resonance frequency as a function of time (reproduced from Ref. 41).

It is considered that Ti represents a true thermod3mamic transition temperature. Adam and Gibbs [19] have developed a modified transition state theory in which the frequency of molecular jumps relates to the cooperative movement of a group of segments of the chain. The number of segments acting cooperatively is then calculated from statistical thermodynamic considerations. 

Upon employment of refined instrumentation, it was possible to estimate the average distance between nanopores in a 60% cross-linked rigid resin and thereby the jump length and the jump frequencies of the molecular water random walk process. These turned out to have the approximate value of 12-13 A and 1.5 x 10 -135x10 Hz for temperature varying between 5 and 90°C (Soles and Yee 2000). ... 

We now argue that, by analogy, the frequency of molecular jumps between two rotational isomeric states of a molecule (Section 1.2.1) is given by... 

Gibbs and Di Marzio [23,24] proposed that the dilatometric Tg is a manifestation of a true equilibrium second-order transition at the temperature T2. In a further development, Adams and Gibbs [21] have shown how the WLF equation can then be derived. On their theory, the frequency of molecular jumps is given by... 

Decorrelated subunits In the first approach, de Gennes suggested that was just the jump frequency of a free subunit of size d, 1/X j(d), times the probability that an extremity of a matrix chain was located within the distance d from the test chain (a necessary condition for the corresponding constraint to be released), i.e., wd ll R(d) (2NJP) If P is the polymerization index of the matrix chains. This makes it possible to evaluate this description, for the particular case a contribution comparable to that of pure reptation (for independent processes, the inverses of the characteristic times add) and has the same scaling behaviour with molecular weight and concentration. 

Eyring theory for molecular mobility supposes that the movement of each molecule (or structural unity) is thermally activated (Figure 6.8). Given a barrier energy A, the jump frequency is written as... 

The system is prepared at t=0 in the quantum state Pik> and the question is how to calculate the probability that at a later time t the system is in the state Fjn>. By construction, these quantum states are solutions of molecular Hamiltonian in absence of the radiation field, Hc->Ho Ho ik> = e k Fik> and H0 Pjn> = Sjn xPJn>. The states are orthogonal. The perturbation driving the jumps between these two states is taken to be H2(p,A)= D exp(icot), where co is the frequency of the incoherent radiation field and D will be a time independent operator. From standard quantum mechanics, the time dependent quantum state is given by ... 

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