Length bias

Green PF (1988) The relationship between track shortening and fission track age reduction in apatite combined influences of inherent instability, annealing anisotropy, length bias and systems calibration. Earth Planet Sci Lett 89 335-352... 

The schematic model is depicted in Fig. 8. As the bias voltage increases, the number of the molecular orbitals available for conduction also increases (Fig. 8) and it results in the step-wise increase in the current. It was also found that the conductance peak plotted vs. the bias voltage decreases and broadens with increasing temperature to ca. 1 K. This fact supports the idea that transport of carriers from one electrode to another can take place through one molecular orbital delocalising over whole length of the CNT, or at least the distance between two electrodes (140 nm). In other words, individual CNTs work as coherent quantum wires. 

For intermediate drift rates (4 < BN < 8), when chain conformations are already distorted, deviates from linear behavior and goes through a maximum at some critical value Bf. of the field, confirming earlier findings by Pandey et al. [103,104]. This critical bias B at which the velocity starts to decrease depends rather weakly on the density Cobs, turns out to be reciprocal to chain length A, implying that only when the total force, /c = B,N 9, acting upon the whole driven molecule, exceeds a certain threshold, which does not depend on the size of the macromolecule, the chains start to get stuck in the medium. 

FIG. 28 Log-normal plot of relaxation time t2 vs bias for three different chain lengths (given as a parameter) and a series of medium densities [21]. 

Now consider the finite sampling systematic error. As discussed in Sect. 6.4.1, the fractional bias error in free energy is related to both the sample size and entropy difference 5e N exp(-AS/kB). With intermediates defined so that the entropy difference for each substage is the same (i.e., AS/n), the sampling length Ni required to reach a prescribed level of accuracy is the same for all stages, and satisfies... 

The smoothing terms have a thermodynamic basis, because they are related to surface gradients in chemical potential, and they are based on linear rate equations. The magnitude of the smoothing terms vary with different powers of a characteristic length, so that at large scales, the EW term should predominate, while at small scales, diffusion becomes important. The literature also contains non-linear models, with terms that may represent the lattice potential or account for step growth or diffusion bias, for example. 

Fig. 6 Schematic representation of an Hg-drop LAJ incorporating SAMs of organic molecules of (a) alkanethiols, (b) oligophenylene thiols and (c) benzylic derivatives of oligophenylene thiols of different length formed on an Ag electrode, (d) Semi-logarithmic plot of measured current at applied bias I 0.5 V vs electrode gap flowing through the a, b, c interfaces...

Fig. 7 Current densities J vs applied bias measured in a polymer-interlayer based LAJs for 1,8-octanedithiol, 1,10-decanedithiol, 1,12-dodecanedithiol and 1,14-tetradecanedithiol. Inset-. InJ as a Junction of the molecular length measured at 0.1 V, 0.3 V and 0.5 V bias. (Reprinted with permission from [80])...

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