Junction model

The constrained-junction model was formulated in order to explain the decrease of the elastic moduli of networks upon stretching. It was first introduced by Ronca and Allegra [39], and Flory [40]. The model assumes that the fluctuations of junctions are diminished below those of the phantom network because of the presence of entanglements and that stretching increases the range of fluctuations back to those of the phantom network. As indicated by the second part of Equation (26), the fluctuations in a phantom network are substantial. For a tetrafunctional network, the mean-square fluctuations of junctions amount to as much as half of the mean-square end-to-end vector of the network chains. The strength of the constraints on these fluctuations is measured by a parameter k, defined as... 

The elastic free energy of the constrained-junction model is given by the expression... 

The elastic free energy of the constrained-junction model, similar to that of the slip-link model, is the sum of the phantom network free energy and that due to the constraints. Both the slip-link and the constrained-junction model free energies reduce to that of the phantom network model when the effect of entanglements diminishes to zero. One important difference between the two models, however, is that the constrained-junction model free energy equates to that of the affine network model in the limit of infinitely strong constraints, whereas the slip-link model free energy may exceed that for an affine deformation, as may be observed from Equation (41). 

According to the arguments based on the constrained-junction model, the term Gch should equate to the phantom network modulus onto which contributions from entanglements are added. 

The effect of restricted junction fluctuations on S(x) is to change the scattering function monotonically from that exhibited by a phantom network to that of the fixed junction model. Network unfolding produces the reverse trend, the change of S(x) with x is even less than that exhibited by a phantom network. Figure 6 illustrates how the scattering function is modified by these two opposing influences. 

In the general case of several electrolytes present in the solutions in contact with the liquid junction, no simple result analogous to (2.6.10) can be obtained. A basic problem stems from the fact that the electrolyte distribution in the liquid junction is dependent on time, so that the liquid-junction potential is also time-dependent. Because of these complications, further discussion will consider only those liquid-junction models where a stationary state has been attained, so that the liquid-junction potential is independent of time. This condition is notably fulfilled in liquid junctions in porous diaphragms. 

In contrast to the Planck solution, the Henderson approximation enjoys considerable use [ 10,11 ]. Henderson s liquid-junction model is based on the assumption that the concentrations of the ions in the liquid junction change linearly withx between values corresponding to the edges of the liquid junction. This assumption is equivalent to the concept of a mixture of electrolytes changing unifonnly between the two edges of the liquid junction. Then... 

While the Planck liquid-junction model corresponds to a junction with restrained flow , for example in a porous diaphragm, fig. 2.2, the Hendersoii model approaches a liquid junction with free diffusion (fig. 2.3). Ives and Janz [13] give inaccuracies in measuring liquid-junction potentials between 1 and 2 mV. 

The behavior predicted by the junction model is shown by the theoretical line. Although there are some unavoidable ambiguities in the values of in... 

Figure 7 The open-circuit photovoltage plotted versus the difference between the work function of the substrate in vaccum, 8 , v.ac> and the solution redox potential, 4>redox- The work function of the substrate in the solution, 8 , the quantity of interest, is difficult to measure directly, but it is related to 8 , vac see the discussion in Ref. 12. The four types of substrates are, from left to right, ITO, Sn02, Au and Pt the filled diamonds are for 0.5 M Lil solution, the open circles are for 0.05 M ferrocene in 0.1 M LiCI04 solution, and the filled triangles are for 0.05 M hydroquinone in 0.1 M LiCI04 solution. The theoretical line shows the behavior predicted by the junction model. (Data from Ref. 12.)...

This refinement of the constrained-junction model is based on re-examination of the constraint problem and evaluation of some neutron-scattering estimates of actual junction fluctuations [158, 159]. It was concluded that the suppression of the fluctuations was over-estimated in the theory, presumably because the entire effect of the inter-chain interactions was arbitrarily placed on the junctions. The theory was therefore revised to make it more realistic by placing the effects of the constraints along the network-chain contours, specifically at their mass centers [4, 160, 161]. This is illustrated in the second portion of Figure 2. Relocating the constraints in this more realistic way provided improved agreement between theory and experiment. 

Use of FPA to Study Helical Dynamics of RNA, with a Junction Model Construct as an Example... 

Fig. 2. Band diagram of two-junction model with photoconductive i layer at (a) open circuit...

A significant theoretical contribution to the potential distribution within nanocrystalline Ti02 films was proposed by Rothenberger et al. [146]. These workers employed a Schottky junction model to derive the potential distribution within the Ti02 particles at negative applied potentials. The model is based on the supposition... 

Empirical Potential Functions. There is one more type of treatment which is properly classified as an empirical model of the H bond. An explicit form is assumed for the potential function associated with the movement of the H atom within the H bond. Nordman and Lipscomb have proposed such a model based upon a Morse function (1521). This proposal will not be explored, however, in favor of the similar and more completely developed potential Junction model of Lippincott and Schroeder (1242, 1815). The potential function is written as a sum of four terms, the first of which has the form... 

The constraining potential represented by virtual chains must be set up so that the fluctuations of junction points are restricted, but the virtual chains must not store any stress. If the number of monomers in each virtual chain is independent of network deformation, these virtual chains would act as real chains and would store elastic energy when the network is deformed. A principal assumption of the constrained-junction model is that the constraining potential acting on junction points changes with network deformation. In the virtual chain representation of this con-... 

The constrained junction model has virtual chains (thin lines) connecting each network junction (circles) to the elastic background (at the crosses). 

The constrained-junction model relies on an additional parameter that determines the strength of the constraining potential, and can be thought of as the ratio of the number of monomers in real network strands and in wirtual chains NjnQ. If this ratio is small, the virtual-chain is relatively long... 

The PT model is frequently used as a minimalistic approximation for more complex models. For instance, it is the mean-field version of the Frenkel Kontorova (FK) model as stressed by D. S. Fisher [29,83] in the context of the motion of charge-density waves. The (mean-field) description of driven, coupled Josephson junctions is also mathematically equivalent to the PT model. This equivalence has been exploited by Baumberger and Carol for a model that, however, was termed the lumped junction model [84] and that attempts to... 

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