Junction point fluctuation

Consider a linear chain consisting of m + jV monomers. The ends of this chain are fixed in space. The x coordinates of the ends are Xf and X2, while the junction point fluctuates with x coordinate R. What is the mean-square x coordinate of the end-to-end vector of the section containing N monomers ((R - Z j) ) ... 

A key assumption of the single molecular theory is that the junction points in the network move affinely with the macroscopic deformation that is, they remain fixed in the macroscopic body. It was soon proposed by James and Guth [9] that this assumption is unnecessarily restrictive. It was considered adequate to assume that the network junction points fluctuate around their most probable positions [9,10] and the chains are portrayed as being able to transect each other. This has been termed the phantom network model. The vector r joining the two junction points is considered as the sum of a time average mean r and the instantaneous fluctuation Ar from the mean so that... 

More recent network models of rubber elasticity by Ronca and Allegra [11] and Hory and Erman [12] are based on the phantom network model but assume that the junction point fluctuations are restricted due to the presence of entanglements. The strength of the constraints is defined by a parameter... 

If restricted junction fluctuations are taken into account, the chain deformation is increased, and is more anisotropic. The effect of increasing k is much more evident in networks of low functionality, since fluctuations of junction points are of minor importance in networks of high functionality. 

Thermal electric noise thermometry 1. Josephson junction point contact 2 Conventional amplifier 0.001-1 4-1400 Mean square voltage fluctuation Nyquist s law oc fegT Other sources of noise serious problem for T > 4 K... 

Assume that each network strand has A monomers. One network strand, shown in Fig. 7.2, has end-to-end vector Rq with projections along the x, y, and z directions of R o, RyQ, and R in the undeformed state. In the affine network model, the positions of the junction points (the ends of the strands) are always fixed at particular points in space by the deformation and not allowed to fluctuate. For affine deformation, the end-to-end vector of the same chain in the deformed state is R (see Fig. 7.2) with projections along the x, y, and z directions of... 

The fluctuations of junction points in a network are quite similar to those of the branch point of an /-arm star polymer. In order to calculate the amplitude of these fluctuations, start with/- 1 strands that are attached at one end to the surface of the network and joined at the other end by a junction point connecting them to a single strand [see the left-most part of Fig. 7.5(a)]. The strands attached to the elastic non-fluctuating network surface are called seniority-zero strands. Each of these /— 1 seniority-zero strands are attached to a single seniority-one strand by a /-functional crosslink [see the left-mostpart of Fig. 7.5(a)]. The seniority of a particular strand is defined by the number of other network strands along the shortest path between it and the network surface. The/- 1 seniority-zero strands... 

The phantom network model assumes there are no interactions between network strands other than their connectivity at the junction points. It has long been recognized that this is an oversimplification. Chains surrounding a given strand restrict its fluctuations, raising the network modulus. This is a very complicated effect involving interactions of many polymer chains, and hence, is most easily accounted for using a mean-field theory. In the... 

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-... 

Consider a comb polymer with p Kuhn monomers of length b between neighbouring points of attachment of n-mer side branches. Assume the number of side branches is large and that the end of each side branch is randomly attached to the elastic non-fluctuating background. Show that the mean-square fluctuation of the junction points is h /pn. 

PAGE 300 End of problem 7.22 add in the limit n p. What is the mean-square fluctuation of the junction point in the opposite limit (p n) ... 

An old point of controversy in rubber elasticity theory deals with the value of the so-called front factor g = Ap which was introduced first in the phantom chain models to connect the number of elastically effective network chains per unit volume and the shear modulus by G = Ar kTv. We use the notation of Rehage who clearly distinguishes between A andp. The factor A is often called the microstructure factor. One obtains A = 1 in the case of affine networks and A = 1 — 2/f (f = functionality) in the opposite case of free-fluctuation networks. The quantity is called the memory factor and is equal to the ratio of the mean square end-to-end distance of chains in the undeformed network to the same quantity for the system with junction points removed. The concept of the memory factor permits proper allowance for changes of the modulus caused by changes of experimental conditions (e.g. temperature, solvent) and the reduction of the modulus to a reference state However, in a number of cases a clear distinction between the two contributions to the front factor is not unambiguous. Contradictory results were obtained even in the classical studies. 

The constrained junction fluctuation theory was modified by Erman and Monnerie [94]. The fundamental difference between the modified and the original models is the adoption of the assumption that constraints affect the centers of mass of the chains rather than the junction points only. They considered two different cases (1) the fluctuations of aU points along the chains in the phantom network are independent of macroscopic strain (constrained chain scheme, CC) and (2) the fluctuations of the points in the phantom network are dependent on the macroscopic strain, only the junctions are invariant to strain (modified constrained chain... 

The theory in the Gaussian limit has been refined greatly to take into account the possible fluctuations of the junction points. In these approaches, the probability of an internal state of the system is the product of the probabilities Win) for each chain. The entropy is deduced by the Boltzmann equation, and the free energy by equation (26). The three main assumptions introduced in the treatment of elasticity of rubber-like materials are that the intermolecular interactions between chains are independent of the configurations of these chains and thus of the extent of deformation (125,126) the chains are Ganssian, freely jointed, and volumeless and the total number of configurations of an isotropic network is the product of the number of configurations of the individual chains. 

For example, the phantom network model of James and Guth (1,2) gave a recipe for predicting the deformation of a polymer network by an applied stress, and allowed predictions of the change in chain dimensions as a function of network expansion or distortion. In an effort to make the phantom model more realistic, and to fit the model to a variety of experimental results, P.J. Flory and collaborators (3,4,5) proposed that the fluctuation of crosslink junction points calculated by the James-Guth method should be very much restricted by chain entanglements. 

Theories have been developed to describe the mechanical properties of amorphous networks and their swelling behavior in terms of an average (1-3). Over the years there have been several modifications in the theories to account for the fluctuations of the junction points, the role of network defects such as dangling chains and loops and the role of trapped entanglements in determining the equilibrium elasticity of a network (4). 

Figure 3.10 shows schematically the difference between the affine network model and the phantom network model. The affine deformation model assumes that the junction points (i.e. the crosslinks) have a specified fixed position defined by the specimen deformation ratio L/Lq, where L is the length of the specimen after loading and Lq is the length of the unstressed specimen). The chains between the junction points are, however, free to take any of the great many possible conformations. The junction points of the phantom network are allowed to fluctuate about their mean values (shown in Fig. 3.10 by the points marked with an A) and the chains between the crosslinks to take any of the great many possible conformations. 

By contrast, electrolyte states are much more limited in their distribution than metal conduction band states so that in many cases electron transfer through surface states may be the dominant process in semiconductor-electrolyte junctions. On the other hand, in contrast to vacuum and insulators, liquid electrolytes allow substantial interaction at the interface. Ionic currents flow, adsorption and desorption take place, solvent molecules fluctuate around ions and reactants and products diffuse to and from the surface. The reactions and kinetics of these processes must be considered in analyzing the behavior of surface states at the semiconductor-electrolyte junction. Thus, at the semiconductor-electrolyte junction, surface states can interact strongly with the electrolyte but from the point of view of the semiconductor the reaction of surface states with the semiconductor carriers should still be describable by equations 1 and 2. 

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