Chain dimensions, probability distribution

Figure 4.4 shows the histogram of R, which corresponds to the probability distribution function of the chain dimension. Information on the distribution was not available from the previous experiments in inverse space. The average radius of gyration, (i xy). was 138, 145, and 143 nm for the PMMA chains in thin films with thickness 15, 50, and 80 nm, respectively. The thickness of 15-80nm is relatively... 

We first require some assumptions. Let the cubes into which the elastomer is decomposed be large relative to the dimensions of a single chain, so that edge and corner effects are negligible. Furthermore, the chains will be represented in the Gaussian approximation, so that for any stretch of a chain containing n segments, the end-to-end separation is described by the probability distribution... 

It is clear from Figures 4 and 5 that the chain dimensions and probabilities have essentially reached their asymptotic values at modest temperatures. We should expect, therefore, that in melts of semiflexible PLCs, unless there are extreme extenuating circumstances (i. e., perfect core orientational ordering), the dimensions of a spacer chain and the conformer probability distributions within a spacer chain should be similar to those quantities in an alkane liquid. The attachment of a spacer at both ends will of course perturb the dimensions and conformer probabilities of the spacer. Clearly the magnitude of this perturbation of the spacer is the key to understanding the role of the spacer in a PLC. 

The renormalization transformation in the problem of polymer chain conformations in the Kadanoff-Wilson fashion forms, in essence, a semigroup. A version of such transformations based on the true group (also called the renormalization group) was applied by Alkhimov (1991, 1994). This method provides an asymptotical solution of the exact equation for the eiid-to-eiid distance probability distribution of a self-avoiding trajectories. The following formula has been obtained for the critical index i/ in d-dimension space ... 

However, when we consider a problem involving a single flexible polymer chain with N effective monomers and take the thermodynamic limit, N—> =, we find a lack of self-averaging when we consider quantities like the mean square end-to-end distance (R ) (for a free chain in solution, = 0 because all direaions for the orientation of the end-to-end veaor are equivalent, of course). For the case of Gaussian chains, we have a probability distribution (in d dimensions)... 

Derivation of the Gaussian Distribution for a Random Chain in One Dimension.—We derive here the probability that the vector connecting the ends of a chain comprising n freely jointed bonds has a component x along an arbitrary direction chosen as the x-axis. As has been pointed out in the text of this chapter, the problem can be reduced to the calculation of the probability of a displacement of x in a random walk of n steps in one dimension, each step consisting of a displacement equal in magnitude to the root-mean-square projection l/y/Z of a bond on the a -axis. Then... 

However, there is another typ>e of confinement that can be imposed on a reactive system, namely, by a reduction in the effective dimensionality. The simplest examples are those in which the motions of the reactive species are confined to a flat surface or a one-dimensional chain. However, in many systems the connectivity of the configuration space is such that it has effectively a fractal dimension d. The Hausdorf dimension is defined from the behavior of the pair distribution function at sufficiently large R, which varies as that is, the probability of finding the pair with a separation between R and R + dR is proportional to dR. The reduction of the encounter problem from d dimensions to the one dimension R is studied in Section VII A. The important case of reactions on surfaces is considered separately in Section VIIB. 

The retractive force in a rubbery material is a direct result of the chain in the extended form trying to regain its most probable, highly coiled conformation. Thus, it is of considerable interest to calculate, in addition to the average dimensions of the polymer chain, the distribution of all the possible shapes available to the molecules experiencing thermal vibrations. 

The need to consider the supply chain adds another dimension to the CE concept. Now it s not just tooling and material that have to be considered but also the distribution channels, inventory policy, and other supply chain issues. CE for the entire supply chain particularly fits in the case of B, C, and E producfs in the table. In these cases, a new product is more likely to be accompanied by a new supply chain. These are probably major projects involving a great number of xmknowns abouf the market and the likelihood of commercial success. 

The same types of expression can be derived for the distribution function in both the y and z directions. It is here assumed that the actual location of the chain end in, for instance, y space does not affect the location of the chain end in the other two dimensions, i.e. F x) = /(or) only. The probability of finding the chain end in the point (or, y, z) in a chain originating at the origin with the other chain end is given by ... 

To establish a useful equation of state for the mechanical behavior of a rubber network, it is necessary to predict the most probable overall dimensions of the molecules under the influence of various externally applied forces. An interesting approach to rubber elasticity consists of simulating network chain configurations (and thus the distribution of end-to-end distances) by the rotational isomeric state technique cited above. Based on the actual chemical structure of the chains, it enables one to circumvent the limitations of the Gaussian distribution function in the high deformation range. Nonetheless, the Gaussian distribution function of the end-to-end distance is very useful. It is obtained from a simple hypothetical model, the so-called freely jointed chain, which can be treated either exactly or at various levels of approximation. 

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