Gaussian conformation distribution

Fig. 15a-d. Simulated ESR spectra for radical —CHj— —C(CH3>C00R with Gaussian conformation distributions about the most probable positions 0, =55° and 0j = 65°, with Half-height widths indicated... 

The second entropic contribution, Asp conf accounts for a decrease in entropy which follows from a change in the chain conformations. The Gaussian conformational distribution found in the homogeneous phase cannot be maintained in the microphase-separated state. Formation of a layer structure leads, for steric reasons, necessarily to a chain stretching which in turn results in a loss in entropy. For a qualitative description we employ the previous Eq. (2.93)... 

Typically in solution, a polymer molecule adopts a conformation in which segments are located away from the centre of the molecule in an approximately Gaussian distribution. It is perfectly possible for any given polymer molecule to adopt a very non-Gaussian conformation, for example an all-trans extended zig-zag. It is, however, not very likely. The Gaussian set of arrangements are known as random coil conformations. 

Most synthetic polymers in which the monomer units are connected via single bonds have rather flexible chains. The bond torsion energy is relatively small and the units can rotate around their bonds [14,30,31]. Each molecule can adopt a large number of energetically equivalent conformations and the resulting molecular geometry is that of a statistical coil, approximately described by a Gaussian density distribution. This coil conformation is the characteristic secondary structure of macromolecules in solution and in the melt. It is entropically favoured because of its... 

Gaussian-like distribution of energy around the energy average. Other ensembles with non-Boltzmann distributions can enhance the sampling considerably for example, in the multi-canonical approach [97, 98], all the conformations are equiprobable in energy in Tsallis statistics [99], the distribution function includes Boltzmann, Lorentzian, and Levy distributions. 

Gaussian segment distribution for PUMA. Since the presence of the naphthalene groups could perturb the average chain conformation,... 

The gap conductance model for conforming rough surfaces was developed, modified, and verified by Yovanovich and co-workers [35,73,100-104]. The gap contact model is based on surfaces having Gaussian height distributions. It also accounts for the mechanical deformation of the contacting surface asperities. The development of the gap conductance model appears in several papers [139,143,147]. 

Now assume that the chain distributions are in their most probable (which is assumed to be Gaussian) conformations at the time the crosslinks are introduced. (This is not always so, as in the case of crosslinking in the swollen state as will be discussed below.) The number, Ny, of displacement vectors of type Ry (in the imstretched state) is... 

Let us suppose that W is the Gaussian distribution commonly used for rubber networks. The change of the network conformational distribution function in the gradient layer can be obtained by Equation (6.41). In the equilibrium of external forces and filler field forces at some extension... 

In a dense melt, the excluded volume of the monomeric units is screened and chains adopt Gaussian conformations on large length scales. In the following, we shall describe the conformations of a polymer as space curves r(r), where the contoiu" parameter r runs from 0 to 1. The probability distribution P[r] of such a path r(r) is given by the Wiener measure... 

A Gaussian chain is self-similar. This means that, for example, a chain of 2N units has the same conformational distribution as an N-chain if (i) we make every other unit of the 2N-chain invisible and (ii) shrink all position vectors of this... 

The latter result, as well as eqn [2], can be rationalized on the basis of profound analogy between conformation of ideal (self-intersecting) polymer chain and a random walk in 3D space. A similar Gaussian probability distribution applies to the vector connecting any pair of segments inside the chain, provided that they are separated by sufficiently large number of segments. 

Here we calculate of a chain with a Gaussian conformation. Using the Gaussian distribution given by Eq. 1.34, ( r - for a given m and n is calculated as... 

The conformational distribution in the undeformed state agrees with that of an uncross-linked melt, i.e, is given by an isotropic Gaussian function. 

The first method is the use of the chain variables R s) directly, and we remember equation (86) for the probability of finding a conformation. If a crosslink fixes segment s from chain i to segment 5 j on chain j we have the constraint Ri s ) = Rj s)) permanently. The probability distribution of the network is given by the constraint averaged over Gaussian conformations of chains. In terms of the Wiener measure this probability can be written as... 

Multiple linear regression is strictly a parametric supervised learning technique. A parametric technique is one which assumes that the variables conform to some distribution (often the Gaussian distribution) the properties of the distribution are assumed in the underlying statistical method. A non-parametric technique does not rely upon the assumption of any particular distribution. A supervised learning method is one which uses information about the dependent variable to derive the model. An unsupervised learning method does not. Thus cluster analysis, principal components analysis and factor analysis are all examples of unsupervised learning techniques. 

There are similar algorithms, also called simulated annealing, that are Monte Carlo algorithms in which the choice conformations obey a Gaussian distribution centered on the lowest-energy value found thus far. The standard deviation of this distribution decreases over the course of the simulation. 

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