Polymer probability density

When Ru-red was used as a catalyst in the presence of a large excess of Ce(IV) oxidant (Scheme 19.1), the rate of 02 evolution was first order with respect to the catalyst concentration, showing that Ru-red is capable of 4-electron oxidation of water. By the decomposition of the Ru-red, N2 was formed its formation rate was second order with respect to the catalyst concentration, showing that the decomposition is bimolecular. The decomposition distance in a polymer (Nafion) matrix was estimated by assuming the random distribution of the catalyst molecule in the matrix. The probability density P(r) of the distance between the nearest neighbor molecules (r nm) is represented by Eq. (19.5) according to the Poisson statistics... 

Two-phase polymerization is modeled here as a Markov process with random arrival of radicals, continuous polymer (radical) growth, and random termination of radicals by pair-wise combination. The basic equations give the joint probability density of the number and size of the growing polymers in a particle (or droplet). From these equations, suitably averaged, one can obtain the mean polymer size distribution. 

We can develop a stochastic model for two-phase polymerization by following the changes in the number and size of the growing polymers (or radicals) with time in an arbitrary particle of a system. [For a more general discussion of probability methods in particulate systems see Ref. 7.] Let us say that at some time t the particle contains m radicals of sizes Xi, x2,. . . xm (in order of their appearance) with probability density pm (xu x2,. .. Xm t). Since a polymer chain is usually long, we take the chain length or polymer size to be a continuous variable. Now, we assume that in a short time interval [t, t + t] changes in the particle occur by these processes ... 

This mean polymer size density is just what would be obtained from the average over all particles in the system. For convenience in computing /(z) we introduce the following probability functions ... 

FIGURE 1.2. Molecular structure of widely used it-conjugated and other polymers (a) poly(para-phenylene vinylene) (PPV) (b) a (solid line along backbone) and it ( clouds above and below the a line) electron probability densities in PPV (c) poly(2-methoxy-5-(2 -ethyl)-hexoxy-l,4-phenylene vinylene) (MEH-PPV) (d) polyaniline (PANI) (d.l) leucoemeraldine base (LEB), (d.2) emeraldine base (EB), (d.3) pernigraniline base (PNB) (e) poly(3,4-ethylene dioxy-2,4-thiophene)-polystyrene sulfonate (PEDOT-PSS) (f) poly(IV-vinyl carbazole) (PVK) (g) poly(methyl methacrylate) (PMMA) (h) methyl-bridged ladder-type poly(jf-phenylene) (m-LPPP) (i) poly(3-alkyl thiophenes) (P3ATs) (j) polyfluorenes (PFOs) (k) diphenyl-substituted frares -polyacetylenes (f-(CH)x) or poly (diphenyl acetylene) (PDPA). 

Doi molecular theory adds a probability density function of molecular orientation to model rigid rodlike polymer molecules. This model is capable of describing the local molecular orientation distribution and nonlinear viscoelastic phenomena. Doi theory successfully predicts director tumbling in the linear regime and two sign changes in the first normal stress difference,as will be discussed later. However, because this theory assumes a uniform spatial structure, it is unable to describe textured LCPs. 

In describing the mechanical response of microstructured fluids, e.g., polymers, emulsions, colloidal dispersions, etc., one needs to determine the pair distribution function - the probability density P(r) for finding a particle at a position r given a particle at the origin in suspensions, or the probability density of the end-to-end vector in polymers, or a measure of the deformation of drops in an emulsion. This probability density satisfies an advection-diffusion or Smoluchowski equation of the following (when suitable approximations have been made) form ... 

As noted in previous sections, the more reactive sites on Cr/silica, which produce low-MW polymer, probably have greater electron deficiency and therefore might be expected to incorporate 1-hexene more easily. The titania-associated sites, however, seem to contradict this view, because they too should have lowered electron density on the Cr center. They do produce a strong increase in MI with the addition of 1-hexene to the reactor, indicating a strong interaction. But they do not create many branches. This curious behavior is observed with Cr/AIPO4 catalysts as well. It may indicate that adsorption or insertion of 1-hexene tends to cause immediate chain transfer. Comonomer on the end or beginning of a chain would not be effective (or measured) as a branch. 

Preparation of blends 13-24, 214, 251, 252, 276, 324, 342-350, 640, 1024-1032,1128, 1151,1337 PRISM (Polymer reference interaction site model) 166,167 Probability density function 166... 

Probability of an open site or bond in Chapter 4 Hydrostatic pressure (mmHg) in Chapters 5 and 6 Persistence length on polymer chain (cm) in Chapter 4 Membrane permeability (cm/s) in Chapter 5 Probability density function Critical probability Peclet number... 

The examples of self-similar functions considered above fall into an especially simple category all are functions of the generalized time or group parameter and functionals (either linear or nonlinear) of the initial condition. However, there are many self-similar physical properties and mathematical objects that depend on additional, unsealed variables. For example, the probability density for the distribution of end-to-end distances of a linear, ideal (phantom) polymer chain is given by the expression... 

Subsequently we show some important properties of the moment generating function. This include the number and weight average of the degree of polymerization and other averages, as well as the distribution function. There is a potential confusing situation in nomenclature. In polymer chemistry, the term cumulative distribution is used for what is called in mathematics the cumulative distribution function or simply distribution function. Further, polymer chemistry uses the term differential distribution for the mathematical term probability density function. 

Equation 3.3 is the basis of the theoretical calculations of global polymer properties which depend on chain conformation. For example, the probability density (nonnalized distribution functicxi) G(R) that the chain has a specified end-to-end distance R is expressed in terms of P( r ) as follows ... 

ZfN) (not to be confused with Z, the number of entanglements on a polymer) is the partition function of a LP with N beads and is indicative of the total number of conformations that it may adopt. As shown in Sections 7.3 and 7.4, static properties of LPs are unaffected by the composition of the CLB hence is assumed to be independent of ( )c. The partition function Zf N) does not enter into calculations explicitly, beyond an additive constant, which cancels out when we calculate MN, c) is the probability with which an LP adopts a conformation in which its ends overlap. The second equality in Equation (7.25) offers a more tractable computational route to calculate P Rc)-Computation of F Rc) thus requires knowledge of (a) X N, ( )c), the probability with which an LP adopts a conformation indistinguishable from lhat of a CP, which occurs when its ends essentially overlap, and (b) P Rc), the probability density function of CPs characterized using the size Rc as the macrostate variable. These quantities are expected to change with the composition of the CLB. 

The universality of the Gaussian polymer is actually a consequence of the Central limit theorem. Suppose we construct a flexible polymer from bonds with independent probability distribution iP t) for a bond vector r. The end-to-end distance is given by R= Ei r, so that the probability density of R is... 

Fig.3. Fluctuation of the longitudinal velocity component for a 100 ppm polymer solution Fig.4. Probability density at Re=9,630. distribution and the mean...

Quantification of polymer entropies is done through the statistics of random walk (Flory, 1953). The model is based on a drunk trying to walk in one dimension Because of his state, the next step the drunk takes could be to the right or to the left with equal probability, but his stride rranains of idmtical length and at every step he waits for the same length of time. The key probability density function is that of end-to-end displacement x, that is, the distance betweai the beginning and the end, which for a linear polymer in one dimraision is Ganssian ... 

The equation above is the Fokker-Planck equation to estimate the evolution of the probability density in space. Various forms of the Fokker-Planck equations result from various expressions of the work done on the systems, and are used in diverse applications, such as reaction diffusion and polymer solutions (Rubi, 2008 Bedeaux et al., 2010 Rubi and Perez-Madrid, 2001). A process may lead to variations in the conformation of the macromolecules that can be described by nonequilibrium thermodynamics. The extension of this approach to the mesoscopic level is called the mesoscopic nonequilibrium thermodynamics, and applied to transport and relaxation phenomena and polymer solutions (Santamaria-Holek and Rubi, 2003). 

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