Fixman

Fixman M 1974 Classical statistical mechanics of constraints a theorem and application to polymers Proc. Natl Acad. Sc/. 71 3050-3... 

The solution of system (6.56) is a very complicated mathematical problem it definitely needs numerical calculations on some stages of processing. At least two successful attempts to overcome these difficulties are well known in the literature. The first method was put forward by Sack and expresses the solution through a continuous fraction. The second was proposed by Fixman and Rider [29], and deals with a kinetic... 

Kluk was the first to obtain these results [269]. They differ drastically from (6.25). When the medium becomes more rarefied, intensity in the central part of the IR spectrum decreases to zero. Hence, tJ j shortens, unlike t, which lengthens. The time t e i behaves in the same manner for the case of weak collisions, though a formula quantitatively analogous to (6.63) is not found for this case. One can refer only to numerical calculations based on the general formulae by Sack or by Fixman and Rider. These calculations provide identical results [85]. Fig. 6.5 shows that, in rarefied media, the difference between weak and strong collision... 

The problem of the influence of intramolecular interactions on the configurations of polymer molecules has been the subject of much controversy, which need not be reviewed here. For treatment of the problem by methods other than the one presented in the following pages, the reader is referred to papers by F. Bueche, J. Chem. Phys., 21, 205 (1953), and B. H. Zimm, W. H. Stockmayer, and M. Fixman, J. Chem. Phys., 21, 1716 (1953). These papers include references to other literature on the subject. 

Figure 1 Distributions for the end-to-end distance of a PDMS chain having n = 20 skeletal bonds of length / = 1.64 A. The Fixman-Alben distribution (dotted curve) and that from a Monte Carlo simulation (solid curve) are compared with the Gaussian approximation (dashed curve).

Among various representations of w(r) for short chains are the Hermite series [31,32], the Fixman-Alben distribution [33], and Monte Carlo simulations [34], The Fixman-Alben distribution is given by... 

The second term in this potential is the so-called Fixman potential. With this potential we simply have... 

This equation is a very important result because it shows that constrained simulations can be used to calculate dA/d . A possible algorithm would consist in running a constrained simulation with the Hamiltonian [which contains the extra Fixman potential 1/(2/ ) In Zc(q)], calculate the rate of change of -Z/J with at each step. Finally by averaging this rate of change the derivative of A can be computed. 

However, several authors took a slightly different route [17, 23, 25-27, 29] and derived an expression which does not explicitly introduce the Fixman potential. 

If the Fixman potential is not added to the Hamiltonian, the system is sampled differently and therefore correction terms must be added to (4.27). First, the weight Ze r in (4.24) must be reintroduced. Second, the Lagrange multiplier is different since the Fixman potential I /(2/3) In is not used. Considering (4.20) for A, it is possible to show that the Lagrange multiplier AF simply needs to be replaced by... 

The Fixman potential is not needed here since we are sampling with Hamiltonian Jtf. The second term is (4.31) can be expressed in terms of... 

Consider the constrained Hamiltonian with the Fixman potential... 

Spectroscopic Methods, [Biological] Applications of Spectroscopy, EPR, Recent Advances in (Smaller). Spectroscopy, Infrared, Use in Biology (Lecomte). Spectroscopy of Transition-Group Complexes (Jorgensen) Statistical-Mechanical Theory of Transport Processes. X. The Heat of Transport in Binary Liquid Systems (Bearman, Kirkwood, Fixman). ... 

Bearman, Kirkwood, Fixman). Transport Processes, Formal Statistical Theory of 1 1... 

The kinetic problem for the intramolecular cross-linking reactions in general form was not yet solved. Only some particular cases, i.e. the cvclization of macromolecules, the intramolecular catalysis and diffusion-controlled collision of two reactive groups were studied theoretically bv Xorawetz, Sisido and Fixman... 

This leads to a system of N Hnear equations from which the unit forces can be obtained. The final result can be expressed in the following general expression derived by Horta and Fixman [73], which does not assume any specific form for the distribution of distances between units, or chain model ... 

The introduction of branching in the Kirkwood formula and the KR calculations can be accomplished in a relatively easy way if Gaussian statistics corresponding to ideal chains are maintained. This description cannot, however, be very accurate in molecules with centers of high functionality because of the presence of cores with a high density of polymer units, which profoundly perturbs the internal distribution of distances. Stockmayer and Fixman [81 ] employed the Kirwood formula and Gaussian statistics to calculate h in the case of uniform stars, obtaining an analytical formula. They also performed a KR evaluation of the viscosity and proposed that g could be evaluated from the approximation... 

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