Flory matrix

The RIS model, coupled with the Flory matrix method, is applied to the calculation of the unperturbed mean-square end-to-end distance in polylcyclohexene sulphone) as a function of several parameters. The calculations are performed for atactic, isotactic and syndiotactic chains the tacticity arises from the two possible ways, D and L, in which the rings can be attached to the main chain, assuming that the C—C bonds are all in the trans conformation, as indicated by dielectric measurements. 

In Fig. 21(a) we plot the variation of R with increasing system density Cobs 3.nd, for comparison, also give the respective change for a system of moving medium (dynamic host matrix) of equal density. This result is in good agreement with recent predictions [89]. If one defines an effective Flory exponent from the scaling relation Rg oc it is then evident from... 

This is the simplest of the models where violation of the Flory principle is permitted. The assumption behind this model stipulates that the reactivity of a polymer radical is predetermined by the type of bothjts ultimate and penultimate units [23]. Here, the pairs of terminal units MaM act, along with monomers M, as kinetically independent elements, so that there are m3 constants of the rate of elementary reactions of chain propagation ka ]r The stochastic process of conventional movement along macromolecules formed at fixed x will be Markovian, provided that monomeric units are differentiated by the type of preceding unit. In this case the number of transient states Sa of the extended Markov chain is m2 in accordance with the number of pairs of monomeric units. No special problems presents writing down the elements of the matrix of the transitions Q of such a chain [ 1,10,34,39] and deriving by means of the mathematical apparatus of the Markov chains the expressions for the instantaneous statistical characteristics of copolymers. By way of illustration this matrix will be presented for the case of binary copolymerization ... 

Here Jta(x) denotes the a-th component of the stationary vector x of the Markov chain with transition matrix Q whose elements depend on the monomer mixture composition in microreactor x according to formula (8). To have the set of Eq. (24) closed it is necessary to determine the dependence of x on X in the thermodynamic equilibrium, i.e. to solve the problem of equilibrium partitioning of monomers between microreactors and their environment. This thermodynamic problem has been solved within the framework of the mean-field Flory approximation [48] for copolymerization of any number of monomers and solvents. The dependencies xa=Fa(X)(a=l,...,m) found there in combination with Eqs. (24) constitute a closed set of dynamic equations whose solution permits the determination of the evolution of the composition of macroradical X(Z) with the growth of its length Z, as well as the corresponding change in the monomer mixture composition in the microreactor. 

Thus, as can be inferred from the foregoing, the calculation of any statistical characteristics of the chemical structure of Markovian copolymers is rather easy to perform. The methods of statistical chemistry [1,3] can reveal the conditions for obtaining a copolymer under which the sequence distribution in macromolecules will be describable by a Markov chain as well as to establish the dependence of elements vap of transition matrix Q of this chain on the kinetic and stoichiometric parameters of a reaction system. It has been rigorously proved [ 1,3] that Markovian copolymers are formed in such reaction systems where the Flory principle can be applied for the description of macromolecular reactions. According to this fundamental principle, the reactivity of a reactive center in a polymer molecule is believed to be independent of its configuration as well as of the location of this center inside a macromolecule. 

For the transformation of the macrocomposite model to a molecular composite model for the ultimate strength of the fibre the following assumptions are made (1) the rods in the macrocomposite are replaced by the parallel-oriented polymer chains or by larger entities like bundles of chains forming fibrils and (2) the function of the matrix in the composite, in particular the rod-matrix interface, is taken over by the intermolecular bonds between the chains or fibrils. In order to evaluate the effect of the chain length distribution on the ultimate strength the monodisperse distribution, the Flory distribution, the half-Gauss and the uniform distribution are considered. 

Ky is the Flory-Huggins interaction parameter between the i and j monomers. In Eq. 6.6, the matrices have a dimension (m) (m). We note that the s-depen-dence of the excluded volume matrix is solely determined by the contribution of the bare susceptibility yoo(Q> ) he invisible matrix component 0 . Finally, combining Eq. 6.6 with Eq. 6.1 the response function in the interacting system is given by ... 

Flory-Huggins parameters Xy do not influence the mobility. For the case of Rouse dynamics, which only depends on the local friction, the bare mobility matrix ... 

The generator matrix treatment of simple chains with excluded volume described earlier S 010) properly reproduces the known chain length dependence of the mean-square dimensions in the limit of infinite chains. The purpose of this paper is to compare the behaviour of finite generator matrix chains with that of Monte-Carlo chains in which atoms participating in long-range interactions behave as hard spheres. The model for the unperturbed chain is that developed by Flory et at. for PE (S 027). 

Angular scattering functions, PIijI, are computed for subchains located in the middle and at the end of a PE chain. The RIS model developed by Flory et at, (S 004 - S 006) is used for the unperturbed chain. Chain expansion is introduced using a matrix treatment which satisfactorily reproduces several configuration-dependent properties of macromolecules perturbed by long-range interactions. 

Ke, the intramolecular excimer equilibrium constant under conditions where the dynamics or rotational Isomeric transitions is suppressed, is examined on the basis of the theory of macrocyclization [Flory, P. J.-, Suter, U. W. Mutter, M. J. Am. Chem. Soc. 1976, 98, 5733, 5740,5745], The conformational averages of the required moments and polynomials are calculated using their exact matrix generation method. The calculations can rationalize the appearance of a maximum near m = 15 In the 1 1 M that have been reported for pyrene-(CH2lrn-pyrene. In order to achieve this agreement, It is Imperative that the theoretical analysis incorporate the probability of a proper angular correlation of the two pyrene ring systems when the separation of their centers Is 0.35 nm. 

Cycllzatlon of polyoxyethylene to form the macrocycle 3x-crown-x is studied for even x from 4 to 20 units using Monte-Carlo methods. POE is assumed to behave in accord with the model developed by Mark and Flory (S 137, S 138). The fraction of acyclic chains satisfying criteria for cyclozation reaches a maximum of 1.2 x 10-3 for x = 6 and falls of to 0.1 x 10-3 when x Is 20. The implication that 18-crown-6 is the most easily formed macrocycle is in harmony with experiment. Conditional probabilities and a priori probabilities are evaluated at 298 K conditional probabilities, in their matrix representations, are ... 

The proton NMR shifts of diad, triad, and tetrad protons in an isolated chain of PP are calculated, taking into account its configurations and conformations using Pople s approximation. The agreement between the observed and calculated results, however, is insufficient with respect to the large difference between syn and anti protons and the order of tetrad protons. Here, a priori probabilities of specified conformations necessary to the estimation of the chemical shifts are calculated using Flory s matrix method. 

Here, again, we start from compressible SCFT formalism described in Section 2.2 and consider a model system in which bulk polymer consists of "free" matrix chains (Ny= 300) and "active" one-sticker chains (Na= 100). Flory-Huggins interaction parameters between various species are summarized in Table 1. This corresponds to the scenario in which surfactants, matrix chains, and functionalized chains are all hydrocarbon molecules (e.g., surfactant is a C12 linear chain, matrix is a 100,000 Da molecular weight polyethylene, and functionalized chain is a shorter polyethylene molecule with one grafted maleic group). The nonzero interaction parameter between voids and hydrocarbon monomers reflects the nonzero surface tension of polyethylene. The interaction parameter between the clay surface and the hydrocarbon monomers, Xac= 10 (a = G, F, A), reflects a very strong incompatibility between the nonpolar polymers and... 

Flory has adopted the use of a matrix to represent the dependency between rotational states of neighboring bonds. For the typical case of a polymer with three rotational states such as trans (t), gauche plus (g+), and gauche minus (g ), the statistical weight matrix is... 

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