Single chain approximation

Frequently function R can be written as a single term having the simple form of equation 1. For Instance, with the aid of the long chain approximation (LCA) and the quasi-steady state approximation ((JSSA), the rate of monomer conversion, I.e., the rate of polymerization, for many chain-addition polymerizations can be written as... 

As a consequence of the single chain approximation the conformation statistics of any two chains are independent of each other. Correspondingly, the mean autocorrelation function is given by... 

Many compounds of formulation Mn(L)(Z) (where L = (161) and Z = a monoanion) have been prepared (see Table 52). Structural evidence upon such compounds is not abundant, but [Mn(salen)-(OAc)] has been shown644 to possess approximately planar Mn(salen) units linked in a polymeric chain by singly bridging acetate ions. The very similar compound, in which the benzene rings of salen are replaced by naphthalene rings, also possesses the same polymeric structure.645... 

A first contribution to J-p is caused by the bonding between two atoms which constitute one molecule. This direct intra-molecular interaction between the atoms is taken into considerations as a f -bond. In addition there are intra-molecular interactions of indirect nature between the both atoms of a molecule. These atoms affect each other indirectly by n point interactions with all remaining atoms and combinations of atoms. The so-called intra-molecular pair cavity function y (ryw) expresses the ensemble of all indirect interactions which appear between the atoms of a molecule in f-bonds [13] and establishes the searched correlation function for all indirect interactions between the atoms inside a molecule. TTie molecular DFT approach evaluates the cluster expansion to calculate y (rjvr) using TPT. This approximation takes into account only presentations with vertices n <— 2, for what reason it is called the single chain approximation (TPT1)[12]. [7,8]... 

Peginterferon alpha-2a is a covalent conjugate of recombinant alpha-2a interferon (approximate MW is 20,000 Da) with a single branched d/s-monomethoxy PEG chain (approximate MW is 40,000 Da). Peginterferon alpha-2a has an approximate MW of 60,000 Da. 

Equation [45] is based on the worm-like chain approximation discussed elsewhere. The free energies of the chains can be summed to obtain the network free energy. The transition from the single chain to the network is made by relating... 

A simpler, analytic approach is to employ the standard idea of replacing the real system with a computationally tractable reference system (polymer chain model in our case), the parameters of which are variation-ally optimized using an approximate single-chain free energy expression. Two such schemes have been explored that differ in both the choice of reference chain model and the form of the approximate free energy. 

In the second-quality we are identifying the connection between the direct correlation function and the functional inverse of the response function for a uniform simple atomic fluid. The reader can verify this result using the Ornstein-Zernike equation [i.e., the Chandler-Andersen equation (8) for an atomic or single site fluid] and Eq. (11). As p(r) - p, Eq. (22) becomes exact. Its use for all p(r), however, is an approximation — the hypernetted chain approximation. 

Similar to the approximation made for in the formalism of Andersen (Eq. 19), only the lowest-order contribntion is retained, and all contribntions with 1 > 0 are neglected. This is the single chain approximation, that yields the following ... 

Single chain approximation— Neglects all graphs with more than one associated cluster. This is TPT, which assumes that the structure of unbonded sites in the fluid is similar to that of the reference fluid. The single chain approximation will fail, for instance, for fluids with a nanatic phase [68]. 

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

Most of the detergent range alcohols used commercially consist of mixtures of alcohols, and a wide variety of products is available. Table 6 shows the approximate carbon chain length composition of both the commonly used mixtures and single carbon materials typical properties are given ia Table 7. 

---