Polymer Boltzmann distribution

The effect of hydrodynamic interactions on polymer collapse has also been studied using MPC dynamics, where the polymer beads are included in the multiparticle collision step [28, 84]. Hydrodynamic interactions can be turned off by replacing multiparticle collisions in the cells by sampling of the particle velocities from a Boltzmann distribution. Collapse occurs more rapidly in the... 

Recently, however, Sanji and coworkers have pointed out that the thermochromic change and increase in absorption intensity is too abrupt to be the result of a Boltzmann distribution between anti and twisted units along the polymer chain23. They proposed instead that the transition is an interconversion between two well-ordered states a tightly-coiled helix, and all-anti . 

The configuration of a polymer chain is analogous to the path r(s) of a diffusing particle with the segment rank s replacing time. At equilibrium, configurations follows the Boltzmann distribution... 

This reptation Monte Carlo algorithm has been incorporated into the NRCC program CLAMPS. Significant chain motions are effectuated by these single moves (17). We have employed this technique to sample the Boltzmann distribution of our polymer systems. 

The heterogeneous nature of polymer melts at Tgtwinkling fractal theory (TFT) [Wool, 2008a,b]. Wool considers Tg to result from the molecular cooperativity that leads to dynamic percolating fractal structures below Tc. He assumes Boltzmann distribution of diatomic oscillators interacting via the Morse anharmonic potential. Integrating the latter from zero to the inflection point, he expresses the T dependence of solidified polymer fraction as... 

The poling thus leads to the alignment of doped molecules in the polymer films according to the Boltzmann distribution law. The oriented molecular dipoles produce a polarization. P Np(cos0), where N is the number density of the molecules, p, the ground state dipole moment, and 0, the angle between the dipole and the direction of the poling field. This method can... 

Radiofrequency pulses are also utilized to measure relaxation times. Three relaxation times have been measured in TPEs, and each is sensitive to different phenomena. Ti, the spin-lattice relaxation time in the laboratory frame, is the relaxation from the nonequilibrium population distribution created by the pulse to the equilibrium Boltzmann distribution. Ti is sensitive to molecular motions that rate in the range of 10 -10 Hz. T2, the spin-spin relaxation time, is the relaxation caused by the establishment of equilibrium between nuclear spins within the system. Spin-spin relaxation measurements also probe motions with rates in the range of 10M0 Hz however, low frequency motions (lOMtPHz) also affect T2. Generally,T2 is one to three orders of magnitude smaller than Ti in solid polymers. Tip, the spin-lattice relaxation time in the rotating frame, probes motions with rates on the order of lO -KfHz. Cross polarization is usually used in Tip measurements. 

Equation (10.28) indicates that the orientational distribution of a polymer at equilibrium is a Boltzmann distribution under the potential Uxt- Thus f/scf is regarded as a mean field potential acting on the polymer. 

The MD simulation is started with a dense amorphous polymer microstructure (see above) with appropriate initial conditions, i.e., the initial position and velocity of every atom must be specified. For the polymer, the initial coordinates are created by the modified RIS method [26], while the velocities are randomly picked from a Maxwell-Boltzmann distribution at the desired temperature. The diffusants are started off in some cavities in the polymer, found by randomly trying to insert them into the structure until a place is found where... 

EOV EOS theory was developed by formulation of canonical function of the Boltzmann distribution of energies and derivation of thermodynamic pressure. The theorem of corresponding states says that the same compressibility factor can be expected for all fluids when compared at reduced temperature and pressure. A two-parameter correlation for compressibility factor, Z, can be derived using the theorem of corresponding states. EOV EOS obeys the corresponding state principle. Characteristic temperature, pressure, and specific volume used in EOV EOS are tabulated for 16 commonly used polymers. 

In simple terms, eqn [52] are diffusion equations in the component densities, which take into account the noise in the system. Dynamics of the molecular ensemble is based on the assumption that for each type of site a the local flux is proportional to the local site concentration 4> . At equilibrium, = constant, which results in the familiar self-consistent field equations for an inhomogeneous polymer system. When the system is not in equilibrium, the negative gradient -V/i,j(r) represents an effective thermodynamic force that drives collective relaxation processes. The time integration of eqn [52] generates an ensemble of density fields with the Boltzmann distribution. 

The Gaussian distribution is derived from the binomial distribution for large N [5]. It is important for statistics, error analysis, diffusion, conformations of polymer chains, and the Maxwell Boltzmann distribution law of gas velocities. 

In the meantime, Scheutjens and Fleer [35] had formulated their now classical numerical self-consistent field (lattice) (SF) fheory for equilibrium adsorption of polymers, which provides a very detailed partition function from which strucfural and fhermodynamic information can be derived. The key idea of a self-consisfenf field approach is that, on the one hand, the interactions between molecules are treated as if they constitute an external field on an individual molecule (fhereby defining potential energies), and that each molecule finds its optimum distribution of conformations and locations as a Boltzmann distribution in that field. On the other hand, the distribution found has to be such that it will indeed produce the field, hence the term self-consistent. An iterative scheme is employed to reach consistency, and as soon as the solution is found, energy and entropy of fhe system are known as well. 

In the previous discussion, a great deal of time has been spent on approaches to surface modification of the clay nanoparticles in order to render the particles to be more compatible with the polymer of interest. This approach mainly concentrates on the enthalpic portion of the Gibb s free energy of intercalation-exfoliation. In order to realize the maximum benefit from a nanocomposite, the exfoliated state is the ultimate goal, since this will present the maximum interfacial interaction between the nanoparticle and the polymer. In reality, a completely exfoliated system probably does not exist, but a Boltzmann distribution of energy states is more likely which invokes some of the entropic terms. In the following... 

In solution, polysaccharide linkages between the individual monomers take conformations and energies according to the Boltzmann distribution. The nomenclature of the resulting random coils is that of classical polymer theory, with the major descriptor of the shape being the characteristic ratio, Coo- The ratio is proportional to the mean square end-to-end distance, normalized for the length of the individual monomeric residues. In the solid state, the polysaccharides often form crystallites that are too small to provide sufficient diffraction information for a full determination of their structure. However, they often yield fiber diffraction patterns that give useful clues to the correct structure. 

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