Linear chain model

Clark, M.B., Zimm,B.H. A linearized chain model for dielectric loss in polymers. ACS Polymer Preprints 12, 116-120 (1971). See also Tobolsky,A.V., DuPre,D.B. Macromolecular relaxation in the damped torsional oscillator and statistical segment models. Advan. Polymer Sci. 6,103-127 (1969). 

Curie-Weiss behaviour from 309 to 83 K with a Weiss constant 6 s 10 K. The 3-carboxylate was studied in the range 275-4.2 K and the data were interpreted in terms of an antiferromagnetic interaction between silver ions using the Ising linear chain model. This gave an exchange energy of / = —30.8 1.0 cm-1 and g = 2.5 0.02 which was somewhat smaller than the experimentally determined (g) of 2.08.513... 

Modeling fat crystal networks started with Van den TempeTs work in 1961, when he proposed the linear chain model (Van den Temple 1961). It was postulated that solid fat particles form linear chains that are held together by two types of bonds— irreversible primary bonds (stronger bonds) and reversible secondary bonds (weaker bonds). In this early model, the shear modulus (G) is predicted to be directly proportional to the volume fraction O of solids and to particle diameter D according to the equation ... 

During the 1920s, the linear chain model appeared as unacceptable because it seemed to imply the existence of free bonds at the end points. The polymer structure was therefore considered as a set of covalent cyclic molecules of low, molecular masses, held together by weak valence bonds. Such a hypothesis led... 

Figure 34. Surface phonon dispersion for 2H-TaSe2. The HAS data are shown as solid circles except for weak points which appear in the TOP spectra as hybridized longitudinal modes that are shown as crosses. All the data were obtained at 60 K, well into the low-temperature phase. The calculated striped and shaded regions, corresponding to transverse and longitudinal polarizations respectively, are the slab-adapted bulk phonon bands, while the solid line is a calculation for the Rayleigh wave based on the Dispersive Linear Chain Model (shown schematically in Fig. 35). The open circles at g = 0 are from Raman scattering experiments. (This figure has been corrected from Fig. 23 in Ref. 54.)...

Figure 35. Schematic drawing of the Dispersive Linear Chain Model applied to 2H-TaSe . The force constants /, and A describe the Ta-Se and Se-Se forces within a TaSej subunit, and / gives the Se-Se force between layers, rf, and are the spring constants between Se atoms and between Ta atoms, respectively, for neighboring subunits in the same layer.

For X and refer to original paper. Cu(CH30-)2 reproduced by a linear chain model. 

In a simple linear-chain model without neutral threefold sites, that is, with no branching and only singly-occupied chain ends, the average number of atoms per chain Ncham... 

The linear chain models evidently have no immediate relevance for relaxation of simple polar molecules in liquids but to the writer at least the results suggest strongly the importance particularly at low temperatures of cooperative interactions of molecules with their neighbors %diich may but need not involve electric dipoles as the source of intermolecular torques. When the energies involved are appreciable relative to k T it seems important to develop and study more realistic but still tractable models for analytic and simulation calculations. There have not yet been many serious attempts or promising results in analytical two and three dimensional theories except for diffusion-like models discussed by other contributors to this volume but a few developments which have some relevance can be mentioned. 

Fig. 35. Mn(Z5,5, 2 -0(Cl)C6H3-CH=N.C6H3(a)0-). Temperature dependence of The full line is the calculated curve of best fit using both binuclear J/hc = — 2.7cm" ) and linear chain models (J/ftc = —1.5cm" ) g = 2.0, Wa=0[71B53].

Fig. 102. Mn(py)jCl2. Temperature dependence of Curves fitted according to anti-ferromagnetic Heisenberg linear-chain model full curve best fit according to Weng s interpolation scheme (J/k= -0.69 0.02 K) dashed curve best fit according to the scaling method...

The linear chain model has also been used to study the role of energy dissipation in atom surface collisions. It is of course too simplistic to approximate the surface by a linear chain of atoms but the model is easily solvable and can give some qualitative insight into the problem. Thus the problem can be formulated by adding the following equation to the set of equations for the spring or chain atoms... 

On the basis of the knowledge of the behavior of the one- and two-parameter models, it is now possible to approximate any Af-parameter linear chain model by a four-parameter model as shown in Figure 3.7. The differential equation of the four-parameter model is... 

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