Expansion coefficient chains

It Is evident that this function is continuous (Fig. 6). Consequently, the following additional correlations may be used for the estimation of expansion coefficients for noncyclic chains ... 

The main notion of the percolation theory is the so-called percolation threshold Cp — minimal concentration of conducting particles C at which a continuous conducting chain of macroscopic length appears in the system. To determine this magnitude the Monte-Carlo method or the calculation of expansion coefficients of Cp by powers of C is used for different lattices in the knots of which the conducting par-... 

Assuming that the chain statistics is an effective Gaussian with the uniform expansion coefficient l, we obtain... 

Finally, it behaves like a liquid provided the chain length is not too long. Just around T some physical properties change distinctively such as the specific volume, the expansion coefficient, the specific heat, the elastic modulus, and the dielectric constant. Determination of the temperature dependence of these quantities can thus be used to determine Tg. 

Since this meta-substituted diamine was much more reactive than OFB, a variety of polyimides could be made, aUowing the synthesis of such structures as PMDA-PFMB, a polyimide with a very low thermal expansion coefficient (CTE) that may provide many advantages for microelectronic fabrication. Linear polyimide structures have generally been found to have low CTEs even when bulky groups were attached to the main chain. 

Negative thermal expansion coefficients along the chain axes have been observed experimentally on many crystalline polymer lattices7,16 -18). Hence, the thermal contraction along the chain axis seems to be a general phenomenon in crystalline polymers. As a result of this conclusion, we are immediately faced with the question... 

Finally, making use the equation of state of Eq. (28) one can demonstrate that the linear thermal expansion coefficient depends on the deformation as follows (for isoenergetic chains)26 211... 

N+/is the number of unit cells of the polymer chain. The LCAO expansion coefficient matrices C(k) and the corresponding energy eigenvalues, e(k), are obtained by solving the eigenvalue problem for each value of k-... 

Using the chain rule for partial derivatives [Eq. (7) of Appendix A] express (dP/dT)r in terms of the thermal expansion coefficient a and the isothermal compressibility k. 

The possibility of negative thermal-expansion coefficient (TEC) values along a direction of strong coupling in layered or chain structures (the so-called membrane effect") was suggested for the first time by Lifshitz [4] for strongly anisotropic compounds. In the phonon spectra of such compounds... 

As an amorphous polymer, lignin undergoes chain segment motion upon heating. This motion, a glass transition, is characteristic of all amorphous polymers, and is indicated by an endothermic shift in the DTA or DSC curves. This glass transition is accompanied by abrupt changes in free volume, heat capacity, and thermal expansion coefficient. 

Based on the free volume theory, Vfj° and txj (thermal expansion coefficient) are assumed to be constant for different molecules. Vfj° has a value of 0.025 and clj has a value of 0.001 for small molecules such as solvent or monomer and 0.00048 for polymer long chains. The subscript j stands for species in the reaction mixture such as monomer, solvent, polymer, etc.. 

Figure 2. The timebehavi< of ttecorrelationfuiictioii (/),Eq. (2.6), with/t>l,

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