Free relaxed

The time of polymer solutions free relaxation (xdetermined experimentally, is the parameter, sensitive to macromolecules sizes and conformation change. It characterizes the rate, with which the optically anisotropic molecules primary orientation disappears, having established under... 

FIGURE 111 The dependence of free relaxation time on macromolecule gyration radius R for solutions of dendrimers PSD in tetrahydrofuran (1) and LPA-3 in chloroform... 

Hence, the free relaxation time of dendrimers macromolecule in solution is a function of its structure that was to be expected by virtue of the general postulate any property of an object is controlled by its structure... 

Kozlov, G. V Dolbin, I. V Tlenkopachev, M. A. The intercommunication of free relaxation time and macromolecule structure for dendrimers solutions. Proceedings of Vl-th International Sci.-Techn. Conf Mathematical and Computer Simulation of Natural Scientific an Social Problems. Penza, PSU, 2012,133-136. 

For free relaxation, the term Cj,E = 0 but the Rapini terms at each surface dramatically influence the relaxation process. The relaxation time of the director at the surface Xj is controlled by the wavevector njb. For the same parameters as above x = 1 - 100 ms that is Xs << i,uik and relaxation process starting... 

An extension of the notion of relaxation is used when the system is not isolated and continuously perturbed by imposition of an external source of energy. In that case one speaks of forced-relaxation in contrast to the free relaxation described above. The model is the same but instead of establishing the dependence upon time of the state variables during their evolution toward equilibrium, the preferred modeling consists of finding transfer functions, that is, impedance or admittance, featuring the behavior of the system independently from the shape of the perturbation signal. 

For this reason, the two aspects of free relaxation of an isolated system and forced relaxation of an open system are developed in detail. This latter case will be treated in the frame of... 

This circuit is treated here in two sitnations free relaxation using large signal models and forced regime nsing small amplitnde AC signals. 

The Formal Graph representing this model is shown in Graph 11.23 (right). This model is quite general (within the frame of homothetic properties) it can be used as well in free relaxation as in forced conditions. 

One begins by discussing the free relaxation permitted by the isolation of the circuit, setting the current to zero, which corresponds to a modified model using one of the individual currents, the capacitive one for instance ... 

The Formal Graph of the free relaxation is given in Graph 11.23 by setting the total current equal to zero. The opposite of the kinetic constant appears as the eigen-value of the time derivation, which means that the eigen-function is the exponential function as demonstrated in Chapter 7 when establishing the influence theory. The solutions for the three state variables of this system are therefore ... 

This is also a general model valid for any shape of system constitutive property operators, but which is only useful when their Fourier transformation can be analytically expressed. It is interesting to proceed in a similar way to the free relaxation case by combining the Fourier transformations of the constitutive properties into a transformed relaxation time because it provides a scaling of the imaginary term... 

FIGURE 11.8 Plots of the potential variation (left) and of the electrical and thermal energies variations (right) during free relaxation of an RC circuit. 

In case of free relaxation, the lineic density of the velocity is equal to zero (owing to the system isolation), so the variation of the relative elongation as a function of time is given by an exponential function... 

G RAPH 11.38 Three Formal Graphs of free relaxation in an isolated dipole assembly with a common flow (left), with a common effort (center), and with a combination of paths into a kinetic operator (right). The left and center ones use the normal representation of two additive contributions, and the right one implements a loop evidencing the irreversibility of the relaxation. 

These three equations are equivalent models of the free relaxation but the last translation has the advantage of allowing the definition of an apparent operator resulting from the combination of the elastance with the conductance ... 

The free relaxation case is retrieved by giving to the imposed flow the shape of a pure impulse, mathematically expressed as the Dirac distribution S(f). In that case the convolution amonnts to the identity operator and the outcome is the exponential function itself giving Eqnation 11.33. (This case is just for checking one of the possibilities of the model and does not justify the usage of the powerful tool of convolution.)... 

Helix unwinding and free relaxation from the unwound state back to the helical structure requires a much longer time (r 1-100 s, within the range Tc a — T = 40-10 °C, respectively) because, in this case, a lot of defects are expelled from the sample and appear again. This process is governed by some apparent flow viscosity which is much higher than 7<. ... 

Fig. 5 Population evolution of selected states for the dissipative dynamics of D2/Ru(0001) using different series of rc-pulses. In panels a-c, the pulses are separated by free relaxation intervals. The percentual population of the states are given at the end of each interval, denoted by the vertical dashed lines. The mode selectivity is reported where relevant. Reproduced with permission from ref. 105.

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