Fluctuation free volume formation

Simultaneously, the data of Figure 1.19 suppose a tight interconnection between the cluster model [13-15] and the fluctuation free volume theory [78, 79] separation of a segment from a cluster means formation of a free volume microvoid, whereas the addition of a segment to a cluster means its collapse . If it is correct, the entropy variation AS due to fluctuation free volume formation (Equation 1.27) must be equal to entropy change at partial decay of thermofluctuational clusters. The value of can be estimated within the frameworks of the Forsman theory [82] according to following relationship ... 

Figure 1.20 The temperature dependences of entropy change due to fluctuation free volume formation AS (1, 2) and partial decay of clusters (3, 4) for PC (1,...

The fluctuation free volume in crosslinking epoxy polymers has a fractal structure and the microvoids formed in the matrix are simulated by Df dimensional spheres. The size of a microvoid is considered as the volume that is necessary for its formation and is a consequence of the accumulation of thermal fluctuations. 

The other approach derives from the free volume theory of diffusion. In this point of view, it is assumed that the fluctuations of local density in the polymer result in free volume or hole formation. When the hole is of sufficiently large size, and if it is formed near the penetrant molecule, the molecule can move or jump into it. The diffusion is assumed to be proportional to the probability of forming such holes of right size. The effect of the diffusing molecules on the free volume formation can be taken into consideration, too. 

In Ref. [25], the asymmetrical periodic function is adduced, showing the dependence of shear stress x on shear strain (Fig. 4.2). As it has been shown before [19], asymmetry of this function and corresponding decrease of the energetic barrier height overcome by macromolecules segments in the elementary yielding act are due to the formation of fluctuation free volume voids during deformation (that is the specific feature of polymers [26]). The data in Fig. 4.2 indicate that in the initial part of periodic curve from zero up to the maximum dependence of x on displacement x can be simulated by a... 

There are some problems of interest in the description of the free volume of polymers. As has already been mentioned, both fluctuation free volume and clusters have a thermofluctuational nature. It is obvious that the energy of thermal fluctuations can be expressed as kT (where k is Boltzmann s constant, T is temperature), while microvoid formation energy is equal to [80] ... 

Hence, the results stated above have shown that fluctuation free volume in epoxy polymers possesses fractal structure. Therefor a microvoid forming it should be simulated by D -dimensional sphere. The size of the microvoids is controlled by the volume which is necessary for accumulation of the thermal fluctuations enei y required for their formation. The absolute values of can serve as characteristic of polymer structure thermodynamic non-equilibrium and for quasi-equilibrium structures the value of coincides with the data obtained according to the William-Landel-Ferry equation. Microvoids of fluctuation free volume form fractal structure, which is a mirror of polymer structure [152-158]. 

In Figure 5.43 the comparison of values of calculated according to Equation 5.77 and experimental values for an epoxy polymer [172] are adduced. Curve 1 was calculated according to the values of and received experimentally (174). As one can see, the shapes of the experimental and theoretical dependences % P) are in agreement, hut the absolute values of received experimentally systematically exceed the calculated ones (hy approximately 3 times). This discrepancy can he removed as follows. It was shown earlier [72, 127] that free volume in crosslinked epoxy polymers consisted of two components -fluctuation free volume fg connected with clusters decay (formation), and the constant component fg ( 0.024) connected with chemical crosslinking nodes (see Figure 5.17). 

The notion was developed earlier that the microhardness of glassy solids is defined by the fluctuation free volume microvoid formation (or collapse) work, ascribed to the microvoid volume unit [1]. Such an approach corresponds to the results in the present section since, as has been shown above, the fluctuation free volume is concentrated in the loosely packed matrix of polymer structure. The relative fraction of the fluctuation free volume can be estimated according to Equation 1.33. As was to be expected, the linear correlation between the values of and calculated according to Equations... 

Thns, the flnctnation free volume in a crosslinking epoxy polymer has a fractal structure and the microvoids, forming are simulated by Df dimensional sphere. The volume that is necessary for accnmulation of the thermal fluctuation energy, sufficient for its formation controls the size of a microvoid. 

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

Consider a homogeneous phase containing N, atoms per unit volume in which in a smaller volume, containing n atoms, the density fluctuates to form a new phase. As discussed in Sec. 9.2, the formation of these embryos results in a local increase in the free energy AG. . So the reason the nuclei form must be related to an increase in the entropy of the system. This increase is configurational and comes about because once the nuclei have formed, it is possible to distribute N embryos on any of N, possible sites. 

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