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Consider the interaction of energy and entropy in the highly elastic materials of an ideal polymeric network.

Now Gibb's free energy cannot be used directly $\Delta H = \Delta G + T \Delta S$ where $\Delta S$ and $\Delta H=U+pV$ for the internal energy $U$ such that

  • $S(R) = -k_{B} \frac{ 3R^{2} }{ 2N l^{2} } + const$

  • $U_{eff}(R) = k_{B} T \frac{ 3R^{2} }{ 2 N l^{2} } + const $

of which I am unsure.

I think the Gibbs free energy may have some nonlinear behaviour with the material such as some correlations between different terms and more passive energy terms due to elesticity -- this may be described by things such as fugacity.

How can you describe the interaction of energy and entropy in highly elastic polymers?

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If I remember correctly, the relationships you have presented are related to the configurational entropy of the polymer chains between cross links of the polymer network. In these equations, R is the spatial distance between cross links, N is the number of chain segments between cross links, and l is the length of each chain segment. The smaller the value of R, the greater the number of configurations that the chain can exhibit (i.e., the greater the entropy). Thus, I believe that there should be a minus sign in your equation for S, because, as R increases, the fewer the number of configurations that the chain can exhibit, and thus the lower the entropy. The parameter you call U is, I believe the Helmholtz free energy, and is a measure of the stored elastic energy of the polymer network. As the R gets longer, the polymer chains have been stretched more, and more elastic energy is stored in the chains. Conceptually, the polymer chains are like springs between the cross links.

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  • $\begingroup$ Thank you for defining the terms +1. Do I understand correctly that fugacity/z-compressability factor only play role in applications with very high pressure profile or very high temperature? I moved my curiosity to new question in chemistry.stackexchange.com/questions/41977/… $\endgroup$ – hhh Dec 10 '15 at 8:37
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Why are fugacity/z-compressability factors used as a substitute for Gibbs free energy criterion in certain applications?

High pressure profile or large heat in which polymers must be designed to work otherwise they can break. Applications contain polymer mud systems such as here related to petroleum industry. More in When fugacity and z-compressability equations instead of Gibbs free energy equations?.

Fugacity of a pure fluid

Fugacity criterion is often used as a substitute for the Gibbs free- energy criterion. The definition for fugacity comes from an analogue with ideal gases that is derived for a closed system under isothermal conditions. Eq. 6 for an isothermal process ($\mathrm{d}T = 0$) is

More in http://petrowiki.org/Equations_of_state.

Macromolecular features relevant to pressure

  1. Fugacity: effective pressure which replaces the true mechanical pressure in accurate chemical equilibrium calculations

  2. Compressability factor

The relationship between fugacity and compressibility is not as straightforward as you might think either. Starting with the expressions for the Gibbs free energy of the real and ideal gases, you can work out the following relationship:

$$\ln{\left(\frac{f}{P}\right)} = \int_0^P\!\frac{Z-1}{P}\,\mathrm{d}P$$

More in https://www.quora.com/Why-is-fugacity-or-z-always-less-than-1.

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