Helmholtz free energy chain

The Helmholtz free energy of the chain F is the energy U minus the product of absolute temperature T and entropy S ... 

The two models (SAFT and PHSC) differ substantially in the way they represent the different contributions to the expression of the residual Helmholtz free energy of a system. In the SAFT model, is the sum of different contributions due to hard spheres, dispersion, chain, and association, respectively ... 

In the PHSC EoS, the Helmholtz free energy is expressed as the sum of two different terms, one is a reference term accounting for chain connectivity and hard sphere interactions, and the other is a perturbation term, which represents the contributions of mean-field forces ... 

Since Chapter 6 presents detailed discussion of Simha-Somcynsky lattice-hole theory, only an outline is provided here. The theory was derived for spherical and chain molecule fluids [Simha and Somcynsky, 1969 Somcynsky and Simha, 1971]. The model lattice contains a volume fraction y of occupied sites and h= —y of nonoc-cupied sites, or holes. From the Helmholtz free energy, F, the S-S equation of state was obtained in the form of coupled equations ... 

Sanchez and Lacombe supposed that in a binary polymer blend, free volume occupied Nq lattice sites, and the bulk polymer density p = NKN + Nq), where N = ILNiri and r, was the chain length of /th fraction, then they developed the lattice fluid theory to calculate Helmholtz free energy (Sanchez and Lacombe 1974 Sanchez 1978), as given by... 

The Helmholtz free energy of a chain can be found by the logarithm of the partition function... 

From the Helmholtz free energy, we can find the entropy S and the average tension f of the chain using the law of thermodynamics ... 

The Equation of State for a Single Polymer Chain. The variation in the Helmholtz free energy is the negative of the work of deformation in isothermal elongation... 

The mean-field Flory approach enables one to derive the latter expression in the most straightforward way by minimizing the (Helmholtz) free energy of the polymer chain ... 

Hie total elastic free energy, or the Helmholtz free energy, of a network consists of the sum of the free energies of the individual chains and the contributions coming from intermolecular correlations. Hie elementary theories of rubber elasticity ignore contributions from intermolecular effects. Improvements in the elementary models have been made by different researchers in different ways. In this section most of the models will be covered. Since the single-chain elasticity forms the basis of rubber elasticity, it will be discussed first in some detail. 

To a certain degree, the contribution from the chain connectivity into can be interpreted as a constraint, which accounts for the configurations of a polymeric molecule. The excess Helmholtz free energy can be decomposed into different contributions. 

Then, the Helmholtz free energy A of the chain is calculated as... 

The Helmholtz Free Energy includes both intermolecular forces and intramolecular forces and also entropic contributions. The intramolecular contributions are the same as those required for the single-chain calculation, but there is more difficulty in producing force fields that include intermolecular contributions. The intermolecular contributions are typically Lennard-Jones type interactions and to obtain plausible values that are satisfactory for a range of different chemical compositions is often debatable. It is, however, possible to obtain some confirmation of their validity in a particular instance by verifying that the calculations predict the correct crystal structure and this must be regarded as a the first step to calculating the elastic constants. 

If a polymer solution is modeled by an assembly of self-avoiding walks on a lattice, a basic physical quantity is the osmotic pressure II. Carrying out a simulation with a fixed number JT of chains of length at a lattice of volume V with one of the dynamic algorithms described in Section 1.2.2, the osmotic pressure is not straightforward to sample. If one had methods that yielded the excess chemical potential and the Helmholtz free energy Fjr, one would find II from the thermodynamic relation... 

A rubber represents an ensemble of polymer chains, each one running between two cross-links. The Helmholtz free energy of a sample is related to the distribution function of the conformational states. Since the chains are... 

This equation relates the change in entropy to the extension ratios and the number of chains between crosslinks per unit volume. It was shown earlier (Section 5.3.1) that there is ideally no change in internal energy U when an elastomer is deformed and since the deformation takes place at constant volume then the change in the Helmholtz free energy per unit volume Ai4 is... 

The work W required to move one end of the chain from a distance r to a distance r=r- - dr with respect to the other end is equal to the change in Helmholtz free energy, that is... 

Previously we have seen a heuristic derivation of TPTl for the special case of a pure polymer with fixed bond length. Actually, TPTl is a much more general theory which allows to describe a wide variety of systems. We will henceforth restrict our attention to the special case of multicomponent mixtures. We will assume that each of the components is a chain molecule, described as an ensemble of A,- identical beads, which interact with each other and with any other bead in the system by means of a site-site potential, Vtj. Furthermore, adjacent beads on the same molecule interact by means of a bonding potential,. The composition of the mixture may be described by means of the total molecular density, p, and the molar fractions of each component, Xj. According to TPTl, the total Helmholtz free energy of the multicomponent mixture may be described in terms of three different contributions [237, 253] ... 

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