A new equation of state based on hole theory
Introduction
Equations of state play an important role in describing the thermodynamic properties of liquids. Liquids are thought to have both gaseous and solid properties. The similarity between solids and liquids is the intermolecular distance. This means that the movement of a molecule or segment is affected by the presence of neighboring ones but unlike gases, liquids do not have long-range order. In other words, from the view point of lattice structure, there exist holes between segments in the liquid. Furthermore, the volume is shared by all of the segments contained in the system 1, 2.
There have been two basic approaches to describe the thermodynamic properties of liquid: one is based on ideal gas state and the other on crystal structure. In the latter approach, the volume is divided into lattices/cells and each segment occupies one of the lattices/cells. There are two branches to formulate thermodynamic relations in the solid-oriented approach: free volume theory and lattice theory. In the free volume theory, vacant cells may be present (hole theory) 3, 4, 5, 6, 7or not (cell theory) 8, 9. In the cell theory, the changes in volume with changes in temperature and pressure can be explained only by the changes in cell size. On the other hand, in the hole theory, a major change in volume is explained by the number of holes and the change in cell size plays a minor role. In the lattice theory, the lattice size is fixed and it explains the change of volume only by the number of vacant sites 10, 11.
As pointed out by Dee and Walsh [9], the lattice theory is more appropriate to depict the thermodynamic properties of the gaseous state. So far, it has been noted that free volume theories, especially hole theories can delineate the thermodynamic properties of liquid well. In the hole theory, the free volume is treated by the concept of linear superposition between `solid-like' term and `gas-like' term 3, 4, 5. In the work of Simha and Somcynsky [5], the free length concept is introduced. This model, though successfully depicting the thermodynamic properties of liquids, cannot account for the pressure effect properly [12].
Zhong et al. [7]indicated that the Simha and Somcynsky (SS) model could not generate sufficient free volume and still lacks fluidity. They modified the SS model by using the open-cell concept [13]. However, to evaluate the number of segments with enough energy to attack neighboring segments, it needs another empirical parameter as spatial parameter. For most polymer systems, the value of the spatial parameter can be set to 0.2. But some systems exhibit deviations from this value.
In this paper, we did not adopt the traditional notion of linear superposition between `gas-like' and `solid-like' terms. Instead, free volume comprises that of the cell itself and volume added by the introduction of the hole. The free volume introduced by the hole is obtained by equally partitioning hole volume into each segment.
Section snippets
Theory
The physical concept of free volume is the space that the center of mass of a given molecule can move under the influence of intermolecular potential generated by neighbor segments around the center of the cell [9].
For an ideal gas, the free volume is cell volume itself. In the cell model (with no vacant cell), it depends on intermolecular potential function. When the square well potential is applied, we can obtain the analytical free volume expression [8]where λ
Equation of state
For hole theories, the configurational partition function can be factorized into three terms: combinatory, free volume, and energy term.where g is the combinatory factor arising from the mixing of segments and holes, c is the external degrees of freedom per chain suggested by Prigogine [8]and E0 is the internal energy when all of the segments are located at the center of the cell. Following the approach of Simha and Somcynsky, in this paper, Flory's combinatory factor is adopted
Results
The EOS developed in this study was applied to the experimental PVT data of polymer melts to test the capability of this equation. A polymer is a long-chain molecule so the end-effect can be ignored. In this case, the parameters in Eq. (21)approach limiting values. External degrees of freedom which are a function of volume, can be related with the number of segmentsfor a long chain molecule, we can fix the value of s/3c to be unity.
The behaviors of EOS derived in this paper are shown in
Conclusion
In this paper, a new free volume expression was suggested by allowing the segments to access all of the volume at a given configuration. The assumption used is that all of the volumes, except hard-core volume can be accessed by segments at a given configuration. With this assumption, the free length added by hole depends on cell volume, not on the hard-core volume.
The new EOS derived in this work could reproduce PVT data of polymeric melt. In particular, this EOS shows good results at low
List of symbols
A Helmholtz free energy 3c external degrees of freedom g combinatory factor k Boltzmann constant lf free length N the number of molecules Ns the number of cells Nh the number of holes P pressure P* characteristic pressure P̃ reduced pressure qz the coordination number of polymer molecules s the number of segments in one molecule T temperature T* characteristic temperature T̃ reduced temperature V volume V* characteristic volume Ṽ reduced volume vf free volume y volume fraction occupied by segment Z configurational partition function z
Acknowledgements
We acknowledge Kumho Chemicals and the Ministry of Trade, Industry, and Energy for their financial support.
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