Lattice theory fluid

For a polymer liquid, the lattice fluid equation-of-state can be written in a reduced form as 

Assuming high molecular weight (r — °°) and near atmospheric pressure (p— o), the chemical potential p is given by 

For polymer liquids, the gradient approximation in conjunction with the lattice fluid model has been used to calculate surface tensions [24,25]. The Cahn-Hilliard relation for surface tension o, in terms of reduced variables, can be expressed as 

A dimensionless constant ic has a priori theoretical value of 0.5 and the subscripts, g and l,in Eq. (13) denote the gas state and the liquid state, respectively. By substituting Eq. (11) into Eq. (14) and then integrating Eq. (13), the surface tension can be obtained. Such a procedure has been previously shown to yield estimates of surface tension for polyethylene melts in good agreement with experiment [26]. 

For simplicity, assuming that the close-packed volume of a PS mer (vps) is equal to that of a PVME mer (vPVME), the binary polymer blend is miscible [20] when 

Flory s equation-of-state theory is quite complex, requiring as it does the manipulation of many interrelated parameters. Sanchez (Sanchez and Larcombe, 1978 Sanchez, 1980) has therefore developed a somewhat simpler theory based on a lattice fluid or Ising model. This approach also proceeds via 

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Since v p is defined as the specific volume at close-packed state and p is equal to e /v, i.e., the cohesive energy density at close-packed state [17], the specific volume at 0 K corresponds to vsp, and the cohesive energy density at 0 K to p. The T is obtained by inserting the values ofp, v, and simulated (T, vsp) data at room temperature into the lattice fluid theory. The absolute values of simulated equation-of-state parameters may not be the same as the experimental ones as shown in Table 1, because the procedures obtaining the parameters are differ-... 

The liquid may be a good or poor solvent for the polymer. For this type of system a theoretical relation can be obtained for K by applying the Flory equation of state theory ( -i) or lattice fluid theory (7-10) of solutions. An important prerequisite for the application of these theories is for the polymer to behave as an equlibrium liquid. This condition is generally valid for a lightly crosslinked, amorphous polymer above its Tg or for the amorphous component of a semi-crystalline polymer above its Tg. 

JI1 Jiang, S., An, L., Jiang, B., and Wolf, B. A., Pressure effects on the thermodynamics of fra 5-decahydronaphthalene/polystyrene polymer solutions apphcation of the Sanchez-Lacombe lattice fluid theory (experimeirtal data by S. Jiang), Macromol. Chem. Phys., 204, 692, 2003. 

The lattice fluid theory predicts miscibility with a high probability if Vl/Vl > I and Tl/Tl > 1 while phase separation is favored if Vl/Vl < 1 and Tl/Tl > 1. 

There exist several approaches for the development of equations of state for polymer systems. A possibility considered at an early stage was to extend the Flory-Huggins theory by introducing holes into the lattice. Here, the number of holes in the lattice is a measure of the system density. Equations of state based on this idea are, for example, the Lattice-Fluid Theory (often called the Sanchez-... 

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... 

One can see that when p = I, this equation can be reduced to the Flory-Huggins-Scott equation for binary polymer blends. The lattice fluid theory can predict both UCST and LOST (lower critical solution temperature) types of phase diagrams for polymer blends, with further considerations of speciflc interactions (Sanchez and Balazs 1989), see more introductions about LOST in Sect. 9.1. 

A number of equation of state theories have been used to model phase behavior of polymers in supercritical fluids. For example the lattice-fluid theory of Sanchez and Lacombe[4U 42] includes holes on the lattice in order to model compressibility. The lattice-fluid theory has been applied to model phase behavior of both homopolymers and copolymers in supercritical fluids[32, 38, 43, 44]. The statistical associating fluid theory (SAFT)[43,45-48] and corresponding state models[49] have also been employed to model compressible polymer-solvent mixtures. Figure 1 gives the pressure-concentration phase diagram for poly(dimethyI siloxane) in CO2 modeled with the lattice-fluid equation of state[50]. 

Some thermodynamic aspects of polymer-polymer compatibility are discussed in this work.The analysis is made with the Lattice-Fluid theory of polymer solutions as modified recently by the author.The theory has been extended to multicomponent and multigroup systems and is,thus,applicable besides others to mixtures of polydisperse polymers and mixtures of random copolymers. The effect of pure component properties,of pressure and of polydispersity on the critical behavior of polymer mixtures are examined.Theoretical estimations are compared with experimental data whenever available.A satisfactory agreement is observed between theory and experiment. 

A thermodynamic model meeting all the above requirements is presented in the next section. It is based on the Lattice-Fluid theory of Sanchez and Lacombe(7) as modified recently by the author (8-12).So far the model has been applied to solvent-homopolymer and homopolymer-homopolymer(both monodisperse) mixtures (1 0), to the gas solubility in polymeric liquids... 

Other recently published correlative methods for predicting Tg include the group interaction modeling (GIM) approach of Porter (42), neural networks (43-45), genetic function algorithms (46), the CODESSA (acronym for Comprehensive Descriptors for Structural and Statistical Analysis ) method (47), the energy, volume, mass (EVM) approach (48,49), correlation to the results of semiempirical quantum mechanical calculations of the electronic structure of the monomer (50), and a method that combines a thermodynamic equation-of-state based on lattice fluid theory with group contributions (51). 

Redlich-Kwong equation of state and Soave modification Peng-Robinson equation of state Tait equation for polymer liquids Flory, Orwoll, and Vrij models Prigogine square-well cell model Sanchez-Lacombe lattice fluid theory... 

The three categories differ in how they represent the compressibility and expansion of the polymer systems under scrutiny. Volumetric changes are restricted to a change in cell volume in cell models. Lattice vacancies are allowed in lattice-fluid theory, and the cell volume is assumed constant. Cell expansion and lattice vacancies are allowed by hole models. The models also differ on the lattice type, such as a face-centered cubic, orthorhombic, hexagonal, and also in their selection of interpolymer/interoligomer potential such as Lennard-Jones potential, hard-sphere, or square-well. 

A pure fluid is completely characterized by the three molecular parameters, v r, and , and the scale factors T P, and p. P v /PP = 1 and Mir = v p = PP p /P. In principle, any thermodynamic property can be utilized to determine these parameters. Saturated vapor pressure data is useful as they are readily available for a variety of fluids. A compendium of such data is available for organic liquids. The lattice fluid theory of Sanchez and Lacombe as described is similar to the van der Waals EOS as discussed in the earlier section for small molecules. The virial form of the EOS of Sanchez and Lacombe can be written as follows ... 

Characteristic Pressure, Temperature for Sanchez-Lacombe Lattice Fluid Theory... 

FIGURE 2.5 Reduced density versus temperature for polyfmethyl methacrylate) (PMMA) at 25°C by Sanchez-Lacombe lattice fluid theory. 

EOS theories for polymer liquids were grouped into three categories (1) lattice-fluid theory, (2) the hole model, and (3) the cell model. The Tait equation was first developed 121 years ago in order to fit compressibility data of freshwater and seawater. The Tait equation is a four-parameter representation of PVT behavior of polymers. Expressions for zero pressure isotherms and Tait parameters were provided, and values of Tait parameters for 16 commonly used polymers were tabulated. 

Sanchez and Lacombe developed the lattice fluid EOS theory using statistical mechanics. Gibbs free energy can be expressed in terms of configurational partition function Z in the pressure ensemble. In the lattice fluid theory development the problem is to determine the number of configurations available for a system of N molecules each of which occupies r sites and vacant sites or holes. Mean field approximation was used to evaluate the partition function. The SL EOS has the capability to account for molecular weight effects, unlike other EOS theories. Characteristic lattice fluid EOS parameters were tabulated for 16 commonly used polymers. 

The EOS developed by Sanchez and Lacombe using lattice fluid theory... 

Find the isentropic volume expansivity for systems described using the Sanchez-Lacombe lattice fluid theory equation of state. The isentropic volume expansivity as defined by Equation (2.80). 

What is the difference in approach between the lattice-fluid theories and hole theories that predict EOS of polymeric liquids ... 

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