Vapor pressure lattice model

Polymer simulations can be mapped onto the Flory-Huggins lattice model. For this purpose, DPD can be considered an off-lattice version of the Flory-Huggins simulation. It uses a Flory-Huggins x (chi) parameter. The best way to obtain % is from vapor pressure data. Molecular modeling can be used to determine x, but it is less reliable. In order to run a simulation, a bead size for each bead type and a x parameter for each pair of beads must be known. 

This type of liquid is characterized by direction independent, relatively weak dispersion forces decreasing with r-6, when r is the distance between neighbouring molecules. A simple model for this type of liquid, which accounts for many properties, was given by Luck 1 2> it is represented by a slightly blurred lattice-like structure, containing hole defects which increase with temperature and a concentration equal to the vapor concentration. Solute molecules are trapped within the holes of the liquid thus reducing their vapor pressure when the latter is negligible. 

The EOS based on the lattice fluid model has also be used to describe thermodynamic properties such as pVT behaviors, vapor pressures and liquid volumes, VLE and LLE of pure normal fluids, polymers and ionic... 

To examine the dynamical aspect of the hysteretic behavior, we consider the system geometry shown in Fig. 4. The porous material of length L in the z-direction is bounded by the gas reservoirs at z = 0 and z = L. Periodic boundary conditions are imposed on the X and y-directions. In typical experimental situations, starting from a (quasi)-equilibrium state, the external vapor pressure of the gas reservoir is instantaneously changed by a small amount, which induces gradual relaxations of the system into a new state. This geometry was used in recent work on dynamics of off-lattice models of adsorption (Sarkisov and Monson,... 

By contrast to the lattice-gas model, the CXC-diameter("7) and non-zero vapor pressure Ps(T) are /-dependent for the whole subcritical range. One may conclude, some paradoxically, that the symme trical map of Ps (T) into the negative pressures i.e. - Ps(T) must exist if the latent particle-hole-type symmetry is in a real fluid. From what has been said above, the FEOS-model confirms this rather unusual possibility. 

Escobedo and de Pablo have proposed some of the most interesting extensions of the method. They have pointed out [49] that the simulation of polymeric systems is often more troubled by the requirements of pressure equilibration than by chemical potential equilibration—that volume changes are more problematic than particle insertions if configurational-bias or expanded-ensemble methods are applied to the latter. Consequently, they turned the GDI method around and conducted constant-volume phase-coexistence simulations in the temperature-chemical potential plane, with the pressure equality satisfied by construction of an appropriate Cla-peyron equation [i.e., they take the pressure as 0 of Eq. (3.3)]. They demonstrated the method [49] for vapor-liquid coexistence of square-well octamers, and have recently shown that the extension permits coexistence for lattice models to be examined in a very simple manner [71]. 

For the condensed phase chemical potential Pc> you can use lattice model Equation (14.8). Use Equation (14.5) for Pi, and set the chemical potentials equal, be = bv< to get the condition for equilibrium in terms of the vapor pressure p,... 

Vaporization equilibria involve a balance of forces. Particles stick together at low temperatures but they vaporize at high temperatures to gain translational entropy. The chemical potential describes the escaping tendency, fjp from the vapor phase, and Pt from the condensed phase. When these escaping tendencies are equal, the system is at equilibrium and no net particle exchange takes place. We used the statistical mechanical model of Chapter 11 for the chemical potential of the gas phase. For the liquid or solid phase, we used a lattice model. Vapor pressure and surface tension equilibria, combined with models, give information about intermolecular interactions. In the next chapter we will go beyond pure phases and consider the equilibria of mixtures. 

To find the relationship between the vapor pressure pb of the B molecules and the solution concentration Xb, use Equation (11.50), pe(gas) = IcTInpe/pB.jnt-for the chemical potential of B in the vapor phase. For the chemical potential of B in the solution, use lattice model Equation (15.15), pB(liquid) = kTlnx + zwbbI2 + kTx,4fl(l - Xb). Substitute these equations into the equilibrium Equation (16.1) and exponentiate to get... 

Figure 16.1 The vapor pressure p/j ofmolecules over a liquid solution having composition Xn, according to the lattice model. Increasing the concentration Xb of B in the liquid increases the vapor pressure of /i. x = 0 represents an ideal solution. B has more escaping tendency when x > arid less when x < 0, for a given Xb-...

Dobson PF, Epstein S, Stolper EM (1989) Hydrogen isotope fractionation between coexisting vapor and silicate glasses and melts at low pressure. Geochim Cosmochim Acta 53 2723-2730 Dove MT, Winkler B, Leslie M, Harris MJ, Salje EKH (1992) A new interatomic model for calcite applications to lattice dynamics studies, phase transition, and isotopic fractionation. Am Mineral 77 244-250... 

From the historical point of view and also from the number of applications in the literature, the common method is to use activity coefficients for the liquid phase, i.e., the polymer solution, and a separate equation-of-state for the solvent vapor phase, in many cases the truncated virial equation of state as for the data reduction of experimental measurements explained above. To this group of theories and models also free-volume models and lattice-fluid models will be added in this paper because they are usually applied within this approach. The approach where fugacity coefficients are calculated from one equation of state for both phases was applied to polymer solutions more recently, but it is the more promising method if one has to extrapolate over larger temperature and pressure ranges. 

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