Equilibrium polymer density

In equation 2.14, the non-equilibrium solute chemical potential is calculated through the use of equation 2.12 and of an appropriate EoS for the polymer-penetrant system under consideration. The pseudo-equilibrium penetrant content in the polymer, can be easily calculated whenever the value of the pseudo-equilibrium polymer density Pp i is known. Such a quantity represents, obviously, a crucial input for the non-equiUbrium approach, since it labels the departure from equilibrium it must be given as a separate independent information, and cannot be calculated simply from temperature and pressure since it depends also on the thermomechanical history of the sample. 

The requirement to specify the polymer density may represent a serious limitation for the practical application of the NELF model. Indeed, the dilation of the polymer matrix at high penetrant pressure could be significant and difficult to estimate without specific experimental data. On the other hand, the use of the non-equilibrium polymer (tensity is actually a powerful tool to represent complex non-equilibrium phenomena. It has been shown, for example, that it allows a description of sorpdon-desorption hysteresis (7) as well as the influence of pretreatments on the solubility isotherms of gases in glassy polymers. For such cases, the different pseudo-equilibrium solubility values at the same prevailing temperature and penetrant fugacity are satisfactorily accounted for by considering die different pseudo-equilibrium polymer densities. 

This approach is a purely thermodynamic approach in the sense that locally, the polymer is at thermal equilibrium. The density inside the layer is homogeneous and is equal to the complex density cC0mp- One of the important experimental results is that, in the multilayers, each layer strongly intermixes with its neighbors but that it keeps its identity in the sense that it does not mix with layers far apart. This freezing of the structure must be due to an extremely slow diffusion between the layers that our model is not able to study. 

However, for membrane applications, where the need to constrain physical dimensions and the mechanical properties required impose limits on the acceptable swelling ratio. Equation 16.1 can be used to predict equilibrium polymer concentration and swelling ratio for a gel with a known cross-link density. Figure 16.10 shows such a prediction for a dextran-based gel with an initial polymer fraction of 0.16 based on a dextran of 500 kDa. Other parameter values are given in Table 16.2. 

Stress Relaxation Polyurethane networks were also polymerized in a mold with a cylindrical cavity. Uniform rings were cut from the cylinders and weighed. The cross sectional area was then derived from the sample diameter and polymer density and approximated 0.05 cm2. Stress relaxation was measured at several strains between 10 and 43% with an Instron tensile tester. During stress relaxation, the samples were immersed in dioxane and swelled to equilibrium. 

In the NET-GP analysis, the glassy polymer-penetrant phases are considered homogeneous, isotropic, and amorphous, and their state is characterized by the classical thermodynamic variables (i.e. composition, temperature, and pressure) with the addition of a single-order parameter, accounting for the departure from equilibrium. The specific volume of the polymer network, or, equivalently, the polymer density Pp, is chosen as the proper order parameter. In other words, the hindered mobility of the glassy polymer chains freezes the material into a non-equilibrium state that can be labeled by the... 

It must be stated clearly that such results have been derived in a completely general manner and are thus independent from the particular EoS model used to describe the Helmholtz free energy or the penetrant chemical potential under equilibrium conditions. Non-equilibrium free energy functions can thus be obtained starting from different EoS such as LF, SAFT, PHSC, just to mention the relevant models considered in this chapter. The non-equilibrium information entering equations 2.11 and 2.12 is represented by the actual value of polymer density in the glassy phase, which must be known from a separate source of information, experimental data, or correlation, and cannot be calculated from the equilibrium EoS. 

For the solubility in glassy phases, the situation is substantially different since the polymer density does not match its equilibrium value p, but it finally reaches an asymptotic value determined by the kinetic constraints acting on the glassy molecules, and is substantially dependent on the past history of the polymer sample. Thus the penetrant concentration in the polymeric phase reflects the pseudo-equilibrium state reached by the system. In view of the NET-GP results, such pseudo-equilibrium condition corresponds to the minimum Gibbs free energy for the system, under the constraint of a fixed value (the pseudo-equilibrium value) of the polymer density in the condensed phase ... 

Where, is volume fraction of the polymer, is molar volume of solvent, X is the polymer solvent interaction parameter. Me is the crosslink density of polymer, pi is solvent density, P2 is polymer density, / is functionality of crosslink, Fequ is equilibrium volume of the hydrogel. 

The pseudo-equilibrium state of the glassy mixture is desaibed by the usual state variables (temperature, pressure and composition) plus the polymer density p2 that accounts for the departure from equilibrium frozen into the glass. 

For equilibrium centrifugation in a density gradient, a mixture of two solvents is prepared, one of which has a lower density than the other, and the density of any component in the polymer product lies between the densities of the two solvents. At equilibrium, a density gradient forms at... 

Figure 9 Snapshots of an equilibrium polymer system at different densities. Left p = 0.34 (isotropic ordering). 

The grand canonical ensemble describes a system of constant volume, but capable of exchanging both energy and particles with its environment. Simulations of open systems under these conditions are particularly useful in the study of adsorption equilibria, surface segregation effects, and nanoscopically confined fluids and polymers. Under these conditions, the temperature and the chemical potentials jti,- of the freely exchanged species are specified, while the system energy and composition are variable. This ensemble is also called the jx VT ensemble. In the case of a one-component system it is described by the equilibrium probability density... 

The magnitude of elastic vibrations is controlled by a parameter of units A. While the original formulation of this method allowed for different parameter values for each atom type (or even every atom), traditionally a single size is used for all atoms in the polymer matrix. The fluctuations impose an additional equilibrium probability density distribution... 

Dai KH, Norton LI, Kramer EJ (1994) Equilibrium segment density distribution of a diblock copolymer segregated to the polymer lymer interface. Macromolecules 27 1949-1956... 

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