Polymer chemical potential

For a polydisperse polymer, analysis of sedimentation equilibrium data becomes complex, because the molecular weight distribution significantly affects the solute distribution. In 1970, Scholte [62] made a thermodynamic analysis of sedimentation equilibrium for polydisperse flexible polymer solutions on the basis of Flory and Huggins chemical potential equations. From a similar thermodynamic analysis for stiff polymer solutions with Eqs. (27) for IT and (28) for the polymer chemical potential, we can show that the right-hand side of Eq. (29) for the isotropic solution of a polydisperse polymer is given, in a good approximation, by Eq. (30) if M is replaced by Mw [41],... 

Xv being a phenomenological interaction parameter for the noncombinatorial part of the solute (polymer) chemical potential, defined on a volume fraction basis. Equations similar to equation (1.9) and (1.12) serve to define x on a segment fraction basis, x -... 

When the cast polymer film is immersed into the precipitation bath, solvent leaves and precipitant enters the film. At the film surface the concentration of the precipitant soon reaches a value resulting in phase separation. In the interior, however, the polymer concentration is still far below the limiting concentration for phase separation. Phase separation therefore occurs initially at the surface of the film, where due to the very steep gradient of the polymer chemical potential on a macroscopic scale, there is a net movement of the polymer perpendicular to the surface. This leads to an increase of the polymer concentration in the surface layer. It is the concentrated surface layer which forms the skin of the membrane. This skin also serves to hinder further precipitant flux into and solvent out of the casting solution. The skin thus becomes the rate-limiting barrier for precipitant transport into the casting solution, and the concentration profiles in the casting solution interior become less steep. Thus, once the precipitated... 

If Ap] is riK asured in a certain concentration range, Gibbs-Durgham s equation enables the polymer chemical potential of mixing to be determined by numerical or graphical integration... 

The notion of osmotic pressure 11 can be extracted from this. A simple way of understanding how n arises is to consider solvent in a container of uniform temperature divided in two portions a and p by membrane permeable to solvent but impermeable to polymer The polymer resides in only side P of the container partition. By the Gibbs-Duhem relation, the solvent chemical potential strives to be the same on both sides, but the polymer chemical potential does not, since it cannot cross the membrane. This requires that but since =1... 

For systems of the present type it is possible to obtain equilibrium information from two sources in the usual manner via the vapor pressmes of the solvent above the solutions within range HI (chemical potential of the solvent) and additionally from the saturatitMi composition w of the polymer (chemical potential of the polymer). The thermodynamic craisistency of these data was documented [52] by predicting w (liquid/soUd equilibrium) from the information of liquid/gas equilibria. This match of thermodynamic information from different sources is a further argument for the suitability to the present approach for the modeling of polymer-containing mixtures. 

Poisson ratio for isotropic heterogeneous composition chemical potential of bound polymer chemical potential chemical potential of free polymer... 

Mayer-Mayer function strength of nematic ordering Onsager s transport coefficients smectic interaction parameter free energy of mixing chemical potential of a polymer chemical potential of a rod Maier-Saupe attractive interaction smectic A order parameter... 

At equilibrium, in order to achieve equality of chemical potentials, not only tire colloid but also tire polymer concentrations in tire different phases are different. We focus here on a theory tliat allows for tliis polymer partitioning [99]. Predictions for two polymer/colloid size ratios are shown in figure C2.6.10. A liquid phase is predicted to occur only when tire range of attractions is not too small compared to tire particle size, 5/a > 0.3. Under tliese conditions a phase behaviour is obtained tliat is similar to tliat of simple liquids, such as argon. Because of tire polymer partitioning, however, tliere is a tliree-phase triangle (ratlier tlian a triple point). For smaller polymer (narrower attractions), tire gas-liquid transition becomes metastable witli respect to tire fluid-crystal transition. These predictions were confinned experimentally [100]. The phase boundaries were predicted semi-quantitatively. 

Hydrophobic fibers are difficult to dye with ionic (hydrophilic) dyes. The dyes prefer to remain in the dyebath where they have a lower chemical potential. Therefore nonionic, hydrophobic dyes are used for these fibers. The exceptions to the rule are polyamide and modified polyacrylonitriles and modified polyester where the presence of a limited number of ionic groups in the polymer, or at the end of polymer chains, makes these fibers capable of being dyed by water-soluble dyes. 

Internal and External Phases. When dyeing hydrated fibers, for example, hydrophUic fibers in aqueous dyebaths, two distinct solvent phases exist, the external and the internal. The external solvent phase consists of the mobile molecules that are in the external dyebath so far away from the fiber that they are not influenced by it. The internal phase comprises the water that is within the fiber infrastmcture in a bound or static state and is an integral part of the internal stmcture in terms of defining the physical chemistry and thermodynamics of the system. Thus dye molecules have different chemical potentials when in the internal solvent phase than when in the external phase. Further, the effects of hydrogen ions (H" ) or hydroxyl ions (OH ) have a different impact. In the external phase acids or bases are completely dissociated and give an external or dyebath pH. In the internal phase these ions can interact with the fiber polymer chain and cause ionization of functional groups. This results in the pH of the internal phase being different from the external phase and the theoretical concept of internal pH (6). 

The subscripts 1,2,3 refer to the main solvent, the polymer, and the solvent added, respectively. The meanings of the other symbols are n refractive index m molarity of respective component in solvent 1 C the concentration in g cm"3 of the solution V the partial specific volume p the chemical potential M molecular weight (for the polymer per residue). The surscript ° indicates infinite dilution of the polymer. 

Conditions of phase equilibrium require that the chemical potential of polymer in each phase and that of solvent in each phase be equal ... 

The computational problem of polymer phase equilibrium is to provide an adequate representation of the chemical potentials of each component in solution as a function of temperature, pressure, and composition. 

Another general type of behavior that occurs in polymer manufacture is shown in Figure 3. In many polymer processing operations, it is necessary to remove one or more solvents from the concentrated polymer at moderately low pressures. In such an instance, the phase equilibrium computation can be carried out if the chemical potential of the solvent in the polymer phase can be computed. Conditions of phase equilibrium require that the chemical potential of the solvent in the vapor phase be equal to that of the solvent in the liquid (polymer) phase. Note that the polymer is essentially involatile and is not present in the vapor phase. 

We have seen above in two instances, those of liquid-liquid phase separation and polymer devolatilization that computation of the phase equilibria involved is essentially a problem of mathematical formulation of the chemical potential (or activity) of each component in the solution. 

The first qualitatively correct attempt to model the relevant chemical potentials in a polymer solution was made independently by Huggins (4, ) and Flory [6). Their models, which are similar except for nomenclature, are now usually called the Flory-Huggins model ( ). 

There is a similar expression for polymer activity. However, if the fluid being sorbed by the polymer is a supercritical gas, it is most useful to use chemical potential for phase equilibrium calculations rather than activity. For example, at equilibrium between the fluid phase (gas) and polymer phase, the chemical potential of the gas in the fluid phase is equal to that in the liquid phase. An expression for the equality of chemical potentials is given by Cheng (12). 

The approach of Rory and Krigbaum was to consider an excess (E) chemical potential that exists arising from the non-ideality of the polymer solution. Then ... 

The term 6 is important it has the same units as temperature and at critical value (0 = T) causes the excess chemical potential to disappear. This point is known as the 6 temperature and at it the polymer solution behaves in a thermodynamically ideal way. 

Differentiation of Eq. (22) with respect to ri2 yields for the chemical potential of the polymeric solute relative to the pure liquid polymer as standard state... 

The chemical potential of the species of size x in a heterogeneous polymer, obtained by differentiating Eq. (23) with respect to is... 

For some purposes, as for example in the treatment of crystallization, it is more convenient to deal with the chemical potential per mole of structural units instead of per mole of polymer, as expressed in the formulas given above. Dividing Eq. (32) by the number of units per polymer molecule, which is xYi/Yu where Vi and v are the molar volumes of the solvent and of the. structural unit, respectively, we obtain for the chemical potential difference per unit... 

In a similar way the chemical potential of the polymer unit readily reduces for large x to... 

Fig. 120.—The chemical potential of the solvent in a binary solution containing polymer at low concentrations vi). Curves have been calculated according to Eq. (XII-26) for a = 1000 and the values of dicated with each curve. ...

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