Gibbs function, polymer thermodynamics

Basic to the thermodynamic description is the heat capacity which is defined as the partial differential Cp = (dH/dT)n,p, where H is the enthalpy and T the temperature. The partial differential is taken at constant pressure and composition, as indicated by the subscripts p and n, respectively A close link between microscopic and macroscopic description is possible for this fundamental property. The integral thermodynamic functions include enthalpy H entropy S, and free enthalpy G (Gibbs function). In addition, information on pressure p, volume V, and temperature T is of importance (PVT properties). The transition parameters of pure, one-component systems are seen as first-order and glass transitions. Mesophase transitions, in general, were reviewed (12) and the effect of specific interest to polymers, the conformational disorder, was described in more detail (13). The broad field of multicomponent systems is particularly troubled by nonequilibrium behavior. Polymerization thermodynamics relies on the properties of the monomers and does not have as many problems with nonequilibrium. 

It would be difficult to enumerate all Ron s scientific achievements in the field of polymer thermodynamic. One can name the generalizations of the Flory-Huggins Gibbs energy leading to the prediction and experimental verification of coexistence of three phases in pseudobinary system with sufficiently broad distribution or, the analysis of the functional form of the interaction term leading to the appearance of off-zero critical concentration , at variance with zero critical concentration associated with theta-temperature. Thanks largely to Ron, polymer scientists realize that the cloud point curve is not the binodal and its maximum or minimum are not identical with the critical temperatures. 

The data obtained were used to calculate enthalpy, entropy and the Gibbs functions for the seq-IPNs synthesis. It was shown that the isotherms of diverse thermodynamic properties of interpenetrating polymer networks plotted versus their composition, in particular the molar fraction of the CPU per conditional mole, can be described by straight lines. This made it possible to estimate the thermodynamic behavior of the seq-IPNs of any compositions at standard pressure within a wide temperature range. It was determined [50] that at molar content > 0.50 of PCN in seq-IPNs studied AG°p (AG° of process) < 0 and this has allowed authors to conclude about thermodynamical miscibility of the components for seq-IPNs of these composition... 

The thermodynamic state of a polymer- solvent system is completely determined, as it was analized before, at fixed temperature and pressure by means of the interaction parameter g. This g is defined through the noncombinatorial part of the Gibbs mixing function, AGm- The more usual interaction parameter, x, is defined similarly but through the solvent chemical potential, A xi, derived from AGm-... 

Description of thermochemical properties of chemical compounds, including that of polymers can be done using a few thermodynamic functions. One basic function is Gibbs free enthalpy that is expressed as follows [1] ... 

A new type of rotational degrees of freedom parameter will be defined for the backbones and side groups of polymers, and correlations for the heat capacity and related thermodynamic functions (enthalpy, entropy and Gibbs free energy) will be developed utilizing both the connectivity indices and the rotational degrees of freedom, in Chapter 4. 

To characterize the thermodynamic behavior of the components in a solution, it is necessary to use the concept of partial molar or partial specific functions. The partial molar quantities most commonly encountered in the thermodynamics of polymer solutions are partial molar volume Vi and partial molar Gibbs free energy Gi. The latter quantity is of special significance since it is identical to the quantity called chemical potential, pi, defined by... 

The second classification has been recently used in a later review article by Meijer and co-workers. This classification is mainly concerned with the mechanism of supramoiecuiar polymerization, which has been defined as the evolution of Gibbs free energy as a function of monomer conversion to polymer (p) from zero to one (p = 0 1) as the concentration, temperature, or some other environmental parameter is altered. This classification has been extremely effective in describing the vast array of examples of SPs, correlating mechanistic similarities with their covalent counterparts, which are widely understood to be classified mechanistically. In this scheme, the authors clearly identify the most fundamental difference between covalent and SPs as the difference in kinetic versus thermodynamic control. The authors argue that it is from this dramatic difference between covalent polymers and SPs, due to the reversibility of the noncovalent interactions, that SPs derive their special properties. This review did not include, however, SPs made from large macromolecular building blocks. 

The measurement of polymer solutions with lower polymer concentrations requires very precise pressure instruments, because the difference in the pure solvent vapor pressure becomes veiy small with decreasing amount of polymer. At least, no one can really answer the question if real thermodynamic equilibrium is obtained or only a fro2en non-equihbrium state. Non-equilibrium data can be detected from imusual shifts of the %-function with some experience. Also, some kind of hysteresis in experimental data seems to point to non-equilibrium results. A common consistency test on the basis of the integrated Gibbs-Duhem equation does not work for vapor pressiue data of binary polymer solutions because the vapor phase is pme solvent vapor. Thus, absolute vapor pressure measurements need very careful handling, plenty of time, and an experienced experimentator. They are not the method of choice for high-viscous polymer solutions. 

Phase diagrams of polymer-polymer blend systems and polymer-solvent mixtures can be constructed as a function of tanperature vs. composition. Two important factors need to be considered when constructing phase diagrams (1) equilibrium and (2) stability. It can be shown from the laws of thermodynamics and reversibility that the equilibrium sate of a closed systan is that state at which the total Gibbs free energy is a minimum with respea to aU possible changes at the given temperature and pressure. At equilibrium ... 

Continuous thermodynamics has also been applied to derive equations for spinodal, critical point and multiple critical points. To do so with continuous thermodynamics is much easier than in usual thermodynamics. Spinodal and critical points may be calculated for very complex systems or for cases in which the segment-molar excess Gibbs energy and depends on some moments of the distribution function. In simple cases (for example, a solution of a polymer in a solvent, where the segment-molar excess Gibbs energy is independent of the distribution function) the equations of the spinodal and the critical point are known from the usual thermodynamic treatment. However, for more complex systems continuous thermodynamics has achieved real progress, for example, for polydisperse copolymer blends, the polydispersity is described by bivariant distribution functions. ... 

Equation (2.8) indicates that in thermodynamics, the internal energy is used as a function of state to characterize the system at constant volume, and also when no work is being performed on or by the system. But the majority of real processes, especially for polymers, take place at constant pressure, because solids and liquids (the only physical states for polymers) are virtually incompressible. For such processes (i.e., those taking place at constant pressure), Gibbs introduced a new function of state, enthalpy H... 

Figure 17.2 Schematic to illustrate the thermodynamics of micellization, with contributions to the Gibbs free energy as a function of aggregate size, showing the driving force (due to the hydrophobic effect), surface-tension correction and head-group term. The sum curve or nucleation barrier of these three parameters has a minimum at the optimum aggregation number. (With kind permission from Springer Science + Business Media Colloid Polymer Science, Supramolecular perspectives in colloid science, 286, 2008, 855-864, M.A.C. Stuart.)...

Phase relationships in equilibrium are determined by the free enthalpy (Gibbs free energy) of the system. The thermodynamic bdiaviour of polymer solutions can be very well described with the free enthalpy of mixing function derived, independently, by Florv (6,7) and Huggins (8—10) on the basis of the lattice theory of the liquid state. For the simplest case conceivable — a solution of a polydisperse polymer in a single solvent quasi-binary system) — we have... 

By finding the equilibrium heat capacities of the solid and liquid [Cp(vibration), Cp(liquid)], as well as the equilibrium transition parameters T, A//f(100%), all thermodynamic functions, enthalpy (//), entropy (5), and Gibbs free energy (G), can be calculated as a function of temperature for equilibrium conditions [3]. All recommended results of equilibrium quantities and parameters, for over 200 polymers, have been collected and organized as part of the ATHAS Data Bank, a part of which is available online [20]. 

Classical polymer solution thermodynamics often did not consider solvent activities or solvent activity coefficients but usually a dimensionless quantity, the so-called Flory-Huggins interaction parameter x The % is not only a function of temperature (and pressure), as was evident from its foimdation, but it is also a function of composition and polymer molecular mass. As pointed out in many papers, it is more precise to call it x function (what is in principle a residual solvent chemical potential function). Because of its widespread use and its possible sources of mistakes and rrusinterpretations, the necessary relations must be included here. Starting from Equation [4.4.1b], the difference between the chemical potentials of the solvent in the mixture and in the standard state belongs to the first derivative of the Gibbs free energy of mixing with respect to the amount of substance of the solvent ... 

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