Polymer solution thermodynamics entropy

The extent to which permeant molecules are sorbed and their mode of sorption in a polymer depend upon the enthalpy and entropy of permeant/polymer mixing, i.e, upon the activity of the permeant within the polymer at equilibrium. Much information on polymer-solution thermodynamics is given elsewhere in this volume. Here, a brief description is given of the most common types of sorption isotherms which have been utilized to describe permeation in polymers. 

In Chap. 8 we discuss the thermodynamics of polymer solutions, specifically with respect to phase separation and osmotic pressure. We shall devote considerable attention to statistical models to describe both the entropy and the enthalpy of mixtures. Of particular interest is the idea that the thermodynamic... 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

These two examples show that regular patterns can evolve but, by definition, dissipative structures disappear once the thermodynamic equilibrium has been reached. When one wants to use dissipative structures for patterning of materials, the dissipative structure has to be fixed. Then, even though the thermodynamic instability that led to and supported the pattern has ceased, the structure would remain. Here, polymers play an important role. Since many polymers are amorphous, there is the possibility to freeze temporal patterns. Furthermore, polymer solutions are nonlinear with respect to viscosity and thus strong effects are expected to be seen in evaporating polymer solutions. Since a macromolecule is a nanoscale object, conformational entropy will also play a role in nanoscale ordered structures of polymers. 

Statistical thermodynamic mean-field theory of polymer solutions, first formulated independently by Flory, Huggins, and Staverman, in which the thermodynamic quantities of the solution are derived from a simple concept of combinatorial entropy of mixing and a reduced Gibbs-energy parameter, the X interaction parameter. 

Nonideal thermodynamic behavior has been observed with polymer solutions in which A Hm is practically zero. Such deviations must be due to the occurrence ofa nonideal entropy, and the first attempts to calculate the entropy change when a long chain molecule is mixed with small molecules were due to Flory [8] and Huggins [9]. Modifications and improvements have been made to the original theory, but none of these variations has made enough impact on practical problems of polymer compatibility to occupy us here. 

The thermodynamic parameters 1/) and K introduced above, pertaining to polymer-solvent interactions in dilute solutions, may be determined from thermodynamic studies of dilute solutions of the polymer, e.g., from osmotic pressure or turbidity measurements at different temperatures. These parameters may also be determined, at least in principle, from viscosity measurements on polymer solutions (see Frictional Propcitics of Polymers). The parameter ij, which is a measure of the entropy of mixing, appears to be related to the spatial or geometrical character of the solvent. For those solvents having cyclic structures, which are relatively compact and symmetrical (e.g., benzene, toluene, and cyclohexane), xp has relatively higher values than for the less symmetrical acyclic solvents capable of assuming a number of different configurations. Cyclic solvents are thus more favorable... 

The solubility parameter approach is a thermodynamically consistent theory and it has some links with other theories such as the van der Waals internal pressure concept, the Lennard-Jones pair potentials between molecules, and entropy of mixing concepts of the lattice theories. The solubility parameter concept has found wide use in industry for nonpolar solvents (i.e. solvent selection for polymer solutions and extraction processes) as well as in academic endeavor (thermodynamics of solutions), but it is unsuccessful for solutions where polar and especially hydrogen-bonding interactions are operating. 

The first application of living polymers in thermodynamic studies was reported by Worsfold and Bywater181) and by McCormick182). In both studies anionic polymerization of a-methyl styrene, initiated in tetrahydrofuran by electron-transfer, was investigated over temperatures ranging from 0°C to -40°C. The results are shown graphically in Fig. 3, and lead to heat of polymerization of — 8 kcal/mol, entropy of polymerization in solution of 29 eu, and to about 1 M equilibrium concentration of a-methyl styrene at ambient temperature. 

Phase separation is frequently observed in polymer solutions and it is mainly due to their low entropy of mixing. At a state of equilibrium each species of the mixture is partitioned between two phases, namely, the supernatant (extremely dilute) and precipitated (moderately dilute) phases [78]. Theoretical models and experimental techniques have been developed to predict the solubility behavior of polymer solutions, polymer blends, and other related systems [79, 80]. Simple theories only permit a rather qualitative description of this phenomenon [78]. Refined and improved theoretical and semiempirical models allow a more accurate prediction of the demixing phenomena and related thermodynamic properties [57, 81]. 

Another specific phenomenon in polymer solutions is the effect of cosolvency when a mixture of two nonsolvents forms a solvent or even a thermodynamically very good solvent for polymer under study. Much less common is the occurence of co-nonsolvency, when a mixture of two solvents for given polymer constitutes its precipitant. Both cosolvency and co-nonsolvency result from solvent - solvent interactions and from the entropy changes of polymer coils induced with their interaction with solvent molecules. It is advisable to consider these unexpected phenomena when working with rrtixed mobile phases. 

In 1956, Hory introduced the semi-flexibility into the classical lattice statistical thermodynamic theory of polymer solutions (Flory 1956). From the classical lattice statistics of flexible polymers, we have derived the total number of ways to arrange polymer chains in a lattice space, as given by (8.15). The first two terms on the right-hand side of that equation are the combinational entropy between polymers and solvent molecules, and the last three terms belong to polymer conformational entropy. Thus the contribution of polymer conformation in the total partition function is... 

It is evident there is a similarity in the expressions for the combinatorial entropy and the free energy of mixing for small and large molecules with a solvent, with mole fractions being replaced by volume fractions for polymer solutions. The solution behaviour is dependent on the volume fraction of polymer in solution so it is possible to relate the thermodynamic changes to the molecular weight of the polymer in solution as follows ... 

Some implicit databases are provided within the Polymer Handbook by Schuld and Wolf or by Orwoll and in two papers prepared earlier by Orwoll. These four sources list tables of Flory s x-fiinction and tables where enthalpy, entropy or volume changes, respectively, are given in the literature for a large number of polymer solutions. The tables of second virial coefficients of polymers in solution, which were prepared by Lechner and coworkers (also provided in the Polymer Handbook), are a valuable source for estimating the solvent activity in the dilute polymer solution. Bonner reviewed vapor-hquid equilibria in concentrated polymer solutions and listed tables containing temperature and concentration ranges of a certain number of polymer solutions. Two CRC-handbooks prepared by Barton list a larger number of thermodynamic data of polymer solutions in form of polymer-solvent interaction or solubility parameters." ... 

For diffusion of polymers, flows through porous media, and the description liquid helium, Fick s and Fourier s laws are generally not applicable, since these laws are based on linear flow-force relations. Extended nonequilibrium thermodynamics is mainly concerned with the nonlinear region and deriving the evolution equations with the dissipative flows as independent variables, beside the usual conserved variables. Typical nonequilibrium variables, such as flows, gradients of intensive properties may contribute to the rate of entropy production. When the relaxation time of these variables differs from the observation time they act as constant parameters. The phenomenon becomes complex when the observation time and the relaxation time are of the same order, and the description of system requires additional variables. Polymer solutions are highly relevant systems for analyses beyond the local-equilibrium theory. 

In the lattice model of polymer solutions, polymer chain is simply represented by a number of consecutively occupied lattice sites, each site corresponding to one chain unit. The rest single sites are assigned to solvents. This simple lattice treatment of polymer solutions allows a very convenient way to calculate thermodynamic properties of flexible and semiflexible polymer solutions from the statistical thermodynamic approach. By the mean-field assumption, the entropy part and the enthalpy part of partition function can be separately calculated. 

A polymier molecule is much larger than a typical small solvent molecule, often by more than a thousandfold. This difference in size has important consequences for the thermodynamics of pol>mier solutions. First, the coUigative attraction of a polymer solution for soWent small-molecule molecules is much higher than would be expected on the basis of the mole fractions. Rather the attraction is proportional to the volume fraction. Second, polymers are typically not miscible with other pol miers because of small mixing entropies. Third, the size as>Tnmetry between a polymer and a small-molecule solvent translates into an asymmetry in miscibility phase diagrams. 

The chemical potential i (i.e. the partial molar free energy) is the most fundamental physical quantity to describe the thermodynamic properties of polymer solutions. If p is represented as functions of absolute temperature (7), pressure (7 ) and the composition (for example, the polymer concentration for a binary mixture), one can determine not only the molecular weight, M, of the solute (the polymer), but also many other thermodynamic quantities, such as the partial molar entropy, the partial molar enthalpy and the partial molar volume of each component. 

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