Thermodynamic diffusion coefficient, polymer

One of the simplest early free-volume diffusion models was formulated in (51,52,60). The concept of this model was considered an advance, because some of the parameters required to describe the concentration dependence of the diffusion coefficient could be obtained from the physico-chemical properties of the polymer and penetrant. The relation proposed for the calculation of the thermodynamic diffusion coefficient, DT, was (51,60) ... 

The concept of free volume in a polymer is an extension of the ideas of Cohen and Turnbull [141], first used to describe the self-diffusion in a liquid of hard spheres. Such theories suggest that the permeant diffuses by a cooperative movement between the permeant and the polymer segments, from one hole to the other within the polymer. The creation of a hole is caused by fluctuations of local density due to thermal motion. Based on the concept of the redistribution of free volume to represent the thermodynamic diffusion coefficient [142], and the standard reference state for free volume [143], Stem and Fang [144] interpreted their permeability data for nonporous membranes, and Fang, Stem, and Frisch [ 145] extended the theory to include the case of permeation of gas and liquid mixtures. 

The diffusion coefficient, sometimes called the diffusivity, is the kinetic term that describes the speed of movement. The solubiHty coefficient, which should not be called the solubiHty, is the thermodynamic term that describes the amount of permeant that will dissolve ia the polymer. The solubiHty coefficient is a reciprocal Henry s Law coefficient as shown ia equation 3. 

The non-bonded interaction energy, the van-der-Waals and electrostatic part of the interaction Hamiltonian are best determined by parametrizing a molecular liquid that contains the same chemical groups as the polymers against the experimentally measured thermodynamical and dynamical data, e.g., enthalpy of vaporization, diffusion coefficient, or viscosity. The parameters can then be transferred to polymers, as was done in our case, for instance in polystyrene (from benzene) [19] or poly (vinyl alcohol) (from ethanol) [20,21]. 

The rapid transport of the linear, flexible polymer was found to be markedly dependent on the concentration of the second polymer. While no systematic studies were performed on these ternary systems, it was argued that the rapid rates of transport could be understood in terms of the dominance of strong thermodynamic interactions between polymer components overcoming the effect of frictional interactions this would give rise to increasing apparent diffusion coefficients with concentration 28-45i. This is analogous to the resulting interplay of these parameters associated with binary diffusion of polymers. 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

The sorption coefficient (K) in Equation (2.84) is the term linking the concentration of a component in the fluid phase with its concentration in the membrane polymer phase. Because sorption is an equilibrium term, conventional thermodynamics can be used to calculate solubilities of gases in polymers to within a factor of two or three. However, diffusion coefficients (D) are kinetic terms that reflect the effect of the surrounding environment on the molecular motion of permeating components. Calculation of diffusion coefficients in liquids and gases is possible, but calculation of diffusion coefficients in polymers is much more difficult. In the long term, the best hope for accurate predictions of diffusion in polymers is the molecular dynamics calculations described in an earlier section. However, this technique is still under development and is currently limited to calculations of the diffusion of small gas molecules in amorphous polymers the... 

Tables 5.4 and 5.5 provides the homogeneous charge transport diffusion coefficients, Dct, for osmium polymers with different loadings in HCIO4 and H2SO4. Further information about the nature of these processes can be obtained by determining the thermodynamic parameters. These parameters are also summarized...

Also, the theory is much more complex than just presented. Bly36 has classified the process into three mechanisms steric exclusion, restricted diffusion, and thermodynamic considerations, and the process has been thoroughly studied. The rate equation is also different for SEC. For polymers, the longer retained peaks have smaller peak dispersivities, H, than early peaks, in direct contrast to normal LC expectations. In part this is due to the fact that the smaller molecules that elute last have higher diffusion coefficients and therefore less mass transfer zone spreading. 

Somewhat closer to the designation of a microscopic model are those diffusion theories which model the transport processes by stochastic rate equations. In the most simple of these models an unique transition rate of penetrant molecules between smaller cells of the same energy is determined as function of gross thermodynamic properties and molecular structure characteristics of the penetrant polymer system. Unfortunately, until now the diffusion models developed on this basis also require a number of adjustable parameters without precise physical meaning. Moreover, the problem of these later models is that in order to predict the absolute value of the diffusion coefficient at least a most probable average length of the elementary diffusion jump must be known. But in the framework of this type of microscopic model, it is not possible to determine this parameter from first principles . 

Third, a serious need exists for a data base containing transport properties of complex fluids, analogous to thermodynamic data for nonideal molecular systems. Most measurements of viscosities, pressure drops, etc. have little value beyond the specific conditions of the experiment because of inadequate characterization at the microscopic level. In fact, for many polydisperse or multicomponent systems sufficient characterization is not presently possible. Hence, the effort probably should begin with model materials, akin to the measurement of viscometric functions [27] and diffusion coefficients [28] for polymers of precisely tailored molecular structure. Then correlations between the transport and thermodynamic properties and key microstructural parameters, e.g., size, shape, concentration, and characteristics of interactions, could be developed through enlightened dimensional analysis or asymptotic solutions. These data would facilitate systematic... 

It has long been a mystery why diffusion coefficients of polymer-diluent systems, especially when the diluent is a good solvent for a given polymer, exhibit so pronounced a concentration dependence that it looks extraordinary. Several proposals have been made for the interpretation of this dependence. Thus Park (1950) attempted to explain it in terms of the thermodynamic non-ideality of polymer-diluent mixtures, but it was found that such an effect was too small to account for the actual data. Fujita (1953) suggested immobilization of penetrant molecules in the polymer network, which, however, was not accepted by subsequent workers. Recently, Barrer and Fergusson (1958) reported that their diffusion coefficient data for benzene in rubber could be analyzed in terms of the zone theory of diffusion due to Barrer (1957). Examination shows, however, that their conclusion is never definitive, since it resorted to a less plausible choice of the value for a certain basic parameter. 

Selectivity and productivity depend on sorption and diffusion. Sorption is dictated by thermodynamic properties, namely, the solubility parameter of the solute(s)/membrane material system. On the other hand, the size, shape, molecular weight of the solute, and the availability of inter/intra molecular free space of the polymer largely govern the second property, the diffusion coefficient. For an ideal membrane, both the sorption and diffusion processes should favor the chosen solute. If one step becomes unfavorable for a given solute the overall selectivity will be poor [28]. 

Separation from mixtures is achieved because the membrane transports one component more readily than the others, even if the driving forces are equal. The effectiveness of pervaporation is measured by two parameters, namely flux, which determines the rate of permeation and selectivity, which measures the separation efficiency of the membrane (controlled by the intrinsic properties of the polymer used to construct it). The coupling of fluxes affecting the permeability of a mixture component can be divided into two parts, namely a thermodynamic part expressed as solubility, and a kinetic part expressed as diffusivity. In the thermodynamic part, the concentration change of one component in the membrane due to the presence of another is caused by mutual interactions between the permeates in the membrane in addition to interactions between the individual components and the membrane material. On the other hand, kinetic coupling arises from the dependence of the concentration on the diffusion coefficients of the permeates in the polymers [155]. 

In conclusion, IGC has shown to be a powerful tool for the evaluation of the thermodynamics of water sorption by polymers at low water uptake levels, as well as for the determination of diffusion coefficients. 

The dimensionless mean retention time, Hi/to is independent of the carrier gas velocity and is only a function of the thermodynamic properties of the polymer-solute system. The dimensionless variance, i2 /tc2. is a function of the thermodynamic and transport properties of the system. The first term of Equation 30 represents the contribution of the slow stationary phase diffusion to peak dispersion. The second term represents the contribution of axial molecular diffusion in the gas phase. At high carrier gas velocities, the dimensionless second moment is a linear function of velocity with the slope inversely proportional to the diffusion coefficient. 

As intensive studies on the ECPs have been carried out for almost 30 years, a vast knowledge of the methods of preparation and the physico-chemical properties of these materials has accumulated [5-17]. The electrochemistry ofthe ECPs has been systematically and repeatedly reviewed, covering many different and important topics such as electrosynthesis, the elucidation of mechanisms and kinetics of the doping processes in ECPs, the establishment and utilization of structure-property relationships, as well as a great variety of their applications as novel electrochemical systems, and so forth [18-23]. In this chapter, a classification is proposed for electroactive polymers and ion-insertion inorganic hosts, emphasizing the unique feature of ECPs as mixed electronic-ionic conductors. The analysis of thermodynamic and kinetic properties of ECP electrodes presented here is based on a combined consideration of the potential-dependent differential capacitance of the electrode, chemical diffusion coefficients, and the partial conductivities of related electronic and ionic charge carriers. 

Equations 4-8 indicate that P is a product of a diffusion coefficient (a kinetic factor) and of a solubility coefficient (a thermodynamic factor). A common method of inferring the mechanism of permeation is to determine the dependence of P on p (or c) and on the temperature. This requires, in turn, a knowledge of the dependence of D and S on these variables. As is shown below, this dependence is very different above and below the glass-transition of the polymer. 

The free-volume model proposed by Vrentas and Duda (67-69) is based on the models of Cohen and Turnbull and of Fujita, while utilizing Bearman s (7j0) relation between the mutual diffusion coefficient and the friction coefficient as well as the entanglement theory of Bueche (71) and Flory s (72) thermodynamic theory. The formulation of Vrentas and Duda relaxes the assumptions deemed responsible for the deficiencies of Fujita s model. Among the latter is the assumption that the molecular weight of that part of the polymer chain involved in a unit "jump" of a penetrant molecule is equal to the... 

Vrentas and Duda s theory formulates a method of predicting the mutual diffusion coefficient D of a penetrant/polymer system. The revised version ( 8) of this theory describes the temperature and concentration dependence of D but requires values for a number of parameters for a binary system. The data needed for evaluation of these parameters include the Tg of both the polymer and the penetrant, the density and viscosity as a function of temperature for the pure polymer and penetrant, at least three values of the diffusivity for the penetrant/polymer system at two or more temperatures, and the solubility of the penetrant in the polymer or other thermodynamic data from which the Flory interaction parameter % (assumed to be independent of concentration and temperature) can be determined. An extension of this model has been made to describe the effect of the glass transition on the free volume and on the diffusion process (23.) ... 

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