Polymers penetration

In sorjDtion experiments, the weight of sorbed molecules scales as tire square root of tire time, K4 t) ai t if diffusion obeys Pick s second law. Such behaviour is called case I diffusion. For some polymer/penetrant systems, M(t) is proportional to t. This situation is named case II diffusion [, ]. In tliese systems, sorjDtion strongly changes tire mechanical properties of tire polymers and a sharjD front of penetrant advances in tire polymer at a constant speed (figure C2.1.18). Intennediate behaviours between case I and case II have also been found. The occurrence of one mode, or tire otlier, is related to tire time tire polymer matrix needs to accommodate tire stmctural changes induced by tire progression of tire penetrant. 

For a classical diffusion process, Fickian is often the term used to describe the kinetics of transport. In polymer-penetrant systems where the diffusion is concentration-dependent, the term Fickian warrants clarification. The result of a sorption experiment is usually presented on a normalized time scale, i.e., by plotting M,/M versus tll2/L. This is called the reduced sorption curve. The features of the Fickian sorption process, based on Crank s extensive mathematical analysis of Eq. (3) with various functional dependencies of D(c0, are discussed in detail by Crank [5], The major characteristics are... 

The permeation technique is another commonly employed method for determining the mutual diffusion coefficient of a polymer-penetrant system. This technique involves a diffusion apparatus with the polymer membrane placed between two chambers. At time zero, the reservoir chamber is filled with the penetrant at a constant activity while the receptor chamber is maintained at zero activity. Therefore, the upstream surface of the polymer membrane is maintained at a concentration of c f. It is noted that c f is the concentration within the polymer surface layer, and this concentration can be related to the bulk concentration or vapor pressure through a partition coefficient or solubility constant. The amount... 

Figure 1 is a schematic diagram illustrating a typical composition dependence of the mutual diffusion coefficient for a polymer-penetrant system [8], Here the penetrant is apparently a good solvent for the polymer since the entire composition range is realized. Note that four regions can be distinguished. In the... 

Figure 1 A schematic diagram illustrating a typical concentration dependence of mutual diffusion coefficient in a polymer-penetrant system. (From Ref. 8.)...

Fujita [12] extended Eq. (17) to polymer-penetrant systems by a proper redefinition. For any polymer-penetrant binary system, the probability of finding... 

In general, mass transfer processes involving polymer-penetrant mixtures are generally analyzed by using a mutual diffusion coefficient. Therefore, a relationship between the mutual diffusion coefficient, D, and self-diffusion coefficients, ZVs is needed. Vrentas et al. [30] proposed an equation relating D to D, for polymer-penetrant systems in which Dx is much larger than Dr. 

Here % is the Flory-Huggins interaction parameter and ( ), is the penetrant volume fraction. In order to use Eqs. (26)—(28) for the prediction of D, one needs a great deal of data. However, much of it is readily available. For example, Vf and Vf can be estimated by equating them to equilibrium liquid volume at 0 K, and Ku/y and K22 - Tg2 can be computed from WLF constants which are available for a large number of polymers [31]. Kn/y and A n - Tg can be evaluated by using solvent viscosity-temperature data [28], The interaction parameters, %, can be determined experimentally and, for many polymer-penetrant systems, are available in the literature. 

Here ZT is calculated from the self-diffusion coefficients of the polymer-penetrant pair, i.e.,... 

JX Li. Case II sorption in glassy polymers Penetration kinetics. PhD Dissertation, University of Toronto, Toronto, Canada, 1998. 

GF Billovits, CJ Durning. Polymer material coordinates for mutual diffusion in polymer-penetrant systems. Chem Eng Commun 82 21-44, 1989. 

The interpretation of A becomes clearer when two plates, originally at very small distance from each other, are separated. At a certain separation, equal to 2A, polymer penetrates into the gap. In dilute solutions, where the chains behave as individual coils, A is expected to be of the order or r, the radius of gyration. However, at concentrations where t e coils overlap, the osmotic pressure of the solution becomes so high that narrower gaps can be entered, and A becomes smaller than Tg. 

This concept also requires that there be no polymer — penetrant interaction to such an extent that significant polymer structural changes affecting s0 and/or K2 occur. Over the experimental gas pressure range normally used (p 0-20 atm), this appears to be so in the case of permanent gases, like N2 but not in the case of condensable gases, like C02. Two effects may be noted here ... 

The conditions for adhesive or coating polymers in nonaqueous solvents to penetrate the fiber wall are less well established. Schneider (14) reviewed the evidence for penetration of various coating polymers into the fiber wall and concluded that most studies did not allow differentation between solvent penetration and polymer penetration. Furuno et al, (15) claimed that methyl methacrylate did penetrate the fiber wall from methanol solution but not from dioxane. As might be expected, all authors report that swelling of the wood is a necessary prerequisite to penetration of the fiber wall by polymers. 

It was shown in the above section that as a rule, at the base of the classical or microscopic diffusion models, there are ad hoc (heuristic) assumptions on a certain molecular behaviour of the polymer penetrant system. The fact that the mathematical formulae developed on such bases often lead to excellent correlations and even semipredictions of diffusion coefficients must be aknowledged. It is true that the classical models are not capable to predict diffusion coefficients only from first principles but this is often not an obstacle to hinder their use in certain types of investigations. Therefore we are quiet sure that this type of diffusion models will certainly be used in the future too for the interpretation of diffusion experiments. 

The problem of diffusion modeling in polymers changes to some degree when one envisages to develop a really atomistic model, with trully predictive capabilities and without making any ad hoc assumption on the molecular behaviour and/or motions in the polymer penetrant system. In principle, a possibility to develop such diffusion modelings, is to simulate theoretically the process of penetrant diffusion in a polymer matrix by computer calculations. 

Based on these encouraging achievements, in the last five or six years, the interest of the researchers shifted from easy-to-compute polymer penetrant systems to those which have interesting technological potentials in such fields as gas barriers (114— 117), gas or liquid separation processes (118-121), molded objects (packagings for example) (122) or swelling of polymers by solvents (123-126). 

However in the packaging sector the large majority of the diffusion processes in polymers imply penetrants with a relative molecular weight ranging between 100 and 1200 daltons and have often quite complex structures. From experiments one knows that these diffusion processes are characterized by D ranging from 10 9 to 10"l2cm2/s or even lower levels (see Appendix I). In (98) it was stated that, to study with MD techniques polymer penetrant systems in which the D are that small, is certainly out of reach for several generations of supercomputers to come. 

One of the central problems in the study of diffusion is to evaluate D for a given system as a function of such parameters as penetrant concentration and temperature. For polymer-penetrant systems with which we are concerned in this article two experimental methods are typical for this purpose. They are the sorption method and the permeation method. 

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