Relaxation rubbery polymers

This relative importance of relaxation and diffusion has been quantified with the Deborah number, De [119,130-132], De is defined as the ratio of a characteristic relaxation time A. to a characteristic diffusion time 0 (0 = L2/D, where D is the diffusion coefficient over the characteristic length L) De = X/Q. Thus rubbers will have values of De less than 1 and glasses will have values of De greater than 1. If the value of De is either much greater or much less than 1, swelling kinetics can usually be correlated by Fick s law with the appropriate initial and boundary conditions. Such transport is variously referred to as diffusion-controlled, Fickian, or case I sorption. In the case of rubbery polymers well above Tg (De < c 1), substantial swelling may occur and... 

We can predict tg for stress relaxation failure, where the failure occurs during stress releocation process under a constant strain. From observation of rubbery polymers, we can approximate the stress relaxation behavior by following ecjuation. 

The differences in the transport and solution behavior of gases in rubbery and glassy polymers are due to the fact that, as mentioned previously, the latter are not in a state of true thermodynamic equilibrium (1,3-8). Rubbery polymers have very short relaxation times and respond very rapidly to stresses that tend to change their physical conditions. Thus, a change in a temperature causes an immediate adjustment to a new equilibrium state (e.g., a new volume). A similar adjustment occurs when small penetrant molecules are absorbed by a rubbery polymer at constant temperature and pressure absorption (solution) equilibrium is very rapidly established. Furthermore, there appears to exist a unique mode of penetrant absorption and diffusion in rubbery polymers (2) ... 

Elastomers are flexible polymers that can be stretched but return to their original state when the stretching force is released. Most amorphous polymers become rubbery beyond their glass transition temperature, but not all rubbery polymers are elastic. Cross links in elastomers limit the extent to which elastomers can be deformed then encourage them to return to their original shape when they are relaxed. 

We have surmised (see Sect 3.2) that the structural relaxation in dense polymers is too slow to notably contribute to the transport of small molecules. This premise is intuitively acceptable for glassy polymers, but we apply it equally to gas motion in both glassy and rubbery polymers [77]. 

Using sorption-desorption kinetics, the extent of non-Fickian behavior decreases as the thickness of the film increases. Presumably, as the ratio of the time scale for diffusion to the time scale for stress-induced relaxation I ID) becomes large, the time-dependent relaxation occurs so rapidly (relatively) that it has a small effect on the diffusion process. In other words, in the limit where molecular-scale responses to swelling stresses are rapid compared to the rate of diffusion, the process appears to be Fickian. This is the case for totally amorphous rubbery polymers that behave as high molecular weight liquids. 

Pick s first and second laws were developed to describe the diffusion process in polymers. Fickian or case I transport is obtained when the local rate of change in the concentration of a diffusing species is controlled by the rate of diffusion of the penetrant. For most purposes, diffusion in rubbery polymers typically follows Fickian law. This is because these rubbery polymers adjust very rapidly to the presence of a penetrant. Polymer segments in their glassy states are relatively immobile, and do not respond rapidly to changes in their conditions. These glassy polymers often exhibit anomalous or non-Fickian transport. When the anomalies are due to an extremely slow diffusion rate as compared to the rate of polymer relaxation, the non-Fickian behaviour is called case II transport. Case II sorption is characterized by a discontinuous boundary between the outer layers of the polymer that are at sorption equilibrium with the penetrant, and the inner layers which are unrelaxed and unswollen. 

Polyvinyl acetate (PVAc) is another rubbery polymer used in mixed-matrix research. Its flexible nature helps to prevent void formation at the solid-polymer interface. Although it may not have practical industrial applications, PVAc aids in developing proof-of-concept associated with mixed-matrix membranes. Zeolite 4A-PVAc membranes have been proven to enhance membrane selectivity in mixed-matrix membranes with only 15 vol% zeolite " however, the permeability is lower than predicted presumably due to matrix rigidification. The rubbery nature of PVAc allows for more polymer relaxation at the solid-polymer interface as compared to the case with traditional, glassy polymers." ... 

Case I, or Fickian transport the diffusion time scale is much longer than the relaxation time scale. (Rubbery polymers usually exhibit such behavior.)... 

The solidity of gel electrolytes results from chain entanglements. At high temperatures they flow like liquids, but on cooling they show a small increase in the shear modulus at temperatures well above T. This is the liquid-to-rubber transition. The values of shear modulus and viscosity for rubbery solids are considerably lower than those for glass forming liquids at an equivalent structural relaxation time. The local or microscopic viscosity relaxation time of the rubbery material, which is reflected in the 7], obeys a VTF equation with a pre-exponential factor equivalent to that for small-molecule liquids. Above the liquid-to-rubber transition, the VTF equation is also obeyed but the pre-exponential term for viscosity is much larger than is typical for small-molecule liquids and is dependent on the polymer molecular weight. 

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