Relaxation-controlled transport diffusion

GWR Davidson IH, NA Peppas. Solute and penetrant diffusion in swellable polymers. V. Relaxation-controlled transport in P(HEMA-co-MMA) copolymers. J Controlled Release 3 243-258, 1986. 

The region of "Case II sorption (relaxation-controlled transport) is separated from the Fickian diffusion region by a region where both relaxation and diffusion mechanisms are operative, giving rise to diffusion anomalies time-dependent or anomalous diffusion. 

Another criterion for predicting if the transport in polymeric gels is controlled by diffusion (Fickian) or by relaxation, is to determine the diffusional Deborah number De), which is a ratio between the characteristic polymer relaxation time of the polymer (2) when it is subject to a swelling stress and a characteristic diffusion time (6), defined as the coefficient between the square of the sample thickness (h) and the coefficient of water diffusion in the polymeric gel... 

While the ingress of liquids in elastomers is diffusion controlled and thus Case I, because the polymer chains are well above the glass transition temperature Tg, liquid transport into glassy polymers is more complex [112, 135]. The swelling behavior below Tg is characterized by a sharp boundary between the remaining polymer core and the swollen region which proceeds with constant velocity. This behavior is also called relaxation controlled because the diffusion is much faster than the segmental relaxation of the polymer. 

Many models have been suggested to describe anomalous (non-Fickian) uptake and a number of the more relevant to structural adhesives will be discussed. Diffusion-relaxation models are concerned with moisture transport when both Case I and Case II mechanisms are present. Berens and Hopfenberg (1978) assumed that the net penetrant uptake could be empirically separated into two parts, a Fickian diffusion-controlled uptake and a polymer relaxation-controlled uptake. The equation for mass uptake using Berens and Hopfenbergs model is shown below. 

The rate and type of release can be analyzed by the expression Mt/Moo=ktn (76). In the case of pure Fickian diffusion n = 0.5, whereas n > 0.5 indicates anomalous transport, i.e., in addition to diffusion another process (or processes) also occurs. If n = 1 (zero order release), transport is controlled by polymer relaxation ("Case II transport") (76). The ln(Mt/Mco) versus In t plots, shown in Figure 4, give n = 0.47 and 0.67 for samples A-9.5-49 and A-4-56, respectively. Evidently theophylline release is controlled by Fickian diffusion in the former network whereas the release is... 

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... 

Porous Membrane DS Devices. The applicability of a simple tubular DS based on a porous hydrophobic PTFE membrane tube was demonstrated for the collection of S02 (dilute H202 was used as the scrubber liquid, and conductometric detection was used) (46). The parameters of available tubular membranes that are important in determining the overall behavior of such a device include the following First, the fractional surface porosity, which is typically between 0.4 and 0.7 and represents the probability of an analyte gas molecule entering a pore in the event of a collision with the wall. Second, wall thickness, which is typically between 25 and 1000 xm and determines, together with the pore tortuosity (a measure of how convoluted the path is from one side of the membrane to the other), the overall diffusion distance from one side of the wall to the other. If uptake probability at the air-liquid interface in the pore is not the controlling factor, then items 1 and 2 together determine the collection efficiency. The transport of the analyte gas molecule takes place within the pores, in the gas phase. This process is far faster than the situation with a hydrophilic membrane the relaxation time is well below 100 ms, and the overall response time may in fact be determined by liquid-phase diffusion in the boundary layer within the lumen of the membrane tube, by liquid-phase dispersion within the... 

Carbon-13 rotating-frame relaxation rate measurements are used to elucidate the mechanism of gas transport in glassy polymers. The nmr relaxation measurements show that antiplasticization-plasticization of a glassy polymer by a low molecular weight additive effects the cooperative main-chain motions of the polymer. The correlation of the diffusion coefficients of gases with the main-chain motions in the polymer-additive blends shows that the diffusion of gases in polymers is controlled by the cooperative motions, thus providing experimental verification of the molecular theory of diffusion. Carbon-13 nmr relaxation... 

Another consideration in choosing a kinetic method is the objective of one s experiments. For example, if chemical kinetics rate constants are to be measured, most batch and flow techniques would be unsatisfactory since they primarily measure transport- and diffusion-controlled processes, and apparent rate laws and rate coefficients are determined. Instead, one should employ a fast kinetic method such as pressure-jump relaxation, electric field pulse, or stopped flow (Chapter 4). 

We have not discussed charged systems. As far as the diffuse parts of double layers are concerned, these relax very rapidly, see sec. II.2.13d, meaning that their dynamics will not be rate-controlling. Double layers do, of course, exert their influence on the diffusional transport rate and on the activation energies for the exchange of ionic surfactants between monolayers and solution (uptake in the monolayer is inhibited, release promoted). 

A second limiting transport process finds the weight gain of penetrant a linear function of time over the entire sorption range. This process has been termed Case II Transport, and is mechanistically quite different from Fickian diffusion. The rate controlling phenomena are penetrant induced polymeric relaxations. A combination of Case I and Case II processes has been... 

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