Swelling Fickian kinetics

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... 

Pure PHEMA gel is sufficiently physically cross-linked by entanglements that it swells in water without dissolving, even without covalent cross-links. Its water sorption kinetics are Fickian over a broad temperature range. As the temperature increases, the diffusion coefficient of the sorption process rises from a value of 3.2 X 10 8 cm2/s at 4°C to 5.6 x 10 7 cm2/s at 88°C according to an Arrhenius rate law with an activation energy of 6.1 kcal/mol. At 5°C, the sample becomes completely rubbery at 60% of the equilibrium solvent uptake (q = 1.67). This transition drops steadily as Tg is approached ( 90°C), so that at 88°C the sample becomes entirely rubbery with less than 30% of the equilibrium uptake (q = 1.51) (data cited here are from Ref. 138). 

Fitting the swelling curves of Fig. 7a to the form Q(t) — kt yields values of a greater than or equal to 0.8. Thus the swelling must be considered anomalous, or non-Fickian. In the absence of ionic interactions, this would not be expected since BMA/DMA 70/30 is initially not far below its Tg at 25 °C. Indeed, swelling measurements of this copolymer in hexane show kinetics that are nearly Fickian (a 0.55), as shown in Fig. 7b. Therefore, the anomalous swelling observed in Fig. 7a must be attributed to ion transport and binding rates in the gel. We will return to this point later. 

Fig. 12. Volume change kinetics of PVME gel (Trial 4) between states above and below the volume transition of 37 °C. Non-Fickian behaviour is observed for both swelling and shrinking. For comparison, lines are calculated which provide the closest fit of Fickian theory to the data. O Swelling (from 50 to 24°C), D = 4.0 x 10-7cm2/s Shrinking (from 23 to 50°C), D = I x 10-5 cm2/s. Reprinted from Polymer (1991) 33 990 by permission of the publishers, Butterworth Heinemann [46]...

After all drug is dissolved, the osmotic pressure decays and the beads shrink to their equilibrium swelling value, as observed for beads with high water swelling (11,12) during this second release phase normal Fickian diffusion kinetics becomes more important, characterized by a very low rate which depends on the hydro-philicity of the polymer. 

For various reasons, almost constant drug release has been observed by some authors and kinetics controlled by the swelling of the polymer have been postulated many times as an explanation. The author [15] has, in fact, proved theoretically that anomalous water transport - a necessary criteria for a non-Fickian solute release - is doubtful with these materials. Let us consider the... 

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. 

In the case of non-swelling tablets, drug release is generally expressed by Fickian diffusion, for which n = 0.5. For most erodible matrices, the drug release follows zero order kinetics, for which n = 1. In the case of swelling tablets, the drug release is due to a combination of swelling and erosion. They follow non-Fickian release behavior. For them, the value of n lies between 0.5 and 1.0 [78]. The value of n was calculated from the slopes of the log(M/M ) vs log t plot in acidic and neutral pH. 

Mt and M q are the absorbed mass at time t and after equilibrium has been reached, respectively, k is a constant and n is the exponent, which indicates the type of diffusion transport in the hydration process. The kinetics was also dependent on the compositions of the prepared formulations as n was close to 0.5 (indicating Fickian type behaviour) for hydrogels containing MMA. When analysing formulations without MMA it was foimd that n was close to unity indicating non-Fickian behaviour and case II water transport mechanism, which is the most desirable kinetic behaviour for a swelling-controlled release material. ... 

For high solubility penetrants, great complexity is often observed in sorption-desorption kinetics or transient permeation due to time-dependent rearrangements of crystalline regions in response to swelling-induced stresses (59,60). These are non-Fickian diffusional behaviors. Protracted approaches to... 

This arises in polymer/permeant systems which obey Fickian diffusion (described later in Section 20.4). The rate of diffusion of permeant molecules is much less than the polymer segment mobility. Under these conditions, the rate at which sorption equilibrium is approached is independent of swelling kinetics. As shown in Figure 4, the sorption curve vs, t ) is linear in the... 

---