Fickian diffusion behaviour

Polymers whose diffusion characteristics can be described by Pick s first and second laws are said to follow Fickian diffusion behaviour. 

Fickian diffusion behaviour is evidenced by a linear relationship between the sorption (or desorption) of penetrant and the square root of the time of diffusion ( )/(Fig. 2.5). 

Non-Fickian diffusion behaviour (also called case II diffusion) is an anomalous diffusion which cannot be described in terms of Fickian laws (cf. section 2.17). It can exist in polymers ... 

Fig. 2.5. Absorption and desorption curves of a given penetrant by a given polymer (Fickian diffusion behaviour).

With regard to the unsteady state behaviour, only the simplest distributed model based on Fickian diffusion with constant effective diffusivities will be considered in this section. However, two important phenomena which are usually neglected in the literature will be included in the unsteady state modelling because of their importance. These are the adsorption mass capacity of the porous catalyst surface and the heat of chemisorption accompanying the steps of the CSD process. 

While many resin systems exhibit Fickian diffusion, others show non-Fickian behaviour. The experimental reabsorption diffusion curve shown in Figure 12.2 represents a smooth process which is linear over the hrst 60% before reaching equilibrium. For the resin system used, Fickian diffusion could only be obtained after preconditioning because often the early stage of moisture absorption deviates from Fickian kinetics. This creates difficulties in the calculation of a precise diffusion constant. It is thought that the non-Fickian behaviour results from the reorganisation of the network as the material becomes plasticised [9—11]. 

Of particular importance is the timescale over which diffusion occurs under various conditions of relative humidity (RH) and temperature. The RH determines the equilibrium moisture concentration, whereas higher temperatures will accelerate the moisture sorption process. In order to predict the moisture profile in a particular structure, it is assumed that Fickian diffusion kinetics operate. It will be seen later that many matrix resins exhibit non-Fickian effects, and other diffusion models have been examined. However, most resin systems in current use in the aerospace industry appear to exhibit Fickian behaviour over much of their service temperatures and times. Since the rate of moisture diffusion is low, it is usually necessary to use elevated temperatures to accelerate test programmes and studies intended to characterize the phenomenon. Elevated temperatures must be used with care though, because many resins only exhibit Fickian diffusion within certain temperature limits. If these temperatures are exceeded, the steady state equilibrium position may not be achieved and the Fickian predictions can then be inaccurate. This can lead to an overestimate of the moisture absorbed under real service conditions. 

In Fickian diffusion, the absorption behaviour is as shown in Fig. 7.6. The absorption (weight gain) and desorption (weight loss) curves can be plotted against (square root of time) and the resulting graphs are linear until 60% of the maximum absorption, has been reached, but later it is concave towards the axis and asymptotically approaches the final equilibrium value. is a constant when the material is fully submerged in a... 

The different behaviour between the two vapours is even more evident if we fit the entire experimental curve directly with Equation (4.6) after expansion into 10 terms (n = 1, 2, 3,. .. 10). The DCM curve yields a nearly perfect fit (Figure 4.8, bottom), indicating that the DCM transport can be described well by the simple Fickian diffusion with a single diffusion constant, independent of time or concentration. In this case the values of D and S are directly obtained from the curve fit and they agree within an error of a few percent with the values in Table 4.3, obtained by the tangent method. 

Several diffusion models have been used to propose transport mechanism of liquid, vapour and gas molecules through the polymer. A model described by Pick s laws is frequently used and known as Case I or Fickian diffusion. The diffusion behaviour in the rubbery polymers, represented by permeation, migration and sorption processes, can be described by the equation of Pick s first law ... 

Ions diffuse into the polymer with a distinct boundary between a fully reduced/ swollen phase and a fully oxidized/contracted phase. The boundary is very sharp on the first reduction cycle but is broader on subsequent cycles. This type of diffusion behaviour is typical of solvent diffusion in polymers that causes a softening phase change and is called Type II (non-Fickian) behaviour [49]. This process occurs in... 

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. ... 

It is well known that insertion of the above effective coefficients Se and Pe or De = Pe/Se into Eqs. (2) or (3) respectively, does not lead to the correct description of transient diffusion. However, the behaviour of the ideal Fickian system defined by Se and Pe or De constitutes a useful standard of reference. Given the appropriate theoretical background, one may then deduce information about S(X), DT(X) from the nature and magnitude of the deviation of suitable observed kinetic parameters from the calculated Fickian values. 

If the penetrant enters the glassy matrix faster than the polymer can adapt itself by volume relaxation, the solvent front advances linearly with time. This behaviour is called case-II diffusion or relaxation-controlled diffusion. It is a special case of anomalous diffusion, where the mean square particle displacement is proportional to t. It commonly applies to polymers in the glassy state [Wei2]. Here the system 1,4-dioxane/PVC is an example (Fig. 10.1.8(b)). Due to the softening of the material behind the diffusion front, the polymer relaxation in the already swollen matrix is fast enough to adapt to a new situation created by further solvent uptake. Therefore, solvent ingress as well as swelling behind the diffusion front is Fickian. 

Figure 3.2 gives an example [4] of a moisture absorption curve for an advanced epoxy resin and the theoretical curve obtained from equation [3.4], This illustrates the difhculties of assessing the diffusion coefficient when deviations from Fickian behaviour occur and an equihbrium moisture... 

Extruded PLA films containing flavonoids like catechin and epicatechin were prepared as a vehicle for the antioxidant release into food. The release kinetics into ethanol 95% was estimated at 20, 30, 40 and 50 °C, displaying Fickian behaviour with diffusion coefficients in the 10 cm s range. The antioxidant activity of the films was also measured, after extraction with... 

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. 

Both types of diffusion i.e. Fickian and non-Fickian (case II), can be superimposed. Some penetrants will show Fickian behaviour in a given polymer whilst others will show non-Fickian (case II) behaviour. By increasing the crosslink density of a given polymer, the diffusion tends to move from Fickian type to non-Fickian (case II) type. 

In Equation (15.3) (only strictly valid if we assume Fickian behaviour, no changes in film thickness and constant absorption coefficient), A is the absorbance of the chosen band at a given time, t, and A is the absorbance at saturation or equilibrium sorption, L is the thickness of the film and D the diffusion coefficient. Lagaron and co-workers [16] carried out measurements for a number of aroma components, namely, limonene, a-pinene and citral and results are given in Table 15.1. 

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