Two-stage diffusion

Two-stage diffusion is considered to correspond to weight gain data sketched by curve B in Fig. 4.2. Such a process accords with fluid ingress that attains an apparent equilibrium level at a shorter duration but subsequently continues to build up until attaining a final stable magnitude. 

This circumstance is commensurate with diffusion coupled with viscoelastic material response. Consequently, all models that attempt to predict this phenomenon incorporate some additional time-dependent aspect within Fick s law. 

It was observed that in some circumstances exposure to ambient fluid is initiated by the formation of a thin molecular layer that adheres to the exposed boundary of the polymer and that the diffusion process involves some time delay before proceeding at full capacity (e.g.. Long and Richman 1960). This observation motivated Long and Richman (1960) to employ a time-dependent boundary condition even under exposure to constant environment, namely 

An improved match against specific data was obtained by extending (5.4) to include a spectrum of retardation times (Cai and Weitsman 1994), namely 

Upon selecting values of S to spread over the time span of interest, the values of mo and m n = 1,2. N) were determined by a numerical scheme that matched the data of M(f) at selected times tkik = 1,2. K). The value of D was selected from the early weight gain data. Results are shown in Fig. 5.1. 

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Example. As a model for two-stage diffusion take i = 1,2 and F as in (7.1). Then 7i,2 = 2 and 72,1 =7i- F°r computing the cross-section for neutron scattering one needs to know the probability density Gs(r, t) that a molecule that, at t = 0, was at r = 0 will, at time t, be at r. The differential cross-section is its double Fourier transform GS(A , co). It is convenient to apply the Fourier transformation in space right away to (7.4) so that both operators Ff reduce to factors,... 

In some cases, a portion of the diffusing molecules may become attached to or trapped within some sites inside the polymeric chains. Such attachment may be temporary and an exchange may occur over time between mobile and bound molecules. This phenomenon is denoted as a Two-Stage Diffusion and, in the one-dimensional case, is formulated as shown below (Carter and Kibler 1978 Gurtin and Yatomi 1979)... 

Coupling Between Diffusion and Polymeric Relaxation Two-Stage Diffusion... 

Both aramid (Kevlar) and glass fibers absorb fluids. Consequently, the overall ingress of fluids is affected by the sorption within two disparate phases, namely the fibers and the polymeric resin. Typically, the combined effect may resemble the weight gain data representative of a two-stage diffusion. 

Curves A and B accord with data collected for both polymers and composites. Of these, A corresponds to a case where weight-gain never attains equihbrium, such as for two phase diffusion, and B represents the circumstance of two-stage diffusion. Weight gain data compatible with those curves are associated with benign fluid effects that are essentially reversible upon drying. 

A different approach attributes two-stage diffusion to the combined effect of two time-dependent phenomena, namely those of diffusion time and polymeric... 

A suitable selection of O should match two-stage diffusion data. If the second stage is barely noticeable until time t = may well be that replacing Mr(0 by — fv)//(f — fv) in (5.10) would provide a yet better agreement. 

In the case of hydrophilic fiber assemblies, vapor diffusion is governed by a non-Fickian, anomalous diffusion [30]. This is a two stage diffusion process. The two stage sorption has been first expressed by Nordon and David [31], The first stage corresponds to a Fickian diffusion with a constant diffusion coefficient but the second stage is non-Fickian diffusion associated with viscoelastic relaxation of the fiber stmcture [32],... 

Berthoud (1912) proposed a model in which growth is divided into two stages, i.e. diffusion through the bulk up to the interface... 

Dissolution of polymers is a very slow process it can take days or even weeks for particularly high relative molar mass substances. Two stages are discernible during the process of dissolution. Firstly, a swollen gel is produced by solvent molecules gradually diffusing into the polymer. Secondly, this gel gradually disintegrates as yet more solvent enters the... 

Extension of the hydrodynamic theory to explain the variation of detonation velocity with cartridge diameter takes place in two stages. First, the structure of the reaction zone is studied to allow for the fact that the chemical reaction takes place in a finite time secondly, the effect of lateral losses on these reactions is studied. A simplified case neglecting the effects of heat conduction or diffusion and of viscosity is shown in Fig. 2.5. The Rankine-Hugoniot curves for the unreacted explosive and for the detonation products are shown, together with the Raleigh line. In the reaction zone the explosive is suddenly compressed from its initial state at... 

The above kinetic scheme of the bimolecular reaction simplifies physical processes that proceed via the elementary bimolecular act. To react, two reactants should (a) meet, (b) be oriented by the way convenient for the elementary act, and (c) be activated to form the TS and then react. Hence, not only translational but also rotational diffusion of particles in the solution and polymer are important for the reaction to be performed. So, the more detailed kinetic scheme of a bimolecular reaction includes the following stages diffusion and encounter the reactants in the cage, orientation of reactants in the cage due to rotational diffusion, and activation of reactants followed by reaction [5,13]. 

In the case of operations ranging between 760 and 10 3 torr, the two-stage rotary pump is adequate. Ultimate vacuum, using the diffusion pump installation, is better than 10-6 torr. 

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