Berens-Hopfenberg model

EXTRACT and O-METHYLATED EXTRACT. The sorption of benzene by the extract and the O-methylated extract is characterized by a rapid, initial uptake followed by a very slow approach to equilibrium. Such sorption behavior is very similar to that of glassy polymers. Thus we have chosen to interpret the sorption curves shown in Figures 2 and 3 in terms of the Berens-Hopfenberg model developed for the sorption of organic vapors into glassy polymers.(lS) By doing so, we attempt to correct the total sorption values for surface adsorption in order to calculate x parameters. 

Finally, we note that the x and solubility parameters of the O-butylated extract are noticeably absent in Table VI. We have attempted to analyze the sorption kinetics according to the Berens-Hopfenberg model in order to correct for adsorption effects, but the treatment yielded unreasonable x parameters. The reason for this is not clear, but we believe it may be due to the fact that di sion into the extract is so rapid. Hole-filling and swelling may have comparable rates so that a separation of the two processes is not possible. 

THE BERENS-HQPFENBERG MODEL. The Berens and Hopfenberg model considers the sorption process in glassy polymers as a linear superposition of independent contributions of a rapid Fickian diffusion into pre-existing holes or vacancies (adsorption) and a slower relaxation of the polymeric network (swelling).(lS) The total amount of sorption per unit weight of polymer may be expressed as... 

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. 

Another semi-empirical diffusion model has been developed (Limm and Hollifield, 1995b) which is based on the simplification of existing diffusion theories by Pace and Datyner (1979) and the generalized trend found by Berens and Hopfenberg (1982) for the diffusion of small molecules in polymers (Chapter 5). From these studies it is possible to arrive at the following relationship ... 

Here m is the ratio of uptake due to relaxation relative to the total uptake, M( ). The quantities fD and fDR are, respectively, the series solution for Fick s second law and a related series which includes the coupling constant. For 1, the behavior reduces to classical Fickian diffusion. For 1, that is relaxation very slow compared to diffusion, the equation reduces to the sum of an independent Fickian and a first order relaxation process. This simple model was originally proposed and widely used by Berens and Hopfenberg (10). 

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