Relaxation-controlled transport

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. 

Penetrant Concentration-Plasticization Polymer Molecular Structure Relaxation-Controlled Transport Applications of Transport Concepts Barrier Materials Devolatilization Additive Migration Dyeing... 

HB Hopfenberg, L Nicolais, E Drioli. Relaxation controlled (case II) transport of lower alcohols in poly(methyl methacrylate). Polymer 17 195-198, 1976. 

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. 

Although blood pressure control follows Ohm s law and seems to be simple, it underlies a complex circuit of interrelated systems. Hence, numerous physiologic systems that have pleiotropic effects and interact in complex fashion have been found to modulate blood pressure. Because of their number and complexity it is beyond the scope of the current account to cover all mechanisms and feedback circuits involved in blood pressure control. Rather, an overview of the clinically most relevant ones is presented. These systems include the heart, the blood vessels, the extracellular volume, the kidneys, the nervous system, a variety of humoral factors, and molecular events at the cellular level. They are intertwined to maintain adequate tissue perfusion and nutrition. Normal blood pressure control can be related to cardiac output and the total peripheral resistance. The stroke volume and the heart rate determine cardiac output. Each cycle of cardiac contraction propels a bolus of about 70 ml blood into the systemic arterial system. As one example of the interaction of these multiple systems, the stroke volume is dependent in part on intravascular volume regulated by the kidneys as well as on myocardial contractility. The latter is, in turn, a complex function involving sympathetic and parasympathetic control of heart rate intrinsic activity of the cardiac conduction system complex membrane transport and cellular events requiring influx of calcium, which lead to myocardial fibre shortening and relaxation and affects the humoral substances (e.g., catecholamines) in stimulation heart rate and myocardial fibre tension. 

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... 

Other parameters of the simulation are specified in subroutine SPECS. The quantity solcon is the solar constant, available here for tuning within observational limits of uncertainty. The quantity diffc is the heat transport coefficient, a freely tunable parameter. The quantity odhc is the depth in the ocean to which the seasonal temperature variation penetrates. In this annual average simulation, it simply controls how rapidly the temperature relaxes into a steady-state value. In the seasonal calculations carried out later in this chapter it controls the amplitude of the seasonal oscillation of temperature. The quantity hcrat is the amount by which ocean heat capacity is divided to get the much smaller effective heat capacity of the land. The quantity hcconst converts the heat exchange depth of the ocean into the appropriate units for calculations in terms of watts per square meter. The quantity secpy is the number of seconds in a year. 

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... 

In electrode kinetics, however, the charge transfer rate coefficient can be externally varied over many orders of magnitude through the electrode potential and kd can be controlled by means of hydrodynamic electrodes so separation of /eapp and kd can be achieved. Experiments under high mass transport rate at electrodes are the analogous to relaxation methods such as the stop flow method for the study of reactions in solution. 

Pressure drop. Good control requires a substantial pressure drop through the valve. For pumped systems, the drop through the valve should be at least 1/3 of the pressure drop in the system, with a minimum of IS psi. When the expected variation in flow is small, this rule can be relaxed. In long liquid transportation lines, for instance, a fully open control valve may absorb less than 1% of the system pressure drop. In systems with centrifugal pumps, the variation of head with capacity must be taken into account when sizing the valve. Example 7.2, for instance, illustrates how the valve drop may vary with flow in such a system. 

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