Tubes models

In the semidilute region, the dominant interaction is caused by the topological constraint that the polymers cannot cross each other. This effect can be treated by the same model as that for flexible polymers. In the situation shown in Fig. 9.1b, the motion along a polymer is almost 

Let us now study how the tube constraint affects the translational and the rotational Brownian motion. 

The main assumptions made in the theory of the reptation theory are as follows (3). 

The primitive chain has a constant contour length L, so fluctuations of the contour length are neglected. 

The primitive chain reptates along itself with a diffusion constant that can be identified as the diffusion coefficient of the Rouse model. Under the action of a force /, the velocity of the polymer in the tube is v =f /, where is the overall friction coefficient of the chain. It is expected that C is related to the friction coefficient of the individual segments, Q, by the expression 

The conformation of the primitive chain becomes Gaussian on a large length scale. This means that if the position of two points on the primitive chain are r(, t) and r(s, t), where 5 and s are the contour lengths measured from the chain end, then 

The three parameters necessary for the characterization of the primitive chain, L, D, and a can be expressed in terms of the Rouse model parameters N, b, and Thus D is given by Eq. (11.14), while La is equal to Nb, the mean square end-to-end distance of the Rouse chain. As a result, the length of the primitive chain can be written as 

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Our company is dedicated solely to metal-ceramic X-ray tubes since 25 years over this time, we have made lots of different tube models especially for tyre inspection systems. The major reasons for the use of metal-ceramic tubes in this inspection technology are robustness, their small and individual shapes, and the frequent need for modifications of their design due to custom designed systems. 

Figure 2-124. The most common molecular graphic representations of biological molecules (lysozyme) a) balls and sticks b) backbone c) cartoon (including the cylinder, ribbon, and tube model) and of inorganic molecules (YBajCujO , d) polyhedral (left) and the same molecule with balls and sticks (right),...

FIGURE 1 6 Molecular models of methane (CH4) (a) Framework (tube) models show the bonds connecting the atoms but not the atoms themselves (b) Ball and stick (ball and spoke) models show the atoms as balls and the bonds as rods (c) Space filling models portray overall molecular size the radius of each sphere approximates the van der Waals radius of the atom (d) An electrostatic potential map of methane... 

FIGURE 113 (a) The framework of bonds shown in the tube model of benzene are cr bonds (b) Each carbon is sp hybridized and has a 2p orbital perpendicular to the cr framework Overlap of the 2p orbitals generates a tt system encompass mg the entire ring (c) Electrostatic potential map of benzene The red area in the center corresponds to the region above and below the plane of the ring where the tt electrons are concentrated... 

Size density surface (top left) space filling model (top right) potential map (bottom left) and tube model (bottom right) for methanol... 

Fig. 4. Laboratory tubing model for supercoiling in closed-circular DNA (a) relaxed DNA, ALK = 0 (b) ALK = ATw -...

The Ball and Wire model is identical to the Wire model, exeept that atom positions are represented by small spheres. This makes it possible to identify all atom locations in all molecules. The Tube model is identical to the Wire model, except that bonds, whether single, double or triple, are represented by single colored tubes. The tubes are useful because they better eonvey the three-dimensional shape of a molecule. The Ball and Spoke model is a variation on the Ibbe model atom positions are represented by colored spheres, making it possible to see all atom locations in all molecules. 

Plug flow A simple convective flow pattern in pipes and tubes that is characterized by a fluid velocity independent of radial position, complete mixing in the radial direction, and no mixing in the axial direction. Also called the parallel tube model or tubular flow. See Eqs. (7) and (8) and Figure 3. 

KS Pang, M Rowland. Hepatic clearance of drugs. I. Theoretical considerations of a well-stirred model and a parallel tube model. Influence of hepatic blood flow, plasma and blood cell binding, and the hepatocellular enzymatic activity on hepatic drug clearance. J Pharmacokin Biopharm 5/6 625-653, 1977. 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

Figure 16 Tube model for reptation of a branched polymer molecule from the work of Blackwell et al. [124]. Reproduced with permission from Blackwell et al. [124]. Copyright 2000, The Society of Rheology, Inc.

Kaiampokis, A., Argyrakis, P., Macheras, P., A heterogeneous tube model of intestinal drug absorption based on probabilistic concepts, Pharm. Res. 1999, 16, 1764-1769. 

Parallel tube model (i.e. complete radial mixing model)]... 

Hydrodynamics Well-stirred model, i.e. uniform concentration inside intestine Parallel tube model, i.e. concentration decrease exponentially down the length of the intestine... 

The intestinal permeability may be determined from the rate of drug appearance in mesenteric blood (i.e. dM/dt) at steady state, using Eq. 2.12. Estimating the term C[ en will again depend on the flow dynamics of the model chosen. The most commonly used experimental procedure is the single-pass perfusion (i.e. parallel tube model) and the luminal concentration can be estimated using the logarithmic mean of inlet and outlet concentrations (i.e. ). 

Keywords. Viscoelasticity, Molecular rheology. Branched polymers. Tube model, Non-Newtonian flow... 

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