Tubes theories

As we conjectured in the introduction, the fundamental role of topology in this approach to entangled polymer dynamics would indicate that changes to the topology of the molecules themselves would radically affect the dynamic response of the melts. In fact rheological data on monodisperse star-branched polymers, in which a number of anionically-polymerised arms are coupled by a multifunctional core molecule, pre-dated the first application of tube theory in the presence of branching [22]. Just the addition of one branch point per molecule has a remarkable effect, as may be seen by comparing the dissipative moduli of comparable linear and star polymer melts in Fig. 5. 

Anderson and Mantzouranis (A5), 1960 Experimental and theoretical study of hold-up and film thicknesses for upward cocurrent gas/film flow in vertical tubes. Theory based on use of universal velocity profile. Numerous experimental data on friction factors, etc. 

In operation, gas flow is converted directly into thermal energy which is used in the initiation process. While the resonance tube theory for the generation of thermal energy is not new, its application in fluidic systems is unique. The device is 0.825 inch long, 0.250 inch in diam and weighs 0.125 ounce Refs 1) EMX Engineering, Inc, 354 Newark-Pompton Turnpike, Wayne, NJ 07470 2) G. Cohn, Edit, Expls Pyrots 7(4), (1.974)... 

Complex Architectures. Perhaps the most significant recent advances in molecular understanding of polymer melts have emerged from the study of branched polymer architectures. We have noted above how a tube theory for star-polymers provided the means to treat fluctuations in entangled path length in linear polymers (see Figure lb). This is simply due to the complete suppression of reptation in star polymers without fluctuation there is no stress-relaxation at all ... 

More recently Morse produced a complete microscopic tube theory for stiff polymers that successfully interpolates between the rigid-rod and flexible chain limits. This theory explains many features of semiflexible polymer rheology, including the two mechanisms for plateau moduli described above (which depend on a comparison of timescales), with the tube diameter being the sole fitting parameter as in the Doi-Edwards theory. More recently, Morse successfully computed a tube diameter from two different approaches (self-consistent binary collision and continuum effective medium) that give similar results, e.g. modulus G p and respectively). An elastic network approximation... 

More recently Morse produced a complete microscopic tube theory for stiff polymers that successfully interpolates between the rigid-rod and flexible chain... 

McLeish, T. C. B. 2002. Tube theory of entangled polymer dynamics. Adv. Phys. 51 1379-1527. 

Theoretical treatment of the statistical properties of linear polymers (LPs), shown schematically in Figure 7.1a, have been in existence since the 1950s (Rouse, 1953), and have undergone continuous refinement. In particular, tube theories (viz. reptation) (de Gennes, 1971 Doi and Edwards, 1986), and subsequent refinements, like constraint release (Viovy et al., 1991) and contour length fluctuation (Frischknecht and Milner, 2000) are some of the greatest... 

Shanbhag, S., Park, S. J., Zhou, Q., and Larson, R. G., 2007. Implications of microscopic simulations of polymer melts for mean-field tube theories. Mol. Phys., 105(2) 249-260. 

Kolupada, S. (1960). John W. Ledoux. Journal of the Hydraulics Division ASCE 86(HY1 42. P Ledoux, J.W. (1913).A mechanism for metering and recording the flow of fluids through Venturi tubes, orifices, or conduits, by integrating velocity head. Trans. ASCE 76 1148-1171. Ledoux, J.W. (1914). The Pitot tube theory. Journal AWWA 1(3) 536-537. 

One may calculate the rate of vapor generation on the basis of Fig. 5. The pressure at the contact surface is Pv = Pi + Ap2. If the saturated vapor is assumed to obey the ideal gas law, one has p = Pv/Rv vt where Ry is the gas constant of the vapor and Ty is the temperature of the saturated vapor at pressure Py. According to the shock tube theory, the flow velocity of a vapor/air mixture is given by the following expression ... 

In the next subsection, we shall discuss the most popular model for entangled polymers called a tube theory (we shall argue that it is not a proper model in a sense we defined in the beginning). After that, we shall discuss two alternative representations of entanglements. At the end, we shall define some new observables that highlight the character of the chain motion in entangled systems. 

One particular difficulty in analyzing the results of the tube theory is that often papers in this area do not start by spetdlying the microscopic equations of motion at all, but rather write down the results for particular physical observables and give... 

An additional difficulty arises when one tries to define the stress in tube models. Indeed, since the forces were not defined in the equation of motion, one carmot use the miaoscopic stress definition (eqn [ 1 ]). Thus, one has to make an additional assumption about the stress tensor expression as a function of the tube coordinates. The tube theory routinely uses an expression... 

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