Chain contour

Effective molecular mass between crosslinks tvtc/kg mol 1 Tensile yield strength cry/MPa Energy release rate Gic/Jm 2 Half crack opening displacement w = 6/2 = Gic/2cry w/pm Chain contour length (Eq. 7.9) lc/nm... 

Of course, the network strands cannot be stretched completely. Stretching ratios of 1.4 for PC [31, 90] and of 1.3 for epoxy polymers [37] have been reported. The chain contour length of the strands is an appropriate measure for a simple estimation of the number of strands that are stretched across the deformation zone. The chain contour length of the strands is assumed to be proportional to... 

Small angle X-ray scattering Rg, chain contour length, L solution conformation and flexibility If M is also known, can provide mass per unit length Ml [5]... 

The chain dimension in the height direction was evaluated as the thickness of the brush layer, I, relative to the chain contour length, io, by atomic force microscopy (AFM). Figure 4.10 shows the solvent dependence of the conformation of the PMMA brush. Whereas the brush chain changes its conformation in response to the solvent quality at the low graft density, the high-density PMMA brush does not show... 

Early-time motion, for segments s such that UgM(s)activated exploration of the original tube by the free end. In the absence of topological constraints along the contour, the end monomer moves by the classical non-Fickian diffusion of a Rouse chain, with spatial displacement f, but confined to the single dimension of the chain contour variable s. We therefore expect the early-time result for r(s) to scale as s. When all prefactors are calculated from the Rouse model [2] for Gaussian chains with local friction we find the form... 

Average projection of the end-to-end vector on the tangent to the chain contour at a chain end in the limit of infinite chain length. 

In the second half of this article, we discuss dynamic properties of stiff-chain liquid-crystalline polymers in solution. If the position and orientation of a stiff or semiflexible chain in a solution is specified by its center of mass and end-to-end vector, respectively, the translational and rotational motions of the whole chain can be described in terms of the time-dependent single-particle distribution function f(r, a t), where r and a are the position vector of the center of mass and the unit vector parallel to the end-to-end vector of the chain, respectively, and t is time, (a should be distinguished from the unit tangent vector to the chain contour appearing in the previous sections, except for rodlike polymers.) Since this distribution function cannot describe internal motions of the chain, our discussion below is restricted to such global chain dynamics as translational and rotational diffusion and zero-shear viscosity. 

Necklace models represent the chain as a connected sequence ctf segments, preserving in some sense the correlation between the spatial relationships among segments and their positions along the chain contour. Simplified versions laid the basis for the kinetic theory of rubber elasticity and were used to evaluate configurational entropy in concentrated polymer solutions. A refined version, the rotational isomeric model, is used to calculate the equilibrium configurational... 

The beads represent entanglement sites which are distributed uniformly along the chain contour the frictional coefficients increase rapidly with distance from the chain ends. The spring constant also depends on contour position, being governed by the mean equilibrium distance of that position from the center of gravity. The resulting spectrum is narrower than the Rouse spectrum, and for E > 1 ... 

This refinement of the constrained-junction model is based on re-examination of the constraint problem and evaluation of some neutron-scattering estimates of actual junction fluctuations [158, 159]. It was concluded that the suppression of the fluctuations was over-estimated in the theory, presumably because the entire effect of the inter-chain interactions was arbitrarily placed on the junctions. The theory was therefore revised to make it more realistic by placing the effects of the constraints along the network-chain contours, specifically at their mass centers [4, 160, 161]. This is illustrated in the second portion of Figure 2. Relocating the constraints in this more realistic way provided improved agreement between theory and experiment. 

The first phenomenological model for description of polymer dynamics in concentrated solutions and melts was proposed in 1971 by P.de Gennes [50]. In this classical work, it was assumed that due to entanglements, the chain motions in the direction normal to the chain contour are blocked up and only tangential ones are possible. This kind of chain motion in the effective tube was called reptation. In the absence of external fields, the chain can escape from the tube by either of the free tube ends. 

The trial chain has coordinates C = X2,X3... Xn,X after the chain units are shifted along the chain contour. This configuration is accepted with probability equal to min(1,exp(-(SE r (Ulj(Xj-X1) - Ulj(Xj-Xi))) where ULj(r) is non-bonding potential (Eq.l) of two beads separated by a distance r. If the move is accepted then the new chain configuration becomes C. Otherwise it remains at C. In any case, the next chain is considered and the selection processes repeated. However, the rejected move is still considered as a configuration in subsequent averaging. 

The cohesive surface description presented here has some similarities to the thermal decohesion model of Leevers [56], which is based on a modified strip model to account for thermal effects, but a constant craze stress is assumed. Leevers focuses on dynamic fracture. The thermal decohesion model assumes that heat generated during the widening of the strip diffuses into the surrounding bulk and that decohesion happens when the melt temperature is reached over a critical length. This critical length is identified as the molecular chain contour. 

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