An arm relaxes on two other arms and , both of which might be connected to other relaxing networks. After relaxes completely, it is replaced by a drag point with a friction , represented as a shaded sphere. Shaded rectangles represent arbitrary networks of connected arms.
Collapsed branch point offers an extra friction . The backbone is modeled as a collection of entropic springs of length connected at the branch points. The th branch point from the free end is still immobile due to unrelaxed side arm.
Schematic behavior of the three concentration functions used: At and , the true unrelaxed fraction (circles) decreases faster than the Rouse relaxation. (dashed line) is held constant at these times until the supertube relaxed fraction (solid line) becomes smaller than . In absence of this constraint-release Rouse events, the three concentration functions are the same.
Data structure used in the computation. (a) Data type arm contains pointers L1, L2, R1, R2 which can be connected to other members of the same data type. Up and down provide pointers to create threaded tree for easy accessibility of all arms belonging to a particular polymer. Type arm holds a number of other variables, some of which are indicated in the figure. They have been explained in the text. (b) Schematic representation of an H polymer. (c) The connectivity of the pointers to represent the same H polymer with data type arm.
Schematic representation of the different arm lengths used. The free-end backbone with arm-length has retracted by an amount (represented by the broken line). At some earlier time, side branches and have collapsed (closed loops). The current time corresponds to the Rouse relaxation time of a segment of length , when the effect of collapsed side branch is mapped to 1D Kramers’ problem. grows linearly in time until it reaches the next branch point position . At that point, the contribution of and the inner branches start contributing. The shaded rectangle represents an arbitrary network of arms.
Different scenarios at arm collapse. Empty loops indicate arms collapsing at the current time, filled loops indicate arms that collapsed at earlier times (ghost arms), and empty rectangles represent arbitrary networks of arms. Wavy, solid, and dashed lines, respectively, represent relaxing, constrained, and arbitrary uncollapsed arms. Filled squares represent friction from collapsed sidearms. In subfigures (d) and (e) the dotted lines show the extent of for a compound arm. Full details are in the text.
Algorithm for calculating the unrelaxed fraction during relaxation between times and , taking care of supertube relaxation.
Flow chart of the algorithm used in the calculation.
and for polyisoprene (asymmetric) star (a), linear (a) and H polymers (b),(c). Circles (triangles) are the experimental . Solid (dashed) lines are from our calculations. For clarity, each subsequent material presented in (a) has been given an additional vertical shift by a factor of 10 compared to the previous material.
and for polyisoprene star-linear blend: (a) Pure star and linear; (b) 30% star; (c) 50% star; and (d) 70% star content in the blend.
and for Polybutadiene combs.
and for metallocene catalyzed polyethylene resins. Circles (solid lines) and triangles (dashed lines), respectively, are experimental (theoretical) values for and . For clarity, the data on HDB2, HDB4, HDB6, and HDL1 have been shifted by a factor of 10 vertically.
Effect of branching and molecular weight on viscosity enhancement compared to linear chains of the same molecular weights as the branched molecules. for (a) and (b) as a function of . (c) Dependence of on at fixed . Experimental resins HDB7, HDB4, and HDB5 are shown with filled diamonds. (d) Number average segmental molecular weight as a function of for .
Effect of branch-on-branch architecture on rheological properties of metallocene-catalyzed polyethylene resins. Number fraction (a) and mass fraction (b) of the molecules which cannot be reduced as a generalized comb molecules as a function of branching probability . Symbols represent actual numbers found in our calculations, while the lines represent estimate from analytical theory. (c) Effect of neglecting branch-on-branch architectures on and . The symbols are experimental data points; the solid (dashed) lines are results from our calculations including (excluding) branch-on-branch architectures. (d) The ratio of zero-shear viscosity for calculations including and excluding branch-on-branch architectures with the molecular weight and branching probability corresponding to the experimental resins (HDB1–HDB7).
Molecular characterization of polyisoprene polymers.
Molecular characterization of polybutadiene comb polymers.
Molecular characteristics of metallocene-catalyzed polyethylene resins.
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