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New theoretical considerations in polymer rheology: Elastic breakdown of chain entanglement network
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View: Figures


Image of FIG. 1.
FIG. 1.

(Color online) (a) Step shear at three amplitudes of to , where “step” means a sudden imposition occurring over a very short time . (b) The Doi-Edwards theory prediction, where the key feature is the kink, i.e., the rapid drop of the shear stress at a time designated by , which is on the order of the Rouse relaxation time according to the theory. Beyond , the time dependence is universal independent of the step amplitude. The quiescent terminal relaxation time, also known as the reptation time for monodisperse entangled polymers, is denoted by .

Image of FIG. 2.
FIG. 2.

(Color online) (a) Shear stress relaxation of a monodisperse 15% PBD solution after strains of 0.15 and 3.0, where the number of entanglements per chain, , is around 66. (b) Shear stress relaxation of a monodisperse SBR melt after strains of 0.15 and 3.0, where the number of entanglements per chain, , is around 76. (c) Step strain extension experiments involving amplitudes of , 1.0, and 1.4. The specimen remains uniform, i.e., intact for . However, higher amplitudes of step strain produce samples that are only temporarily stable. A subsequent breakup causes a sharp decline in the measured force.

Image of FIG. 3.
FIG. 3.

(Color online) Shear stress growth as a function of time for various applied shear rates for the same 15% PBD solution, as described in Fig. 2(a), where the numbers indicate the shear rates in the unit of .

Image of FIG. 4.
FIG. 4.

(Color online) The peak shear stress as a function of the strain at the overshoot point, read from Fig. 3. The inset shows the dependence of the coordinate of the yield point on , revealing an exponent of for .

Image of FIG. 5.
FIG. 5.

(Color online) The measured tensile force resulting from five discrete continuous uniaxial stretching experiments, expressed in the form of the engineering stress , where the numbers indicates the Hencky rate of extension .

Image of FIG. 6.
FIG. 6.

(Color online) Production of retraction forces in each strand between entanglements due to chain deformation by the externally imposed strain. On the average, the retraction force within a strand is proportional to the external shear strain for affine deformation.

Image of FIG. 7.
FIG. 7.

(Color online) In quiescence, each Gaussian strand enjoys maximum conformational entropy on the order of by remaining hooked (i.e., immersed) with other chains.

Image of FIG. 8.
FIG. 8.

(Color online) For a coiled chain to change from conformation (a) to (b) would require a waiting time proportional to , typically much longer than the reptation time for . Thus, reptation dynamics are dominant in quiescence.

Image of FIG. 9.
FIG. 9.

(Color online) Depiction of sequential disappearance of entanglement points after a large imposed deformation from the chain ends (with filled dots). Depending on how many strands straighten, there will be one, two, or more (not depicted) entanglement points eradicated by the retraction force . The coils (dashed lines) at the entanglement junction(s) undergo straightening (i.e., to become the solid lines) when the available in each strand is sufficiently high such that .

Image of FIG. 10.
FIG. 10.

(Color online) Depiction of force growth (double-log scale) during continuous deformation at three rates. At the yield point, the peak stress scales with time as according to the actual experimental data from Figs. 3 and 5.

Image of FIG. 11.
FIG. 11.

(Color online) Stress growth as a function of strain on linear scale, where appears to be around unity, coinciding with the critical strain for a strained sample to break up during relaxation as discussed in Section II A. The observed linear relationship between and at the yield point introduces a cohesive modulus that is actually found (Refs. 52 and 53) to be of the same order of magnitude as the elastic plateau modulus .

Image of FIG. 12.
FIG. 12.

(Color online) Depiction of continuous uniaxial extension, where the entangled strand between the two “rings” near the chain end is shown to undergo stretching, due to at the entanglement junctions (open rings), leading to . Removal of this entanglement requires straightening the dangling strand on the right-hand side of the junction. In other words, there is a force of resisting dissolution of the entanglement point.

Image of FIG. 13.
FIG. 13.

(Color online) Force imbalance at the yield point between the monotonically growing retraction force and the declining driving force , for three different rates of deformation that introduce three different experimental time scales.

Image of FIG. 14.
FIG. 14.

Conjectured scaling behavior of the intermolecular locking force as a function of the rescaled time by so that the dimensionless terminal relaxation time is .


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: New theoretical considerations in polymer rheology: Elastic breakdown of chain entanglement network