^{1,a)}, Sham Ravindranath

^{1}, Yangyang Wang

^{1}and Pouyan Boukany

^{1}

### Abstract

Recent experimental evidence has motivated us to present a set of new theoretical considerations and to provide a rationale for interpreting the intriguing flow phenomena observed in entangled polymer solutions and melts [P. Tapadia and S. Q. Wang, Phys. Rev. Lett.96, 016001 (2006);96, 196001 (2006);S. Q. Wang *et al.*, *ibid.*97, 187801 (2006)]. Three forces have been recognized to play important roles in controlling the response of a strained entanglement network. During flow, an intermolecular locking force arises and causes conformational deformation in each load-bearing strand between entanglements. The chain deformation builds up a retractive force within each strand. Chain entanglement prevails in quiescence because a given chain prefers to stay interpenetrating into other chains within its pervaded volume so as to enjoy maximum conformational entropy. Since each strand of length has entropy equal to , the disentanglement criterion is given by in the case of interrupted deformation. This condition identifies as a cohesive force. Imbalance among these forces causes elastic breakdown of the entanglement network. For example, an entangled polymer yields during continuous deformation when the declining cannot sustain the elevated . This opposite trend of the two forces is at the core of the physics governing a “cohesive” breakdown at the yield point (i.e., the stress overshoot) in startup flow. Identifying the yield point as the point of force imbalance, we can also rationalize the recently observed striking scaling behavior associated with the yield point in continuous deformation of both shear and extension.

The current version represents great improvement over a previous one as a result of addressing a comment by P. G. de Gennes on an earlier draft of this manuscript about whether chain entanglement really provides a cohesive force that holds up an entangled polymer as a solid on time scales much shorter than the quiescent terminal relaxation time. One of the authors (S.-Q.W) also acknowledges useful conversations with K. Schweizer, A. Gent, and J. Douglas. This work is supported, in part, by a Small Grant for Exploratory Research of the National Science Foundation (DMR-0603951) and an ACS grant (PRF No. 40596-AC7). This paper is dedicated to the memory of Pierre-Gilles de Gennes.

I. INTRODUCTION

II. ASSESSMENT ON CURRENT THEORETICAL UNDERSTANDING OF LINEAR VISCOELASTIC BEHAVIOR

A. Single-chain versus interactive/colletive phases in “terminal” regime

B. Various models for chain entanglement

1. Onset entanglement molecular weight determined from linear viscoelasticity

2. Chain entanglement models

C. Stress due to bonded versus nonbonded forces

III. NEW EXPERIMENTAL OBSERVATIONS CONCERNING NONLINEAR BEHAVIOR

A. Cohesive collapse after step shear and step extension

B. Yielding during continuous deformation

IV. NEW THEORETICAL PICTURE IN NONLINEAR POLYMER RHEOLOGY: CHAIN DISENTANGLEMENT, FORCE IMBALANCE, AND ELASTIC BREAKUP

A. Step shear: Overcoming entanglement cohesion

1. Retraction force

2. Entanglement network in quiescence

3. Chain disentanglement after large step strain

B. Yield behavior in continuous deformation: Stress overshoot and force imbalance

1. Intermolecular locking force

2. Imbalance of forces at yield point

C. Scaling behavior in elastic deformation regime: or

D. Scaling behavior in viscoelastic deformation regime: or

E. Terminal flow regime: or

V. CONCLUSIONS

### Key Topics

- Polymers
- 58.0
- Elasticity
- 40.0
- Retraction
- 24.0
- Shear deformation
- 24.0
- Polymer flows
- 23.0

## Figures

(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 .

(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 .

(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.

(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.

(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 .

(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 .

(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 .

(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 .

(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 .

(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 .

(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.

(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.

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

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

(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.

(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.

(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 .

(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 .

(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.

(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.

(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 .

(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 .

(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.

(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.

(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.

(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.

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

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

Article metrics loading...

Full text loading...

Commenting has been disabled for this content