^{1,a)}, Henrik Koblitz Rasmussen

^{2}and Ole Hassager

^{3}

### Abstract

The stress in the startup of uniaxial elongational flow until steady state, followed by stress relaxation, has been measured for a narrow molar mass distribution polystyrene melt with a molecular weight of . The experiments are conducted on a filament stretching rheometer, where a closed loop control of the midfilament diameter ensures controlled uniaxial extension. The closed loop control algorithm is extended to apply to the stress relaxation part of the experiment. It ensures a constant midfilament diameter, by controlling the motion of the end plates. By dividing the measured stress with the theoretically predicted stress from the Doi and Edwards model during relaxation, the stretch factors corresponding to each imposed stretch rate are obtained. These stretch factors converge to a unique envelope and eventually converge to unity for long times for all measured elongational rates.

The authors gratefully acknowledge the financial support of the Graduate School of Polymer Science from the Danish Research Training Council, and the Danish Technical Research Council to the Danish Polymer Center.

I. INTRODUCTION

II. MATERIAL

A. Mechanical spectroscopy and linear viscoelasticity

III. ELONGATIONAL TECHNIQUE

IV. REPTATION-BASED CONSTITUTIVE MODELS

V. ELONGATIONAL MEASUREMENTS

VI. STRESS RELAXATION

VII. CONCLUSION

### Key Topics

- Stress relaxation
- 35.0
- Viscosity
- 12.0
- Polymers
- 8.0
- Relaxation times
- 8.0
- Polymer melts
- 7.0

## Figures

Measured loss, (open circles; ) and storage moduli, (bullets; ) both as a function of the angular frequency, . The data are obtained from small angle oscillatory shear rheometry. The experiments have been performed at 120, 130, and , and shifted to a reference temperature of . The solid lines (—–) are the least-squares fitting to the BSW model in Eq. (2).

Measured loss, (open circles; ) and storage moduli, (bullets; ) both as a function of the angular frequency, . The data are obtained from small angle oscillatory shear rheometry. The experiments have been performed at 120, 130, and , and shifted to a reference temperature of . The solid lines (—–) are the least-squares fitting to the BSW model in Eq. (2).

Evolution of filament diameter, , and plate separation, , for an elongational rate of at , stretched for and then relaxed. During the relaxation, the diameter is kept constant by the closed loop controller in the experiment in the left figure, and the distance between the end plates is kept constant in the experiment in the figure to the right. The initial filament diameter and the initial plate separation for both the controlled and uncontrolled experiments are: and .

Evolution of filament diameter, , and plate separation, , for an elongational rate of at , stretched for and then relaxed. During the relaxation, the diameter is kept constant by the closed loop controller in the experiment in the left figure, and the distance between the end plates is kept constant in the experiment in the figure to the right. The initial filament diameter and the initial plate separation for both the controlled and uncontrolled experiments are: and .

Quenched polystyrene filaments after cessation of flow. The elongational rate at startup is , the strain at relaxation is , and the temperature in all performed experiments. The samples are relaxed, respectively ( ), before the quenching. The ruler to the right shows the length in millimeters.

Quenched polystyrene filaments after cessation of flow. The elongational rate at startup is , the strain at relaxation is , and the temperature in all performed experiments. The samples are relaxed, respectively ( ), before the quenching. The ruler to the right shows the length in millimeters.

Measured stress (at ) as a function of time at startup and relaxation of elongational flow, as in Fig. 2. The diameter is kept constant by the closed loop controller in one of the experiments . In the two other curves is the plate separation, , stopped at . The boxes are the calculation of the true stress , where the open circles curve are calculated with the assumption of a constant midfilament diameter during the relaxation as .

Measured stress (at ) as a function of time at startup and relaxation of elongational flow, as in Fig. 2. The diameter is kept constant by the closed loop controller in one of the experiments . In the two other curves is the plate separation, , stopped at . The boxes are the calculation of the true stress , where the open circles curve are calculated with the assumption of a constant midfilament diameter during the relaxation as .

Figure (a) shows the measured startup and relaxation viscosity performed at an elongational rate of . In one experiment the filament is stretched for and then relaxed (pluses, ), and in the other experiment the filament is stretched for and then relaxed, (boxes, ). The two dotted lines (- - -) are the linear viscoelastic prediction for startup and relaxation after . Figure (b) is the evolution of filament diameter, , and plate separation, , for the experiment with elongational rate of at , stretched for and then relaxed. The diameter is kept constant by the closed loop controller in this experiment.

Figure (a) shows the measured startup and relaxation viscosity performed at an elongational rate of . In one experiment the filament is stretched for and then relaxed (pluses, ), and in the other experiment the filament is stretched for and then relaxed, (boxes, ). The two dotted lines (- - -) are the linear viscoelastic prediction for startup and relaxation after . Figure (b) is the evolution of filament diameter, , and plate separation, , for the experiment with elongational rate of at , stretched for and then relaxed. The diameter is kept constant by the closed loop controller in this experiment.

The measured corrected startup and relaxation viscosity performed at rates of at . In all cases the flow is stopped at an extension of and allowed to relax for , . The solid lines (——) are the Doi–Edwards predictions of the transient elongational viscosity, Eq. (8) with . The dashed lines (- - - -) are the MSF model prediction from Eq. (10) with a tube diameter relaxation time of .

The measured corrected startup and relaxation viscosity performed at rates of at . In all cases the flow is stopped at an extension of and allowed to relax for , . The solid lines (——) are the Doi–Edwards predictions of the transient elongational viscosity, Eq. (8) with . The dashed lines (- - - -) are the MSF model prediction from Eq. (10) with a tube diameter relaxation time of .

The steady stress divided with the plateau modulus against the Marrucci-Deborah number for , , , and at . The value of is at . is the zero-shear viscosity.

The steady stress divided with the plateau modulus against the Marrucci-Deborah number for , , , and at . The value of is at . is the zero-shear viscosity.

The stretch, , calculated using the measured relaxation viscosity at in Fig. 6, as a function of the time, from the start of the stress relaxation. The stretch is found using Eq. (11) based on the independent alignment approximation, Eq. (8). The solid lines are the stretch calculated using Eq. (13), with a tube diameter relaxation time and stretch of the fully extended molecule of .

The stretch, , calculated using the measured relaxation viscosity at in Fig. 6, as a function of the time, from the start of the stress relaxation. The stretch is found using Eq. (11) based on the independent alignment approximation, Eq. (8). The solid lines are the stretch calculated using Eq. (13), with a tube diameter relaxation time and stretch of the fully extended molecule of .

## Tables

Linear viscoelastic and molecular weight properties of the NMMD polystyrene melts at . The and constants in the BSW model are obtained from Jackson and Winter (1995) and from Bach *et al.* (2003a). The two parameters and are the time constants for specific models for the nonlinear relaxation of stretch.

Linear viscoelastic and molecular weight properties of the NMMD polystyrene melts at . The and constants in the BSW model are obtained from Jackson and Winter (1995) and from Bach *et al.* (2003a). The two parameters and are the time constants for specific models for the nonlinear relaxation of stretch.

At Prestretch values.

At Prestretch values.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content