Article pubs.acs.org/Macromolecules
Extensional Step Strain Rate Experiments on an Entangled Polymer Solution P. K. Bhattacharjee,*,† D. A. Nguyen,§ Y. Masubuchi,‡ and T. Sridhar§ †
School of Civil, Environmental and Chemical Engineering, RMIT University, Melbourne, Victoria, Australia National Composite Center, Nagoya University, Nagoya, Japan § Department of Chemical Engineering, Monash University, Clayton, Victoria, Australia ‡
S Supporting Information *
ABSTRACT: The dynamics of chain entanglement/disentanglement in concentrated polymer solutions are matters of current debate. We report the results of step strain rate experiments in uniaxial extensional flow on a well-characterized polymer solution containing about 22 entanglements per chain which allows these dynamics to be probed. In these experiments the polymer solution is subjected to homogeneous stretching at a given strain rate up until a predetermined value of strain. After this strain is reached, the strain rate is changed to a new value, and stretching is continued until steady state is acquired in the extensional stress. The strain rates are increased in step-up experiments and decreased in stepdown experiments. The strain rates are adjusted so that both the orientation and the stretching dynamics can be probed. Additionally, the predictions of two recent single-mode molecular models are evaluated against the experimental data. These models include the effects of entanglement dynamics in their predictions in an ad hoc manner. This leads to only a qualitative improvement in the predictive capacity of one of the models and reduces that of the other. Slip-link simulations using the primitive chain network model are also included for further insight into the underlying dynamics. The models are found to yield qualitative predictions of the experimental observations.
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INTRODUCTION The dynamics of entangled polymeric melts and solutions are generally considered using the framework of the Doi−Edwards model (also called the tube model).1 The model envisages that a test chain in a polymer solution or melt is confined to a tubelike region that is defined by the proximity of the neighboring chains. The lateral movement of a test chain is restricted, and following deformation the chain can only relieve stress by moving along the backbone of the tube. This original structure has since been modified to introduce several new physics. In extensional flow, the current version of the model suggests three regimes. In Regime 1, when the strain rate is less than the inverse of the longest relaxation time of the chains (ε̇ < 1/τd), the extensional viscosity ηE = 3η0, where η0 is the zeroshear rate viscosity. In Regime 2 where 1/τd < ε̇ < 1/τR, where τR = τd/3Z is the Rouse time of the chain and Z is the average number of entanglements per chain at equilibrium, ηE ∼ ε̇−0.5. In Regime 3, when ε̇ > 1/τR, the extensional viscosity rises above 3η0, until the chains are fully extended, and the extensional viscosity becomes independent of the applied strain rate in Regime 4. The level at which the extensional viscosity saturates depends on λmax and Z where λmax is the ratio to the maximum length of the polymer chain segments to its equilibrium length. A generic flow curve is shown in Figure 1. Data on polystyrene melts,2 surprisingly, did not exhibit the expected increase in Regime 3, and the extensional viscosity © XXXX American Chemical Society
Figure 1. Typical graph of Trouton ratio (ηE/η0) against ε̇ showing the different regimes.
continued to decrease. This difference between solutions and melts posed a direct challenge to the theoretical framework outlined above and indicated that some physics was missing.3 To account for this discrepancy, it was suggested that the friction coefficient is not constant but is dependent on the Kuhn segment configuration. The friction coefficient decreases when the Kuhn segments are aligned with the flow. Thus, the Received: April 20, 2016 Revised: October 14, 2016
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DOI: 10.1021/acs.macromol.6b00823 Macromolecules XXXX, XXX, XXX−XXX
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and Leq(t) is the equilibrium length of the chain under flow conditions. Leq(t) is different than Leq, where the latter is the initial equilibrium length. Additionally, the MBP model also accounts for the fact that reorientation processes will be reduced by the close packing of the highly oriented chains through the use of the term (1 − SKuhn) where SKuhn is a function of (Λ/Λmax) and Λmax is the maximum value of Λ under flow conditions. In the DCR-CS-E, however, constraint release is moderated by the stretch of the chains (see eq 1 in ref 4b), and if the deformation is faster than the relaxation of chain stretch, the rate of loss of the original tube segments can go to zero. Also in both the models the entanglements are created through a diffusive process at the chain ends. The changes in the drag affect the relaxation times and change the spacing of the relaxation times from that existing at rest. Additionally both are single mode, differential models and are expected to only give qualitative agreement with experimental data. In this paper we consider experiments where an entangled polystyrene solution is subjected to sudden changes in stretching rates (strain rates) in extensional flow. The extensional flow experiments described below invokes both orientation effects and effects due to chain stretch. The stepstrain rate experiments discussed here can provide information on transient dynamics that can be useful in evaluating the validity of the amendments proposed in the model through changes in entanglement numbers and relaxation times. Additionally performing experiments on entangled polymer solutions allows for the study of the entanglement dynamics in isolation, since the drag coefficient is known to remain unaffected in these systems under flow conditions. We expect that these experiments will be able to examine the robustness of the physics that underpin the recent modifications of the tube model framework.
idea of configuration dependent friction coefficient (CDFC) provides a possible explanation for the observed differences between polystyrene solutions and melts. Recent studies4 of polymers blended with oligomeric diluents have evaluated how the Kuhn length of the diluent affects the extensional flow behavior by changes in friction and supports the CDFC idea although data are available that document more complex behavior in Regime 3.3,5 In polymer solutions that use molecular solvents like diethyl phthalate, this anisotropy is diluted away6 due to the smaller isotropic solvent molecules. Therefore, while in the case of entangled melts the CDFC affects both the Rouse and the relaxation time associated with entanglement dynamics, in the systems used in this work the effects CDFC do not influence the dynamics as the solvent is isotropic. Even with the incorporation of CDFC, tube models are not complete. Simulations indicate that the number of entanglements changes with deformation. This paper attempts to illuminate this, without the effects of CDFC. Another idea that has recently been proposed is that the number of entanglements changes under deformation. This study focuses on the effect of entanglement/disentanglement dynamics by changes in step strain rate to specifically bring out the effects of entanglement/disentanglement. The work is in part motivated by a previous investigation conducted in steady shear flow.7 In the standard models the extensional stress typically has the form σ(t) = GS(t)λ(t)F(λ(t)) where σ is the stress, G is the modulus that depends on the number of entanglements per unit volume, S is the orientation tensor, and λ is the stretch ratio which is the ratio of the instantaneous length of the entanglement segment to the equilibrium length. The nondimensional factor F(λ) is the non-Gaussian correction to the spring tension and is a function of λ.8 Indeed, all these quantities are dependent on time t except at steady state. In the standard version of the model the number of entanglements per chain (Z) remains constant at the equilibrium value. However, simulations9 show a decrease in the number of entanglements due to deformation. The loss of entanglements results from constraint release and new entanglements are formed from a diffusive process at the chain ends. Currently there is insufficient experimental evidence to confirm the disentanglement phenomenon although several simulations have shown that the effective number of entanglements per chain can decrease in strong flows.9 In this work we isolate the effects of changes in entanglement density from that of CDFC by using solutions that do not exhibit CDFC. A model that accounts for the effects of entanglement dynamics (ED) has been proposed by Mead et al.6 (MBP), and another model which has been suggested by Ianniruberto and Marrucci10 is a version of the standard model (DCR-CS) suggested earlier8 and modified to accommodate entanglement dynamics. We call the latter model DCR-CS-E (E for entanglement dynamics). Both the models allow the dynamics of the entanglements to influence the orientation and stretch of the chains. These processes ultimately influence the relaxation time scales although the effect of entanglement dynamics on the relaxation times is more pronounced for the MBP model than for the DCR-CS-E. This is one of the features that differentiate the two models. The other key feature that distinguishes the MBP model is the use of a switch function (1/ Λ) where Λ = [L(t)/Leq(t)], which reduces the effect of constraint release on the test chain in highly oriented systems. Here L(t) is the instantaneous length of a tube contour length,
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MATERIALS AND METHODS
We have used an entangled polymer solution of nearly monodisperse (Mw/Mn = 1.01) polystyrene (Mw = 1.95 × 106 g/mol) in diethyl phthalate. The solution was prepared by dissolving the polymer (15 wt %) in diethyl phthalate in the presence of a cosolvent dichloromethane and subsequently evaporating the cosolvent out of the solution over a period of a week, until constant weight is obtained. The linear viscoelastic response of the solution was measured using a laboratory rheometer (RFS II, Rheometrics, USA) at 21 °C. The linear viscoelastic response was fitted to the predictions of the Milner− McLeish model.11 The model requires estimates of the average number of entanglements per chain (Z) and the plateau modulus (G0N). These number of entanglements per chain was estimated as Z = Mw/Mesoln where Mesoln = 13300(c/ρ)−1, where c is the concentration of the polymer in solution and ρ is the density of polystyrene. Also the plateau modulus was estimated as G0N = 0.8cRT/Mesoln, where R is the universal gas constant and T the absolute temperature. Details of these procedures are available elsewhere.12 Estimated in this way Z ≈ 22 entanglements per chain at equilibrium, and the plateau modulus (G0N) is 3815 Pa. The extensional flow experiments were conducted using the filament stretching rheometer also at 21 °C. In these experiments the sample is stretched between two end-plates moving in opposite directions in a prescribed manner. The force is measured at the top end-plate, and the midfilament diameter is monitored using a laser micrometer and high-speed camera (Phantom V5, vision Research, USA). Details of the instrument and procedures are available elsewhere.13 The instrument allows rapid ( 1. On the contrary, for the DCR-CS-E model, the predictions improve quite remarkably with the above adjustment. When these results are considered along with those in Figure 2, it appears that the current implementation of the entanglement dynamics and indeed the use of the switch function require further development in future work. To look further into entanglement dynamics, we have conducted simulations of the step-strain rate experiments using the dual-slip-link paradigm using the primitive chain network model. This simulation technique is able to capture the disentanglement process in fast flows. Details of the simulations can be found in the Supporting Information. In Figure 9 we
up experiments and the broken lines the step-down experiments. The black lines are the predictions of the MBP model, and the gray lines those of the DCR-CS-E model. For the stepup experiments, the model predictions of the stretch ratio at steady state are significantly higher compared to those noticed in Figure 5b since the value of WiR is greater than unity. For the step-down experiments, however, the stretch ratio decreases monotonically from its peak value after the imposition of the step as before, and the time taken to reach a steady-state value is roughly the same for the two types of experiments in these predictions. We also plot the predicted evolution of Z1, Z2, and N as a function of time in Figures 7c and 7d, as was done in Figure 5. The qualitative observation remains the same with the overshoot in Z2 conspicuous in both the step-up and stepdown experiments and reinforcing the need to reconsider the implementation of constrain release in the current form of the MBP model. The comparisons of the theoretical predictions provided by the models discussed above demonstrate that the predictions are only in qualitative agreement with experimental observations. However, some indication regarding the origin of the discrepancies can be obtained as follows. In Figures 8a and
Figure 8. Combined figures of step strain rate experiments conducted in the step-up and step-down modes, with the corresponding predictions of the MBP (black line) and the DCR-CS-E (gray line) models. The experimental data and the predictions are normalized by the stress at which the change in the strain rate is applied. (a) Experiments in the extension thinning region (WiR < 1) with strain rates ε̇ = 1 s−1 → 3 s−1 and ε̇ = 3 s−1 → 1 s−1. (b) Experiments in the extension thickening region (WiR > 1) with strain rates ε̇ = 3 s−1 → 10 s−1 and ε̇ = 10 s−1 → 3 s−1.
Figure 9. Steady-state extensional flow behavior of the entangled polystyrene solution (filled symbols) and the predictions of various models (lines). Simulation results are represented by unfilled circles.
replot the steady-state data in Figure 2 along with the predictions from the single-mode models and include the results of the simulations. It can be observed from Figure 9 that the simulations overpredict the experimental results, particularly in the extension thinning region, although the agreement improves in the chain-stretching region. The simulation result in the extension thinning region can be improved if the value of entanglement per chain Z is increased. However, in such a case the prediction in the chain-stretching region becomes worse due to the decrease of the maximum stretch ratio. For the effect of maximum stretch, refer to the earlier study in which reasonable agreement of the simulation with the data for entangled solutions was reported.15 Interestingly, the MBP model performs the best in the extension thinning region where the entanglement dynamics are expected to influence the flow process most. In Figures 10a−d a comparison is made between the experimental observations and the results of simulations for the step strain rate experiments. The experimental data are plotted as symbols, and the simulations are shown as lines. Figure 10a shows the results of the step-up experiments when the strain rate is changed from 1 to 3 s−1 and Figure 9b the corresponding results of the step-down experiments (3 to 1 s−1). It can be seen
8b we plot the same data shown in Figures 4 and 6 with the stress normalized by the stress (σ+E,P) at which the step was imposed. It can be observed from Figures 8a and 8b that the predictions change, particularly for the step-down experiments, indicating that the nature of the physics discussed is probably accurate. The results in Figure 8a, which discusses experiments conducted in the extension thinning region, shows that upon normalization both the models underpredict of the final steady state in stress, the DCR-CS-E more than the MBP. In stepdown experiments, however, the predictions are better for the DCR-CS-E model than for the MBP model. These results, when compared to those presented in Figures 4a and 4b, indicate that the apparent success of the MBP model in the latter figures are likely to be due to small but finite overpredictions of the initial response as it approaches the transition point. The normalization eliminates these small discrepancies, which might result from the single-mode nature of the two models and allows for a better evaluation of the model predictions. In Figure 8b the experiments in the extension thickening region is discussed. The MBP model G
DOI: 10.1021/acs.macromol.6b00823 Macromolecules XXXX, XXX, XXX−XXX
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Figure 10. Comparison of simulation results with experimental observations. Lines are predictions from the simulation and symbols are experimental observations: (a) step-up experiments (ε̇ = 1 s−1 to ε̇ = 3 s−1); (b) step-down experiments (ε̇ = 3 s−1 to ε̇ = 1 s−1); (c) stepup experiments (ε̇ = 3 s−1 to ε̇ = 10 s−1); (d) step-down experiments (ε̇ = 10 s−1 to ε̇ = 3 s−1).
that the simulations overpredict the steady-state values in both Figures 10a and 10b. Similar trends are noticed in Figures 10c and 10d as well where the simulation results are compared against observations made in experiments where the strain rate is changed from 3 s−1 to 10 s−1 (Figure 10c) and from 10 s−1 to 3 s−1 (Figure 10d). Finally, for completion, we compare the predictions of the two single-mode models with those of the simulations results in Figures 11 and 12. The results of the simulations are shown as symbols, and those of the single-mode models are shown as lines. The black line is for the MBP model, and the gray line is for the DCR-CS-E model. Figures 11a and 11b consider the comparisons with the single-mode models in step-up and stepdown modes, respectively, when the strain rates are changed between 1 and 3 s−1. It can be observed that the simulation results agree with the predictions of the single-mode models only initially. However, the steady-state predictions are higher and the rate of stress growth is also higher in the simulations compared to those predicted by the single-mode models. Figures 11c and 11d compare the predictions of the changes in the number of entanglements, and Figures 11e and 11f compare the predictions of the evolution of the stretch ratio in the step-up and step-down modes for the above experiments. It can be observed from Figures 11c and 11d that while the trends are qualitatively similar, the simulations predict a much greater and quicker decrease in the entanglements than those predicted by the single-mode models. The qualitative similarity is also noticed in the case of Figures 11e and 11f. However, in this case the degree of the predicted change is smaller for the simulations than for the single-mode models. In Figures 12a−f a similar comparison is made between the simulations and the single-mode models for the experiments at strain rates of 3 and 10 s−1 in step-up and step-down modes. Figures 12a,b compare the transient stresses, Figures 12c,d the evolution of the number of entanglements, and Figures 12e,f the evolution of the stretch ratio for the experiments
Figure 11. Comparison of simulation results with predictions from the single-model models. Lines are predictions from the single-mode models (black: MBP, gray: DCR-CS-E), and symbols are the simulation results. (a) Comparison in the step-up experiment. (b) Comparison the step-down experiment. (c) Predicted evolution of entanglements (step-up experiment). (d) Predicted evolution of entanglements (step-down experiment). (e) Predicted evolution of the stretch ratio (step-up experiment). (f) Predicted evolution of stretch ratio in the step-down experiment.
conducted. Once again the predicted behavior is qualitatively similar. However, the stresses grow to a lower magnitude in simulations, and the extents to which the entanglements reduce are larger than those predicted by the single-mode models. Also, the rates and magnitudes of changes in the stretch ratio predicted by the simulations are lower than those predicted by the single-mode models. The comparisons shown in Figures 11 and 12 demonstrate several discrepancies as well as qualitative consistency between the simulation and the models. For the stress, the growth is slower and the steady value is larger for the simulation than those for the tube models. These discrepancies can possibly be accommodated for if the unit time of the simulation is further tuned. Indeed, as mentioned in the Supporting Information, the values of τR and τd are 30% smaller for the simulation than those for the tube models. If the unit time is reduced, it causes a faster stress growth. More importantly, a reduction of unit time corresponds to a reduction of normalized flow rate that suppresses the stress. The underestimation of the unit time explains the discrepancy in the number of entanglements as well.15 In this respect, the entanglement dynamics incorporated in the tube models and that implemented in the multichain H
DOI: 10.1021/acs.macromol.6b00823 Macromolecules XXXX, XXX, XXX−XXX
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the inclusion of the effects of entanglement dynamics and configuration-dependent changes in the drag coefficient, the experimental system chosen in this work allows for the evaluation of the entanglement dynamics in isolation. We have compared the model predictions to experimental data at steady state as well as with the transients observed in stepextensional strain rate experiments. However, in the present form the inclusion of the new physics results in only a qualitative improvement of the predictive capacity of the MBP model and reduces that of the DCR-CS-E model. We have also performed slip-link simulations using the primitive chain network models to have a better understanding of the underlying dynamics. We find that in the current form the simulations overpredict experimental observations, again suggesting that entanglement dynamics is not well captured.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b00823. Evidence on reproducibility of experiments and details of slip-link simulations (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] (P.K.B.). ORCID
Y. Masubuchi: 0000-0002-1306-3823 Notes
Figure 12. Comparison of simulation results with predictions from the single-model models. Lines are predictions from the single mode models (black: MBP, gray: DCR-CS-E), and symbols are the simulation results. (a) Comparison in the step-up experiment. (b) Comparison the step-down experiment. (c) Predicted evolution of entanglements (step-up experiment). (d) Predicted evolution of entanglements (step-down experiment). (e) Predicted evolution of the stretch ratio (step-up experiment). (f) Predicted evolution of stretch ratio in the step-down experiment.
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank the referees whose comments have helped in making this paper better. REFERENCES
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simulation are fairly consistent with each other. Nevertheless, further assessment would be interesting against finer grained molecular models for which the entanglement number has been reported under fast shear.16 The reduction of stretch in the simulation is consistent with the smaller number of entanglements than the tube models. It is also noted that the force balance around entanglement reduces the local stretch as well.
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CONCLUSION We have used step-extensional strain rate experiments to investigate the response of a well-characterized entangled polymer solution. We have conducted experiments where the strain rate is increased (step-up) or decreased (step-down) after the liquid is stretched at a given strain rate. We observed that after the imposition of the step, the time taken to reach a new value of steady state decreases as the strain at which the step is applied is increased. We have conducted these experiments in the range of strain rates where WiR < 1 and also where WiR > 1. We have also evaluated the predictive capacity of two singlemode versions of the state-of-the art tube model in describing the observations. While both versions of the model allow for I
DOI: 10.1021/acs.macromol.6b00823 Macromolecules XXXX, XXX, XXX−XXX
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