Torsional Linearity in Nonlinear Stress-Optical Regimes for Polymeric

Letter pubs.acs.org/macroletters

Torsional Linearity in Nonlinear Stress-Optical Regimes for Polymeric Materials Chunggi Baig* School of Energy and Chemical Engineering, Ulsan National Institute of Science and Technology (UNIST), Ulsan 689-798, South Korea S Supporting Information *

ABSTRACT: The birefringence on anisotropic materials under flow has been tremendously useful for understanding the physical states of nonequilibrium materials. It is, however, well-known that the linear relationship between the anisotropy of the stress and that of the birefringence breaks down above a certain flow strength, severely limiting its practical applicability. Here we present an encouraging result which helps overcome this limitation and extends our knowledge beyond the conventional the stress-optical rule regime. Through a detailed molecular-level analysis of the stress-optical behaviors of various polyethylene melts under shear via direct atomistic nonequilibrium molecular dynamics simulations, we found a universal feature in the stress-optical behaviors of polymeric materials that there exists a strictly linear relationship between the birefringence tensor and the stress tensor contributed solely by the bond-torsional interaction in the whole range of flow strength. While a limited range of chain length and architecture (unentangled and moderately entangled linear and H-shaped polymer melts) under shear flow was investigated in this study, the main features and conclusions as drawn are considered to be general and valid regardless of (i) the chain length and molecular architecture, (ii) flow types, and (iii) force fields of polymeric materials. They are further useful as guidance in developing an accurate coarse-grained polymer model.

O

optical behaviors of polymer melt systems under shear spanning from linear up to highly nonlinear flow regimes, as attained through full atomistic nonequilibrium molecular dynamics (NEMD) simulations. Three representative polyethylene (PE) melts were investigated in this work: (a) short (unentangled) linear C50H102, (b) long (entangled) linear C400H802, and (c) H-shaped branched (denoted by H_78_25, with each chain containing 78 and 25 carbon atoms in the backbone and each of its four branches, respectively). Canonical NEMD simulations of planar Couette flow (the x and y axes representing the flow and the velocity gradient direction, respectively) were conducted at a constant temperature T = 450 K and density ρ = 0.7438 g/cm3 for C50H102, ρ = 0.7640 g/cm3 for C400H802, and ρ = 0.7818 g/ cm3 for H_78_25; these simulations were implemented using a Nosé−Hoover thermostat9,10 and the Lees-Edwards slidingbrick boundary conditions.11 The set of evolution equations was numerically integrated using the reversible reference system propagator algorithm (r-RESPA)12 with two different time scales for an MD step: 0.48 fs for three bonded (bondstretching, bond-bending, and bond-torsional) interactions and 2.39 fs for two nonbonded (inter- and intramolecular LennardJones (LJ)) interactions (see ref 13 for simulation details and

ptical measurements on anisotropic materials under an external flow field have been tremendously useful not only for understanding the physical states of nonequilibrium materials but also for establishing a general knowledge of the structure−property linkage. Among such measurements, the flow birefringence measurement has proven to be particularly informative for characterizing the structure−stress correlation of flowing polymeric materials.1 Numerous experiments1−6 and simulations7,8 have shown that, in general, the linear relationship between the stress tensor and the birefringence tensor [the stress-optical rule (SOR)] of polymeric materials under elongation or shear starts to break down at a critical stress. This behavior can be readily understood because the degree of molecular bond alignment beyond a certain field strength is not linearly proportional to the imposed stress; it eventually attains a limiting value (in elongation) or a maximum plateau (in shear) under a very strong flow regime. Would it then be ultimately futile to devote efforts toward building a rheo-optical connection for predicting nonlinear rheological properties of polymers, restricting such a useful, noninvasive optical method to only a linear regime in the study of the structure−property relationship? Clearly, it would be greatly beneficial if we could understand the molecular origins of nonlinear rheo-optical behaviors and identify intrinsic features preserving a simple (i.e., linear) linkage between structure and property in the entire range of the flow regime. In this Letter, we report an encouraging result that guides us toward this end. Our findings are based on a detailed molecular-level analysis of the stress© XXXX American Chemical Society

Received: November 17, 2015 Accepted: February 4, 2016

273

DOI: 10.1021/acsmacrolett.5b00826 ACS Macro Lett. 2016, 5, 273−277

Letter

ACS Macro Letters

dynamical momentum transfer. Further, we note that the linear relationship appears to be valid slightly beyond the shearthinning regime of material functions (e.g., viscosity and normal stress coefficients),18 indicating that polymeric materials continue to exhibit a part of their nonlinear viscoelastic characteristics linearly with structural deformation. In contrast, at medium to high stress levels, a breakdown of the SOR is evident in both the xy and the (xx−yy) components for all the systems studied, indicating a nonlinear structural and dynamical correlation. Such rheo-optical nonlinearity is further manifested in the variation of the chain orientation (or extinction) angle19 χ with respect to the flow strength (Figure 1b). It is clearly seen from the figure that whereas chains tend to align to the flow direction in proportion to an applied stress in a weak flow regime, beyond a certain field strength, the orientation angles represented by the birefringence and stress tensors no longer coincide with each other. All these results apparently indicate the general inadequacy of the stress tensor for directly quantifying the degree of structural deformation occurring in polymeric systems over a wide range of flow strengths, which severely limits the application of such versatile stress-optical methods to only a weak flow regime. Should we then forego entirely such a useful linkage between molecular structure and stress in the study of rheo-optically nonlinear material properties? This dilemma has led us to determine the molecular origins underlying the breakdown of the stress-birefringence linear relationship. Using an additive property of force fields, we conducted a detailed analysis by separately calculating the contribution to the total stress from each individual molecular interaction mode (i.e., bond-stretching, bond-bending, bond-torsional, intermolecular LJ, and intramolecular LJ). Figure 2 shows the χ values of the

the Supporting Information therein for the RESPA formula). The well-known (Siepmann-Karaborni-Smit) SKS14 and (Transferrable Potentials for Phase Equilibria) TraPPE15 united-atom potential models were employed in the simulations for linear and branched polymers, respectively.16 A sufficient number of molecules were employed for each system (i.e., 120, 192, and 162 chains for C50H102, C400H802, and H_78_25, respectively) in a rectangular simulation box enlarged in the flow direction to avoid system-size effects, especially under strong flow fields where molecules become highly stretched and aligned. For a thorough examination of the system response from linear up to highly nonlinear flow regimes, a wide range of flow strengths spanning the interval 0.2 ≤ Wi ≤ 3000 was considered, where Wi denotes the Weissenberg number, defined as the product of the longest relaxation time λ of the system and the imposed strain rate (λ = 0.56 ± 0.03 ns for C50H102, λ = 218 ± 10 ns for C400H802, and λ = 33 ± 1.5 ns for H_78_25 in this work). The standard stress-optical behaviors under shear are shown in Figure 1a for the C400H802 linear and H_78_25 branched

Figure 2. Comparison of the orientation angle χ as a function of the Wi number between the birefringence tensor n and the stress contributed by each individual interaction mode for the H_78_25 PE melt. The inset shows the corresponding results for the C50H102 linear PE melt.

Figure 1. (a) Plot of the birefringence tensor n vs the stress tensor σ and (b) comparison of the orientation angle χ as a function of Wi number between n (red triangles) and σ (black circles), for the C400H802 linear polyethylene (PE) melt. The insets display the corresponding results for the H_78_25 PE melt.

individual interaction modes as a function of flow strength. First, it is observed that while the orientation angles computed by all the interaction modes coincide with each other at low shear rates, quantitatively (as well as qualitatively) distinct behaviors between different modes are evident above a certain intermediate flow strength (i.e., Wi ≥ 10 for the H_78_25 polymer melt), indicating dissimilar degrees of the dynamical response of each mode to an external field. Further analysis of the result reveals that beyond a rather narrow linear flow regime, regardless of the chain length and molecular

polyethylene melts [similar results (not shown here) are observed for linear C50H102]. The plots of both the xy and the (xx−yy) components of the birefringence tensor17 (n) versus the stress tensor (σ) exhibit typical linear proportionality of the structural deformation in response to an applied field strength at a low stress level for both the linear and the branched polymers, indicating the existence of a simple correlation between the local-to-global molecular structure and the 274

DOI: 10.1021/acsmacrolett.5b00826 ACS Macro Lett. 2016, 5, 273−277

Letter

ACS Macro Letters architecture, the bond-stretching and nonbonded LJ modes always predict a larger value of χ at the same flow strength than the birefringence, whereas the bond-bending mode exhibits a smaller value. Furthermore, the discrepancy of each mode from the birefringence is observed to increase with flow strength. These results indicate that the dissipative shear stress σxy develops relatively stronger (weaker) than the restoring elastic stress (σxx−σyy) for the bond-stretching and nonbonded LJ interaction modes (bond-bending interaction mode). Most interestingly, as is clearly seen in Figure 2, the orientation angles exhibited by the torsional interaction mode are found to quantitatively match those of the birefringence over the entire range of flow strength covering highly nonlinear flow regimes for all the linear and branched systems studied, indicating the existence of a simple one-to-one orientational correlation between the torsional interaction mode and the birefringence. This remarkable result, together with systematic experimental analyses, can be very useful in a general study of the structure−property relationship of deformed polymeric systems by linearly linking the local-to-global structural deformation of system with the average torsional distributions of polymer chains. It should be emphasized that the overall trends conveyed by each individual interaction mode are essentially the same, regardless of the chain length, molecular architecture, and force fields of the polymeric materials studied in the present work. On the basis of all these observations, we carried out an overall comparison between the birefringence tensor and the stress tensor contributed by each individual interaction mode. As is evident in Figure 3, the bond-torsional interaction mode reveals a general strictly linear relationship between its own stress tensor and the birefringence tensor over the entire range of field strength in both the xy and the (xx−yy) components for all the linear and branched polymers, while the other interaction modes exhibit their contributions to the breakdown of the SOR above a certain intermediate stress. In practice, this information might provide a statistically more reliable estimate of the stress-optical coefficient (as it is customarily calculated by plotting the total stress versus the birefringence within only the rheo-optically linear flow regime), by evaluating the slope of the torsional stress tensor versus the birefringence tensor throughout the entire flow regime together with the relative ratio of the bond-torsional stress to the total stress in the linear regime. Moreover, we can use information on the birefringence tensor to obtain the torsional stress and the bond-torsional distribution and, thus, estimate a large-scale polymer structure even in highly nonlinear flow regimes. Based on the width of the range of Wi numbers in which the birefringence versus the stress contributed by each mode maintain their linear relationship in both the xy (shear) and (xx−yy) (normal) components, the present results reveals that the linearity of the SOR holds well in the following order: bond-torsional > bond-stretching ≈ intramolecular LJ > bondbending > intermolecular LJ. We note that fairly reliable measures for predicting structural deformation are offered by the bond-stretching and intramolecular LJ modes through their linear correlations with the birefringence up to intermediate field strength; this information can be useful for analyzing the structural deformation occurring in a variety of dilute and semidilute solutions where the intramolecular structure is essential and also for developing a coarse-grained (CG) polymer model where CG bead−bead stretching is crucial. Finally, we examine the generalized shear compliance

Figure 3. Plots of the birefringence tensor n vs the stress tensor σ contributed by each individual interaction mode for (a) the xy component of the H_78_25 polyethylene (PE) melt and (b) the (xx− yy) component of the C400H802 linear PE melt.

(representing the ratio of the principal elastic stress to the squared shear stress) as a function of flow strength (Figure 4). The relative ratio represented by the birefringence tensor is shown to increase with the field strength, which is not well reproduced by the total stress. Consistent with all the results described above, the general trend of the bond-torsional mode is found to exactly match that of the birefringence, while the

Figure 4. Plots of the nonequilibrium shear compliance Je = (xx − yy)/ 2(xy)2 as a function of the Wi number between the birefringence tensor n and the stress contributed by each individual interaction mode for the C400H802 linear polyethylene (PE) melt. The inset shows the corresponding results for the H_78_25 PE melt. For comparison, the Je values of the birefringence tensor have been rescaled by a factor of 10−3. 275

DOI: 10.1021/acsmacrolett.5b00826 ACS Macro Lett. 2016, 5, 273−277

Letter

ACS Macro Letters

properties via primitive path analysis29 of the obtained atomistic trajectories as a function of time. (iii) The conclusions drawn from this study would not be readily applicable to exceedingly nonflexible (e.g., rod-like) polymer systems or short-chain branched polymers where the branches are too short to generate a sufficient torsional flexibility; nonetheless, they are supposed to be valid for structurally highly intricate biopolymers (e.g., double-helix DNA) involved in complicated force-field interactions. In addition, the torsional linearity found in this study appears to be qualitatively consistent with experimental reports30,31 for the linear dependence of twistinduced circular birefringence on twist rate in optical fibers. We thus expect a potential usefulness of the present findings in the study of viscoelastic properties of DNA molecules32 and in the development of electronic devices. (iv) In general, any physical interaction mode that faithfully represents local-to-global scale molecular structures can be a good measure in the domain of the structure−property relationship. One such mode is the bond-torsional mode in the case of polymeric materials. As such, a thorough knowledge of the torsional stress would be useful for understanding a variety of rheological properties and phenomena of polymeric systems under various flow conditions by means of a simple, linear structure−property relationship. It would be also beneficial for expanding our rheo-optical knowledge to extend the present study to general condensed matters, including solid materials.

other modes exhibit qualitative discrepancies, especially in strong flow regimes. Overall, the bond-torsional interaction mode has been found to offer a remarkably accurate measure in the overall stressstructure connection in the entire linear and nonlinear flow regimes, which is, in conjunction with experiments, considered to be very beneficial for our exploration of the structure− property relationship in polymer physics and rheology. The main features found in this study are further expected to be applicable to other flow types (e.g., elongational or mixed flows) and polymer solutions,20,21 because the inherent natures of individual interaction modes are independent of flow type or solvent characteristics. The following points are worth mentioning. (i) Further inspection of the present results indicates that the breakdown of the SOR is roughly synchronized with the time when the intermolecular LJ stress triggers its nonlinear stress-optical behavior. This is supposed to stem from the strong dynamical collisions between molecules at high shear fields; in comparison, the intramolecular LJ mode is observed to bear the linear relationship quite faithfully up to intermediate field strength. In this regard, it would be interesting to examine how each individual stress component and the nonlinear stress-optical behaviors are related to chain rotation and tumbling dynamics (as closely associated with intermolecular collisions) typically observed in polymeric systems above an intermediate flow strength. We also note that excessively strong molecular collisions at extremely high field strengths would eventually disrupt any linear relationship between structure and stress, even in the bond-torsional mode. (ii) On the basis of the present work, the capability of a particle-based CG model,22,23 where certain groups of chemically connected atoms are lumped into superatoms and thus the microscopic details of the atomistic interactions (the bond-stretching, bond-bending, and bondtorsional modes) are combined into an effective synthetic CG interaction potential (e.g., bead−spring models), is expected to depend on how accurately the CG model accounts for the bond-torsional interaction. In other words, an adequately constructed CG model in regard to the bond-torsional interaction should be able to predict a well-defined linear SOR behavior (apart from nonbonded interactions included in the total stress) even up to highly nonlinear flow regimes. We mention in the passing that recently there appeared a controversial result between two different CG simulations24,25 in conjunction with previous experimental results26 for the validity of SOR for entangled polymer systems undergoing a start-up shear in nonlinear flow regimes in the range of τ−1 d < γ̇ , where τ and τ are, respectively, the reptation and Rouse < τ−1 d d R time of the system in the tube theory.27 Although the origin is not so clear at the moment, the debate might be able to be resolved by a detailed analysis of the CG results in the context of the present findings together with the fact28 that a CG model is susceptible to overestimation of large-scale chain deformation in comparison with the corresponding atomistic model, especially at strong flow fields, due to relatively weak potential energies for local bonded interactions in CG representations. To resolve the issue, it would be certainly beneficial to conduct extensive atomistic NEMD simulations for entangled polymer melts under a start-up shear and carry out a detailed analysis of the total and individual stress tensor in comparison with the structural (e.g., radius of gyration and conformation tensor) and topological (e.g., the primitive path contour length, the number of entanglements, and the entanglement strand vector)

■

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.5b00826. Plots of the stress tensor σ (the xy and the (xx−yy) components) contributed by each individual interaction mode as a function of Wi number for the C50H102 linear, C400H802 linear, and H_78_25 PE melts (PDF).

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +82-52-217-2538. Fax: +8252-217-2649. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS This work was supported by the National Research Foundation of Korea (NRF-2013R1A1A2007749). REFERENCES

(1) Janeschitz-Kriegl, H. Polymer Melt Rheology and Flow Birefringence; Springer-Verlag: New York, 1983. (2) Matsumoto, T.; Bogue, D. C. J. Polym. Sci., Polym. Phys. Ed. 1977, 15, 1663−1674. (3) Inoue, T.; Okamoto, H.; Osaki, K. Macromolecules 1991, 24, 5670−5675. (4) Muller, R.; Froelich, D. Polymer 1985, 26, 1477−1482. (5) Venerus, D. C.; Zhu, S.-H.; Ö ttinger, H. C. J. Rheol. 1999, 43, 795−813. (6) Luap, C.; Müller, C.; Schweizer, T.; Venerus, D. C. Rheol. Acta 2005, 45, 83−91. (7) Kröger, M.; Loose, W.; Hess, S. J. Rheol. 1993, 37, 1057−1079. (8) Baig, C.; Edwards, B. J.; Keffer, D. J. Rheol. Acta 2007, 46, 1171− 1186. 276

DOI: 10.1021/acsmacrolett.5b00826 ACS Macro Lett. 2016, 5, 273−277

Letter

ACS Macro Letters (9) Nosé, S. Mol. Phys. 1984, 52, 255−268. (10) Hoover, W. G. Phys. Rev. A: At., Mol., Opt. Phys. 1985, 31, 1695−1697. (11) Evans, D. J.; Morriss, G. P. Statistical Mechanics of Nonequilibrium Liquids; Academic Press: New York, 1990. (12) Tuckerman, M.; Berne, B. J.; Martyna, G. J. J. Chem. Phys. 1992, 97, 1990−2001. (13) Baig, C.; Mavrantzas, V. G.; Kröger, M. Macromolecules 2010, 43, 6886−6902. (14) Siepmann, J. I.; Karaborni, S.; Smit, B. Nature 1993, 365, 330− 332. (15) Martin, M. G.; Siepmann, J. I. J. Phys. Chem. B 1999, 103, 4508− 4517. (16) The SKS and TraPPE united-atom potential models are known to produce adequately rheological (i.e., stress, birefringence, etc.) data in reasonable agreement with experimental data, e.g., refs 8, 13, and the following: Cui, S. T.; et al. J. Chem. Phys. 1996, 105, 1214−220. and Mavrantzas, V. G.; Theodorou, D. N. Comput. Theor. Polym. Sci. 2000, 10, 1−13. (17) See ref 8 for the calculation details of the birefringence tensor. (18) Detailed rheological behaviors of material functions can be found in ref 8 for the linear C50H102, ref 13 for the linear C400H802, and for the H_78_25 PE melts, Baig, C.; Mavrantzas, V. G. J. Chem. Phys. 2010, 132, 014904. (19) The orientation angle χ is the angle formed between the flow direction and the director which is the eigenvector corresponding to the largest eigenvalue of a second-rank tensor (i.e., stress tensor and birefringence tensor). Figure 2 shows the results of χ calculated for each stress tensor contributed by the individual interaction modes. (20) Bhandar, A. S.; Wiest, J. M. Macromolecules 2001, 34, 7113− 7120. (21) Jain, S.; Dalal, I. S.; Orichella, N.; Shum, J.; Larson, R. G. Chem. Eng. Sci. 2009, 64, 4566−4571. (22) Reith, D.; Meyer, H.; Müller-Plathe, M. Macromolecules 2001, 34, 2335−2345. (23) Harmandaris, V. A.; Kremer, K. Macromolecules 2009, 42, 791− 802. (24) Lu, Y.; An, L.; Wang, S.; Wang, Z. ACS Macro Lett. 2014, 3, 569−573. (25) Masubuchi, Y.; Watanabe, H. ACS Macro Lett. 2014, 3, 1183− 1186. (26) Pearson, D. S.; Kiss, A. D.; Fetters, L. J.; Doi, M. J. Rheol. 1989, 33, 517−535. (27) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: New York, 1986. (28) Baig, C.; Harmandaris, V. G. Macromolecules 2010, 43, 3156− 3160. (29) Everaers, R.; Sukumaran, S. K.; Grest, G. S.; Svaneborg, C.; Sivasubramanian, A.; Kremer, K. Science 2004, 303, 823−826. Kröger, M. Comput. Phys. Commun. 2005, 168, 209−232. Tzoumanekas, C.; Theodorou, D. N. Macromolecules 2006, 39, 4592−4604. (30) Ulrich, R.; Simon, A. Appl. Opt. 1979, 18, 2241−2251. (31) Barlow, A.; Payne, D. N. IEEE J. Quantum Electron. 1983, 19, 834−839. (32) Strick, T.; Allemand, J.-F.; Croquette, V.; Bensimon, D. Prog. Biophys. Mol. Biol. 2000, 74, 115−140.

277

DOI: 10.1021/acsmacrolett.5b00826 ACS Macro Lett. 2016, 5, 273−277