1. Introduction

Optical measurement of the anisotropy of materials has proven to be very informative and useful in characterizing the structure and stress of molecular crystals^1,2 and as a non-invasive tool in the study of rheology of polymeric fluids.^3,4 There exist generally two types of birefringence, form birefringence and flow birefringence. The former originates from the difference in the intrinsic (average) polarizability between the solvent and polymer and is important in dilute polymer solutions. The latter results from anisotropic orientation of polymer chain bonds induced by the flow⁵ and makes the dominant influence in the concentrated polymer solutions or melts. In this study, we investigate in detail the stress-optical behavior of a linear, short-chain polyethylene melt, of C₅₀H₁₀₂, under shear. We also present a derivation of the generalized Clausius-Mossotti formula for anisotropic media. We further investigate the relationship between the birefringence and two structural properties that have been regarded as very important in a coarse-grained level of description:⁶ one is the conformation tensor, , and the other the orientation tensor, . Therefore, four important second-rank tensors, the stress tensor, , birefringence tensor, , conformation tensor, and orientation tensor, are calculated directly from simulations at various shear rates and compared with each other. 2. Technical approach

The melt studied in this work is C₅₀H₁₀₂, which is sufficient in length in order to see the flow effect on birefringence and, at the same time, computationally feasible across a broad range of shear rates. With this melt, we have performed NVT canonical nonequilibrium molecular dynamics (NEMD) simulations using the SLLOD equations of motion for a homogeneous shear flow.⁷ To maintain a constant temperature, we employed the Nosé-Hoover thermostat. As regards the potential model, we employed the well-known SKS united-atom model developed by Siepmann et al.⁸ for the bond-bending, bond-torsional, and inter-atomic interactions, but replaced the rigid bond by a flexible one for the bond-stretching interaction. We employed 120 molecules of C₅₀H₁₀₂ in a rectangular box, enlarged in the X-direction, with dimensions (XxYxZ) of 93x45x45 Ĺ³. The X-dimension was chosen to be sufficiently large in order to avoid any undesirable system-size effects at high shear rates where chains are quite extended and oriented in the flow direction. The temperature and density were chosen as 450 K and 0.7438 g/cm³. We used 18 different shear rates covering a large range of dimensionless shear rate,  = 0.0005 ~ 1.0 in reduced units. (This corresponds to  = 2.1x10⁷ ~ 4.3x10¹¹ s^-1 in real units). 3. Results and Discussion

the present system, the critical shear stress for shear-thinning and the breakdown of the SOR were found to be 3.2 MPa and 5.5 MPa, respectively. Thus, a linear relationship between the stress and birefringence, the so-called the stress-optical rule (SOR) appears to be valid up to a certain region beyond the incipient point of shear thinning. The slope of the birefringence vs. stress curve appeared to decrease with increasing stress for all three quantities (xx-yy, yy-zz, and xy), consistent with many existing experimental results.^4,9,10 The orientation angles obtained from each of the four tensors (, , , ) were shown to be close to each other at low strain rates, but became more and more distinct as shear rate increased. This implies that the principal frame of reference of each tensor does not coincide with that of other tensors, in general, except for  and , thus indicating a narrow Gaussian distribution of the chain end-to-end distance. Rather surprisingly at first, even  and  (also  as well) were shown to be nonlinear at high shear stress values. The critical stress value at which nonlinearity began was approximately the same as that at which breakdown of the SOR occurred. Furthermore, the customary view that the SOR breaks down due to the saturation of chain extension and orientation was demonstrated to be incorrect under shear, since the failure of the SOR was observed to occur at a much earlier stage in both chain extension and orientation. Specifically, the chain extension at the point of breakdown of the SOR was about 27% of the full extension and the orientation angle of the birefringence was 23^o; this does not seem to be close to the saturated condition at all, compared with 12^o at high shear rates. 4. References

¹M. F. Vuks, Opt Spectrosc (USSR) 20, 361 (1966).

²S. de Jong, F. Groeneweg, and F. van Voorst Vader, J. Appl. Cryst. 24,171 (1991).

³F. A. Morrison, Understanding Rheology (Oxford University Press, New York, 2001).

⁴H. Janeschitz-Kriegl, Polymer melt rheology and flow birefringence, (Springer-Verlag, Berlin Heidelberg New York, 1983).

⁵M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford Univerisity Press, New York, 1986).

⁶A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems, (Oxford University Press, New York, 1994).

⁷D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Liquids (Academic Press, New York, 1990).

⁸J. I. Siepmann, S. Karaborni, and B. Smit, Nature 365, 330 (1993).

⁹T. Matsumoto T and D. C. Bogue, J. Polym. Sci. B Polym. Phys. Ed. 15, 1663 (1977).

¹⁰D. C. Venerus, S.-H. Zhu, and H. C. Öttinger, J. Rheol. 43, 795 (1999).