(250f) A Molecular Dynamics Study of the Stress-Optical Behavior of a Linear Short-Chain Polyethylene Melt under Shear

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 ų.
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).