Numerous experimental studies of liquid crystal polymers(LCP) undergoing shear flow have revealed that they generally adopt a texture of randomly oriented nematic microdomains separated by singular structures, known as disclinations in the orientation field. As a consequence their rheological and optical properties are largely those of an ordinary isotropic polymer. As was found by Larson and Mead, under shear, the microstructural and rheological behavior of LCPs can be very rich with multiple transitions in texture.

This is quite different from the situation found in small nematics. When subject to shear flow, small nematics generally adopt a fixed orientation near the flow direction, and disclinations are much less common. Their material properties are generally that of a nematic phase.

A complete understanding of the dynamics of texture evolution and its interaction with flow is still missing. Several attempts to analyze these experimental observations have been made via numerical calculations using the Leslie-Ericksen(LE) theory. However, the LE theory is accurate only for small shear rates, and in principle cannot handle disclinations. The Doi theory uses a much richer order parameter, the orientational distribution functions. It takes into account the effects of flow, Brownian dynamics and intermolecular interaction. However, it does not include the so-called distortional elasticity which is the elastic penalty for spatial distortions. Therefore the Doi theory is only valid for spatially homogeneous systems. Indeed the work of Nayak shows that even though the Doi model leads readily to disclination and microstructure formation, the simulations eventually break down since spatial interaction is not properly accounted for.

Marrucci and Greco proposed the incorporation of long-range distortional elasticity into the Doi model via a generalization of the Maier-Saupe mean-field potential. Their model was recently extended into a new constitutive theory for LCP flows. This theory is a mixture of the macroscopic order parameter description and the description using orientational distribution function. It is therefore still desirable to develop a theory that is based solely on the Doi approach using orientational distribution function.

In our work, we account for the distortional elasticity by properly taking into account the spatial interactions between the rod. This allow us to study the long time dynamics of liquid crystal polymer flow, including domain formation, dynamics of disclinations between domains, and transition between different dynamic regimes such as flow aligning, wagging and tumbling. we also propose a new method to deal with anchoring condition for Fokker-Planck equation.

1+1 Case

• First "1": dimention of macroscopical flow.
• Second "1": DOF of microcosmic molecular orientation.

The disclination first generated in $y\approxeq 0.16, 0.84$, and then propagate to the both sides, become weaker and weaker, 50s later it meets with another disclination and get the strongest at $y\approxeq 0.45, 0,55$

2+1 Case

Under the flow, the orientations of the rods in the whole region present a polydomain structure. The texture is produced where the directors abruptly change

• M. Doi and S.F. Edwards, The theory of polymer dynamics, Oxford University Press, Oxford, 1986.



• R.B. Bird, O. Hassager, R.C Armstrong and C.F. Curtiss, Dynamics of polymeric liquids; Vol. 2: kinetic theory (2nd ed.), Wiley-Interscience, New York, 1987.



• J.D. Ferry, Viscoelastic properties of polymers (3rd edition), Wiley, New York, 1980.



• P.G. de Gennes and J. Prost, The physics of liquid crystal, Clarendon Press, Oxford, 1993.



• P.J. Flory, Principles of polymer chemistry, Cornell University Press, Ithaca, NY, 1953.



• J.K.C. Suen, Y.L. Joo and R.C. Armstrong, Molecular orientation effects in viscoelasticity, Annu. Rev. Fluid Mech. 34(2002), 417–444.



• M. Doi, Introduction to polymer physics, Clarendon Express, Oxford, 1996.



• P.G. de Gennes, Introduction to polymer dynamics, Cambridge University Press, Cambridge, 1990.



• R.G. Larson, The structure and reology of complex fluids, Oxford University Press, Oxford, 1969.