Effects of strong anchoring on the dynamic moduli of heterogeneous nematic polymers

Abstract

We focus on the linear viscoelastic response of heterogeneous nematic polymers to small amplitude oscillatory shear, paying special attention to the macroscopic influence of strong plate anchoring conditions. The model consists of the Stokes hydrodynamic equations with viscous and nematic stresses, coupled to orientational dynamics and structure driven by the flow gradient, an excluded-volume potential, and a two-constant distortional elasticity potential. We show that the dynamical response simplifies when plate anchoring is either tangential or homeotropic, recovering explicitly solvable Leslie–Ericksen–Frank behavior together with weakly varying order parameters across the plate gap. With these plate conditions, we establish “model consistency” so that all experimental driving conditions (plate-controlled velocity [strain] or shear stress, imposed oscillatory pressure) yield identical dynamic moduli for the same material parameters and anchoring conditions, eliminating the culpability of device influence in scaling behavior. Two physical predictions emerge that imply significant macroscopic elastic and viscous effects controlled by plate anchoring relative to flow geometry: (1) The storage modulus is enhanced by two to three orders of magnitude for homeotropic relative to parallel anchoring, across all frequencies. (2) The loss modulus exhibits enhancement of a factor of two to three for homeotropic over tangential anchoring, restricted to low frequencies. We further deduce a scaling law for the dynamic moduli versus anisotropy of the distortional elasticity potential.

Appendix

The monodomain solution of Eqs. 20–24 is the restriction that s ⁽¹⁾, β ⁽¹⁾, and ψ ⁽¹⁾ are constant with respect to y and that $\frac{{\partial \upsilon _x^{\left( 1 \right)} }}{{\partial y}} = \cos \omega t$. Ignoring the long-time effects discussed in Choate and Forest (2006), this leads us to the storage and loss moduli

$$\begin{aligned} G\prime \left( \omega \right) = \sin ^2 2\psi _0 \frac{{a^2 \left( {1 - s_0 } \right)a_1 a_2 }}{6}\frac{{\omega ^2 \left[ {a_2 + 3a_1 \left( {1 + 2s_0 } \right) + \left( {a_1 + 3a_2 \left( {1 + 2s_0 } \right)} \right)\omega ^2 } \right]}}{{\left( {a_1^2 + \omega ^2 } \right)\left( {a_2^2 + \omega ^2 } \right)}} \\G\prime \prime \left( \omega \right) = \left( {\frac{{\mu _1 \left( {s_0 + 2} \right)}}{6} + \frac{{\mu _2 s_0^2 }}{4}\sin ^2 2\psi _0 + \frac{{\mu _3 }}{2}} \right)\omega \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \sin ^2 2\psi _0 \frac{{a^2 \left( {1 - s_0 } \right)}}{6}\frac{{\omega \left[ {2a_1^2 a_2^2 \left( {2 + 3s_0 } \right) + \left( {a_1^2 + 3a_2^2 \left( {1 + 2s_0 } \right)} \right)\omega ^2 } \right]}}{{\left( {a_1^2 + \omega ^2 } \right)\left( {a_2^2 + \omega ^2 } \right)}} \\ \end{aligned} $$

(58)

where a ₁ = 6αNs ₀ and a ₂ = −2α(6 −  N(2 + s ₀)).