Polar order parameter

A polar order parameter is a vector whose direction distinguishes head from tail. In an isotropic medium without an external polar field, spatial inversion makes the free energy even under $\mathbf{p}\mapsto -\mathbf{p}$.

The polar molecular field is the functional derivative of the free energy with respect to the polar order parameter. In the positive-derivative convention, dissipative dynamics is $D_t\mathbf{p}=-\Gamma \mathbf{h}$ with positive kinetic coefficient. For a quartic local term and an isotropic gradient penalty,Some authors instead define the molecular field with a minus sign, which changes the associated stress and evolution conventions.

The material derivative of a polar order parameter contains rigid rotation with the local vorticity and a material-dependent alignment response to the symmetric strain-rate tensor.

In simple shear flow, a persistent polar order parameter with $|\xi |<1$ has no fixed angle and rotates continuously. Integrating $\dot{\theta }=g(\xi \cos 2\theta -1)/2$ through $2\pi$ gives the displayed period. An unoriented nematic director repeats after half that time. At $\xi =0$, the magnitude relaxes independently while the angular speed is $-g/2$.

For a uniform polar order parameter in simple shear flow $\mathbf{v}=(gy,0)$, the angular equation is $\dot{\theta }=g(\xi \cos 2\theta -1)/2$. A fixed orientation for nonzero shear therefore requires $|\xi |\ge 1$. For $|\xi |>1$, its angular stability condition is $g\xi \sin 2\theta >0$; equality at $|\xi |=1$ requires nonlinear analysis. The ordered magnitude must separately satisfy the radial stationarity equation and be nonzero.

An advected order parameter contributes stress because a velocity gradient changes its free energy. For a polar vector this includes distortion, antisymmetric rotational, and symmetric flow-alignment stresses involving the molecular field.

Using the positive polar molecular field $h_i=\delta F/\delta p_i$ and velocity gradient ${\partial }_i{v}_j$, the reversible order-parameter stress consists of a distortion contribution and rotational and alignment contributions:For a local free-energy density, a representative is ${\Sigma }_{ij}^{(1)}=(f-p_k{h}_k){\delta }_{ij}-f_{{\partial }_i{p}_k}{\partial }_j{p}_k$. The Euler-Lagrange equation verifies its divergence. Integration by parts gives $\delta F=\int {\Sigma }_{ij}^p{\partial }_i{u}_j$ for an incompressible displacement without surface work; the corresponding mechanical power has the opposite sign. Pressure contributions may be reassigned in an incompressible flow.