Effective free energy of dowser domains
This protocol is extracted from research article:
Sculpting stable structures in pure liquids
Sci Adv, Feb 15, 2019; DOI: 10.1126/sciadv.aav4283

The time derivative of the dowser orientation can also be viewed as a relaxation under the effective potential U that depends on the velocity field. The equation of motion in that case isEmbedded Image(22)

The master equation for Embedded Image is recovered forEmbedded Image(23)

Integrating the above equation over ϕ leads toEmbedded Image(24)which can be written in a covariant formEmbedded Image(25)where a constant C is not dependent on ϕ. C has to vanish at v = 0 and must be at least quadratic in v to preserve the invariant form. Since we are interested only in the linear response to the velocity field, we can set C = 0.

Using the effective potential U, we can phenomenologically construct an effective free energy F of a dowser state in microfluidic confinement in contact with a homeotropic nematic state (with ansatz n = ez)Embedded Image(26)where T is the line tension of a nematic disclination (18). Specifically, we are interested in the free energy of circular dowser domains with homogeneous alignment of the dowser vector d in a flow field that is homogeneous in the xy plane. This substantially simplifies the expression for the free energyEmbedded Image(27)where Embedded Image. The dynamics of the loop growth or annihilation is given byEmbedded Image(28)where the viscosity parameter γr is due to a drag force on a moving disclination line. The line tension contribution becomes dominant at small radii, leading to universal annihilation behavior of shrinking loops (fig. S1).

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