FR22 · Read a field of directions

Nematic Defect Microscope

How can a direction turn through half a circle around a defect? A nematic direction has no arrowhead: turning a rod halfway around returns the same orientation. Around a half defect, the direction changes by half a turn and joins smoothly to itself. Counting the signed turn distinguishes the two defect shapes.

Follow the rods around a bright core, then bring two opposite defects together. The microscope locates each defect from the sampled directions; a coarse grid can miss a close pair.

Positive half defectNegative half defect

Rod hue shows headless direction; marker locations are accurate to one sampling cell.

Positive half defects

2

Negative half defects

2

Total charge

0

Mean nematic order

0.992

The direction turns as you circle a core

Amber and violet curves follow one positive and one negative detected core. Horizontal position is progress around a closed loop; vertical position is accumulated doubled-angle winding, in full turns.

Equations, approximations and sources

The analytic complex texture has four charge-balanced zeros on a torus. The detector wraps neighboring doubled-angle differences around each cell; close pairs and noise can change what the sampled field resolves.

The coarse browser model isolates coupling and sampling. It omits passive elastic stress and material calibration; its trajectories are numerical examples. Export includes raw tensor samples, velocity samples, seeds, parameters and tracked charges.

Giomi (2015), active nematodynamics, equations 1–2 and Figure 1

Continue through the active-matter course

Read the active matter course: from persistent motion to defects and flow

Start with the essentials

Focus question: How can a direction turn through half a circle around a defect?
One-sentence intuition: Count the change of the doubled direction around a closed loop, then divide its winding by two.

Core mathematical model

A direction without an arrowhead

Q_{xx}=\frac{S}{2}\cos(2\theta),\qquad Q_{xy}=\frac{S}{2}\sin(2\theta)

The tensor has the same value after a half turn. Its magnitude also falls near a defect core.

Charge from sampled directions

s=\frac{1}{4\pi}\sum_{\square}\operatorname{wrap}\!\left(\Delta\arg(Q_{xx}+iQ_{xy})\right)

Each phase difference is wrapped into the interval from minus pi to pi. A phase winding of one turn gives a half defect.

Common difficulties

Two meanings of motion

Typical misconceptionA bright marker is a particle with a prescribed speed.

Better mental modelMarkers are found by winding the sampled tensor field around each cell; dragging prepares a new texture.

Model scale

Typical misconceptionThe grid is a calibrated sample of a biological material.

Better mental modelThe starting texture is an analytic, periodic alignment field with two positive and two negative half defects. This microscope samples a prescribed texture, with optional angular noise. It does not evolve a fluid or infer defects from an experimental image.

Run the experiment

01
Follow one core
Choose Four half-turns. Trace the rods around an amber or violet marker.
What to observe: The two colors have opposite signed winding. The four detected charges add to zero on the periodic square.
02
Bring the pair closer
Decrease Defect separation parameter, then choose A coarse camera.
What to observe: A pair closer than a sampling cell can cancel within that cell and escape detection. Resolve it with a finer grid.
03
Save the measured field
Drag a highlighted core, then export the sampled record.
What to observe: The export contains the sampled tensor values, detected charges and tracking history.

Nematic Defect Microscope

The direction turns as you circle a core