Liquid Crystal — Nematic Director Field & Fréedericksz Transition

Observe how rod-like molecules self-organise into a director field n̂. Control temperature and electric field to trigger phase transitions, optical switching, and topological defects.

Director field texture (line = average molecular orientation)
Order parameter S and optical transmittance vs reduced temperature T/Tc

Controls

Fréedericksz transition: Electric field (E > Ec = π√(K/ε₀Δε)/d) realigns molecules from planar to homeotropic configuration, switching cell from opaque to transparent.
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Physical State

Order param. S— Phase— Ec threshold (V/μm)— Field vs Ec— Director tilt θ— Transmittance T— Birefringence Δn— Elastic energy F—
Frank Elastic Energy

F = ½K₁(∇·n̂)² + ½K₂(n̂·∇×n̂)² + ½K₃|n̂×∇×n̂|²

One-constant approx: K₁=K₂=K₃=K. The order parameter S = ½⟨3cos²θ−1⟩ drops to 0 at Tc (isotropic transition).

Topological defects at ±½ (disclinations) are stable because π/2 rotation around loop returns n̂ to same state.

About this simulation

This simulation models a nematic liquid crystal as a grid of rod-like molecules, each represented by a director angle that relaxes toward its neighbours' average orientation (a finite-difference approximation of the Frank elastic energy). An order parameter S falls smoothly to zero as reduced temperature T/Tc approaches the isotropic transition, while an applied electric field competes with elastic alignment above a critical Fréedericksz threshold Ec. The resulting director tilt sets a birefringence Δn, which is combined with cell thickness to compute optical transmittance through crossed polarizers, exactly as in a real LCD pixel.

🔬 What it shows

The left canvas draws a 28×22 grid of director lines whose length and colour encode local order S; the right canvas plots S(T) and transmittance versus T/Tc alongside your current temperature marker. Three presets seed different starting textures: uniform planar alignment (Nematic 5CB), a helical twist across the cell (Twisted nematic), and a pair of ±½ topological defects (Defects) that the relaxation loop then evolves frame by frame.

🎮 How to use

Raise Temperature T/Tc toward 1.0 to watch order collapse into the isotropic phase, or drop the Electric field E above the computed Ec threshold to reorient molecules from planar to homeotropic and flip the cell's transmittance. The Elastic constant K and Cell thickness d sliders change Ec itself (Ec = π√(K/ε₀Δε)/d), and the live Physical State panel reports S, phase, tilt angle θ, transmittance, birefringence Δn and Frank elastic energy F in real time.

💡 Did you know?

The ±½ disclinations in the Defects preset are stable precisely because a nematic director has no arrowhead — rotating by only half a turn (π, not 2π) around the defect core brings the director field back to the same physical state, unlike full vector fields such as magnetic spins.

Frequently asked questions

What is a nematic liquid crystal?

A nematic liquid crystal is a phase of matter between solid and liquid in which rod-like molecules have no positional order (they flow like a liquid) but do have long-range orientational order, tending to point along a common local direction called the director n̂. This simulation represents that director as an angle at each point of a grid and relaxes it toward its neighbours to mimic elastic alignment.

What does the order parameter S actually measure?

The order parameter S = ½⟨3cos²θ−1⟩ quantifies how tightly the molecules cluster around the average director direction, ranging from S ≈ 0.4-0.5 in a well-ordered nematic down to S = 0 in the disordered isotropic liquid. In the simulation S is computed from reduced temperature T/Tc and directly controls line length, colour saturation and birefringence in the display.

What is the Fréedericksz transition?

It is the field-induced reorientation of liquid crystal molecules once an applied electric (or magnetic) field exceeds a critical threshold Ec = π√(K/ε₀Δε)/d, where K is the elastic constant and d is the cell thickness. Below Ec the elastic torque keeps the director in its original planar configuration; above it, the field torque wins and molecules tilt toward homeotropic (perpendicular) alignment, which is exactly how an LCD pixel switches between light and dark states.

How does the simulation turn director orientation into an image on screen?

Once the director field is known, the simulation computes a birefringence Δn proportional to S, then a phase retardation φ = πΔn·d/λ for light of wavelength 550 nm passing through the cell. Transmittance through crossed polarizers follows T = sin²(φ) scaled down by the field-induced tilt, which is the same optical mechanism a real twisted-nematic LCD uses to control pixel brightness.

Why are the ±½ defects in the "Defects" preset stable?

Unlike vector fields with a definite arrow, a nematic director is unoriented (n̂ and −n̂ describe the same physical state), so a disclination only needs to rotate the director by π, not 2π, to return to the same configuration going around the defect core. That allows stable half-integer topological charges (±½) which real nematics also display as characteristic four-brush textures under a polarizing microscope.