Time Cholesteric Dynamics

• Time cholesteric is a class of liquid crystals characterized by dynamic control of helical order, defect evolution, and pitch switching via external stimuli.
• Experimental studies using electric-field and UV-induced photochemistry reveal non-equilibrium responses, including metastable defect states and branch selection.
• Continuum models based on Landau–de Gennes theory capture the interplay of elastic, anchoring, and field effects, elucidating temporal switching kinetics and defect transport.

Searching arXiv for recent and relevant papers on cholesteric systems with temporal switching, time-dependent fields, and pitch transitions. “Time cholesteric” is an Editor’s term for cholesteric liquid-crystalline systems in which the helical state, defect population, or effective pitch is explicitly driven as a function of time by external control fields. In the literature considered here, the temporal degree of freedom is introduced through electric-field switching and rotation in cholesteric droplets and colloid-doped hosts, and through UV-induced photochemical conversion that changes the equilibrium pitch in planar cells. Across these cases, the central problem is not the static cholesteric texture alone, but the kinetics of branch selection, defect transport, metastable trapping, and relaxation under time-dependent forcing (Fadda et al., 2017).

1. Temporal control of cholesteric order

Cholesterics are characterized by a preferred twist $q_0=2\pi/p_0$ or $q_0=2\pi/P_0$, depending on notation, and by a competition between bulk elastic preference, surface anchoring, and external fields. In the droplet formulation, the concentration field $\phi$ distinguishes a cholesteric droplet from the surrounding isotropic fluid, while the tensor order parameter $Q_{\alpha\beta}$ resolves orientational order and defects. In the photosensitive planar-cell formulation, the key variable is the evolving pitch $P(t)$, inferred from polarimetry and compared with the equilibrium pitch $P_0(t)$ set by photochemistry. In colloidal dispersions, time-dependent fields reorganize both defect topology and particle positions (Orlova et al., 2013).

The common physical pattern is a driven mismatch between the instantaneous preferred twist and the twist or defect arrangement actually realized by the system. In a droplet under a uniform or rotating electric field, the mismatch appears through elastic deformation, defect dragging, and ON–OFF hysteresis. In a photosensitive cell, the mismatch is between the free twisting number $\nu_0=2D/P_0$ and the realized half-turn number $\nu=2D/P$, with jump-like transitions controlled by anchoring energy. In colloidal cholesterics, the mismatch is generated by time-dependent electric forcing that repeatedly unwinds and rewinds the host, producing a sequence of metastable equilibria rather than a unique steady state (D'Adamo et al., 2015).

A plausible implication is that “time cholesteric” behavior is best understood as a non-equilibrium cholesteric response problem in which temporal forcing selects among many elastic and topological states rather than merely perturbing a single equilibrium texture.

2. Continuum descriptions and governing equations

The droplet and colloidal studies both employ Landau–de Gennes $Q$-tensor theory with dielectric coupling to the electric field. For a cholesteric droplet of volume $V$ immersed in an isotropic fluid, the free energy is written as (Fadda et al., 2017)

$q_0=2\pi/P_0$0

$q_0=2\pi/P_0$1

Here $q_0=2\pi/P_0$2 denotes planar anchoring and $q_0=2\pi/P_0$3 homeotropic anchoring. The same dielectric coupling term,

$q_0=2\pi/P_0$4

also appears in the colloidal study, where the total free energy is decomposed into bulk, elastic, and field contributions (D'Adamo et al., 2015).

The droplet dynamics are governed by the Beris–Edwards equation

$q_0=2\pi/P_0$5

with molecular field

$q_0=2\pi/P_0$6

and co-rotational term

$q_0=2\pi/P_0$7

where $q_0=2\pi/P_0$8. The fluid is incompressible and obeys Navier–Stokes with $q_0=2\pi/P_0$9 (Fadda et al., 2017).

By contrast, the colloidal field-driven study uses purely relaxational dynamics for $\phi$0,

$\phi$1

while colloid centers obey overdamped Langevin dynamics with Stokes drag $\phi$2 (D'Adamo et al., 2015). This distinction separates hydrodynamically coupled switching in droplets from defect-mediated assembly in a model where particle motion is overdamped.

Several dimensionless groups recur. In the droplet problem, the Ericksen number is

$\phi$3

and the dimensionless field strength is

$\phi$4

The chirality parameter satisfies $\phi$5 for $\phi$6–0.01 and $\phi$7 or $\phi$8 (Fadda et al., 2017). In the colloidal study,

$\phi$9

and a bulk cholesteric–nematic reorientation occurs when $Q_{\alpha\beta}$0 exceeds a threshold $Q_{\alpha\beta}$1 (D'Adamo et al., 2015).

3. Switching kinetics in cholesteric droplets

The droplet study resolves an ON–OFF protocol: the system starts from equilibrium at $Q_{\alpha\beta}$2, the field is switched ON at $Q_{\alpha\beta}$3, the system is allowed to reach a new steady ON state after $Q_{\alpha\beta}$4, the field is then switched OFF at $Q_{\alpha\beta}$5, and relaxation is monitored over $Q_{\alpha\beta}$6. For typical cases with $Q_{\alpha\beta}$7, $Q_{\alpha\beta}$8, and $Q_{\alpha\beta}$9, field orientation relative to the cholesteric axis is decisive (Fadda et al., 2017).

When the field is parallel to the cholesteric axis, $P(t)$0–5 gives $P(t)$1, and OFF relaxation is $P(t)$2. Under these conditions, metastable defect structures form and the OFF-state elastic free energy satisfies $P(t)$3. When the field is perpendicular to the cholesteric axis, defect motion is negligible, the director is nearly nematic uniform, and both $P(t)$4 and $P(t)$5 are $P(t)$6 (Fadda et al., 2017).

The basic hydrodynamic and diffusive scales contextualize these switching times. With $P(t)$7, $P(t)$8, and $P(t)$9, the characteristic hydrodynamic time is $P_0(t)$0, while the diffusive time for $P_0(t)$1 is $P_0(t)$2. Rotational viscosity $P_0(t)$3 Poise corresponds to $P_0(t)$4 in simulation units (Fadda et al., 2017). The fact that switching and recovery occur over $P_0(t)$5-scale intervals rather than $P_0(t)$6-scale intervals suggests that elastic restructuring and defect kinetics, rather than only viscous momentum relaxation, dominate the temporal response.

A compact summary of representative switching outcomes is given below.

Anchoring	Field direction	Outcome
homeotropic	$P_0(t)$7	new metastable
homeotropic	$P_0(t)$8	near-surface defects return partially
tangential	$P_0(t)$9	partial twist reversal
free	$\nu_0=2D/P_0$0	cholesteric axis rotated by $\nu_0=2D/P_0$1

These outcomes indicate that field direction and anchoring do not merely alter switching speed; they determine whether the droplet returns toward equilibrium, rotates its cholesteric axis, or becomes trapped in a defect-rich metastable state (Fadda et al., 2017).

4. Rotating fields, periodic response, and defect transport

Under a rotating electric field of varying frequency, the droplet and its defects rotate as well, typically at lower speed than the field because of the inertia of the liquid crystal. If the surface anchoring is homeotropic, a periodic motion is found (Fadda et al., 2017). The principal control parameter is the field frequency $\nu_0=2D/P_0$2, which enters through the Ericksen number $\nu_0=2D/P_0$3.

For $\nu_0=2D/P_0$4, the measured droplet angular velocity $\nu_0=2D/P_0$5 depends strongly on anchoring. At $\nu_0=2D/P_0$6, $\nu_0=2D/P_0$7 for tangential anchoring and $\nu_0=2D/P_0$8 for homeotropic anchoring; at $\nu_0=2D/P_0$9, the corresponding values are $\nu=2D/P$0 and $\nu=2D/P$1; at $\nu=2D/P$2, $\nu=2D/P$3 and $\nu=2D/P$4; and at $\nu=2D/P$5, $\nu=2D/P$6 and $\nu=2D/P$7. The associated Ericksen numbers are $\nu=2D/P$8, $\nu=2D/P$9, $Q$0, and $Q$1 (Fadda et al., 2017).

For $Q$2, the response is linear, $Q$3, with $Q$4 for tangential anchoring and $Q$5 for homeotropic anchoring. Above $Q$6, inertial and frictional effects saturate $Q$7 (Fadda et al., 2017). This separates a low-$Q$8 regime in which the droplet approximately tracks the forcing from a high-$Q$9 regime in which elastic and dissipative limitations cap the rotational response.

A similar temporal logic appears in the colloidal study, but there the field waveform itself becomes a design parameter. The electric field is prescribed as $V$0, with square-wave or sine-wave modulation. Because $V$1, the even-harmonic content of the waveform affects cholesteric–nematic switching and the defect free-energy landscape. For a pulsed field of duration $V$2 along $V$3, a near-threshold field partially unwinds the host and nucleates disconnected twist-lines, whereas a strong field yields a nearly perfect nematic host with a Saturn-ring $V$4 disclination around the particle. After field removal, the system does not fully rewind to the initial cholesteric configuration but becomes trapped in metastable states, including stacked loops or a glassy amorphous network of disclinations (D'Adamo et al., 2015).

In steady cyclic driving, a square wave produces a two-state oscillation between a Saturn-ring state and a stacked-loop state, with negligible colloid motion, whereas a sine wave generates transient three-hoop configurations and a more intricate three-dimensional defect network growing and dissolving each cycle (D'Adamo et al., 2015). This establishes a broader temporal-control principle: the waveform, not only the amplitude, is a selector of accessible topological pathways.

5. Anchoring as a selector of temporal branches

Surface anchoring is the principal selector of defect topology and reversibility. In droplets, homeotropic anchoring yields many surface-pinned $V$5 and twist disclinations; under a field parallel to the cholesteric axis, these are dragged into the bulk and can form hyperbolic hedgehogs. The OFF states are strongly metastable and not easily restored. Tangential anchoring produces fewer defects, such as $V$6 or $V$7 pairs; under a parallel field, bulk bend stripes appear in the ON state, but relaxation returns almost fully to equilibrium. Free surfaces are defect-free, and ON–OFF cycles simply rotate the cholesteric axis by $V$8 (Fadda et al., 2017).

In planar cells, the same anchoring issue appears in a different mathematical form. The Rapini–Papoular surface anchoring potential is

$V$9

and the actual twist $q_0=2\pi/P_0$00 minimizes

$q_0=2\pi/P_0$01

For strong symmetric anchoring, the half-turn number is locked to an integer,

$q_0=2\pi/P_0$02

For semistrong anchoring, the equilibrium condition becomes

$q_0=2\pi/P_0$03

with stability condition

$q_0=2\pi/P_0$04

These relations predict continuous branches of $q_0=2\pi/P_0$05 versus $q_0=2\pi/P_0$06 that terminate in fold-type instabilities, forcing jump-like pitch transitions (Orlova et al., 2013).

For the asymmetric semistrong cell, one substrate is strong and the other has $q_0=2\pi/P_0$07, giving $q_0=2\pi/P_0$08 and $q_0=2\pi/P_0$09. The fitted stable branches satisfy

$q_0=2\pi/P_0$10

Successive jumps occur only between adjacent integer branches, $q_0=2\pi/P_0$11 (Orlova et al., 2013). In the symmetric strong–strong cell, by contrast, the system follows the integer-step model $q_0=2\pi/P_0$12, with jumps when $q_0=2\pi/P_0$13 crosses a half-integer average of neighboring branches (Orlova et al., 2013).

Taken together, these results show that anchoring does not simply alter boundary conditions; it sets the admissible temporal branches along which the cholesteric can evolve. In droplets, this branch selection appears as defect-rich metastable recovery versus near-complete restoration. In planar cells, it appears as integer locking versus fold-limited continuous evolution.

6. Photochemical pitch dynamics and measurement of $q_0=2\pi/P_0$14

The photosensitive cholesteric system of Orlova et al. introduces time dependence through UV-driven isomerization of the chiral dopant. The two relevant species are 7-DHC, with concentration $q_0=2\pi/P_0$15 and right-handed chirality, and tachysterol, with concentration $q_0=2\pi/P_0$16 and left-handed chirality. Under UV irradiation at $q_0=2\pi/P_0$17, the dominant process is essentially irreversible photo-isomerization,

$q_0=2\pi/P_0$18

so that

$q_0=2\pi/P_0$19

The instantaneous equilibrium pitch follows the helical-twisting-power relation

$q_0=2\pi/P_0$20

with $q_0=2\pi/P_0$21 for 7-DHC and $q_0=2\pi/P_0$22 for tachysterol (Orlova et al., 2013).

The actual pitch $q_0=2\pi/P_0$23 is then extracted polarimetrically. For normally incident linearly polarized He–Ne light at $q_0=2\pi/P_0$24, the transmitted field becomes elliptically polarized. In the circular basis,

$q_0=2\pi/P_0$25

where $q_0=2\pi/P_0$26, $q_0=2\pi/P_0$27 is the misalignment between rubbing and incident polarization, and $q_0=2\pi/P_0$28 is the rotating-wave transmission matrix. The ellipticity is

$q_0=2\pi/P_0$29

Experimentally, one measures $q_0=2\pi/P_0$30 after each UV dose and inverts the numerically computed function $q_0=2\pi/P_0$31 through a lookup table generated by exact transfer-matrix calculations, thereby obtaining $q_0=2\pi/P_0$32 (Orlova et al., 2013).

The physical interpretation given in the source is explicit: as the UV-driven reaction shifts the equilibrium twist $q_0=2\pi/P_0$33 from large positive to large negative, the cell passes from right-handed helices through the unwound nematic state to left-handed helices. Strong anchoring enforces integer half-turn numbers and therefore discrete jumps, while finite anchoring allows continuous distortion on each branch until a fold instability triggers a first-order transition (Orlova et al., 2013). This is a canonical realization of time-dependent cholesteric behavior in which the temporal driver is chemical rather than electrical.

7. Metastability, assembly, and device-relevant implications

Across droplets, planar cells, and colloid-doped cholesterics, the recurring motif is a free-energy landscape containing many competing metastable equilibria. In droplets with homeotropic anchoring and a field parallel to the cholesteric axis, the system becomes stuck in metastable states rich in topological defects; when the field is perpendicular to the cholesteric axis, the effect on defect dynamics is usually negligible (Fadda et al., 2017). In colloidal hosts, time-dependent electric fields drive the system reproducibly out of equilibrium through different kinetic pathways, generating states that range from Saturn rings to amorphous defect networks and stacks of disclination loops (D'Adamo et al., 2015).

The colloidal work further shows that these non-equilibrium pathways can reposition particles. In a colloidal dimer, each field switch produces an impulsive defect-mediated force of order $q_0=2\pi/P_0$34–$q_0=2\pi/P_0$35 pN, and after a few cycles the particles line up along the field direction in a columnar stack even though this has higher free energy than the starting state. In a dilute suspension of many colloids, square-wave cycles along $q_0=2\pi/P_0$36 assemble straight columns, and subsequent cycles along $q_0=2\pi/P_0$37 and $q_0=2\pi/P_0$38 reorganize those columns into planar sheets (D'Adamo et al., 2015). The proposed mechanism is a competition between the field-on impulse, which creates Saturn-ring or figure-eight structures and pulls particles in the plane orthogonal to the field, and the field-off impulse, which releases stored elastic energy, produces multiple loops and network segments, and preferentially displaces particles along the field axis.

Several design rules stated in the source define the operational regime. The threshold coupling for cholesteric-to-nematic switching is $q_0=2\pi/P_0$39, and robust Saturn-ring formation requires $q_0=2\pi/P_0$40. The pulse-on time should exceed the cholesteric relaxation time $q_0=2\pi/P_0$41, while the pulse-off time should be long enough, $q_0=2\pi/P_0$42, to allow loop nucleation. Square-wave driving yields the most reproducible two-state hopping, whereas sine waves populate more intermediate metastables and can impart stronger impulses. Hydrodynamic simulations confirm the same two competing impulses and show that viscous backflow can enhance net displacement (D'Adamo et al., 2015).

A common misconception is that time-dependent forcing simply accelerates equilibration to the nearest cholesteric state. The cited studies show the opposite in several regimes: forcing can stabilize defect-rich OFF states, rotate the cholesteric axis by $q_0=2\pi/P_0$43, induce branch-jump hysteresis in the half-turn number, or assemble colloids into structures of higher free energy than the initial configuration (Fadda et al., 2017). This suggests that temporal control in cholesterics is fundamentally a problem of path dependence and state selection. In that sense, “time cholesteric” behavior denotes a class of driven cholesteric phenomena in which the chronology of the stimulus—its direction, amplitude, frequency, waveform, and anchoring context—is part of the state definition itself.