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Theory and modeling
This protocol is extracted from research article:
Tunable structure and dynamics of active liquid crystals
Sci Adv, Oct 12, 2018; DOI: 10.1126/sciadv.aat7779

Theoretical model. The bulk free energy of the nematic LC, F, is defined asEmbedded Image(1)where fLdG is the short-range free energy, fel is the long-range elastic energy, and fsurf is the surface free energy due to anchoring. fLdG is given by a Landau–de Gennes expression of the form (30, 43)Embedded Image(2)

Parameter U controls the magnitude of q0, namely, the equilibrium scalar order parameter via Embedded Image. The elastic energy fel is written as (Qij,k means ∂kQij)Embedded Image(3)

If the system is uniaxial, the above equation is equivalent to the Frank-Oseen expressionEmbedded Image(4)

The L values in Eq. 3 can then be mapped to the K values in Eq. 4 viaEmbedded Image(5)

By assuming a one elastic constant K11 = K22 = K33 = K24K, one has Embedded Image and L2 = L3 = L4 = 0. Point-wise, n is the eigenvector associated with the greatest eigenvalue of the Q-tensor at each lattice point.

To simulate the LC’s nonequilibrium dynamics, a hybrid lattice Boltzmann method was used to simultaneously solve a Beris-Edwards equation and a momentum equation, which accounts for the hydrodynamic effects. By introducing a velocity gradient Wij = ∂jui, strain rate A = (W + WT)/2, vorticity Ω = (WWT)/2, and a generalized advection termEmbedded Image(6)one can write the Beris-Edwards equation (44) according toEmbedded Image(7)

The constant ξ is related to the material’s aspect ratio, and Γ is related to the rotational viscosity γ1 of the system by Embedded Image (45). The molecular field H, which drives the system toward thermodynamic equilibrium, is given byEmbedded Image(8)where […]st is a symmetric and traceless operator. When velocity is absent, that is, u(r) ≡ 0, Beris-Edwards equation (Eq. 7) reduces to Ginzburg-Landau equationEmbedded Image

To calculate the static structures of ±1/2 defects, we adopted the above equation to solve for the Q-tensor at equilibrium.

Degenerate planar anchoring is implemented through a Fournier-Galatola expression (46) that penalizes out-of-plane distortions of the Q tensor. The associated free energy expression is given byEmbedded Image(9)where Embedded Image and Embedded Image. Here, P is the projection operator associated with the surface normal ν as P = Iνν. The evolution of the surface Q-field at one-constant approximation is governed by (47)Embedded Image(10)where Γs = Γ/ξN with Embedded Image, namely, nematic coherence length.

Using an Einstein summation rule, the momentum equation for the nematics can be written as (45, 48)Embedded Image(11)

The stress Π = Πp + Πa consists of a passive and an active part. The passive stress Πp is defined asEmbedded Image(12)where η is the isotropic viscosity, and the hydrostatic pressure P0 is given by (49)Embedded Image(13)

The temperature T is related to the speed of sound cs by Embedded Image. The active stress reads (50)Embedded Image(14)

in which α is the activity in the simulation. The stress becomes extensile when α > 0 and contractile when α < 0.

Numerical details. We solve the evolution Eq. 7 using a finite difference method. The momentum Eq. 11 is solved simultaneously via a lattice Boltzmann method over a D3Q15 grid (51). The implementation of stress follows the approach proposed by Guo et al. (52). The units are chosen as follows: The unit length a is chosen to be a = ξN = 1 μm, characteristic of the filament length, the characteristic viscosity is set to γ1 = 0.1 Pa∙s, and the force scale is made to be F0 = 10−11 N. Other parameters are chosen to be A0 = 0.1, K = 0.1, ξ = 0.8, Γ = 0.13, η = 0.33, and U = 3.5, leading to q0 ≈ 0.62. The simulation was performed in a rectangular box. The boundary conditions in the xy plane are periodic with size [Nx, Ny] = [250, 250]. Two confining walls were introduced in the z dimension, with strong degenerate planar anchoring, ensuring a quasi-2D system with z dimension 7 ≤ Nz ≤ 11. We refer the reader to (47) for additional details on the numerical methods used here.

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