2.9. Order Parameters

2.9.1. Order parameter initialisations Composition \(\phi\)

The following initialisations are available.

phi_initialisation          spinodal    # spinodal
phi0                        0.0         # mean
noise                       0.05        # noise amplitude
random_seed                 102839      # +ve integer

Suitable for initialising isothermal spinodal decomposition, the order parameter may be set at random at each position via \(\phi = \phi_0 + A(\phi_r - 1/2)\) with the random variate \(\phi_r\) selected uniformly on the interval \([0,1)\). For symmetric quenches (mean order parameter \(\phi_0 = 0\) and \(\phi^\star = \pm 1\)), a value of \(A\) in the range 0.05-0.1 is usually appropriate.

For off-symmetric quenches, larger patches of fluid may be required to initiate decomposition:

phi_initialisation          patches     # patches of phi = +/- 1
phi_init_patch_size         2           # patch size
phi_init_patch_vol          0.1         # volume fraction phi = -1 phase
random_seed                 13          # +ve integer

The initialises cubics patches of fluid of given size with \(\phi= \pm 1\) at random. The requested overall volume fractions may be met approximately.

A uniform value of the order parameter may be apprpropriate for some situations. This is arranged using a single value \(\phi_0\):

phi_initialisation          uniform     # same everywhere
phi0                        0.2         # the uniform value

A spherical drop can be initialised at the centre of the system.

phi_initialisation          drop        # spherical droplet
phi_init_drop_radius        16.0        # radius
phi_init_drop_amplitude    -1.0         # phi value inside

The drop is initialised with a \(\tanh(r/\xi)\) profile where the interfacial width \(\xi\) is computed via the current free energy parameters.

A pair of plane interfaces at \(z = L_z/4\) and \(z=3L_z/4\) may be initialised via

phi_initialisation          block

The interfacial width is again set via the current free energy parameters. The centre of the system has order parameter \(\phi = +\phi^\star\).

For restarted simulations, the default position is to read order parameter information from file

phi_initialisation          from_file

in which case a file or files for the appropriate time step should be present in the working directory.

2.9.2. Tensor order parameter

A number of different initialisations are available for the liquid crystal order parameter \(Q_{\alpha\beta}\). Some care may be required to ensure consistency between the choice and the free energy parameters, the system size, and so on (particularly for the blue phases).

A summary of choices is:

lc_q_initialisation   nematic          # uniform nematic...
lc_init_nematic       1.0_0.0_0.0      # ...with given director

lc_q_initialisation   cholesteric_x    # cholesteric with helical axis x
lc_q_initialisation   cholesteric_y    # cholesteric with helical axis y
lc_q_initialisation   cholesteric_z    # cholesteric with helical axis z

lc_q_initialisation   o8m              # BPI high chirality limit
lc_q_initialisation   o2               # BPII high chirality limit
lc_q_initialisation   o5
lc_q_initialisation   h2d              # 2d hexagonal
lc_q_initialisation   h3da             # 3d hexagonal BP A
lc_q_initialisation   h3db             # 3d hexagonal BP B
lc_q_initialisation   dtc              # double twist cylinders

lc_q_initialisation   bp3

lc_q_initialisation   cf1_x            # cholesteric ``finger'' axis x
lc_q_initialisation   cf1_y            # cholesteric ``finger'' axis y
lc_q_initialisation   cf1_z            # cholesteric ``finger'' axis z

lc_q_initialisation   cf1_fluc_x       # as cf1_x with random perterbations
lc_q_initialisation   cf1_fluc_y       # as cf1_y with random perturbations
lc_q_initialisation   cf1_flux_z       # as cf1_z with random perturbations

lc_q_initialisation   random           # with randomly chosen unit director
lc_q_initialisation   random_xy        # random nematic in (x,y) plane

Note many of the initialiations require an initial amplitude of order, which should be set via

lc_q_init_amplitude   0.01             # initial amplitude of order A

For example, if an initial uniform nematic is requested with unit director \(n_\alpha\), the corresponding initial tensor will be

\[Q_{\alpha\beta} = {\textstyle \frac{1}{2}} A (3 n_\alpha n_\beta - \delta_{\alpha\beta}).\]