2.9. Order Parameters

2.9.1. Order parameter initialisations

2.9.1.1. 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

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

2.9.3. Order parameter gradients

Free energies involving an order parameter typically require computing the spatial gradients of that order parameter. There are various different stencils available for this computation.

If surface wetting (binary/ternary fluids) or anchoring (liquid crystals) are required, the gradient computation must take account of the surface. A particular gradient computation will be necessary.

If Lees-Edwards sliding periodic boundary conditions are required, the gradient computation must again take account of the co-ordinate transformations involved. A subset of the available stencils support the Lees-Edwards transformation.

If a two-dimensional system is used with \(L_z = 1\), it is recommended to use a two-dimensional gradient stencil (usually with “2d” in the name). This may become mandatory in the future.

More generally, the width of the system should be at least twice the halo width required for the free energy to ensure correct operation. This is generally computed internally, so should be correct. However, care may be required for “thin” three-dimensional systems.