Order Parameters ---------------- .. contents:: In this section :depth: 1 :local: :backlinks: none Order parameter initialisations ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Composition :math:`\phi` """""""""""""""""""""""" The following initialisations are available. .. code-block:: none 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 :math:`\phi = \phi_0 + A(\phi_r - 1/2)` with the random variate :math:`\phi_r` selected uniformly on the interval :math:`[0,1)`. For symmetric quenches (mean order parameter :math:`\phi_0 = 0` and :math:`\phi^\star = \pm 1`), a value of :math:`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: .. code-block:: none 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 :math:`\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 :math:`\phi_0`: .. code-block:: none phi_initialisation uniform # same everywhere phi0 0.2 # the uniform value A spherical drop can be initialised at the centre of the system. .. code-block:: none 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 :math:`\tanh(r/\xi)` profile where the interfacial width :math:`\xi` is computed via the current free energy parameters. A pair of plane interfaces at :math:`z = L_z/4` and :math:`z=3L_z/4` may be initialised via .. code-block:: none phi_initialisation block The interfacial width is again set via the current free energy parameters. The centre of the system has order parameter :math:`\phi = +\phi^\star`. For restarted simulations, the default position is to read order parameter information from file .. code-block:: none phi_initialisation from_file in which case a file or files for the appropriate time step should be present in the working directory. Tensor order parameter ^^^^^^^^^^^^^^^^^^^^^^ A number of different initialisations are available for the liquid crystal order parameter :math:`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: .. code-block:: none 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 .. code-block:: none lc_q_init_amplitude 0.01 # initial amplitude of order A For example, if an initial uniform nematic is requested with unit director :math:`n_\alpha`, the corresponding initial tensor will be .. math:: Q_{\alpha\beta} = {\textstyle \frac{1}{2}} A (3 n_\alpha n_\beta - \delta_{\alpha\beta}). 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 :math:`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.