2.3. The free energy

2.3.1. Available free energies

The choice of free energy is determined as follows:

free_energy   none

The default value is none, i.e., a simple Newtonian fluid is used. Possible values of the free_energy key are:

#  none                     Newtonian fluid [DEFAULT]
#  symmetric                Binary fluid (finite difference)
#  symmetric_lb             Binary fluid (two distributions)
#  brazovskii               Brazovskii smectics
#  polar_active             Polar active gels
#  lc_blue_phase            Liquid crystal (nematics, ...)
#  lc_droplet               Liquid crystal emulsions
#  fe_electro               Single fluid electrokinetics
#  fe_electro_symmetric     Binary fluid electrokinetics
#  ternary                  Three-component fluid

The choice of free energy will control automatically a number of factors related to choice of order parameter, the degree of parallel communication required, and so on. Each free energy has a number of associated parameters discussed in the following sections.

Details of general (Newtonian) fluid parameters, such as viscosity, are discussed in Simulation parameters.

2.3.2. Symmetric binary fluid

We recall the free energy is a functional of composition \(\phi\) and its density may be written:

\[f(\phi) = {\textstyle \frac{1}{2}} A \phi^2 + {\textstyle \frac{1}{4}} B \phi^4 + {\textstyle \frac{1}{2}}\kappa (\partial_\alpha \phi)^2.\]

Parameters are set in the input file via

# Binary fluid
free_energy            symmetric         # Use free energy
symmetric_a            -0.0625           # Bulk term          [required]
symmetric_b            +0.0625           # Bulk term          [required]
symmetric_kappa        +0.04             # Interfacial term   [required]
symmetric_c             0.00             # Surface term       [optional]
symmetric_h             0.00             # Surface term       [optional]

Common usage has \(A < 0\) and \(B = -A\) so that the separated phase has values \(\phi^\star = (-A/B)^{1/2} = \pm 1\). The parameter \(\kappa\) (key K) controls the interfacial energy penalty and is usually positive. The combination of parameters determines the interfacial width \(\xi = (-2\kappa/A)^{1/2}\) and the interfacial tension \(\sigma = 4\kappa\phi^{\star 2}/3\xi\).

The surface terms are discussed further in Plane walls.

In this approach, the fluid is treated using lattice Boltzmann, while the order parameter evolves according to a Cahn-Hilliard equation treated numerically via finite difference. For historical interest, the symmetric free energy problem can also be treated using two lattice Boltzmann distributions:

free_energy   symmetric_lb

Other parameters have the same meaning. This approach was used in an earlier implementation, is and discussed at some length in work including [Kendon2001].

[Kendon2001]

V.M. Kendon, M.E. Cates, I. Pagonabarraga, J.-C. Desplat, and P. Bladon, Inertial effects in three-dimensional spinodal decomposition of a symmetric binary fluid mixture: a lattice Boltzmann study, J. Fluid Mech., 440 147-203 (2001).

2.3.3. Brazovskii smectics

The free energy density is a function of the composition

\[f(\phi) = {\textstyle \frac{1}{2}} A \phi^2 + {\textstyle \frac{1}{4}} B \phi^4 + {\textstyle \frac{1}{2}} \kappa (\partial_\alpha \phi)^2 + {\textstyle \frac{1}{2}} C (\partial_\alpha^2 \phi)^2\]

Parameters are introduced via the keys

free_energy  brazovskii
A             -0.0005                        # Default: 0.0
B             +0.0005                        # Default: 0.0
K             -0.0006                        # Default: 0.0
C             +0.00076                       # Default: 0.0

For \(A < 0\), phase separation occurs with a result depending on \(\kappa\): one gets two symmetric phases for \(\kappa >0\) (cf. the symmetric case) or a lamellar phase for \(\kappa < 0\). Typically, \(B = -A\) and the parameter in the highest derivative \(C > 0\).

2.3.4. Polar active gels

The free energy density is a function of vector order parameter \(P_\alpha\):

\[f(P_\alpha) = {\textstyle \frac{1}{2}} A P_\alpha P_\alpha + {\textstyle \frac{1}{4}} B (P_\alpha P_\alpha)^2 + {\textstyle \frac{1}{2}} \kappa (\partial_\alpha P_\beta)^2\]

There are no default parameters:

free_energy        polar_active
polar_active_a    -0.1                       # Default: 0.0
polar_active_b    +0.1                       # Default: 0.0
polar_active_k     0.01                      # Default: 0.0

It is usual to choose \(B > 0\), in which case \(A > 0\) gives an isotropic phase, whereas \(A < 0\) gives a polar nematic phase. The elastic constant \(\kappa\) (key polar_active_k) is positive.

2.3.5. Liquid crystals

The free energy density is a function of tensor order parameter \(Q_{\alpha\beta}\):

\[f(Q_{\alpha\beta}) = {\textstyle\frac{1}{2}} A_0 (1 - \gamma/3)Q^2_{\alpha\beta} - {\textstyle\frac{1}{3}} A_0 \gamma Q_{\alpha\beta}Q_{\beta\delta}Q_{\delta\alpha} + {\textstyle\frac{1}{4}} A_0 \gamma (Q^2_{\alpha\beta})^2 + {\textstyle\frac{1}{2}} \big( \kappa_0 (\epsilon_{\alpha\delta\sigma} \partial_\delta Q_{\sigma\beta} + 2q_0 Q_{\alpha\beta})^2 + \kappa_1(\partial_\alpha Q_{\alpha\beta})^2 \big)\]

The corresponding free_energy value, despite its name, is suitable for nematics and cholesterics, and not just blue phases:

free_energy      lc_blue_phase
lc_a0            0.01                       # Default: 0.0
lc_gamma         3.0                        # Default: 0.0
lc_q0            0.19635                    # Default: 0.0
lc_kappa0        0.00648456                 # Default: 0.0
lc_kappa1        0.00648456                 # Default: 0.0

The bulk free energy parameter \(A_0\) is positive and controls the energy scale (key lc_a0); \(\gamma\) is positive and influences the position in the phase diagram relative to the isotropic/nematic transition (key lc_gamma). The two elastic constants must be equal, i.e., we enforce the single elastic constant approximation (both keys lc_kappa0 and lc_kappa1 must be specified).

Other important parameters in the liquid crystal picture are:

lc_xi            0.7                         # Default: 0.0
lc_Gamma         0.5                         # Default: 0.0

The first is \(\xi\) (key lc_xi) is the effective molecular aspect ratio and should be in the range \(0 < \xi < 1\). The rotational diffusion constant is \(\Gamma\) (key lc_Gamma; not to be confused with lc_gamma).

2.3.5.1. Liquid crystal activity

There exists an option to model contractile or extensile active fluids via the addition of an “active” stress in the liquid crystal free energy. For historical reasons, this is additional active stress written as

\[S_{\alpha\beta} = \zeta_0 \delta_{\alpha\beta} - \zeta_1 Q_{\alpha\beta} - \zeta_2 (\partial_\beta P_\alpha + \partial_\alpha P_\beta)\]

where \(P_\alpha = Q_{\alpha\gamma} \partial_\beta Q_{\beta\gamma}\). The first term in \(\zeta_0\) is included for completeness: it should only influence compressability and one may safely leave \(\zeta_0 = 0\). The second term models active force dipoles and sets the force density (\(\zeta_1 < 0\) for a contractile fluid, and \(\zeta_1 > 0\) for an extensile fluid). The third term in \(\zeta_2\) is experimental.

The relevant input keys and values are, e.g.,

lc_activity      yes                         # Required for activity
lc_active_zeta0  0.0                         # Default: 0.0
lc_active_zeta1  0.001                       # Default: 0.0
lc_active_zeta2  0.0                         # Default: 0.0

2.3.5.2. Liquid crystal anchoring

Different types of anchoring are available at solid surfaces, with one or two related free energy parameters depending on the type. The type of anchoring may be set independently for walls and colloids. Please see the apprpropriate section for details:

• Liquid crystal anchoring at colloid surfaces

• Anchoring for liquid crystals at plane walls

2.3.6. Liquid crystal emulsion

This an interaction free energy which combines the symmetric and liquid crystal free energies. The liquid crystal free energy constant \(\gamma\) becomes a function of composition via \(\gamma(\phi) = \gamma_0 + \delta(1 + \phi)\). Typically, we might choose \(\gamma_0\) and \(\delta\) so that \(\gamma(-\phi^\star) < 2.7\) and the \(-\phi^\star\) phase is isotropic, while \(\gamma(+\phi^\star) > 2.7\) and the \(+\phi^\star\) phase is ordered (nematic, cholesteric, or blue phase). Experience suggests that a suitable choice is \(\gamma_0 = 2.5\) and \(\delta = 0.25\).

A coupling term is added to the free energy density:

\[W Q_{\alpha\beta} \partial_\alpha \phi \partial_\beta \phi.\]

For anchoring constant \(W > 0\), the liquid crystal anchoring at the interface is planar, while for \(W < 0\) the anchoring is normal. This is set via key lc_droplet_W.

Relevant keys (with default values) are:

free_energy            lc_droplet

A                      -0.0625
B                      +0.0625
K                      +0.053

lc_a0                   0.1
lc_q0                   0.19635
lc_kappa0               0.007
lc_kappa1               0.007

lc_droplet_gamma        2.586                # Default: 0.0
lc_droplet_delta        0.25                 # Default: 0.0
lc_droplet_W           -0.05                 # Default: 0.0

Note that key lc_gamma is not used in this case.

2.3.6.1. Liquid crystal emulsion activity

An option for an additional active stress in the case of an emulsion is present. The form of the stress allows for activity in the ordered phase (\(\phi = +1\)):

\[S_{\alpha\beta} = {\textstyle\frac{1}{2}} (1 + \phi) [ \zeta_0 \delta_{\alpha\beta} - \zeta_1 Q_{\alpha\beta} ].\]

The meaning of the active terms is the same as for the bare (active) liquid crystal case.

The relevant input keys are:

lc_droplet_active_zeta0    0.0       # Default 0.0
lc_droplet_active_zeta1    0.001     # Default 0.0

Note that these are separate from the bare liquid crystal activity parameters (which are not used at the same time).

2.3.6.2. Liquid crystal in external electric field

A term the the free energy density arising from an external electric field is available, being written

\[f(Q_{\alpha\beta}) = - \frac{\epsilon_a}{12\pi} E_\alpha Q_{\alpha\beta} E_\beta\]

where \(E_\alpha\) is the external field, and \(\epsilon_a\) is the dielectric anistropy. These terms may be specified via

lc_dielectric_anisotropy    41.4            # Should be +ve; default 0
electric_e0                 0.01_0.0_0.0

It is convenient to render the free energy density dimensionless, in which case a reduced, or effective electric field strength is defined:

\[e^2 = \frac{27 \epsilon_a}{32 \pi A_0 \gamma} E_\alpha E_\alpha.\]

The quantity \(e\) is computed and reported by the code when an external field is present.

2.3.7. Ternary free energy

An implementation of the ternary model following [Semprebon] is available. This uses a lattice Boltzmann density \(\rho\) coupled to two scalar order parameters \(\phi\) and \(\psi\) to give three components. The two scalar order parameters each evolve via a Cahn-Hilliard equation treated by finite difference.

The basic free energy parameters are:

free_energy               ternary            # Select ternary free energy

ternary_kappa1            0.01               # Interfacial parameter > 0
ternary_kappa2            0.02               # Interfacial parameter > 0
ternary_kappa3            0.05               # Interfacial parameter > 0
ternary_alpha             1.00               # Interfical width

ternary_mobility_phi      0.15               # Mobility for phi
ternary_mobility_psi      0.10               # Mobility for psi

All the parameters must be specified.

As the description is rather involved, we do not repeat it here.

[Semprebon]

C. Semprebon, T. Krueger, and H. Kusumaatmaja, Ternary free-energy lattice Boltzmann model with tunable contact angles, Phys. Rev. E, 93 033305 (2016).

2.3.8. Electrokinetic free energy

An implementation of a free energy appropriate for charged fluids, the electrokinetic problem, is available following [Capuani].

The approach features two or more charged species (usually two), and solves the Nernst-Planck equation for charge transport (advection and diffusion for charge) coupled to the Navier-Stokes equations. A solver for the Poisson equation which relates the charge density to the electric potential is also required.

The basic free energy parameters are, here for two charged species:

free_energy               fe_electro         # Electrokinetics

electrokinetcs_eunit      1.0                # The unit charge
electrokinetics_z0        +1                 # Valency species one
electrokinetics_z1        -1                 # Valency species two
electrokinetics_d0        0.0125             # Diffusivity species one
electrokinetics_d1        0.0075             # Diffusivity species two
electrockinetcs_epsilon   3300.0             # Permittivity

Further details of the electrockinetic problem and some simple examples appear in the Electrokinetics tutorial section.

[Capuani]

F. Capuani, I. Pagonabarraga, and D. Frenkel, Discrete solution of the electrokinetic equations, j. Chem. Phys. 121 973–986 (2004).