\(\renewcommand{\AA}{\text{Å}}\)

# pair_style lubricate command

Accelerator Variants: lubricate/omp

# pair_style lubricate/poly command

Accelerator Variants: lubricate/poly/omp

## Syntax

    pair_style style mu flaglog flagfld cutinner cutoff flagHI flagVF

- style = lubricate or lubricate/poly

- mu = dynamic viscosity (dynamic viscosity units)

- flaglog = 0/1 to exclude/include log terms in the lubrication approximation

- flagfld = 0/1 to exclude/include Fast Lubrication Dynamics (FLD) effects

- cutinner = inner cutoff distance (distance units)

- cutoff = outer cutoff for interactions (distance units)

- flagHI (optional) = 0/1 to exclude/include 1/r hydrodynamic interactions

- flagVF (optional) = 0/1 to exclude/include volume fraction corrections in the long-range isotropic terms

## Examples

(all assume radius = 1)

    pair_style lubricate 1.5 1 1 2.01 2.5
    pair_coeff 1 1 2.05 2.8
    pair_coeff * *

    pair_style lubricate 1.5 1 1 2.01 2.5
    pair_coeff * *
    variable mu equal ramp(1,2)
    fix 1 all adapt 1 pair lubricate mu * * v_mu

## Description

Styles lubricate and lubricate/poly compute hydrodynamic interactions between mono-disperse finite-size spherical particles in a pairwise fashion. The interactions have 2 components. The first is Ball-Melrose lubrication terms via the formulas in (Ball and Melrose)

\[\begin{split}W & = - a_{sq} | (v_1 - v_2) \bullet \mathbf{nn} |^2 - a_{sh} | (\omega_1 + \omega_2) \bullet (\mathbf{I} - \mathbf{nn}) - 2 \Omega_N |^2 - \\ & a_{pu} | (\omega_1 - \omega_2) \bullet (\mathbf{I} - \mathbf{nn}) |^2 - a_{tw} | (\omega_1 - \omega_2) \bullet \mathbf{nn} |^2 \qquad r < r_c \\ & \\ \Omega_N & = \mathbf{n} \times (v_1 - v_2) / r\end{split}\]

which represents the dissipation W between two nearby particles due to their relative velocities in the presence of a background solvent with viscosity mu. Note that this is dynamic viscosity which has units of mass/distance/time, not kinematic viscosity.

The Asq (squeeze) term is the strongest and is included if flagHI is set to 1 (default). It scales as 1/gap where gap is the separation between the surfaces of the 2 particles. The Ash (shear) and Apu (pump) terms are only included if flaglog is set to 1. They are the next strongest interactions, and the only other singular interaction, and scale as log(gap). Note that flaglog = 1 and flagHI = 0 is invalid, and will result in a warning message, after which flagHI will be set to 1. The Atw (twist) term is currently not included. It is typically a very small contribution to the lubrication forces.

The flagHI and flagVF settings are optional. Neither should be used, or both must be defined.

Cutinner sets the minimum center-to-center separation that will be used in calculations irrespective of the actual separation. Cutoff is the maximum center-to-center separation at which an interaction is computed. Using a cutoff less than 3 radii is recommended if flaglog is set to 1.

The other component is due to the Fast Lubrication Dynamics (FLD) approximation, described in (Kumar), which can be represented by the following equation

\[F^{H} = -R_{FU}(U-U^{\infty}) + R_{FE}E^{\infty}\]

where U represents the velocities and angular velocities of the particles, \(U^{\infty}\) represents the velocity and the angular velocity of the undisturbed fluid, and \(E^{\infty}\) represents the rate of strain tensor of the undisturbed fluid with viscosity mu. Again, note that this is dynamic viscosity which has units of mass/distance/time, not kinematic viscosity. Volume fraction corrections to R_FU are included as long as flagVF is set to 1 (default).

Note

When using the FLD terms, these pair styles are designed to be used with explicit time integration and a correspondingly small timestep. Thus either fix nve/sphere or fix nve/asphere should be used for time integration. To perform implicit FLD, see the pair_style lubricateU command.

Style lubricate requires monodisperse spherical particles; style lubricate/poly allows for polydisperse spherical particles.

The viscosity mu can be varied in a time-dependent manner over the course of a simulation, in which case in which case the pair_style setting for mu will be overridden. See the fix adapt command for details.

If the suspension is sheared via the fix deform command then the pair style uses the shear rate to adjust the hydrodynamic interactions accordingly. Volume changes due to fix deform are accounted for when computing the volume fraction corrections to R_FU.

When computing the volume fraction corrections to R_FU, the presence of walls (whether moving or stationary) will affect the volume fraction available to colloidal particles. This is currently accounted for with the following types of walls: wall/lj93, wall/lj126, wall/colloid, and wall/harmonic. For these wall styles, the correct volume fraction will be used when walls do not coincide with the box boundary, as well as when walls move and thereby cause a change in the volume fraction. Other wall styles may still work, but they will result in the volume fraction being computed based on the box boundaries. Several wall styles are not compatible with these pair styles and using them will result in an error.

Since lubrication forces are dissipative, it is usually desirable to thermostat the system at a constant temperature. If Brownian motion (at a constant temperature) is desired, it can be set using the pair_style brownian command. These pair styles and the brownian style should use consistent parameters for mu, flaglog, flagfld, cutinner, cutoff, flagHI and flagVF.

The following coefficients must be defined for each pair of atoms types via the pair_coeff command as in the examples above, or in the data file or restart files read by the read_data or read_restart commands, or by mixing as described below:

- cutinner (distance units)

- cutoff (distance units)

The two coefficients are optional. If neither is specified, the two cutoffs specified in the pair_style command are used. Otherwise both must be specified.

Styles with a gpu, intel, kk, omp, or opt suffix are functionally the same as the corresponding style without the suffix. They have been optimized to run faster, depending on your available hardware, as discussed on the Accelerator packages page. The accelerated styles take the same arguments and should produce the same results, except for round-off and precision issues.

These accelerated styles are part of the GPU, INTEL, KOKKOS, OPENMP, and OPT packages, respectively. They are only enabled if LAMMPS was built with those packages. See the Build package page for more info.

You can specify the accelerated styles explicitly in your input script by including their suffix, or you can use the -suffix command-line switch when you invoke LAMMPS, or you can use the suffix command in your input script.

See the Accelerator packages page for more instructions on how to use the accelerated styles effectively.

## Mixing, shift, table, tail correction, restart, rRESPA info

For atom type pairs I,J and I != J, the two cutoff distances for this pair style can be mixed. The default mix value is geometric. See the “pair_modify” command for details.

This pair style does not support the pair_modify shift option for the energy of the pair interaction.

The pair_modify table option is not relevant for this pair style.

This pair style does not support the pair_modify tail option for adding long-range tail corrections to energy and pressure.

This pair style writes its information to binary restart files, so pair_style and pair_coeff commands do not need to be specified in an input script that reads a restart file.

This pair style can only be used via the pair keyword of the run_style respa command. It does not support the inner, middle, outer keywords.

## Restrictions

These styles are part of the COLLOID package. They are only enabled if LAMMPS was built with that package. See the Build package page for more info.

Only spherical monodisperse particles are allowed for pair_style lubricate.

Only spherical particles are allowed for pair_style lubricate/poly.

These pair styles will not restart exactly when using the read_restart command, though they should provide statistically similar results. This is because the forces they compute depend on atom velocities. See the read_restart command for more details.

## Related commands

pair_coeff, pair_style lubricateU

## Default

The default settings for the optional args are flagHI = 1 and flagVF = 1.

(Ball) Ball and Melrose, Physica A, 247, 444-472 (1997).

(Kumar) Kumar and Higdon, Phys Rev E, 82, 051401 (2010). See also his thesis for more details: A. Kumar, “Microscale Dynamics in Suspensions of Non-spherical Particles”, Thesis, University of Illinois Urbana-Champaign, (2010). (https://www.ideals.illinois.edu/handle/2142/16032)