.. index:: pair_style colloid
.. index:: pair_style colloid/gpu
.. index:: pair_style colloid/omp

pair_style colloid command
==========================

Accelerator Variants: *colloid/gpu*, *colloid/omp*

Syntax
""""""

.. code-block:: LAMMPS

   pair_style colloid cutoff

* cutoff = global cutoff for colloidal interactions (distance units)

Examples
""""""""

.. code-block:: LAMMPS

   pair_style colloid 10.0
   pair_coeff * *  25 1.0 10.0 10.0
   pair_coeff 1 1 144 1.0 0.0 0.0 3.0
   pair_coeff 1 2  75.398 1.0 0.0 10.0 9.0
   pair_coeff 2 2  39.478 1.0 10.0 10.0 25.0

Description
"""""""""""

Style *colloid* computes pairwise interactions between large colloidal
particles and small solvent particles using 3 formulas.  A colloidal
particle has a size > sigma; a solvent particle is the usual
Lennard-Jones particle of size sigma.

The colloid-colloid interaction energy is given by

.. math::

   U_A = & - \frac{A_{cc}}{6} \left[
   \frac{2 a_1 a_2}{r^2-\left(a_1+a_2\right)^2}
   + \frac{2 a_1 a_2}{r^2 - \left(a_1 - a_2\right)^2}
     + \mathrm{ln}
       \left(
  \frac{r^2-\left(a_1+a_2\right)^2}{r^2-\left(a_1-a_2\right)^2}
   \right)
  \right] \\
    & \\
    U_R = & \frac{A_{cc}}{37800}  \frac{\sigma^6}{r}
    \biggl[ \frac{r^2-7r\left(a_1+a_2\right)+6\left(a_1^2+7a_1a_2+a_2^2\right)}
  {\left(r-a_1-a_2\right)^7} \\
   &\qquad              +\frac{r^2+7r\left(a_1+a_2\right)+6\left(a_1^2+7a_1a_2+a_2^2\right)}
  {\left(r+a_1+a_2\right)^7}  \\
  &\qquad               -\frac{r^2+7r\left(a_1-a_2\right)+6\left(a_1^2-7a_1a_2+a_2^2\right)}
  {\left(r+a_1-a_2\right)^7} \\
  &\qquad       \left.  -\frac{r^2-7r\left(a_1-a_2\right)+6\left(a_1^2-7a_1a_2+a_2^2\right)}
  {\left(r-a_1+a_2\right)^7}
  \right]  \\
  & \\
  U = & U_A + U_R, \qquad r < r_c

where :math:`A_{cc}` is the Hamaker constant, :math:`a_1` and :math:`a_2` are the
radii of the two colloidal particles, and :math:`r_c` is the cutoff.  This
equation results from describing each colloidal particle as an
integrated collection of Lennard-Jones particles of size sigma and is
derived in :ref:`(Everaers) <Everaers1>`.

The colloid-solvent interaction energy is given by

.. math::

   U = \frac{2 ~ a^3 ~ \sigma^3 ~ A_{cs}}{9 \left( a^2 - r^2 \right)^3}
   \left[ 1 - \frac{\left(5 ~ a^6+45~a^4~r^2+63~a^2~r^4+15~r^6\right) \sigma^6}
   {15 \left(a-r\right)^6 \left( a+r \right)^6} \right], \quad r < r_c

where :math:`A_{cs}` is the Hamaker constant, *a* is the radius of the colloidal
particle, and :math:`r_c` is the cutoff.  This formula is derived from the
colloid-colloid interaction, letting one of the particle sizes go to
zero.

The solvent-solvent interaction energy is given by the usual
Lennard-Jones formula

.. math::

   U = \frac{A_{ss}}{36} \left[ \left( \frac{\sigma}{r}
        \right)^{12} - \left( \frac{ \sigma}{r} \right)^6 \right], \quad
        r < r_c

with :math:`A_{ss}` set appropriately, which results from letting both
particle sizes go to zero.

When used in combination with :doc:`pair_style yukawa/colloid
<pair_yukawa_colloid>`, the two terms become the so-called DLVO potential,
which combines electrostatic repulsion and van der Waals attraction.

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

* A (energy units)
* :math:`\sigma` (distance units)
* d1 (distance units)
* d2 (distance units)
* cutoff (distance units)

A is the Hamaker energy prefactor and should typically be set as
follows:

* :math:`A_{cc}` = colloid/colloid = :math:`4 \pi^2 = 39.5`
* :math:`A_{cs}` = colloid/solvent = :math:`\sqrt{A_{cc} A_{ss}}`
* :math:`A_{ss}` = solvent/solvent = 144 (assuming epsilon = 1, so that 144/36 = 4)

:math:`\sigma` is the size of the solvent particle or the constituent
particles integrated over in the colloidal particle and should typically
be set as follows:

* :math:`\sigma_{cc}` = colloid/colloid = 1.0
* :math:`\sigma_{cs}` = colloid/solvent = arithmetic mixing between colloid :math:`\sigma` and solvent :math:`\sigma`
* :math:`\sigma_{ss}` = solvent/solvent = 1.0 or whatever size the solvent particle is

Thus typically :math:`\sigma_{cs} = 1.0`, unless the solvent particle's size !=
1.0.

D1 and d2 are particle diameters, so that d1 = 2\*a1 and d2 = 2\*a2 in
the formulas above.  Both d1 and d2 must be values >= 0.  If d1 > 0
and d2 > 0, then the pair interacts via the colloid-colloid formula
above.  If d1 = 0 and d2 = 0, then the pair interacts via the
solvent-solvent formula.  I.e. a d value of 0 is a Lennard-Jones
particle of size :math:`\sigma`.  If either d1 = 0 or d2 = 0 and the other is
larger, then the pair interacts via the colloid-solvent formula.

Note that the diameter of a particular particle type may appear in
multiple pair_coeff commands, as it interacts with other particle
types.  You should ensure the particle diameter is specified
consistently each time it appears.

The last coefficient is optional.  If not specified, the global cutoff
specified in the pair_style command is used.  However, you typically
want different cutoffs for interactions between different particle
sizes.  E.g. if colloidal particles of diameter 10 are used with
solvent particles of diameter 1, then a solvent-solvent cutoff of 2.5
would correspond to a colloid-colloid cutoff of 25.  A good
rule-of-thumb is to use a colloid-solvent cutoff that is half the big
diameter + 4 times the small diameter.  I.e. 9 = 5 + 4 for the
colloid-solvent cutoff in this case.

.. note::

   When using pair_style colloid for a mixture with 2 (or more)
   widely different particles sizes (e.g. sigma=10 colloids in a
   background sigma=1 LJ fluid), you will likely want to use these
   commands for efficiency: :doc:`neighbor multi <neighbor>` and
   :doc:`comm_modify multi <comm_modify>`.

----------

.. include:: accel_styles.rst

----------

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

For atom type pairs I,J and I != J, the A, sigma, d1, and d2
coefficients and cutoff distance for this pair style can be mixed.  A
is an energy value mixed like a LJ epsilon.  D1 and d2 are distance
values and are mixed like sigma.  The default mix value is
*geometric*\ .  See the "pair_modify" command for details.

This pair style supports the :doc:`pair_modify <pair_modify>` shift
option for the energy of the pair interaction.

The :doc:`pair_modify <pair_modify>` table option is not relevant
for this pair style.

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

This pair style writes its information to :doc:`binary restart files <restart>`, 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
:doc:`run_style respa <run_style>` command.  It does not support the
*inner*, *middle*, *outer* keywords.

----------

Restrictions
""""""""""""

This style is part of the COLLOID package.  It is only enabled if
LAMMPS was built with that package.  See the :doc:`Build package <Build_package>` page for more info.

Normally, this pair style should be used with finite-size particles
which have a diameter, e.g. see the :doc:`atom_style sphere <atom_style>` command.  However, this is not a requirement,
since the only definition of particle size is via the pair_coeff
parameters for each type.  In other words, the physical radius of the
particle is ignored.  Thus you should ensure that the d1,d2 parameters
you specify are consistent with the physical size of the particles of
that type.

Per-particle polydispersity is not yet supported by this pair style;
only per-type polydispersity is enabled via the pair_coeff parameters.

Related commands
""""""""""""""""

:doc:`pair_coeff <pair_coeff>`

Default
"""""""

none

----------

.. _Everaers1:

**(Everaers)** Everaers, Ejtehadi, Phys Rev E, 67, 041710 (2003).
