.. index:: fix ttm
.. index:: fix ttm/grid
.. index:: fix ttm/mod

fix ttm command
===============

fix ttm/grid command
====================

fix ttm/mod command
===================

Syntax
""""""

.. parsed-literal::

   fix ID group-ID ttm seed C_e rho_e kappa_e gamma_p gamma_s v_0 Nx Ny Nz keyword value ...
   fix ID group-ID ttm/mod seed init_file Nx Ny Nz keyword value ...

* ID, group-ID are documented in :doc:`fix <fix>` command
* style = *ttm* or *ttm/grid* or *ttm/mod*
* seed = random number seed to use for white noise (positive integer)
* remaining arguments for fix ttm or fix ttm/grid

  .. parsed-literal::

       C_e  = electronic specific heat (energy/(electron\*temperature) units)
       rho_e = electronic density (electrons/volume units)
       kappa_e = electronic thermal conductivity (energy/(time\*distance\*temperature) units)
       gamma_p = friction coefficient due to electron-ion interactions (mass/time units)
       gamma_s = friction coefficient due to electronic stopping (mass/time units)
       v_0 = electronic stopping critical velocity (velocity units)
       Nx = number of thermal solve grid points in the x-direction (positive integer)
       Ny = number of thermal solve grid points in the y-direction (positive integer)
       Nz = number of thermal solve grid points in the z-direction (positive integer)

* remaining arguments for fix ttm/mod:

  .. parsed-literal::

       init_file = file with the parameters to TTM
       Nx = number of thermal solve grid points in the x-direction (positive integer)
       Ny = number of thermal solve grid points in the y-direction (positive integer)
       Nz = number of thermal solve grid points in the z-direction (positive integer)

* zero or more keyword/value(s) pairs may be appended
* keyword = *set* or *infile* or *outfile*

  .. parsed-literal::

       *set* value = Tinit
         Tinit = initial electronic temperature at all grid points (temperature units)
       *infile* value = file.in with grid values for electronic temperatures
       *outfile* values = Nout file.out
         Nout = dump grid temperatures every this many timesteps
         file.out = filename to write grid temperatures to

Examples
""""""""

.. code-block:: LAMMPS

   fix 2 all ttm 699489 1.0 1.0 10 0.1 0.0 2.0 1 12 1 infile initial outfile 1000 T.out
   fix 3 all ttm/grid 123456 1.0 1.0 1.0 1.0 1.0 5.0 5 5 5 infile Te.in
   fix 4 all ttm/mod 34277 parameters.txt 5 5 5 infile T_init outfile 10 T_out

Example input scripts using these commands can be found in examples/ttm.

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

Use a two-temperature model (TTM) to represent heat transfer through
and between electronic and atomic subsystems.  LAMMPS models the
atomic subsystem as usual with a molecular dynamics model and the
classical force field specified by the user.  The electronic subsystem
is modeled as a continuum, or a background "gas", on a regular grid
which overlays the simulation domain.  Energy can be transferred
spatially within the grid representing the electrons.  Energy can also
be transferred between the electronic and atomic subsystems.  The
algorithm underlying this fix was derived by D. M.  Duffy
and A. M. Rutherford and is discussed in two J Physics: Condensed
Matter papers: :ref:`(Duffy) <Duffy>` and :ref:`(Rutherford)
<Rutherford>`.  They used this algorithm in cascade simulations where
a primary knock-on atom (PKA) was initialized with a high velocity to
simulate a radiation event.

The description in this sub-section applies to all 3 fix styles:
*ttm*, *ttm/grid*, and *ttm/mod*.

Fix *ttm/grid* distributes the regular grid across processors consistent
with the sub-domains of atoms owned by each processor, but is otherwise
identical to fix ttm.  Note that fix *ttm* stores a copy of the grid on
each processor, which is acceptable when the overall grid is reasonably
small.  For larger grids you should use fix *ttm/grid* instead.

Fix *ttm/mod* adds options to account for external heat sources (e.g. at
a surface) and for specifying parameters that allow the electronic
heat capacity to depend strongly on electronic temperature.  It is
more expensive computationally than fix *ttm* because it treats the
thermal diffusion equation as non-linear.  More details on fix *ttm/mod*
are given below.

Heat transfer between the electronic and atomic subsystems is carried
out via an inhomogeneous Langevin thermostat.  Only atoms in the fix
group contribute to and are affected by this heat transfer.

This thermostatting differs from the regular Langevin thermostat
(:doc:`fix langevin <fix_langevin>`) in three important ways.  First,
the Langevin thermostat is applied uniformly to all atoms in the
user-specified group for a single target temperature, whereas the TTM
fixes apply Langevin thermostatting locally to atoms within the
volumes represented by the user-specified grid points with a target
temperature specific to that grid point.  Second, the Langevin
thermostat couples the temperature of the atoms to an infinite heat
reservoir, whereas the heat reservoir for the TTM fixes is finite and
represents the local electrons.  Third, the TTM fixes allow users to
specify not just one friction coefficient, but rather two independent
friction coefficients: one for the electron-ion interactions
(*gamma_p*), and one for electron stopping (*gamma_s*).

When the friction coefficient due to electron stopping, *gamma_s*, is
non-zero, electron stopping effects are included for atoms moving
faster than the electron stopping critical velocity, *v_0*.  For
further details about this algorithm, see :ref:`(Duffy) <Duffy>` and
:ref:`(Rutherford) <Rutherford>`.

Energy transport within the electronic subsystem is solved according
to the heat diffusion equation with added source terms for heat
transfer between the subsystems:

.. math::

  C_e \rho_e \frac{\partial T_e}{\partial t} =
  \bigtriangledown (\kappa_e \bigtriangledown T_e) -
  g_p (T_e - T_a) + g_s T_a'

where C_e is the specific heat, rho_e is the density, kappa_e is the
thermal conductivity, T is temperature, the "e" and "a" subscripts
represent electronic and atomic subsystems respectively, g_p is the
coupling constant for the electron-ion interaction, and g_s is the
electron stopping coupling parameter.  C_e, rho_e, and kappa_e are
specified as parameters to the fix.  The other quantities are derived.
The form of the heat diffusion equation used here is almost the same
as that in equation 6 of :ref:`(Duffy) <Duffy>`, with the exception that the
electronic density is explicitly represented, rather than being part
of the specific heat parameter.

Currently, the TTM fixes assume that none of the user-supplied
parameters will vary with temperature. Note that :ref:`(Duffy)
<Duffy>` used a tanh() functional form for the temperature dependence
of the electronic specific heat, but ignored temperature dependencies
of any of the other parameters.  See more discussion below for fix
ttm/mod.

..note::

  These fixes do not perform time integration of the atoms in the fix
  group, they only rescale their velocities.  Thus a time integration
  fix such as :doc:`fix nve <fix_nve>` should be used in conjunction
  with these fixes.  These fixes should not normally be used on atoms
  that have their temperature controlled by another thermostatting
  fix, e.g. :doc:`fix nvt <fix_nh>` or :doc:`fix langevin
  <fix_langevin>`.

..note::

  These fixes require use of an orthogonal 3d simulation box with
  periodic boundary conditions in all dimensions.  They also require
  that the size and shape of the simulation box do not vary
  dynamically, e.g. due to use of the :doc:`fix npt <fix_nh>` command.
  Likewise, the size/shape of processor sub-domains cannot vary due to
  dynamic load-balancing via use of the :doc:`fix balance
  <fix_balance>` command.  It is possible however to load balance
  before the simulation starts using the :doc:`balance <balance>`
  command, so that each processor has a different size sub-domain.

Periodic boundary conditions are also used in the heat equation solve
for the electronic subsystem.  This varies from the approach of
:ref:`(Rutherford) <Rutherford>` where the atomic subsystem was
embedded within a larger continuum representation of the electronic
subsystem.

The *set* keyword specifies a *Tinit* temperature value to initialize
the value stored on all grid points.

The *infile* keyword specifies an input file of electronic temperatures
for each grid point to be read in to initialize the grid.  By default
the temperatures are all zero when the grid is created.  The input file
is a text file which may have comments starting with the '#' character.
Each line contains four numeric columns: ix,iy,iz,Temperature.  Empty
or comment-only lines will be ignored. The
number of lines must be equal to the number of user-specified grid
points (Nx by Ny by Nz).  The ix,iy,iz are grid point indices ranging
from 0 to nxnodes-1 inclusive in each dimension.  The lines can appear
in any order.  For example, the initial electronic temperatures on a 1
by 2 by 3 grid could be specified in the file as follows:

.. parsed-literal::

   # UNITS: metal COMMENT: initial electron temperature
   0 0 0 1.0
   0 0 1 1.0
   0 0 2 1.0
   0 1 0 2.0
   0 1 1 2.0
   0 1 2 2.0

where the electronic temperatures along the y=0 plane have been set to
1.0, and the electronic temperatures along the y=1 plane have been set
to 2.0.  If all the grid point values are not specified, LAMMPS will
generate an error. LAMMPS will check if a "UNITS:" tag is in the first
line and stop with an error, if there is a mismatch with the current
units used.

..note::

  The electronic temperature at each grid point must be a non-zero
  positive value, both initially, and as the temperature evovles over
  time.  Thus you must use either the *set* or *infile* keyword or be
  restarting a simulation that used this fix previously.

The *outfile* keyword has 2 values.  The first value *Nout* triggers
output of the electronic temperatures for each grid point every Nout
timesteps.  The second value is the filename for output which will
be suffixed by the timestep.  The format of each output file is exactly
the same as the input temperature file. It will contain a comment in
the first line reporting the date the file was created, the LAMMPS
units setting in use, grid size and the current timestep.

Note that the atomic temperature for atoms in each grid cell can also
be computed and output by the :doc:`fix ave/chunk <fix_ave_chunk>`
command using the :doc:`compute chunk/atom <compute_chunk_atom>`
command to create a 3d array of chunks consistent with the grid used
by this fix.

----------

**Additional details for fix ttm/mod**

Fix *ttm/mod* uses the heat diffusion equation with possible external
heat sources (e.g. laser heating in ablation simulations):

.. math::

  C_e \rho_e \frac{\partial T_e}{\partial t} =
  \bigtriangledown (\kappa_e \bigtriangledown T_e) -
  g_p (T_e - T_a) + g_s T_a' + \theta (x-x_{surface})I_0 \exp(-x/l_{skin})

where theta is the Heaviside step function, I_0 is the (absorbed)
laser pulse intensity for ablation simulations, l_skin is the depth
of skin-layer, and all other designations have the same meaning as in
the former equation. The duration of the pulse is set by the parameter
*tau* in the *init_file*.

Fix ttm/mod also allows users to specify the dependencies of C_e and
kappa_e on the electronic temperature. The specific heat is expressed
as

.. math::

  C_e = C_0 + (a_0 + a_1 X + a_2 X^2 + a_3 X^3 + a_4 X^4) \exp (-(AX)^2)

where *X* = T_e/1000, and the thermal conductivity is defined as
kappa_e = D_e\*rho_e\*C_e, where D_e is the thermal diffusion
coefficient.

Electronic pressure effects are included in the TTM model to account
for the blast force acting on ions because of electronic pressure
gradient (see :ref:`(Chen) <Chen>`, :ref:`(Norman) <Norman>`).  The total force
acting on an ion is:

.. math::

  {\vec F}_i = - \partial U / \partial {\vec r}_i + {\vec
  F}_{langevin} - \nabla P_e/n_{ion}

where F_langevin is a force from Langevin thermostat simulating
electron-phonon coupling, and nabla P_e/n_ion is the electron blast
force.

The electronic pressure is taken to be P_e = B\*rho_e\*C_e\*T_e

The current fix ttm/mod implementation allows TTM simulations with a
vacuum. The vacuum region is defined as the grid cells with zero
electronic temperature. The numerical scheme does not allow energy
exchange with such cells. Since the material can expand to previously
unoccupied region in some simulations, the vacuum border can be
allowed to move. It is controlled by the *surface_movement* parameter
in the *init_file*. If it is set to 1, then "vacuum" cells can be
changed to "electron-filled" cells with the temperature *T_e_min* if
atoms move into them (currently only implemented for the case of
1-dimensional motion of flat surface normal to the X axis). The
initial borders of vacuum can be set in the *init_file* via *lsurface*
and *rsurface* parameters. In this case, electronic pressure gradient
is calculated as

.. math::

  \nabla_x P_e = \left[\frac{C_e{}T_e(x)\lambda}{(x+\lambda)^2} +
  \frac{x}{x+\lambda}\frac{(C_e{}T_e)_{x+\Delta
  x}-(C_e{}T_e)_{x}}{\Delta x} \right]

where lambda is the electron mean free path (see :ref:`(Norman) <Norman>`,
:ref:`(Pisarev) <Pisarev>`)

The fix ttm/mod parameter file *init_file* has the following syntax/
Every line with the odd number is considered as a comment and
ignored. The lines with the even numbers are treated as follows:

.. parsed-literal::

   a_0, energy/(temperature\*electron) units
   a_1, energy/(temperature\^2\*electron) units
   a_2, energy/(temperature\^3\*electron) units
   a_3, energy/(temperature\^4\*electron) units
   a_4, energy/(temperature\^5\*electron) units
   C_0, energy/(temperature\*electron) units
   A, 1/temperature units
   rho_e, electrons/volume units
   D_e, length\^2/time units
   gamma_p, mass/time units
   gamma_s, mass/time units
   v_0, length/time units
   I_0, energy/(time\*length\^2) units
   lsurface, electron grid units (positive integer)
   rsurface, electron grid units (positive integer)
   l_skin, length units
   tau, time units
   B, dimensionless
   lambda, length units
   n_ion, ions/volume units
   surface_movement: 0 to disable tracking of surface motion, 1 to enable
   T_e_min, temperature units

----------

Restart, fix_modify, output, run start/stop, minimize info
"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

These fixes write the state of the electronic subsystem and the energy
exchange between the subsystems to :doc:`binary restart files
<restart>`.  See the :doc:`read_restart <read_restart>` command for
info on how to re-specify a fix in an input script that reads a
restart file, so that the operation of the fix continues in an
uninterrupted fashion.  Note that the restart script must define the
same size grid as the original script.

Because the state of the random number generator is not saved in the
restart files, this means you cannot do "exact" restarts with this
fix, where the simulation continues on the same as if no restart had
taken place.  However, in a statistical sense, a restarted simulation
should produce the same behavior.

None of the :doc:`fix_modify <fix_modify>` options are relevant to
these fixes.

These fixes compute 2 output quantities stored in a vector of length
2, which can be accessed by various :doc:`output commands
<Howto_output>`.  The first quantity is the total energy of the
electronic subsystem.  The second quantity is the energy transferred
from the electronic to the atomic subsystem on that timestep. Note
that the velocity verlet integrator applies the fix ttm forces to the
atomic subsystem as two half-step velocity updates: one on the current
timestep and one on the subsequent timestep.  Consequently, the change
in the atomic subsystem energy is lagged by half a timestep relative
to the change in the electronic subsystem energy. As a result of this,
users may notice slight fluctuations in the sum of the atomic and
electronic subsystem energies reported at the end of the timestep.

The vector values calculated are "extensive".

No parameter of the fixes can be used with the *start/stop* keywords
of the :doc:`run <run>` command.  The fixes are not invoked during
:doc:`energy minimization <minimize>`.

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

All these fixes are part of the EXTRA-FIX package. They are only
enabled if LAMMPS was built with that package.  See the :doc:`Build
package <Build_package>` page for more info.

As mentioned above, these fixes require 3d simulations and orthogonal
simulation boxes periodic in all 3 dimensions.

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

:doc:`fix langevin <fix_langevin>`, :doc:`fix dt/reset <fix_dt_reset>`

Default
"""""""

none

----------

.. _Duffy:

**(Duffy)** D M Duffy and A M Rutherford, J. Phys.: Condens. Matter, 19,
016207-016218 (2007).

.. _Rutherford:

**(Rutherford)** A M Rutherford and D M Duffy, J. Phys.:
Condens. Matter, 19, 496201-496210 (2007).

.. _Chen:

**(Chen)** J Chen, D Tzou and J Beraun, Int. J. Heat
Mass Transfer, 49, 307-316 (2006).

.. _Norman:

**(Norman)** G E Norman, S V Starikov, V V Stegailov et al., Contrib.
Plasma Phys., 53, 129-139 (2013).

.. _Pisarev:

**(Pisarev)** V V Pisarev and S V Starikov, J. Phys.: Condens. Matter, 26,
475401 (2014).
