# fix ttm/mod command¶

## Syntax¶

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 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

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:

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

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¶

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: (Duffy) and (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 (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 (Duffy) and (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:

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

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 (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 (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 (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:

# 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 fix ave/chunk command using the compute chunk/atom command to create a 3d array of chunks consistent with the grid used by this fix.

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

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

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

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

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 (Chen), (Norman)). The total force acting on an ion is:

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

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

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

where lambda is the electron mean free path (see (Norman), (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:

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 binary restart files. See the 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 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 output commands. 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 run command. The fixes are not invoked during energy minimization.

## Restrictions¶

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

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

## Default¶

none

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

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

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

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

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