257 lines
9.4 KiB
ReStructuredText
257 lines
9.4 KiB
ReStructuredText
.. index:: compute heat/flux
|
|
|
|
compute heat/flux command
|
|
=========================
|
|
|
|
Syntax
|
|
""""""
|
|
|
|
.. code-block:: LAMMPS
|
|
|
|
compute ID group-ID heat/flux ke-ID pe-ID stress-ID
|
|
|
|
* ID, group-ID are documented in :doc:`compute <compute>` command
|
|
* heat/flux = style name of this compute command
|
|
* ke-ID = ID of a compute that calculates per-atom kinetic energy
|
|
* pe-ID = ID of a compute that calculates per-atom potential energy
|
|
* stress-ID = ID of a compute that calculates per-atom stress
|
|
|
|
Examples
|
|
""""""""
|
|
|
|
.. code-block:: LAMMPS
|
|
|
|
compute myFlux all heat/flux myKE myPE myStress
|
|
|
|
Description
|
|
"""""""""""
|
|
|
|
Define a computation that calculates the heat flux vector based on
|
|
contributions from atoms in the specified group. This can be used by
|
|
itself to measure the heat flux through a set of atoms (e.g., a region
|
|
between two thermostatted reservoirs held at different temperatures),
|
|
or to calculate a thermal conductivity using the equilibrium
|
|
Green-Kubo formalism.
|
|
|
|
For other non-equilibrium ways to compute a thermal conductivity, see
|
|
the :doc:`Howto kappa <Howto_kappa>` doc page. These include use of
|
|
the :doc:`fix thermal/conductivity <fix_thermal_conductivity>` command
|
|
for the Muller-Plathe method. Or the :doc:`fix heat <fix_heat>` command
|
|
which can add or subtract heat from groups of atoms.
|
|
|
|
The compute takes three arguments which are IDs of other
|
|
:doc:`computes <compute>`. One calculates per-atom kinetic energy
|
|
(\ *ke-ID*\ ), one calculates per-atom potential energy (\ *pe-ID)*, and the
|
|
third calculates per-atom stress (\ *stress-ID*\ ).
|
|
|
|
.. note::
|
|
|
|
These other computes should provide values for all the atoms in
|
|
the group this compute specifies. That means the other computes could
|
|
use the same group as this compute, or they can just use group "all"
|
|
(or any group whose atoms are superset of the atoms in this compute's
|
|
group). LAMMPS does not check for this.
|
|
|
|
In case of two-body interactions, the heat flux :math:`\mathbf{J}` is defined as
|
|
|
|
.. math::
|
|
\mathbf{J} &= \frac{1}{V} \left[ \sum_i e_i \mathbf{v}_i - \sum_{i} \mathbf{S}_{i} \mathbf{v}_i \right] \\
|
|
&= \frac{1}{V} \left[ \sum_i e_i \mathbf{v}_i + \sum_{i<j} \left( \mathbf{F}_{ij} \cdot \mathbf{v}_j \right) \mathbf{r}_{ij} \right] \\
|
|
&= \frac{1}{V} \left[ \sum_i e_i \mathbf{v}_i + \frac{1}{2} \sum_{i<j} \bigl( \mathbf{F}_{ij} \cdot \left(\mathbf{v}_i + \mathbf{v}_j \right) \bigr) \mathbf{r}_{ij} \right]
|
|
|
|
:math:`e_i` in the first term of the equation
|
|
is the per-atom energy (potential and kinetic).
|
|
This is calculated by the computes *ke-ID*
|
|
and *pe-ID*. :math:`\mathbf{S}_i` in the second term is the
|
|
per-atom stress tensor calculated by the compute *stress-ID*.
|
|
See :doc:`compute stress/atom <compute_stress_atom>`
|
|
and :doc:`compute centroid/stress/atom <compute_stress_atom>`
|
|
for possible definitions of atomic stress :math:`\mathbf{S}_i`
|
|
in the case of bonded and many-body interactions.
|
|
The tensor multiplies :math:`\mathbf{v}_i` by a :math:`3\times3` matrix
|
|
to yield a vector.
|
|
Note that as discussed below, the :math:`1/V` scaling factor in the
|
|
equation for :math:`\mathbf{J}` is **not** included in the calculation
|
|
performed by these computes; you need to add it for a volume appropriate to the
|
|
atoms included in the calculation.
|
|
|
|
.. note::
|
|
|
|
The :doc:`compute pe/atom <compute_pe_atom>` and
|
|
:doc:`compute stress/atom <compute_stress_atom>`
|
|
commands have options for which
|
|
terms to include in their calculation (pair, bond, etc). The heat
|
|
flux calculation will thus include exactly the same terms. Normally
|
|
you should use :doc:`compute stress/atom virial <compute_stress_atom>`
|
|
or :doc:`compute centroid/stress/atom virial <compute_stress_atom>`
|
|
so as not to include a kinetic energy term in the heat flux.
|
|
|
|
.. warning::
|
|
|
|
The compute *heat/flux* has been reported to produce unphysical
|
|
values for angle, dihedral, improper and constraint force contributions
|
|
when used with :doc:`compute stress/atom <compute_stress_atom>`,
|
|
as discussed in :ref:`(Surblys2019) <Surblys3>`, :ref:`(Boone) <Boone>`
|
|
and :ref:`(Surblys2021) <Surblys4>`. You are strongly advised to
|
|
use :doc:`compute centroid/stress/atom <compute_stress_atom>`,
|
|
which has been implemented specifically for such cases.
|
|
|
|
.. warning::
|
|
|
|
Due to an implementation detail, the :math:`y` and :math:`z`
|
|
components of heat flux from :doc:`fix rigid <fix_rigid>`
|
|
contribution when computed via :doc:`compute stress/atom <compute_stress_atom>`
|
|
are highly unphysical and should not be used.
|
|
|
|
The Green--Kubo formulas relate the ensemble average of the
|
|
auto-correlation of the heat flux :math:`\mathbf{J}`
|
|
to the thermal conductivity :math:`\kappa`:
|
|
|
|
.. math::
|
|
\kappa = \frac{V}{k_B T^2} \int_0^\infty \langle J_x(0) J_x(t) \rangle \, \mathrm{d} t = \frac{V}{3 k_B T^2} \int_0^\infty \langle \mathbf{J}(0) \cdot \mathbf{J}(t) \rangle \, \mathrm{d}t
|
|
|
|
----------
|
|
|
|
The heat flux can be output every so many timesteps (e.g., via the
|
|
:doc:`thermo_style custom <thermo_style>` command). Then as a
|
|
post-processing operation, an auto-correlation can be performed, its
|
|
integral estimated, and the Green--Kubo formula above evaluated.
|
|
|
|
The :doc:`fix ave/correlate <fix_ave_correlate>` command can calculate
|
|
the auto-correlation. The trap() function in the
|
|
:doc:`variable <variable>` command can calculate the integral.
|
|
|
|
An example LAMMPS input script for solid argon is appended below. The
|
|
result should be an average conductivity
|
|
:math:`\approx 0.29~\mathrm{W/m \cdot K}`.
|
|
|
|
----------
|
|
|
|
Output info
|
|
"""""""""""
|
|
|
|
This compute calculates a global vector of length 6.
|
|
The first three components are the :math:`x`, :math:`y`, and :math:`z`
|
|
components of the full heat flux vector
|
|
(i.e., :math:`J_x`, :math:`J_y`, and :math:`J_z`).
|
|
The next three components are the :math:`x`, :math:`y`, and :math:`z`
|
|
components of just the convective portion of the flux (i.e., the
|
|
first term in the equation for :math:`\mathbf{J}`).
|
|
Each component can be accessed by indices 1--6. These values can be used by any
|
|
command that uses global vector values from a compute as input.
|
|
See the :doc:`Howto output <Howto_output>` documentation for an overview of
|
|
LAMMPS output options.
|
|
|
|
The vector values calculated by this compute are "extensive", meaning
|
|
they scale with the number of atoms in the simulation. They can be
|
|
divided by the appropriate volume to get a flux, which would then be an
|
|
"intensive" value, meaning independent of the number of atoms in the
|
|
simulation. Note that if the compute group is "all", then the
|
|
appropriate volume to divide by is the simulation box volume. However,
|
|
if a group with a subset of atoms is used, it should be the volume
|
|
containing those atoms.
|
|
|
|
The vector values will be in energy\*velocity :doc:`units <units>`. Once
|
|
divided by a volume the units will be that of flux, namely
|
|
energy/area/time :doc:`units <units>`
|
|
|
|
Restrictions
|
|
""""""""""""
|
|
none
|
|
|
|
Related commands
|
|
""""""""""""""""
|
|
|
|
:doc:`fix thermal/conductivity <fix_thermal_conductivity>`,
|
|
:doc:`fix ave/correlate <fix_ave_correlate>`,
|
|
:doc:`variable <variable>`
|
|
|
|
Default
|
|
"""""""
|
|
|
|
none
|
|
|
|
----------
|
|
|
|
Example Input File
|
|
------------------
|
|
|
|
.. code-block:: LAMMPS
|
|
|
|
# Sample LAMMPS input script for thermal conductivity of solid Ar
|
|
|
|
units real
|
|
variable T equal 70
|
|
variable V equal vol
|
|
variable dt equal 4.0
|
|
variable p equal 200 # correlation length
|
|
variable s equal 10 # sample interval
|
|
variable d equal $p*$s # dump interval
|
|
|
|
# convert from LAMMPS real units to SI
|
|
|
|
variable kB equal 1.3806504e-23 # [J/K] Boltzmann
|
|
variable kCal2J equal 4186.0/6.02214e23
|
|
variable A2m equal 1.0e-10
|
|
variable fs2s equal 1.0e-15
|
|
variable convert equal ${kCal2J}*${kCal2J}/${fs2s}/${A2m}
|
|
|
|
# setup problem
|
|
|
|
dimension 3
|
|
boundary p p p
|
|
lattice fcc 5.376 orient x 1 0 0 orient y 0 1 0 orient z 0 0 1
|
|
region box block 0 4 0 4 0 4
|
|
create_box 1 box
|
|
create_atoms 1 box
|
|
mass 1 39.948
|
|
pair_style lj/cut 13.0
|
|
pair_coeff * * 0.2381 3.405
|
|
timestep ${dt}
|
|
thermo $d
|
|
|
|
# equilibration and thermalization
|
|
|
|
velocity all create $T 102486 mom yes rot yes dist gaussian
|
|
fix NVT all nvt temp $T $T 10 drag 0.2
|
|
run 8000
|
|
|
|
# thermal conductivity calculation, switch to NVE if desired
|
|
|
|
#unfix NVT
|
|
#fix NVE all nve
|
|
|
|
reset_timestep 0
|
|
compute myKE all ke/atom
|
|
compute myPE all pe/atom
|
|
compute myStress all stress/atom NULL virial
|
|
compute flux all heat/flux myKE myPE myStress
|
|
variable Jx equal c_flux[1]/vol
|
|
variable Jy equal c_flux[2]/vol
|
|
variable Jz equal c_flux[3]/vol
|
|
fix JJ all ave/correlate $s $p $d &
|
|
c_flux[1] c_flux[2] c_flux[3] type auto file J0Jt.dat ave running
|
|
variable scale equal ${convert}/${kB}/$T/$T/$V*$s*${dt}
|
|
variable k11 equal trap(f_JJ[3])*${scale}
|
|
variable k22 equal trap(f_JJ[4])*${scale}
|
|
variable k33 equal trap(f_JJ[5])*${scale}
|
|
thermo_style custom step temp v_Jx v_Jy v_Jz v_k11 v_k22 v_k33
|
|
run 100000
|
|
variable k equal (v_k11+v_k22+v_k33)/3.0
|
|
variable ndens equal count(all)/vol
|
|
print "average conductivity: $k[W/mK] @ $T K, ${ndens} /A\^3"
|
|
|
|
----------
|
|
|
|
.. _Surblys3:
|
|
|
|
**(Surblys2019)** Surblys, Matsubara, Kikugawa, Ohara, Phys Rev E, 99, 051301(R) (2019).
|
|
|
|
.. _Boone:
|
|
|
|
**(Boone)** Boone, Babaei, Wilmer, J Chem Theory Comput, 15, 5579--5587 (2019).
|
|
|
|
.. _Surblys4:
|
|
|
|
**(Surblys2021)** Surblys, Matsubara, Kikugawa, Ohara, J Appl Phys 130, 215104 (2021).
|