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<p><span class="math notranslate nohighlight">\(\renewcommand{\AA}{\text{Å}}\)</span></p>
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<section id="calculate-thermal-conductivity">
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<h1><span class="section-number">8.3.6. </span>Calculate thermal conductivity<a class="headerlink" href="#calculate-thermal-conductivity" title="Link to this heading"></a></h1>
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<p>The thermal conductivity <span class="math notranslate nohighlight">\(\kappa\)</span> of a material can be measured in at
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least 4 ways using various options in LAMMPS. See the <code class="docutils literal notranslate"><span class="pre">examples/KAPPA</span></code>
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directory for scripts that implement the 4 methods discussed here for
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a simple Lennard-Jones fluid model. Also, see the <a class="reference internal" href="Howto_viscosity.html"><span class="doc">Howto viscosity</span></a> page for an analogous discussion
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for viscosity.</p>
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<p>The thermal conductivity tensor <span class="math notranslate nohighlight">\(\mathbf{\kappa}\)</span> is a measure of the propensity
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of a material to transmit heat energy in a diffusive manner as given
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by Fourier’s law</p>
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<div class="math notranslate nohighlight">
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\[J = -\kappa \cdot \text{grad}(T)\]</div>
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<p>where <span class="math notranslate nohighlight">\(J\)</span> is the heat flux in units of energy per area per time and
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<span class="math notranslate nohighlight">\(\text{grad}(T)\)</span> is the spatial gradient of temperature. The thermal
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conductivity thus has units of energy per distance per time per degree
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K and is often approximated as an isotropic quantity, i.e. as a
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scalar.</p>
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<p>The first method is to setup two thermostatted regions at opposite
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ends of a simulation box, or one in the middle and one at the end of a
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periodic box. By holding the two regions at different temperatures
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with a <a class="reference internal" href="Howto_thermostat.html"><span class="doc">thermostatting fix</span></a>, the energy added to
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the hot region should equal the energy subtracted from the cold region
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and be proportional to the heat flux moving between the regions. See
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the papers by <a class="reference internal" href="#howto-ikeshoji"><span class="std std-ref">Ikeshoji and Hafskjold</span></a> and
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<a class="reference internal" href="#howto-wirnsberger"><span class="std std-ref">Wirnsberger et al</span></a> for details of this idea. Note
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that thermostatting fixes such as <a class="reference internal" href="fix_nh.html"><span class="doc">fix nvt</span></a>, <a class="reference internal" href="fix_langevin.html"><span class="doc">fix langevin</span></a>, and <a class="reference internal" href="fix_temp_rescale.html"><span class="doc">fix temp/rescale</span></a> store the cumulative energy they
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add/subtract.</p>
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<p>Alternatively, as a second method, the <a class="reference internal" href="fix_heat.html"><span class="doc">fix heat</span></a> or
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<a class="reference internal" href="fix_ehex.html"><span class="doc">fix ehex</span></a> commands can be used in place of thermostats
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on each of two regions to add/subtract specified amounts of energy to
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both regions. In both cases, the resulting temperatures of the two
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regions can be monitored with the “compute temp/region” command and
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the temperature profile of the intermediate region can be monitored
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with the <a class="reference internal" href="fix_ave_chunk.html"><span class="doc">fix ave/chunk</span></a> and <a class="reference internal" href="compute_ke_atom.html"><span class="doc">compute ke/atom</span></a> commands.</p>
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<p>The third method is to perform a reverse non-equilibrium MD simulation
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using the <a class="reference internal" href="fix_thermal_conductivity.html"><span class="doc">fix thermal/conductivity</span></a>
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command which implements the rNEMD algorithm of Muller-Plathe.
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Kinetic energy is swapped between atoms in two different layers of the
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simulation box. This induces a temperature gradient between the two
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layers which can be monitored with the <a class="reference internal" href="fix_ave_chunk.html"><span class="doc">fix ave/chunk</span></a> and <a class="reference internal" href="compute_ke_atom.html"><span class="doc">compute ke/atom</span></a> commands. The fix tallies the
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cumulative energy transfer that it performs. See the <a class="reference internal" href="fix_thermal_conductivity.html"><span class="doc">fix thermal/conductivity</span></a> command for
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details.</p>
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<p>The fourth method is based on the Green-Kubo (GK) formula which
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relates the ensemble average of the auto-correlation of the heat flux
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to <span class="math notranslate nohighlight">\(\kappa\)</span>. The heat flux can be calculated from the fluctuations of
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per-atom potential and kinetic energies and per-atom stress tensor in
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a steady-state equilibrated simulation. This is in contrast to the
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two preceding non-equilibrium methods, where energy flows continuously
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between hot and cold regions of the simulation box.</p>
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<p>The <a class="reference internal" href="compute_heat_flux.html"><span class="doc">compute heat/flux</span></a> command can calculate
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the needed heat flux and describes how to implement the Green_Kubo
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formalism using additional LAMMPS commands, such as the <a class="reference internal" href="fix_ave_correlate.html"><span class="doc">fix ave/correlate</span></a> command to calculate the needed
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auto-correlation. See the page for the <a class="reference internal" href="compute_heat_flux.html"><span class="doc">compute heat/flux</span></a> command for an example input script
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that calculates the thermal conductivity of solid Ar via the GK
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formalism.</p>
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<hr class="docutils" />
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<p id="howto-ikeshoji"><strong>(Ikeshoji)</strong> Ikeshoji and Hafskjold, Molecular Physics, 81, 251-261
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(1994).</p>
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<p id="howto-wirnsberger"><strong>(Wirnsberger)</strong> Wirnsberger, Frenkel, and Dellago, J Chem Phys, 143, 124104
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(2015).</p>
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