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<p><span class="math notranslate nohighlight">\(\renewcommand{\AA}{\text{Å}}\)</span></p>
<section id="calculate-viscosity">
<h1><span class="section-number">8.3.7. </span>Calculate viscosity<a class="headerlink" href="#calculate-viscosity" title="Link to this heading"></a></h1>
<p>The shear viscosity <span class="math notranslate nohighlight">\(\eta\)</span> of a fluid can be measured in at least 6 ways
using various options in LAMMPS. See the <code class="docutils literal notranslate"><span class="pre">examples/VISCOSITY</span></code> directory
for scripts that implement the 5 methods discussed here for a simple
Lennard-Jones fluid model and 1 method for SPC/E water model.
Also, see the <a class="reference internal" href="Howto_kappa.html"><span class="doc">page on calculating thermal conductivity</span></a>
for an analogous discussion for thermal conductivity.</p>
<p><span class="math notranslate nohighlight">\(\eta\)</span> is a measure of the propensity of a fluid to transmit momentum in
a direction perpendicular to the direction of velocity or momentum
flow. Alternatively it is the resistance the fluid has to being
sheared. It is given by</p>
<div class="math notranslate nohighlight">
\[J = -\eta \cdot \text{grad}(V_{\text{stream}})\]</div>
<p>where <span class="math notranslate nohighlight">\(J\)</span> is the momentum flux in units of momentum per area per time.
and <span class="math notranslate nohighlight">\(\text{grad}(V_{\text{stream}})\)</span> is the spatial gradient of the velocity of the fluid
moving in another direction, normal to the area through which the
momentum flows. Viscosity thus has units of pressure-time.</p>
<p>The first method is to perform a non-equilibrium MD (NEMD) simulation
by shearing the simulation box via the <a class="reference internal" href="fix_deform.html"><span class="doc">fix deform</span></a>
command, and using the <a class="reference internal" href="fix_nvt_sllod.html"><span class="doc">fix nvt/sllod</span></a> command to
thermostat the fluid via the SLLOD equations of motion.
Alternatively, as a second method, one or more moving walls can be
used to shear the fluid in between them, again with some kind of
thermostat that modifies only the thermal (non-shearing) components of
velocity to prevent the fluid from heating up.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>A recent (2017) book by <a class="reference internal" href="#daivis-viscosity"><span class="std std-ref">(Daivis and Todd)</span></a>
discusses use of the SLLOD method and non-equilibrium MD (NEMD)
thermostatting generally, for both simple and complex fluids,
e.g. molecular systems. The latter can be tricky to do correctly.</p>
</div>
<p>In both cases, the velocity profile setup in the fluid by this
procedure can be monitored by the <a class="reference internal" href="fix_ave_chunk.html"><span class="doc">fix ave/chunk</span></a>
command, which determines <span class="math notranslate nohighlight">\(\text{grad}(V_{\text{stream}})\)</span> in the equation above.
E.g. the derivative in the y-direction of the <span class="math notranslate nohighlight">\(V_x\)</span> component of fluid
motion or <span class="math notranslate nohighlight">\(\text{grad}(V_{\text{stream}}) = \frac{\text{d} V_x}{\text{d} y}\)</span>. The <span class="math notranslate nohighlight">\(P_{xy}\)</span> off-diagonal component of
the pressure or stress tensor, as calculated by the <a class="reference internal" href="compute_pressure.html"><span class="doc">compute pressure</span></a> command, can also be monitored, which
is the <span class="math notranslate nohighlight">\(J\)</span> term in the equation above. See the <a class="reference internal" href="Howto_nemd.html"><span class="doc">Howto nemd</span></a> page for details on NEMD simulations.</p>
<p>The third method is to perform a reverse non-equilibrium MD simulation
using the <a class="reference internal" href="fix_viscosity.html"><span class="doc">fix viscosity</span></a> command which implements
the rNEMD algorithm of Muller-Plathe. Momentum in one dimension is
swapped between atoms in two different layers of the simulation box in
a different dimension. This induces a velocity gradient which can be
monitored with the <a class="reference internal" href="fix_ave_chunk.html"><span class="doc">fix ave/chunk</span></a> command.
The fix tallies the cumulative momentum transfer that it performs.
See the <a class="reference internal" href="fix_viscosity.html"><span class="doc">fix viscosity</span></a> command for details.</p>
<p>The fourth method is based on the Green-Kubo (GK) formula which
relates the ensemble average of the auto-correlation of the
stress/pressure tensor to <span class="math notranslate nohighlight">\(\eta\)</span>. This can be done in a fully
equilibrated simulation which is in contrast to the two preceding
non-equilibrium methods, where momentum flows continuously through the
simulation box.</p>
<p>Here is an example input script that calculates the viscosity of
liquid Ar via the GK formalism:</p>
<div class="highlight-LAMMPS notranslate"><div class="highlight"><pre><span></span><span class="c"># Sample LAMMPS input script for viscosity of liquid Ar</span>
<span class="k">units</span><span class="w"> </span><span class="n">real</span>
<span class="k">variable </span><span class="nv nv-Identifier">T</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="m">200.0</span><span class="w"> </span><span class="c"># run temperature</span>
<span class="k">variable </span><span class="nv nv-Identifier">Tinit</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="m">250.0</span><span class="w"> </span><span class="c"># equilibration temperature</span>
<span class="k">variable </span><span class="nv nv-Identifier">V</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="n">vol</span>
<span class="k">variable </span><span class="nv nv-Identifier">dt</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="m">4.0</span>
<span class="k">variable </span><span class="nv nv-Identifier">p</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="m">400</span><span class="w"> </span><span class="c"># correlation length</span>
<span class="k">variable </span><span class="nv nv-Identifier">s</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="m">5</span><span class="w"> </span><span class="c"># sample interval</span>
<span class="k">variable </span><span class="nv nv-Identifier">d</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="nv">$p</span><span class="o">*</span><span class="nv">$s</span><span class="w"> </span><span class="c"># dump interval</span>
<span class="c"># convert from LAMMPS real units to SI</span>
<span class="k">variable </span><span class="nv nv-Identifier">kB</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="m">1.3806504e-23</span><span class="w"> </span><span class="c"># [J/K] Boltzmann</span>
<span class="k">variable </span><span class="nv nv-Identifier">atm2Pa</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="m">101325.0</span>
<span class="k">variable </span><span class="nv nv-Identifier">A2m</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="m">1.0e-10</span>
<span class="k">variable </span><span class="nv nv-Identifier">fs2s</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="m">1.0e-15</span>
<span class="k">variable </span><span class="nv nv-Identifier">convert</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="nv">${atm2Pa}</span><span class="o">*</span><span class="nv">${atm2Pa}</span><span class="o">*</span><span class="nv">${fs2s}</span><span class="o">*</span><span class="nv">${A2m}</span><span class="o">*</span><span class="nv">${A2m}</span><span class="o">*</span><span class="nv">${A2m}</span>
<span class="c"># setup problem</span>
<span class="k">dimension</span><span class="w"> </span><span class="m">3</span>
<span class="k">boundary</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="n">p</span>
<span class="k">lattice</span><span class="w"> </span><span class="n">fcc</span><span class="w"> </span><span class="m">5.376</span><span class="w"> </span><span class="n">orient</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="n">orient</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="n">orient</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="m">1</span>
<span class="k">region </span><span class="nv nv-Identifier">box</span><span class="w"> </span><span class="n">block</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="m">4</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="m">4</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="m">4</span>
<span class="k">create_box </span><span class="nv nv-Identifier">1</span><span class="w"> </span><span class="nv nv-Identifier">box</span>
<span class="k">create_atoms</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="n">box</span>
<span class="k">mass</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="m">39.948</span>
<span class="k">pair_style</span><span class="w"> </span><span class="n">lj</span><span class="o">/</span><span class="n">cut</span><span class="w"> </span><span class="m">13.0</span>
<span class="k">pair_coeff</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">0.2381</span><span class="w"> </span><span class="m">3.405</span>
<span class="k">timestep</span><span class="w"> </span><span class="nv">${dt}</span>
<span class="k">thermo</span><span class="w"> </span><span class="nv">$d</span>
<span class="c"># equilibration and thermalization</span>
<span class="k">velocity</span><span class="w"> </span><span class="n">all</span><span class="w"> </span><span class="n">create</span><span class="w"> </span><span class="nv">${Tinit}</span><span class="w"> </span><span class="m">102486</span><span class="w"> </span><span class="n">mom</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="n">rot</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="n">dist</span><span class="w"> </span><span class="n">gaussian</span>
<span class="k">fix </span><span class="nv nv-Identifier">NVT</span><span class="w"> </span><span class="nv nv-Identifier">all</span><span class="w"> </span><span class="n">nvt</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="nv">${Tinit}</span><span class="w"> </span><span class="nv">${Tinit}</span><span class="w"> </span><span class="m">10</span><span class="w"> </span><span class="n">drag</span><span class="w"> </span><span class="m">0.2</span>
<span class="k">run</span><span class="w"> </span><span class="m">8000</span>
<span class="c"># viscosity calculation, switch to NVE if desired</span>
<span class="k">velocity</span><span class="w"> </span><span class="n">all</span><span class="w"> </span><span class="n">create</span><span class="w"> </span><span class="nv">$T</span><span class="w"> </span><span class="m">102486</span><span class="w"> </span><span class="n">mom</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="n">rot</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="n">dist</span><span class="w"> </span><span class="n">gaussian</span>
<span class="k">fix </span><span class="nv nv-Identifier">NVT</span><span class="w"> </span><span class="nv nv-Identifier">all</span><span class="w"> </span><span class="n">nvt</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="nv">$T</span><span class="w"> </span><span class="nv">$T</span><span class="w"> </span><span class="m">10</span><span class="w"> </span><span class="n">drag</span><span class="w"> </span><span class="m">0.2</span>
<span class="c">#unfix NVT</span>
<span class="c">#fix NVE all nve</span>
<span class="k">reset_timestep</span><span class="w"> </span><span class="m">0</span>
<span class="k">variable </span><span class="nv nv-Identifier">pxy</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="n">pxy</span>
<span class="k">variable </span><span class="nv nv-Identifier">pxz</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="n">pxz</span>
<span class="k">variable </span><span class="nv nv-Identifier">pyz</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="n">pyz</span>
<span class="k">fix </span><span class="nv nv-Identifier">SS</span><span class="w"> </span><span class="nv nv-Identifier">all</span><span class="w"> </span><span class="n">ave</span><span class="o">/</span><span class="n">correlate</span><span class="w"> </span><span class="nv">$s</span><span class="w"> </span><span class="nv">$p</span><span class="w"> </span><span class="nv">$d</span><span class="w"> </span><span class="o">&amp;</span>
<span class="w"> </span><span class="n">v_pxy</span><span class="w"> </span><span class="n">v_pxz</span><span class="w"> </span><span class="n">v_pyz</span><span class="w"> </span><span class="n">type</span><span class="w"> </span><span class="n">auto</span><span class="w"> </span><span class="n">file</span><span class="w"> </span><span class="n">S0St.dat</span><span class="w"> </span><span class="n">ave</span><span class="w"> </span><span class="n">running</span>
<span class="k">variable </span><span class="nv nv-Identifier">scale</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="nv">${convert}</span><span class="o">/</span><span class="nv">(${kB}*$T)</span><span class="o">*</span><span class="nv">$V</span><span class="o">*</span><span class="nv">$s</span><span class="o">*</span><span class="nv">${dt}</span>
<span class="k">variable </span><span class="nv nv-Identifier">v11</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="n">trap</span><span class="nv">(f_SS[3])</span><span class="o">*</span><span class="nv">${scale}</span>
<span class="k">variable </span><span class="nv nv-Identifier">v22</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="n">trap</span><span class="nv">(f_SS[4])</span><span class="o">*</span><span class="nv">${scale}</span>
<span class="k">variable </span><span class="nv nv-Identifier">v33</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="n">trap</span><span class="nv">(f_SS[5])</span><span class="o">*</span><span class="nv">${scale}</span>
<span class="k">thermo_style</span><span class="w"> </span><span class="n">custom</span><span class="w"> </span><span class="n">step</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="n">press</span><span class="w"> </span><span class="n">v_pxy</span><span class="w"> </span><span class="n">v_pxz</span><span class="w"> </span><span class="n">v_pyz</span><span class="w"> </span><span class="n">v_v11</span><span class="w"> </span><span class="n">v_v22</span><span class="w"> </span><span class="n">v_v33</span>
<span class="k">run</span><span class="w"> </span><span class="m">100000</span>
<span class="k">variable </span><span class="nv nv-Identifier">v</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="nv">(v_v11+v_v22+v_v33)</span><span class="o">/</span><span class="m">3.0</span>
<span class="k">variable </span><span class="nv nv-Identifier">ndens</span><span class="w"> </span><span class="n">equal</span><span class="w"> </span><span class="n">count</span><span class="nv">(all)</span><span class="o">/</span><span class="n">vol</span>
<span class="k">print</span><span class="w"> </span><span class="s">&quot;average viscosity: $v [Pa.s] @ $T K, ${ndens} atoms/A^3&quot;</span>
</pre></div>
</div>
<p>The fifth method is related to the above Green-Kubo method,
but uses the Einstein formulation, analogous to the Einstein
mean-square-displacement formulation for self-diffusivity. The
time-integrated momentum fluxes play the role of Cartesian
coordinates, whose mean-square displacement increases linearly
with time at sufficiently long times.</p>
<p>The sixth is the periodic perturbation method, which is also a non-equilibrium MD method.
However, instead of measuring the momentum flux in response to an applied velocity gradient,
it measures the velocity profile in response to applied stress.
A cosine-shaped periodic acceleration is added to the system via the
<a class="reference internal" href="fix_accelerate_cos.html"><span class="doc">fix accelerate/cos</span></a> command,
and the <a class="reference internal" href="compute_viscosity_cos.html"><span class="doc">compute viscosity/cos</span></a> command is used to monitor the
generated velocity profile and remove the velocity bias before thermostatting.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>An article by <a class="reference internal" href="#hess3"><span class="std std-ref">(Hess)</span></a> discussed the accuracy and efficiency of these methods.</p>
</div>
<hr class="docutils" />
<p id="daivis-viscosity"><strong>(Daivis and Todd)</strong> Daivis and Todd, Nonequilibrium Molecular Dynamics (book),
Cambridge University Press, <a class="reference external" href="https://doi.org/10.1017/9781139017848">https://doi.org/10.1017/9781139017848</a>, (2017).</p>
<p id="hess3"><strong>(Hess)</strong> Hess, B. The Journal of Chemical Physics 2002, 116 (1), 209-217.</p>
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