# 8.4.6. Calculate viscosity

The shear viscosity eta of a fluid can be measured in at least 5 ways using various options in LAMMPS. See the examples/VISCOSITY directory for scripts that implement the 5 methods discussed here for a simple Lennard-Jones fluid model. Also, see the Howto kappa doc page for an analogous discussion for thermal conductivity.

Eta 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

J = -eta grad(Vstream)

where J is the momentum flux in units of momentum per area per time. and grad(Vstream) 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.

The first method is to perform a non-equilibrium MD (NEMD) simulation by shearing the simulation box via the fix deform command, and using the fix nvt/sllod 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.

Note

A recent (2017) book by (Daivis and Todd) 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.

In both cases, the velocity profile setup in the fluid by this procedure can be monitored by the fix ave/chunk command, which determines grad(Vstream) in the equation above. E.g. the derivative in the y-direction of the Vx component of fluid motion or grad(Vstream) = dVx/dy. The Pxy off-diagonal component of the pressure or stress tensor, as calculated by the compute pressure command, can also be monitored, which is the J term in the equation above. See the Howto nemd doc page for details on NEMD simulations.

The third method is to perform a reverse non-equilibrium MD simulation using the fix viscosity 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 fix ave/chunk command. The fix tallies the cumulative momentum transfer that it performs. See the fix viscosity command for details.

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 eta. 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.

Here is an example input script that calculates the viscosity of liquid Ar via the GK formalism:

# Sample LAMMPS input script for viscosity of liquid Ar units real variable T equal 86.4956 variable V equal vol variable dt equal 4.0 variable p equal 400 # correlation length variable s equal 5 # 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 atm2Pa equal 101325.0 variable A2m equal 1.0e-10 variable fs2s equal 1.0e-15 variable convert equal ${atm2Pa}*${atm2Pa}*${fs2s}*${A2m}*${A2m}*${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 # viscosity calculation, switch to NVE if desired #unfix NVT #fix NVE all nve reset_timestep 0 variable pxy equal pxy variable pxz equal pxz variable pyz equal pyz fix SS all ave/correlate $s $p $d & v_pxy v_pxz v_pyz type auto file S0St.dat ave running variable scale equal ${convert}/(${kB}*$T)*$V*$s*${dt} variable v11 equal trap(f_SS[3])*${scale} variable v22 equal trap(f_SS[4])*${scale} variable v33 equal trap(f_SS[5])*${scale} thermo_style custom step temp press v_pxy v_pxz v_pyz v_v11 v_v22 v_v33 run 100000 variable v equal (v_v11+v_v22+v_v33)/3.0 variable ndens equal count(all)/vol print "average viscosity: $v [Pa.s] @ $T K, ${ndens} /A^3"

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.

**(Daivis and Todd)** Daivis and Todd, Nonequilibrium Molecular Dynamics (book),
Cambridge University Press, https://doi.org/10.1017/9781139017848, (2017).