Heat Flux¶

Introduction¶

It is possible to use DL_POLY_5 to calculate the heat flux of a material with two-body interactions in an MD simulation. The heat flux can subsequently be used as a means to calculate the thermal conductivity of a material via the Green-Kubo relation (see Section Correlation Functions).

It is also possible to calculate the momentum density, which may be required to apply corrections to the thermal-conductivity in multi-component systems when comparing to experimental data.

To enable the calculation of heat flux add heat_flux On into the CONTROL file. For the momentum density supply a list of atom types to calculate the values for e.g. momentum_density [Li F] to compute for Li and F atoms.

Heat flux is currently supported for: direct and tabulated VDW interactions (see Table (13)), SPME interactions, and direct and tabulated metal potentials (see Table (14)). Tersoff, three, and four body potentials are currently unsupported.

Theory¶

The heat flux for two-body interactions is defined as:

(159)¶\[\mathbf{J} = \frac{1}{V} \sum\limits^{N}_{i} \left[ e_{i} \mathbf{v}_{i} + \frac{1}{2}\sum\limits_{j\neq i} \mathbf{f}_{ij}\cdot\mathbf{v}_{i}\mathbf{r}_{ij} \right]\]

where \(\mathbf{\mathbf{\textbf{J}}}\) is the heat flux, \(V\) is the volume of the cell, \(N\) is the number of particles, \(e_{i}\) is the energy, \(\mathbf{v}_{i}\) is the velocity, \(\mathbf{f}_{ij}\) is the force of j on i. All subscripts \(i\) refer to a particle.

The thermal conductivity can then be calculated as an auto-correlation of the heat flux

\[\kappa = \frac{V}{k_{B} T^{2}} \int\limits_{0}^{\infty} \langle \mathbf{J}(0) \mathbf{J}(t) \rangle \, \mathrm{d}t\]

where \(\kappa\) is the thermal conductivity, \(V\) is the volume of the cell, \(k_{B}\) is the Boltzmann constant.

In mixed systems where components significantly differ in mass it may be necessary to correct for mass transfer effects when calculating thermal-conductivity ⁸. In order to facilitate these corrections the momentum density may also be calculated. This is defined for components (atomic types) C as,

\[\mathbf{q}_{C} = \frac{1}{V}\sum_{i\in C}m_{i}\mathbf{v}_{i}.\]

Where \(C\) is the set of atoms.

Implementation¶

For the purposes of calculating per-particle SPME interactions, the long-range electrostatics forces and energies are calculated differently for the heat flux case. From the SPME equations, we can calculate a per particle contribution via:

\[\begin{split}\begin{gathered}\Omega ^{\mathit{ABC}}=\sum _j\omega _j^{\mathit{ABC}}\\\omega _j^{\mathit{ABC}}=\frac 1{2\pi V}\sum _{n_1n_2n_3=-\infty }^{\infty }\sum _{k_1}^{K_1-1}\sum _{k_2}^{K_2-1}\sum _{k_3}^{K_3-1}Q_j^{\mathit{ABC}}(\mathbf k,\mathbf n)\sum _{\mathbf m\neq 0}\left[\prod _{\mu }b_{\mu }(\mathbf m)e^{2\pi i\frac{m_{\mu }k_{\mu }}{K_{\mu }}}\right]f(\mathbf m)\\Q_j^{\mathit{ABC}}(\mathbf k,\mathbf n)=q_j\left[\frac{K_{\alpha }}{2\pi i}\right]^A\frac{\partial ^A}{\partial u_{\alpha j}^A}M_n(u_{\alpha j}-k_{\alpha }-n_{\alpha }K_{\alpha })\times \\\left[\frac{K_{\beta }}{2\pi i}\right]^B\frac{\partial ^B}{\partial u_{\beta j}^B}M_n(u_{\beta j}-k_{\beta }-n_{\beta }K_{\beta })\times \\\left[\frac{K_{\gamma }}{2\pi i}\right]^C\frac{\partial ^C}{\partial u_{\gamma j}^C}M_n(u_{\gamma j}-k_{\gamma }-n_{\gamma }K_{\gamma })\end{gathered}\end{split}\]

where \(\omega\) is the per-particle contribution for particle \(j\), \(q\) is the charge, other values are defined in the SPME section (see: Smoothed Particle Mesh Ewald)

File¶

The heat flux method creates a file called HEATFLUX which contains the relevant data structured as 6 Reals per line: STEP TEMPERATURE VOLUME HEAT-FLUX (x, y, z).

If momentum_density is specified with some atom types listed then the file will include additional data for each atom type’s momentum density on each line after the HEAT-FLUX data. Given momentum_density [Li] in a Li-F simulation the data will be structured as 9 Reals per line and a Character: STEP TEMPERATURE VOLUME HEAT-FLUX (x, y, z) Li MOMENTUM-DENSITY-Li (x, y, z).