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Thermal Accommodation Coefficients

In the free-molecular regime heat conduction between a surface and a gas depends on the thermal accommodation coefficient, α, which specifies the energy transferred between a surface composed of thermally-vibrating atoms and a colliding gas molecule. The thermal accommodation coefficient is defined as

$\alpha \equiv \dfrac{{\left\langle {{E_o} - {E_i}} \right\rangle }}{{\left\langle {{E_{o,max }} - {E_i}} \right\rangle }} $

where <Eo−Ei> is the increase in energy between incident and scattered molecular streams and <Eo,max−Ei> is the maximum energy increase allowed by thermodynamics, which would occur every gas molecules equilibrated with the surface. This equation can be rewritten as

$\alpha \equiv \dfrac{{\left\langle {{E_o} - {E_i}} \right\rangle }}{{(2+\zeta_{int}/2) k_B (T_s-T_g)}} $

where kB is Boltzmann's constant, Ts and Tg are the surface and gas temperatures, respectively, and ζint is the number of active internal degrees of freedom of the gas molecule.

Thermal Accommodation Coefficients for Soot/Graphite

We elucidated the physics of gas-surface scattering in LII experiments by measuring α between soot and different monatomic and polyatomic gases. Soot was extracted from an ethylene laminar diffusion flame and entrained into a range of monatomic and polyatomic carrier gases using a venturi eductor. The soot-laden aerosol then flowed into an LII sample chamber, and was finally exhausted to the environment.

The results of these experiments1 showed that α increases monotonically with the gas molecule/surface atomic mass ratio, μ, for monatomic gases, because the lighter gas molecules generate more surface phonons when they impact the uppermost graphene sheet, which propagate energy away from the collision site. In contrast, heavier gas molecules collide more slowly and retain some of their incident energy by "trampolining." The results also show that α becomes smaller with increasing structural complexity for polyatomic gases having similar atomic masses. This is because surface energy is transferred preferentially to the translational mode of the gas molecule over the internal energy modes.  Furthermore, identical α values for molecules having the same mass and structure but different bond energies is strong evidence that molecules remain intact throughout the collision process.

Molecular dynamics provides further insight into gas-surface scattering in LII experiments2,3. Pairwise potentials are defined between the surface atoms and between the surface atoms and the colliding gas molecules. Graphite potentials4 were used as a surrogate for soot, and the pairwise gas/surface potential was represented with pairwise Lennard-Jones 6-12 potentials,

${U_{gs}}\left( r \right) = 4{\varepsilon _{gs}}\left[ {{{\left( {\dfrac{{{\sigma _{gs}}}}{r}} \right)}^{12}} - {{\left( {\dfrac{{{\sigma _{gs}}}}{r}} \right)}^6}} \right]$

with parameters found by fitting to published desoprtion energies and equilibrium distances5 and/or by using the Lorentz-Berthelot combining rules,

${\sigma _{gs}} = \dfrac{{\sigma {}_{gg} + {\sigma _{ss}}}}{2},\quad {\varepsilon _{gs}} = \sqrt {{\varepsilon _{gg}}{\varepsilon _{ss}}}$

where [σss, εss] and [σgg, εgg] are the L-J 6-12 parameters for the homogeneous surface and gas systems.

Molecular trajectories are found by integrating Newton's equations of motion. In each simulation the initial gas molecular velocity is sampled from an equilibrium distribution at 300 K and the surface is warmed up to 3000 K. The MD results agree well with the experimental data.

Thermal accommodation coefficients derived from experiment and molecular dynamics (left); Molecular dynamics simulation of a N2O molecule scattering from a hot graphite surface (right).
experimental and MD TACs for soot/graphene

Thermal Accommodation Coefficients for Non-carbonaceous Nanoparticles

More recent work has focused on developing thermal accommodation coefficients to support TiRe-LII characterization of non-carbonaceous nanoparticles. Our research commenced on metallic nanoparticles, focusing on nickel6, iron7, and molybdenum7. Results showed that the Lorentz-Berthelot combining rules severely overestimate the gas-surface potential well. Consequently, we partnered with Prof Mikko Karttunen and Dr. John Titantah, who calculated the ground state energies for gas molecules at various heights above the nanoparticle surface using ab initio techniques (density functional theory). We then fit a pairwise Morse potential to the ground state energies,

${U_{gs}}\left( {r} \right) = {D_e}{\left\{ {1 - \exp \left[ { - a\left({{r} - {r_e}} \right)} \right]} \right\}^2} - {D_e}$

These potentials were used to calculate the trajectories of gas molecules scattering from the nanoparticle surface. A similar procedure was used to estimate the thermal accommodation coefficients for silicon8 nanoparticles in argon and helium, to support a TiRe-LII study carried out in collaboration with the Center for Nanointegration Duisburg-Essen.

A comparative study of these results show that α increases with the potential well depth, D, as well as the reduced mass, μ=mg/ms. Again, the MD-derived thermal accommodation coefficients match experimentally-derived values.

Further insight into the gas/surface scattering is found by plotting the average change between incident and scattered molecular kinetic energies. In the case of silicon, for example, there is a clear change with surface temperature corresponding with the melting point of silicon. This result highlights the fact that the surface phase influences the motion of the surface atoms, and the nature of thermal accommodation.

In the case of metal nanoparticles the Lorentz-Berthelot combining rules severely overestimate the gas/surface potential. A more accurate model is developed by fitting a pairwise Morse potential to ground state energies found with density functional theory.

The energy transferred between a silicon nanoparticle and a scattering molecule dramatically increases as the silicon surface phase changes from solid to liquid (left). Molecular dynamics simulation of an argon molecule scattering from silicon (right).


  1. K. J. Daun, G. J. Smallwood, F. Liu, 2008, "Investigation of thermal accommodation coefficients using laser-induced incandescence," Journal of Heat Transfer, 130, pp. 121201.
  2. K. J. Daun, G. J. Smallwood, F. Liu, 2009, "Molecular dynamics simulations of translational thermal accommodation coefficients for time-resolved LII," Applied Physics B, 94, 39.
  3. K. J. Daun, 2009, "Thermal accommodation coefficients between polyatomic gas molecules and soot in iaser-induced incandescence experiments," International Journal of Heat and Mass Transfer, 52, 5081.
  4. M. B. Någård, P.U. Andersson, N. Markovic, J.B.C. Pettersson, 1998, "Scattering and trapping dynamics of gas–surface interactions: theory and experiments for the Xe–graphite system," J. Chem. Phys., 109, pp. 10339
  5. G. Vidali, G. Ihm, H.-Y. Kim, M. W. Cole, 1991, "Potentials of physical adsorption"Surface Science Reports, 12, pp. 135.
  6. K. J. Daun, J. T. Titantah, M. Karttunen, 2012, "Molecular Dynamics Simulation of Thermal Accommodation Coefficients for Laser-Induced Incandescence Sizing of Nickel Particles", Applied Physics B, 107, pp. 221.
  7. K. J. Daun, T. A., Sipkens, J. T. Titantah, M. Karttunen, M, 2013, "Thermal Accommodation Coefficients for Laser-Induced Incandescence Sizing of Metal Nanoparticles in Monatomic Gases", Applied Physics B, 112, pp. 599.
  8. T. A. Sipkens, R. Mansmann, K. J. Daun, N. Petermann, J. T. Titantah, M Karttunen, H. Wiggers, T. Dreier, C. Schulz, 2014, "In Situ Nanoparticle Size Measurements of Gas-Borne Silicon Nanoparticles by Time-Resolved Laser-Induced Incandescence," Applied Physics B, 116, pp. 623.