We have considered two simple candidate models to explain the linear rate of mass transport observed in the H2 experiments. The first is ballistic transport of a 2D surface gas. This model is motivated by the behavior at high temperature where the hole is filled in by ballistic transport from the 3D gas. In this regime, the filling rate is linear because the sticking coefficient on the thin region is large but the reevaporation rate is very small, so the filling rate is determined by the rate of incident gas molecules. As the coverage approaches the equilibrium value, the reevaporation rate approaches the deposition rate, as dictated by the principle of detailed balance.

The 2D analog of this type of process requires a gas of adatoms whose concentration n is determined by the equality of the chemical potentials of the 2D and 3D gases. In the 2D gas,

where ,and E_b is the binding energy of the adatom to the surface. The chemical potential is approximately -L in the low temperature regime considered here. The adatom concentration is therefore

The flux of ballistic transport is proportional to the product of the concentration and the characteristic thermal velocity , which is approximately 10⁴ cm/sec for H₂ at T=2K. The dominant temperature dependence of the flux is due to factor in the concentration, and the activation energy determined in the experiment is E_b-L within the context of this model. This implies that E_b is approximately 95K-19K = 76K for H₂. When this value of E_b is used to compute the surface concentration, the derived mass flux is nearly five orders of magnitude higher than observed. The mass flux can be reduced by introducing a mean free path, but then the transport becomes diffusive and scales as , rather than t.

Another possible model for mass transport in a film is viscous flow at a constant velocity which results from a balance of a driving force and viscous dissipation. The driving force is the large difference in surface free energy DW between the thin region of the film and the regions with equilibrium thickness. The film will flow in response to this energy gradient with a characteristic velocity determined by dimensional considerations to be of order (DW)/h, where h is the viscosity of the material. This type of hydrodynamic analysis which leads to a fluid front moving at nearly constant velocity, is commonly used to describe spreading of viscous liquids on a surface, and is known as Darcy's law [18]. If this model is applicable to flow of solid hydrogen films, the activation energy we observe must be interpreted as the energy which characterizes the Arrhenius behavior of the viscosity . The viscosity can be related to the diffusion coefficient D through the Einstein relation [19], which gives , where a is a molecular length.

Comparison with previous work and Conclusion

Several recent experiments have investigated the behavior of quench condensed films of solid hydrogen formed by condensing gas on to a substrate held at 1.5K. The thickness of these films is determined by the deposition process and is typically much larger than the equilibrium thickness. As the temperature is raised above 1.5K, the films coarsen and become rough as the film phase separates into regions with the equilibrium thickness and regions with bulk crystallites. The temperature dependence of the roughening process has been monitored using plasmon [5,7], the mobility of surface electrons [6] and surface acoustic wave scattering [4]. Classen et al [4] have obtained activation energies of 23 K and 47 K for H₂ and D₂ respectively. In those experiments, the film approaches equilibrium from a state that is too thick, while in our experiments, equilibrium was approached from an initial state that was too thin. It is noteworthy that the activation energy for both types of processes are similar.

The surface mobility of solid HD of equilibrium thickness adsorbed on MgO(100) powder has been investigated using quasi-elastic neutron scattering [8] for temperatures between 7 and 15K. These temperatures are far too high to use our hole-burning technique, since the transport is completely dominated by gas phase processes. The width of the neutron scattering line is a measure of the surface mobility, which was found to be thermally activated with an activation energy of 16.1 K. The high value of the surface mobility was explained in terms of a quasi-liquid layer (QLL) at the surface of the solid film. This type of surface melted layer is known to exist in a variety of materials near the triple point temperature T₃. In most non-quantum systems, the quasi-liquid surface layer is expected to vanish below 0.7 T₃ [20]. In HD, the activated behavior persisted down to at least 0.5T₃ (8K), but Maruyama et al [8] suggest that the large zero-point energy of a quantum solid may tend to stabilize the QLL down to anomalously low temperature. If the QLL is responsible for activated surface mobility, our observations would imply the existence of the QLL at temperatures as low as 0.1T₃.

The results of previous experiments have been interpreted in terms of enhanced surface diffusion at low temperature. The most surprising result of our work is that simple models of surface diffusion do not explain the time dependence of the surface mass transport. Our results suggest that the surface of solid hydrogen behaves in some ways like a highly viscous liquid. The high surface mobility of solid hydrogen places constraints on attempts to dope solid hydrogen with atomic impurities, which is of interest for applications as a high energy density material.

Acknowledgments

This work was supported by AFOSR grant F49620-95-1-0213.