Diffusing "Uphill" Against the Concentration Gradient and the Saturation of Stagnant Lakes on Titan

Diffusive Saturation of Titan’s Lakes

In our everyday lives and laboratories, molecules diffuse from relatively high concentrations to relatively low concentrations, as is stated by Fick’s Law of Diffusion (diffusive molar flux is proportional to the concentration gradient).  However, Fick’s Law is an empirical law derived from Fourier’s mathematical description of heat flow (Fick, 1855), which only spontaneously flows from hot to cold. Fick literally notes that: “One only needs to swap the word ‘quantity of heat’ with ‘quantity of dissolved substance’ and the word ‘temperature’ with ‘concentration’” (Fick, 1855, p.65). However, it is now known that diffusing molecules follow gradients in chemical potential, rather than gradients in concentration. 

For many systems, including all that Fick considered, these two quantities are effectively analogous and behave interchangeably. Nevertheless, deviations from Fick’s Law, including the well-studied phenomenon of “uphill diffusion” against concentration gradients(e.g., Krishna, 2015), can occur when equilibrium thermodynamic conditions nevertheless sufficiently vary across a system to significantly affect the chemical potentials throughout the system.

Such is the case within Titan’s lakes, where the pressure gradient within the lake due to hydrostatic pressure causes these deviations. Titan’s lakes are composed primarily of methane-ethane mixtures that dissolve atmospheric nitrogen, with higher pressures promoting higher solubility of N2. According to Fickian diffusion, these lakes would dissolve N2 until the entire lake matches the concentration of N2 at the surface in Vapor-Liquid Equilibrium (VLE). However, this ignores the higher solubility of N2 at depth, which “pulls” additional N2 downward to greater depths to equilibrate the chemical potentials. Thus, as chemical potentials equilibrate, N2 concentrations at depth will increase above those at the surface, leading to N2 diffusing “uphill” against the concentration gradient.

Energy Minimization

Upon dissolution of atmospheric nitrogen, Titan’s lakes will release energy to the environment until saturation is achieved. Imagine a lake on Titan not yet saturated in nitrogen (N2 may even be increasing with depth, but is not yet saturated) and consider a nitrogen molecule in the lake that randomly diffuses to a deeper layer, opening up room for a molecule from a higher layer to diffuse down. This ultimately enables another N2 molecule to dissolve from the atmosphere (releasing its latent heat of condensation). Over time, the lake releases this latent heat until it becomes saturated in nitrogen.

 

Entropy Maximization

In statistical mechanics, the entropy of a system is proportional to the natural log of the number of microscopic states accessible to the system, with each state equally probable. Thus, maximizing the number of unique microscopic states maximizes the entropy of the system.  Because nitrogen concentrations at saturation increase with depth, the number of available microscopic states for N2 similarly increases with depth.  Thus, were a nitrogen molecule to randomly diffuse to a deeper layer and enable additional N2 molecules to dissolve into the lake, the system would suddenly have access to a much larger number of states: nearly the same number of microscopic states that it previously had access multiplied by the number of states that this newly dissolved molecule can be found in. Thus, the lake would continue dissolving atmospheric N2, driving uphill diffusion until it fully saturates.

 

Random Walk

In a three-dimensional random walk, a particle has an equal probability of moving in any direction at each step; in diffusion, these steps are motion between collisions with other molecules. Eventually, a particle diffusing in an inert box is equally likely to be in any position in that box; this is a direct result of the particle having no preferential interaction with the box. This is not the case for N2 diffusing through a methane lake, which interacts ever more strongly with nitrogen at depth. Under pressure, methane  “holds on” to the nitrogen more strongly, causing any diffusing N2 molecule to spend more time at depth than near the surface, and thus be preferentially found at depth.  With these same dynamics playing out over a grand canonical ensemble of particles, the effect is increasing N2 concentrations with depth.

This random walk framework further allows us to estimate the timescale of this process.  In a three-dimensional random walk, the standard deviation (σ) of the average distance (d) traveled from the starting point is 

Where N is the number of steps in the random walk and λ is the mean free path of a nitrogen molecule 

Where n is the molecular density of the liquid and R is the scattering radius of the molecule. The kinetic diameter of a nitrogen molecule is 3.64 angstroms, and the molecular densities of liquid nitrogen (1.7x10²⁸ molecules/m³) and liquid methane (2.2x10²⁸ molecules/m³) correspond to a mean free path of ~3-4 angstroms for liquid nitrogen and methane.

Thus, the number of steps needed for the standard deviation (σ) of this random walk to equal a distance d is

Also consider that the time required for each step is just the time required for a nitrogen molecule to move a distance λ while traveling at the thermal speed

which is ~260 m/s for a nitrogen molecule at 90 K. Thus, by multiplying the number of steps by  τ, we obtain the timescale τ_diffusive required for the standard deviation of diffusing nitrogen to grow to a depth d is:

This shows that Titan’s deepest (~100m lakes) diffusively saturate over timescales of ~1-10 millennia in the absence of a mechanism changing the system, such as overturn, which can degas the lake. This timescale drops to centuries for lakes ~40 m deep and decades for lakes ~10 m deep. Thus stagnant lakes on Titan will diffusively saturate over relatively short timescales.

 

References

Fick, A. (1855) Ueber Diffusion. Annalen der Physik 170, 59-86

Krishna, R. (2015) Uphill Diffusion in Multicomponent Mixtures. Chemical Society Reviews 44, 2812-2836