Fluid mixtures in nanotubes

The aim of the paper is the study of fluid mixtures in nanotubes by the methods of continuum mechanics. The model starts from a statistical distribution in mean-field molecular theory and uses a density expansion of Taylor series. We get a continuous expression of the volume free energy with density's spatial derivatives limited at the second order. The nanotubes can be filled with liquid or vapor according to the chemical characteristics of the walls and of liquid or vapor mixture bulks. An example of a two-fluid mixture constituted of water and ethanol inside carbon nanotubes at $20{}^{\circ }\mathrm{C}$ is considered. When diameters are small enough, nanotubes are filled with a liquid mixture whatever are the liquid or vapor mixture bulks. The carbon wall influences the ratio of the fluid components in favor of ethanol. The fluid mixture flows across nanotubes can be much more important than classical ones and if the external bulk is vapor, then the flow can be several hundred thousand times larger than Poiseuille flow.

Figure 1

Two-phase fluid component in a nanotube: The nanotube is simultaneously filled with two phases liquid and vapor of fluid components. The two phases (a) and (b) are separated by a cylindrical interface.

Figure 2

Case $c=0.3$ (corresponding to the volume proportion between water and ethanol in the mixture bulk outside the carbon nanotube). The different graphs represent the change of the ratio between volume proportion of water and ethanol versus $r/R$ (representing the normalized distance to the carbon wall) when the radius nanotube is successively equal to $R=0.5$ at 50 nm. We recall that to draw all the diameter cases on the same figure we have normalized the diameter with respect to radius $R$ and, therefore, the $x$ axis is drawn between 0 and 1 for any real value of the nanotube diameter. The graph (a) shows the density profiles between the nanotube axis and the carbon wall ($r/R\in [0,1]$). The second graph (b) enlarges the water density near the wall ($r/R\in [0.8,1]$). The third graph (c) enlarges the ethanol density near the wall ($r/R\in [0.8,1]$). In the second and third graphs, the curved arrows indicate the different graphs for increasing values of $R$ and for the eight nanotube radius-values corresponding to $2R\in \{1,2,3,5,10,20,50,100\}$ in nanometers.

Figure 3

Case $c=0.5$ (corresponding to the volume proportion between water and ethanol in the mixture bulk outside the carbon nanotube). Comments are similar to those for the caption to Fig. 2.

Figure 4

Case $c=0.7$ (corresponding to the volume proportion between water and ethanol in the mixture bulk outside the carbon nanotube). Comments are similar to those for the caption to Fig. 2.