Theory of nanobubble formation and induced force in nanochannels

This paper presents a fundamental theory of nanobubble formation and induced force in confined nanochannels. It is shown that nanobubble formation between hydrophobic plates can be predicted from their surface tension and geometry, with estimated values for the surface free energy and the force acting on the plates in good agreement with the results of molecular dynamics simulation and experimentation. When a bubble is formed between two plates, vertical attractive force and horizontal retract force due to the shifted plates are applied to the plates. The net force exerted on the plates is not dependent on the distance between them. The short-range force between hydrophobic surfaces due to hydrophobic interaction appears to correspond to the force estimated by our theory. We compared between experimental and theoretical values for the binding energy of a molecular motor system to validate our theory. The tendency that the binding energy increases as the size of the protein increases is consistent with the theory.

Figure 1

(a) Two solid plates in the bulk water. The size of the plates is represented by $a$ (width) $\times$ $b$ (height) $\times$ $c$ (depth). The lower plate is fixed in (0, 0) and the upper plate ($x$, $y$) assumed to move the $x$ and $y$ direction. (b) Surface free energy $E$ in between two plates and force $F$ exerted on the plate. When the phase in between the two plates is vapor (or bubble), $y$ $<$ $y_{\mathrm{c}}$, the surface free energy is $E_{\mathrm{vap}}$. On the other hand, when water is filled in between the two plates, $y$ $>$ $y_{\mathrm{c}}$, the surface free energy is $E_{\mathrm{liq}}$.

Figure 2

Sliced snapshots of the bubble formation in between shifted plates. These snapshots are viewed from the $z$ direction and their depths are 20 Å. We measured in four shifted lengths $l_{\mathrm{s}}$ = (a) 8.8 Å, (b) 17.6 Å, (c) 26.3 Å, (d) 35.1 Å, (e) 43.9 Å, and (f) 52.7 Å. The bubble formation occurrs at only the part in which two nanoassemblies overlap each other.

Figure 3

Forces in the $x$ and $y$ directions for the shifted plates. They are expressed as a function of $x$ position of the upper plate. Solid lines show theoretical forces using Eqs. (8) and (9). Circle and square points show the force acting shifted plates in the molecular dynamics simulation.

Figure 4

Gibbs free energy ($\mathrm{\Delta }G$) evaluated by the potential mean force [9] and the surface free-energy ($\mathrm{\Delta }{E}_{\mathrm{vap}}$) curves. The solid and dashed lines are $\mathrm{\Delta }G/k_{\mathrm{B}}T$ and $\mathrm{\Delta }{E}_{\mathrm{vap}}/k_{\mathrm{B}}T$, respectively.

Figure 5

Area-dependent binding energy for various proteins. The dotted curve is our theory [Eq. (3)]. We assumed the temperature is a human internal body temperature ($\sim 310 \mathrm{K}$) and a distance between plates is 3 Å, which is similar as the size of water molecules. Circles are experimental data for a kinesin, a myosin, an amino acid, and a formic acid.