Thermodynamic Stability of Nanobubbles Phil Attard [email protected], Sydney, Australia (Dated: 3.14 15) The observed stability of nanobubbles contradicts the well-known result in classical nucleation the- ory, that the critical radius is both microscopic and thermodynamically unstable. Here nanoscopic stability is shown to be the combined result of two non-classical mechanisms. It is shown that the surface tension decreases with increasing supersaturation, and that this gives a nanoscopic critical radius. Whilst neither a free spherical bubble nor a hemispherical bubble mobile on an hydrophobic surface are stable, it is shown that an immobilized hemispherical bubble with a pinned contact rim is stable and that the total entropy is a maximum at the critical radius. I. INTRODUCTION: NANOBUBBLE TROUBLE 2.E+08 Experimental evidence for nanobubbles was first pub- 1.E+08 lished in 1994,1 and since then their existence has 0.E+00 been confirmed from various features of the mea- B k 2–4 / sured forces between hydrophobic surfaces, includ- tot S ing a reduced attraction in de-aerated water,5–11 and S -1.E+08 from images obtained with tapping mode atomic force microscopy.3,4,9,12–17 For a recent review of theory and -2.E+08 experiment, see Ref. 18. In the fields of nucleation science and bubble research, -3.E+08 19,20 three concepts are important. First, the critical ra- 0.0E+00 5.0E-07 1.0E-06 1.5E-06 2.0E-06 2.5E-06 dius, which is the point at which the derivatives of the to- R (m) tal entropy (equivalently the free energy) vanish. Second, the Laplace-Young equation, which says that the inter- FIG. 1: Total entropy of a spherical air bubble in water as a nal gas pressure of the bubble is larger than the pressure function of radius for a supersaturation ratio of 2 and a surface of the surrounding liquid by an amount equal to twice tension of 0.072 N/m. The solid curve follows the mechanical the surface tension divided by radius. Third, for the case equilibrium path, ∂Stot/∂R = 0, the dashed curve follows of heterogenous nucleation and bubbles adhered to sur- the diffusive equilibrium path, ∂Stot/∂N = 0, and the dash- faces, the Young equation, which says that the cosine dotted curve follows the constant number path, N = Ncrit. of the contact angle is equal to the difference in surface The dotted line marks the critical radius. energies of the solid divided by the liquid-vapor surface tension. The trouble with nanobubbles is that or days) of nanobubbles that it calls into question con- ventional nucleation thermodynamics itself. The second 1. the critical radius is an unstable equilibrium point point represents a quantitative discrepancy between the (it is a saddle point), whereas nanobubbles appear length scales for bubbles predicted by thermodynamics to be stable using conventional parameter values and the measured 2. the conventional critical radius for realistic param- length scales of nanobubbles. The third point says that eters is an order of magnitude or so larger than the there is a strong thermodynamic gradient between the typical nanobubble radius of curvature gas inside the nanobubble and the atmosphere, which should lead to the runaway dissolution of a nanobubble arXiv:1503.04365v1 [physics.chem-ph] 15 Mar 2015 3. the Laplace-Young equation predicts that typical on time-scales orders of magnitude shorter than the ex- nanobubbles have an internal gas pressure one or perimentally observed lifetimes. The fourth point again two orders of magnitude larger than atmospheric appears to prove that conventional macroscopic ther- pressure modynamics, both bulk and surface, is inapplicable on nanoscopic length scales, which broad point is at vari- 4. the measured contact angle for typical nanobubbles ance with diverse experimental measurements in other is significantly larger than the contact angle mea- systems, molecular-level computer simulation data, and sured for macroscopic droplets on the same surface conventional wisdom. These four points are unlikely to Any explanation for the existence of nanobubbles must be independent of each other. reconcile these four points with conventional thermody- Figure 1 shows the total entropy for a nucleating spher- namic principles and realistic physical models. ical bubble. A stable equilibrium point corresponds to a The first point is such a strong qualitative contradic- local maximum of the entropy. (In this article all thermo- tion with the experimentally observed stability (hours dynamic results are cast in terms of entropy. These can 2 be translated into free energy by multiplying throughout The total system is in thermal equilibrium at tempera- by minus the temperature. Hence maximum total en- ture T . Here N is the number of molecules in the bubble, tropy is the same as minimum free energy.) It can be V =4πR3/3 is its volume, A =4πR2 is its area, and R is seen that when the growth of the bubble occurs along a its radius. The pressure of the reservoir is p, which equals path of either mechanical or diffusive equilibrium, the to- atmospheric pressure, the chemical potential of the reser- tal entropy is a minimum at the critical radius. However voir is µ, and the liquid-vapor surface tension is γ. As if the change in size occurs at either constant number shown in Appendix A, it would be more precise to set (shown) or constant radius (not shown), then the total the external reservoir pressure equal to the partial vapor † entropy is a maximum at the critical radius. Hence the pressure of air at saturation, p = p 0.8patm, but for total entropy has a saddle point at the critical radius, simplicity this is not done here. ≈ which means that it is a point of unstable equilibrium. In the capillarity approximation, the bubble is taken The critical radius for these parameters is about an order to be an ideal gas19,21 of magnitude larger than the radius of curvature typically measured for nanobubbles. NΛ3 S (N,V,T )= k N 1 ln , (2.2) The physical reason for the instability of the critical s B − V radius is readily understood. At the critical radius the diffusive (number) and mechanical (radius) gradients are where kB is Boltzmann’s constant and Λ is the thermal zero. A fluctuation to a larger radius along the path of wave length. Although the entropy of a real gas differs mechanical equilibrium (mechanical equilibrium occurs from this quantitatively, the ideal gas model is not ex- faster than diffusive equilibrium with the reservoir be- pected to affect the results qualitatively. yond the immediate vicinity of the surface) reduces the A bubble can only be in equilibrium if the water is su- internal pressure and hence the gas chemical potential persaturated with air (see below). The supersaturation inside the bubble. There is now a thermodynamic gradi- ratio, s, is the ratio of the actual concentration of dis- ent that forces dissolved gas from the reservoir into the solved air to the saturation concentration, which equals bubble, which increases the internal pressure. This cre- the ratio of the air pressures that would give the two † ates a mechanical force imbalance, which acts to increase concentrations, s = pss/p , where the dagger here and the volume of the bubble at the expense of the volume below denotes saturation. The chemical potential can be of the reservoir. This increases the radius, magnifying obtained from the supersaturation ratio by the standard the original fluctuation. The opposite occurs if the ini- ideal gas result21 tial fluctuation decreases the radius, in which case the bubble is driven to zero radius. sp†Λ3 µ = kBT ln . (2.3) An apparently trivial but essential point is easily over- kBT looked in this explanation: The instability at the critical radius relies upon the fact that the volume of the bubble In this work the saturation air pressure will be taken † increases with increasing radius. It will turn out that this to be atmospheric pressure, p = p. This neglects the simple geometric fact holds the key to understanding the vapor pressure of water, which is about one fifth of an thermodynamic stability of nanobubbles. atmosphere. The purpose of this paper is to show what it takes The number derivative is to make the critical radius of a bubble both stable and ∂S (N,V,T ) µ µ nanoscopic. tot = s + , (2.4) ∂N − T T which vanishes when µs = µ. Using the ideal gas equa- II. CONVENTIONAL NUCLEATION tion this is THERMODYNAMICS sp ρ =Λ−3eµ/kBT = , (2.5) k T Consider an air bubble in water. Only the air need B be explicitly accounted for, as the water solvent can be where ρ = N/V is the number density of air inside the considered always at equilibrium (see Appendix A). The bubble. This is the equation for diffusive equilibrium total entropy is taken to be the sum of three terms: the between the air inside the bubble and the air dissolved bulk entropy of the sub-system, which is the bubble, the in the water. bulk entropy of the reservoir, which is the air dissolved The volume derivative is in water, and the interfacial entropy of their boundary,21 ∂S (N,V,T ) p p 2γ tot = s . (2.6) Stot(N,V,T ) ∂V T − T − T R γA = Ss(N,V,T )+ Sr(Nr, Vr,T ) This vanishes when − T pV µN 4πγR2 2γ = S (N,V,T ) + . (2.1) p = p + , (2.7) s − T T − T s R 3 which is the Laplace-Young equation. This is the equa- 1.E+08 tion for mechanical equilibrium for the bubble.